General Relativity
Benjamin Crowell
Fullerton, California
Copyright 2009
Benjamin Crowell
rev. October 30, 2013
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Brief Contents
A Geometrical Theory of Spacetime
Geometry of Flat Spacetime 41
Differential Geometry 87
Tensors 123
Curvature 159
Vacuum Solutions 201
Symmetries 241
Sources 261
Gravitational Waves 331
1 A geometrical theory of spacetime
1.1 Time and causality . . . . . . . . . . . . . . . .
1.2 Experimental tests of the nature of time . . . . . . . .
The Hafele-Keating experiment, 15.—Muons, 16.—Gravitational
red-shifts, 16.
1.3 Non-simultaneity and the maximum
effect . . . . . . . . . . . . . . .
1.4 Ordered geometry. . . . . . . .
1.5 The equivalence principle. . . . .
speed of cause and
. . . . . . . . . 17
. . . . . . . . . 18
. . . . . . . . . 20
Proportionality of inertial and gravitational mass, 21.—Geometrical
treatment of gravity, 21.—Eötvös experiments, 22.—The equivalence principle, 23.—Gravitational red-shifts, 32.—The PoundRebka experiment, 34.
Problems . . . . . . . . . . . . . . . . . . . . . .
2 Geometry of flat spacetime
2.1 Affine properties of Lorentz geometry. . . . . . . . .
Parallelism and measurement, 42.—Vectors, 46.
2.2 Relativistic properties of Lorentz geometry . . . . . .
2.3 The light cone . . . . . . . . . . . . . . . . . .
Velocity addition, 66.—Logic, 68.
2.4 Experimental tests of Lorentz geometry . . . . . . . .
Dispersion of the vacuum, 69.—Observer-independence of c, 69.—
Lorentz violation by gravitational forces, 71.
2.5 Three spatial dimensions . . . . . . . . . . . . . .
Lorentz boosts in three dimensions, 72.—Gyroscopes and the equivalence principle, 74.—Boosts causing rotations, 75.—An experimental test: Thomas precession in hydrogen, 82.
Problems . . . . . . . . . . . . . . . . . . . . . .
3 Differential geometry
3.1 Tangent vectors . . . . . . . . . . . . . . . . . .
3.2 Affine notions and parallel transport . . . . . . . . .
The affine parameter in curved spacetime, 89.—Parallel transport,
3.3 Models . . . . . . . . . . . . . . . . . . . . .
3.4 Intrinsic quantities . . . . . . . . . . . . . . . . .
Coordinate independence, 96.
3.5 The metric . . . . . . . . . . . . . . . . . . . .
The Euclidean metric, 100.—The Lorentz metric, 105.—Isometry,
inner products, and the Erlangen program, 106.—Einstein’s carousel,
3.6 The metric in general relativity. . . . . . . . . . . . 114
The hole argument, 114.—A Machian paradox, 115.
3.7 Interpretation of coordinate independence. . . . . . . 116
Is coordinate independence obvious?, 116.—Is coordinate indepen-
dence trivial?, 117.—Coordinate independence as a choice of gauge,
Problems . . . . . . . . . . . . . . . . . . . . . . 119
4 Tensors
4.1 Lorentz scalars . . . . . . . . . . . . . . . . . . 123
4.2 Four-vectors . . . . . . . . . . . . . . . . . . . 124
The velocity and acceleration four-vectors, 124.—The momentum
four-vector, 126.—The frequency vector and the relativistic Doppler
shift, 133.—A non-example: electric and magnetic fields, 136.—
The electromagnetic potential four-vector, 137.
4.3 The tensor transformation laws . . . . . . . . . . . 138
4.4 Experimental tests . . . . . . . . . . . . . . . . 142
Universality of tensor behavior, 142.—Speed of light differing from
c, 142.—Degenerate matter, 143.
4.5 Conservation laws. . . . . . . . . . . . . . . . . 148
No general conservation laws, 148.—Conservation of angular momentum and frame dragging, 149.
4.6 Things that aren’t quite tensors . . . . . . . . . . . 151
Area, volume, and tensor densities, 151.—The Levi-Civita symbol,
153.—Spacetime volume, 155.—Angular momentum, 155.
Problems . . . . . . . . . . . . . . . . . . . . . . 156
5 Curvature
Tidal curvature versus curvature caused by local sources
The stress-energy tensor . . . . . . . . . . . . . .
Curvature in two spacelike dimensions . . . . . . . .
Curvature tensors . . . . . . . . . . . . . . . . .
Some order-of-magnitude estimates . . . . . . . . .
The geodetic effect, 170.—Deflection of light rays, 171.
5.6 The covariant derivative . . . . . . . . . . . . . . 172
The covariant derivative in electromagnetism, 173.—The covariant
derivative in general relativity, 174.
5.7 The geodesic equation . . . . . . . . . . . . . . . 178
Characterization of the geodesic, 178.—Covariant derivative with
respect to a parameter, 179.—The geodesic equation, 179.—Uniqueness,
5.8 Torsion . . . . . . . . . . . . . . . . . . . . . 181
Are scalars path-dependent?, 181.—The torsion tensor, 183.—Experimental
searches for torsion, 184.
5.9 From metric to curvature . . . . . . . . . . . . . . 187
Finding the Christoffel symbol from the metric, 187.—Numerical
solution of the geodesic equation, 188.—The Riemann tensor in
terms of the Christoffel symbols, 190.—Some general ideas about
gauge, 190.
5.10 Manifolds . . . . . . . . . . . . . . . . . . . . 193
Why we need manifolds, 193.—Topological definition of a manifold,
194.—Local-coordinate definition of a manifold, 196.
Problems . . . . . . . . . . . . . . . . . . . . . . 199
6 Vacuum solutions
6.1 Event horizons . . . . . . . . . . . . . . . . . . 201
The event horizon of an accelerated observer, 201.—Information
paradox, 203.—Radiation from event horizons, 204.
6.2 The Schwarzschild metric . . . . . . . . . . . . . 205
The zero-mass case, 206.—Geometrized units, 208.—A large-r limit,
208.—The complete solution, 209.—Geodetic effect, 212.—Orbits,
215.—Deflection of light, 220.
6.3 Black holes. . . . . . . . . . . . . . . . . . . . 223
Singularities, 223.—Event horizon, 224.—Expected formation, 225.—
Observational evidence, 225.—Singularities and cosmic censorship,
227.—Hawking radiation, 230.—Black holes in d dimensions, 232.
6.4 Degenerate solutions . . . . . . . . . . . . . . . 234
Problems . . . . . . . . . . . . . . . . . . . . . . 238
7 Symmetries
7.1 Killing vectors. . . . . . . . . . . . . . . . . . . 241
Inappropriate mixing of notational systems, 245.—Conservation
laws, 246.
7.2 Spherical symmetry . . . . . . . . . . . . . . . . 247
7.3 Static and stationary spacetimes. . . . . . . . . . . 248
Stationary spacetimes, 248.—Isolated systems, 249.—A stationary field with no other symmetries, 250.—A stationary field with
additional symmetries, 251.—Static spacetimes, 251.—Birkhoff’s
theorem, 251.—The gravitational potential, 254.
7.4 The uniform gravitational field revisited . . . . . . . . 255
Closed timelike curves, 258.
Problems . . . . . . . . . . . . . . . . . . . . . . 260
8 Sources
8.1 Sources in general relativity . . . . . . . . . . . . . 261
Point sources in a background-independent theory, 261.—The Einstein field equation, 262.—Energy conditions, 275.—The cosmological constant, 285.
8.2 Cosmological solutions. . . . . . . . . . . . . . . 288
Evidence for the finite age of the universe, 289.—Evidence for
expansion of the universe, 289.—Evidence for homogeneity and
isotropy, 290.—The FRW cosmologies, 292.—A singularity at the
Big Bang, 297.—Observability of expansion, 300.—The vacuumdominated solution, 307.—The matter-dominated solution, 312.—
The radiation-dominated solution, 315.—Local effects of expansion,
315.—Observation, 319.
8.3 Mach’s principle revisited . . . . . . . . . . . . . . 322
The Brans-Dicke theory, 323.—Predictions of the Brans-Dicke theory,
326.—Hints of empirical support, 327.—Mach’s principle is false.,
Problems . . . . . . . . . . . . . . . . . . . . . . 329
9 Gravitational waves
9.1 The speed of gravity . . . . . . . . . . . . . . . . 331
9.2 Gravitational radiation . . . . . . . . . . . . . . . 332
Empirical evidence, 332.—Energy content, 334.—Expected properties,
336.—Some exact solutions, 338.—Rate of radiation, 340.
Problems . . . . . . . . . . . . . . . . . . . . . . 344
Appendix 1: Excerpts from three papers by Einstein . . . . . . . . . 345
“On the electrodynamics of moving bodies” . . . . . . . . . . . . . . . . . . .345
“Does the inertia of a body depend upon its energy content?” 358
“The foundation of the general theory of relativity” . . . . . . . . . . . 360
Appendix 2: Hints and solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Chapter 1
A geometrical theory of
“I always get a slight brain-shiver, now [that] space and time appear
conglomerated together in a gray, miserable chaos.” – Sommerfeld
This is a book about general relativity, at a level that is meant
to be accessible to advanced undergraduates.
This is mainly a book about general relativity, not special relativity. I’ve heard the sentiment expressed that books on special
relativity generally do a lousy job on special relativity, compared to
books on general relativity. This is undoubtedly true, for someone
who already has already learned special relativity — but wants to
unlearn the parts that are completely wrong in the broader context
of general relativity. For someone who has not already learned special relativity, I strongly recommend mastering it first, from a book
such as Taylor and Wheeler’s Spacetime Physics.
In the back of this book I’ve included excerpts from three papers
by Einstein — two on special relativity and one on general relativity.
They can be read before, after, or along with this book. There are
footnotes in the papers and in the main text linking their content
with each other.
I should reveal at the outset that I am not a professional relativist. My field of research was nonrelativistic nuclear physics until
I became a community college physics instructor. I can only hope
that my pedagogical experience will compensate to some extent for
my shallow background, and that readers who find mistakes will be
kind enough to let me know about them using the contact information provided at
1.1 Time and causality
Updating Plato’s allegory of the cave, imagine two super-intelligent
twins, Alice and Betty. They’re raised entirely by a robotic tutor
on a sealed space station, with no access to the outside world. The
robot, in accord with the latest fad in education, is programmed to
encourage them to build up a picture of all the laws of physics based
on their own experiments, without a textbook to tell them the right
answers. Putting yourself in the twins’ shoes, imagine giving up
all your preconceived ideas about space and time, which may turn
out according to relativity to be completely wrong, or perhaps only
approximations that are valid under certain circumstances.
Causality is one thing the twins will notice. Certain events result in other events, forming a network of cause and effect. One
general rule they infer from their observations is that there is an
unambiguously defined notion of betweenness: if Alice observes that
event 1 causes event 2, and then 2 causes 3, Betty always agrees that
2 lies between 1 and 3 in the chain of causality. They find that this
agreement holds regardless of whether one twin is standing on her
head (i.e., it’s invariant under rotation), and regardless of whether
one twin is sitting on the couch while the other is zooming around
the living room in circles on her nuclear fusion scooter (i.e., it’s also
invariant with respect to different states of motion).
You may have heard that relativity is a theory that can be interpreted using non-Euclidean geometry. The invariance of betweenness is a basic geometrical property that is shared by both Euclidean
and non-Euclidean geometry. We say that they are both ordered
geometries. With this geometrical interpretation in mind, it will
be useful to think of events not as actual notable occurrences but
merely as an ambient sprinkling of points at which things could happen. For example, if Alice and Betty are eating dinner, Alice could
choose to throw her mashed potatoes at Betty. Even if she refrains,
there was the potential for a causal linkage between her dinner and
Betty’s forehead.
Betweenness is very weak. Alice and Betty may also make a
number of conjectures that would say much more about causality.
For example: (i) that the universe’s entire network of causality is
connected, rather than being broken up into separate parts; (ii) that
the events are globally ordered, so that for any two events 1 and 2,
either 1 could cause 2 or 2 could cause 1, but not both; (iii) not only
are the events ordered, but the ordering can be modeled by sorting
the events out along a line, the time axis, and assigning a number t,
time, to each event. To see what these conjectures would entail, let’s
discuss a few examples that may draw on knowledge from outside
Alice and Betty’s experiences.
Example: According to the Big Bang theory, it seems likely that
the network is connected, since all events would presumably connect
Chapter 1
A geometrical theory of spacetime
back to the Big Bang. On the other hand, if (i) were false we might
have no way of finding out, because the lack of causal connections
would make it impossible for us to detect the existence of the other
universes represented by the other parts disconnected from our own
Example: If we had a time machine,1 we could violate (ii), but
this brings up paradoxes, like the possibility of killing one’s own
grandmother when she was a baby, and in any case nobody knows
how to build a time machine.
Example: There are nevertheless strong reasons for believing
that (ii) is false. For example, if we drop Alice into one black hole,
and Betty into another, they will never be able to communicate
again, and therefore there is no way to have any cause and effect
relationship between Alice’s events and Betty’s.2
Since (iii) implies (ii), we suspect that (iii) is false as well. But
Alice and Betty build clocks, and these clocks are remarkably successful at describing cause-and-effect relationships within the confines of the quarters in which they’ve lived their lives: events with
higher clock readings never cause events with lower clock readings.
They announce to their robot tutor that they’ve discovered a universal thing called time, which explains all causal relationships, and
which their experiments show flows at the same rate everywhere
within their quarters.
“Ah,” the tutor sighs, his metallic voice trailing off.
“I know that ‘ah’, Tutorbot,” Betty says. “Come on, can’t you
just tell us what we did wrong?”
“You know that my pedagogical programming doesn’t allow that.”
“Oh, sometimes I just want to strangle whoever came up with
those stupid educational theories,” Alice says.
The twins go on strike, protesting that the time theory works
perfectly in every experiment they’ve been able to imagine. Tutorbot gets on the commlink with his masters and has a long, inaudible
argument, which, judging from the hand gestures, the twins imagine
to be quite heated. He announces that he’s gotten approval for a
field trip for one of the twins, on the condition that she remain in a
sealed environment the whole time so as to maintain the conditions
of the educational experiment.
The possibility of having time come back again to the same point is often
referred to by physicists as a closed timelike curve (CTC). Kip Thorne, in his
popularization Black Holes and Time Warps, recalls experiencing some anxiety
after publishing a paper with “Time Machines” in the title, and later being
embarrassed when a later paper on the topic was picked up by the National
“CTC” is safer because nobody but physicists know what it means.
This point is revisited in section 6.1.
Section 1.1
Time and causality
“Who gets to go?” Alice asks.
“Betty,” Tutorbot replies, “because of the mashed potatoes.”
“But I refrained!” Alice says, stamping her foot.
“Only one time out of the last six that I served them.”
The next day, Betty, smiling smugly, climbs aboard the sealed
spaceship carrying a duffel bag filled with a large collection of clocks
for the trip. Each clock has a duplicate left behind with Alice. The
clock design that they’re proudest of consists of a tube with two
mirrors at the ends. A flash of light bounces back and forth between
the ends, with each round trip counting as one “tick,” one unit of
time. The twins are convinced that this one will run at a constant
rate no matter what, since it has no moving parts that could be
affected by the vibrations and accelerations of the journey.
Betty’s field trip is dull. She doesn’t get to see any of the outside
world. In fact, the only way she can tell she’s not still at home is that
she sometimes feels strong sensations of acceleration. (She’s grown
up in zero gravity, so the pressing sensation is novel to her.) She’s
out of communication with Alice, and all she has to do during the
long voyage is to tend to her clocks. As a crude check, she verifies
that the light clock seems to be running at its normal rate, judged
against her own pulse. The pendulum clock gets out of synch with
the light clock during the accelerations, but that doesn’t surprise
her, because it’s a mechanical clock with moving parts. All of the
nonmechanical clocks seem to agree quite well. She gets hungry for
breakfast, lunch, and dinner at the usual times.
When Betty gets home, Alice asks, “Well?”
“Great trip, too bad you couldn’t come. I met some cute boys,
went out dancing, . . . ”
“You did not. What about the clocks?”
“They all checked out fine. See, Tutorbot? The time theory still
holds up.”
“That was an anticlimax,” Alice says. “I’m going back to bed
“Bed?” Betty exclaims. “It’s three in the afternoon.”
The twins now discover that although all of Alice’s clocks agree
among themselves, and similarly for all of Betty’s (except for the
ones that were obviously disrupted by mechanical stresses), Alice’s
and Betty’s clocks disagree with one another. A week has passed
for Alice, but only a couple of days for Betty.
1.2 Experimental tests of the nature of time
Chapter 1
A geometrical theory of spacetime
1.2.1 The Hafele-Keating experiment
In 1971, J.C. Hafele and R.E. Keating3 of the U.S. Naval Observatory brought atomic clocks aboard commercial airliners and went
around the world, once from east to west and once from west to east.
(The clocks had their own tickets, and occupied their own seats.) As
in the parable of Alice and Betty, Hafele and Keating observed that
there was a discrepancy between the times measured by the traveling clocks and the times measured by similar clocks that stayed
at the lab in Washington. The result was that the east-going clock
lost an amount of time ∆tE = −59 ± 10 ns, while the west-going
one gained ∆tW = +273 ± 7 ns. This establishes that time is not
universal and absolute.
Nevertheless, causality was preserved. The nanosecond-scale effects observed were small compared to the three-day lengths of the
plane trips. There was no opportunity for paradoxical situations
such as, for example, a scenario in which the east-going experimenter
arrived back in Washington before he left and then proceeded to
convince himself not to take the trip.
Hafele and Keating were testing specific quantitative predictions
of relativity, and they verified them to within their experiment’s
error bars. At this point in the book, we aren’t in possession of
enough relativity to be able to make such calculations, but, like
Alice and Betty, we can inspect the empirical results for clues as to
how time works.
a / The clock took up two seats,
and two tickets were bought for it
under the name of “Mr. Clock.”
The opposite signs of the two results suggests that the rate at
which time flows depends on the motion of the observer. The eastgoing clock was moving in the same direction as the earth’s rotation,
so its velocity relative to the earth’s center was greater than that of
the ones that remained in Washington, while the west-going clock’s
velocity was correspondingly reduced.4 The signs of the ∆t’s show
that moving clocks were slower.
On the other hand, the asymmetry of the results, with |∆tE | =
|∆tW |, implies that there was a second effect involved, simply due
to the planes’ being up in the air. Relativity predicts that time’s
rate of flow also changes with height in a gravitational field. The
deeper reasons for such an effect are given in section 1.5.6 on page
Although Hafele and Keating’s measurements were on the ragged
edge of the state of the art in 1971, technology has now progressed
to the point where such effects have everyday consequences. The
Hafele and Keating, Science, 177 (1972), 168
These differences in velocity are not simply something that can be eliminated
by choosing a different frame of reference, because the clocks’ motion isn’t in
a straight line. The clocks back in Washington, for example, have a certain
acceleration toward the earth’s axis, which is different from the accelerations
experienced by the traveling clocks.
Section 1.2
Experimental tests of the nature of time
satellites of the Global Positioning System (GPS) orbit at a speed
of 1.9 × 103 m/s, an order of magnitude faster than a commercial
jet. Their altitude of 20,000 km is also much greater than that of
an aircraft. For both these reasons, the relativistic effect on time is
stronger than in the Hafele-Keating experiment. The atomic clocks
aboard the satellites are tuned to a frequency of 10.22999999543
MHz, which is perceived on the ground as 10.23 MHz. (This frequency shift will be calculated in example 12 on page 59.)
1.2.2 Muons
Although the Hafele-Keating experiment is impressively direct,
it was not the first verification of relativistic effects on time, it did
not completely separate the kinematic and gravitational effects, and
the effect was small. An early experiment demonstrating a large and
purely kinematic effect was performed in 1941 by Rossi and Hall,
who detected cosmic-ray muons at the summit and base of Mount
Washington in New Hampshire. The muon has a mean lifetime of
2.2 µs, and the time of flight between the top and bottom of the
mountain (about 2 km for muons arriving along a vertical path)
at nearly the speed of light was about 7 µs, so in the absence of
relativistic effects, the flux at the bottom of the mountain should
have been smaller than the flux at the top by about an order of
magnitude. The observed ratio was much smaller, indicating that
the “clock” constituted by nuclear decay processes was dramatically
slowed down by the motion of the muons.
1.2.3 Gravitational red-shifts
The first experiment that isolated the gravitational effect on time
was a 1925 measurement by W.S. Adams of the spectrum of light
emitted from the surface of the white dwarf star Sirius B. The gravitational field at the surface of Sirius B is 4 × 105 g, and the gravitational potential is about 3,000 times greater than at the Earth’s
surface. The emission lines of hydrogen were red-shifted, i.e., reduced in frequency, and this effect was interpreted as a slowing of
time at the surface of Sirius relative to the surface of the Earth. Historically, the mass and radius of Sirius were not known with better
than order of magnitude precision in 1925, so this observation did
not constitute a good quantitative test.
The first such experiment to be carried out under controlled
conditions, by Pound and Rebka in 1959, is analyzed quantitatively
in example 7 on page 129.
The first high-precision experiment of this kind was Gravity
Probe A, a 1976 experiment5 in which a space probe was launched
vertically from Wallops Island, Virginia, at less than escape velocity, to an altitude of 10,000 km, after which it fell back to earth and
crashed down in the Atlantic Ocean. The probe carried a hydro5
Chapter 1
Vessot at al., Physical Review Letters 45 (1980) 2081
A geometrical theory of spacetime
b / Gravity Probe A.
gen maser clock which was used to control the frequency of a radio
signal. The radio signal was received on the ground, the nonrelativistic Doppler shift was subtracted out, and the residual blueshift
was interpreted as the gravitational effect effect on time, matching
the relativistic prediction to an accuracy of 0.01%.
1.3 Non-simultaneity and the maximum speed
of cause and effect
We’ve seen that time flows at different rates for different observers.
Suppose that Alice and Betty repeat their Hafele-Keating-style experiment, but this time they are allowed to communicate during
the trip. Once Betty’s ship completes its initial acceleration away
from Betty, she cruises at constant speed, and each girl has her own
equally valid inertial frame of reference. Each twin considers herself
to be at rest, and says that the other is the one who is moving.
Each one says that the other’s clock is the one that is slow. If they
could pull out their phones and communicate instantaneously, with
no time lag for the propagation of the signals, they could resolve
the controversy. Alice could ask Betty, “What time does your clock
read right now ?” and get an immediate answer back.
By the symmetry of their frames of reference, however, it seems
that Alice and Betty should not be able to resolve the controversy
during Betty’s trip. If they could, then they could release two radar
beacons that would permanently establish two inertial frames of
reference, A and B, such that time flowed, say, more slowly in B
than in A. This would violate the principle that motion is relative,
Section 1.3
Non-simultaneity and the maximum speed of cause and effect
and that all inertial frames of reference are equally valid. The best
that they can do is to compare clocks once Betty returns, and verify
that the net result of the trip was to make Betty’s clock run more
slowly on the average.
Alice and Betty can never satisfy their curiosity about exactly
when during Betty’s voyage the discrepancies accumulated or at
what rate. This is information that they can never obtain, but
they could obtain it if they had a system for communicating instantaneously. We conclude that instantaneous communication is
impossible. There must be some maximum speed at which signals
can propagate — or, more generally, a maximum speed at which
cause and effect can propagate — and this speed must for example
be greater than or equal to the speed at which radio waves propagate. It is also evident from these considerations that simultaneity
itself cannot be a meaningful concept in relativity.
1.4 Ordered geometry
Let’s try to put what we’ve learned into a general geometrical context.
Euclid’s familiar geometry of two-dimensional space has the following axioms,6 which are expressed in terms of operations that can
be carried out with a compass and unmarked straightedge:
E1 Two points determine a line.
E2 Line segments can be extended.
E3 A unique circle can be constructed given any point as its center
and any line segment as its radius.
E4 All right angles are equal to one another.
E5 Parallel postulate: Given a line and a point not on the line,
no more than one line can be drawn through the point and
parallel to the given line.7
The modern style in mathematics is to consider this type of
axiomatic system as a self-contained sandbox, with the axioms, and
any theorems proved from them, being true or false only in relation
to one another. Euclid and his contemporaries, however, believed
them to be empirical facts about physical reality. For example, they
considered the fifth postulate to be less obvious than the first four,
because in order to verify physically that two lines were parallel,
one would theoretically have to extend them to an infinite distance
These axioms are summarized for quick reference in the back of the book on
page 388.
This is a form known as Playfair’s axiom, rather than the version of the
postulate originally given by Euclid.
Chapter 1
A geometrical theory of spacetime
and make sure that they never crossed. In the first 28 theorems of
the Elements, Euclid restricts himself entirely to propositions that
can be proved based on the more secure first four postulates. The
more general geometry defined by omitting the parallel postulate is
known as absolute geometry.
What kind of geometry is likely to be applicable to general relativity? We can see immediately that Euclidean geometry, or even
absolute geometry, would be far too specialized. We have in mind
the description of events that are points in both space and time.
Confining ourselves for ease of visualization to one dimension worth
of space, we can certainly construct a plane described by coordinates (t, x), but imposing Euclid’s postulates on this plane results
in physical nonsense. Space and time are physically distinguishable
from one another. But postulates 3 and 4 describe a geometry in
which distances measured along non-parallel axes are comparable,
and figures may be freely rotated without affecting the truth or
falsehood of statements about them; this is only appropriate for a
physical description of different spacelike directions, as in an (x, y)
plane whose two axes are indistinguishable.
We need to throw most of the specialized apparatus of Euclidean
geometry overboard. Once we’ve stripped our geometry to a bare
minimum, then we can go back and build up a different set of equipment that will be better suited to relativity.
The stripped-down geometry we want is called ordered geometry,
and was developed by Moritz Pasch around 1882. As suggested by
the parable of Alice and Betty, ordered geometry does not have
any global, all-encompassing system of measurement. When Betty
goes on her trip, she traces out a particular path through the space
of events, and Alice, staying at home, traces another. Although
events play out in cause-and-effect order along each of these paths,
we do not expect to be able to measure times along paths A and B
and have them come out the same. This is how ordered geometry
works: points can be put in a definite order along any particular
line, but not along different lines. Of the four primitive concepts
used in Euclid’s E1-E5 — point, line, circle, and angle — only the
non-metrical notions of point (i.e., event) and line are relevant in
ordered geometry. In a geometry without measurement, there is no
concept of measuring distance (hence no compasses or circles), or of
measuring angles. The notation [ABC] indicates that event B lies
on a line segment joining A and C, and is strictly between them.
The axioms of ordered geometry are as follows:8
The axioms are summarized for convenient reference in the back of the book
on page 388. This is meant to be an informal, readable summary of the system,
pitched to the same level of looseness as Euclid’s E1-E5. Modern mathematicians
have found that systems like these actually need quite a bit more technical
machinery to be perfectly rigorous, so if you look up an axiomatization of ordered
geometry, or a modern axiomatization of Euclidean geometry, you’ll typically
Section 1.4
Ordered geometry
O1 Two events determine a line.
O2 Line segments can be extended: given A and B, there is at
least one event such that [ABC] is true.
O3 Lines don’t wrap around: if [ABC] is true, then [BCA] is false.
a / Axioms O2 (left) and O3
O4 Betweenness: For any three distinct events A, B, and C lying
on the same line, we can determine whether or not B is between
A and C (and by statement 3, this ordering is unique except
for a possible over-all reversal to form [CBA]).
O1-O2 express the same ideas as Euclid’s E1-E2. Not all lines
in the system will correspond physically to chains of causality; we
could have a line segment that describes a snapshot of a steel chain,
and O3-O4 then say that the order of the links is well defined. But
O3 and O4 also have clear physical significance for lines describing
causality. O3 forbids time travel paradoxes, like going back in time
and killing our own grandmother as a child; figure a illustrates why a
violation of O3 is referred to as a closed timelike curve. O4 says that
events are guaranteed to have a well-defined cause-and-effect order
only if they lie on the same line. This is completely different from
the attitude expressed in Newton’s famous statement: “Absolute,
true and mathematical time, of itself, and from its own nature flows
equably without regard to anything external . . . ”
If you’re dismayed by the austerity of a system of geometry without any notion of measurement, you may be more appalled to learn
that even a system as weak as ordered geometry makes some statements that are too strong to be completely correct as a foundation
for relativity. For example, if an observer falls into a black hole, at
some point he will reach a central point of infinite density, called a
singularity. At this point, his chain of cause and effect terminates,
violating O2. It is also an open question whether O3’s prohibition
on time-loops actually holds in general relativity; this is Stephen
Hawking’s playfully named chronology protection conjecture. We’ll
also see that in general relativity O1 is almost always true, but there
are exceptions.
b / Stephen
1.5 The equivalence principle
find a much more lengthy list of axioms than the ones presented here. The
axioms I’m omitting take care of details like making sure that there are more
than two points in the universe, and that curves can’t cut through one another
without intersecting. The classic, beautifully written book on these topics is
H.S.M. Coxeter’s Introduction to Geometry, which is “introductory” in the sense
that it’s the kind of book a college math major might use in a first upper-division
course in geometry.
Chapter 1
A geometrical theory of spacetime
1.5.1 Proportionality of inertial and gravitational mass
What physical interpretation should we give to the “lines” described in ordered geometry? Galileo described an experiment (which
he may or may not have actually performed) in which he simultaneously dropped a cannonball and a musket ball from a tall tower. The
two objects hit the ground simultaneously, disproving Aristotle’s assertion that objects fell at a speed proportional to their weights. On
a graph of spacetime with x and t axes, the curves traced by the two
objects, called their world-lines, are identical parabolas. (The paths
of the balls through x − y − z space are straight, not curved.) One
way of explaining this observation is that what we call “mass” is really two separate things, which happen to be equal. Inertial mass,
which appears in Newton’s a = F/m, describes how difficult it is
to accelerate an object. Gravitational mass describes the strength
with which gravity acts. The cannonball has a hundred times more
gravitational mass than the musket ball, so the force of gravity acting on it is a hundred times greater. But its inertial mass is also
precisely a hundred times greater, so the two effects cancel out, and
it falls with the same acceleration. This is a special property of the
gravitational force. Electrical forces, for example, do not behave
this way. The force that an object experiences in an electric field
is proportional to its charge, which is unrelated to its inertial mass,
so different charges placed in the same electric field will in general
have different motions.
a / The cannonball and the musketball have identical parabolic
On this type of
space-time plot, space is conventionally shown on the horizontal
axis, so the tower has to be
depicted on its side.
1.5.2 Geometrical treatment of gravity
Einstein realized that this special property of the gravitational
force made it possible to describe gravity in purely geometrical
terms. We define the world-lines of small9 objects acted on by gravity to be the lines described by the axioms of the geometry. Since
we normally think of the “lines” described by Euclidean geometry
and its kin as straight lines, this amounts to a redefinition of what
it means for a line to be straight. By analogy, imagine stretching a
piece of string taut across a globe, as we might do in order to plan
an airplane flight or aim a directional radio antenna. The string
may not appear straight as viewed from the three-dimensional Euclidean space in which the globe is embedded, but it is as straight as
possible in the sense that it is the path followed by a radio wave,10
or by an airplane pilot who keeps her wings level and her rudder
straight. The world-“line” of an object acted on by nongravitational forces is not considered to be a straight “line” in the sense of
The reason for the restriction to small objects is essentially gravitational
radiation. The object should also be electrically neutral, and neither the object nor the surrounding spacetime should contain any exotic forms of negative
energy. This is discussed in more detail on p. 280. See also problem 1 on p. 344.
Radio waves in the HF band tend to be trapped between the ground and
the ionosphere, causing them to curve over the horizon, allowing long-distance
Section 1.5
b / A piece of string held taut
on a globe forms a geodesic
from Mexico City to London.
Although it appears curved, it
is the analog of a straight line
in the non-Euclidean geometry
confined to the surface of the
Earth. Similarly, the world-lines of
figure a appear curved, but they
are the analogs of straight lines
in the non-Euclidean geometry
used to describe gravitational
fields in general relativity.
The equivalence principle
O1-O4. When necessary, one eliminates this ambiguity in the overloaded term “line” by referring to the lines of O1-O4 as geodesics.
The world-line of a low-mass object acted on only by gravity is one
type of geodesic.11
c / Loránd Eötvös (1848-1919).
We can now see the deep physical importance of statement O1,
that two events determine a line. To predict the trajectory of a
golf ball, we need to have some initial data. For example, we could
measure event A when the ball breaks contact with the club, and
event B an infinitesimal time after A.12 This pair of observations can
be thought of as fixing the ball’s initial position and velocity, which
should be enough to predict a unique world-line for the ball, since
relativity is a deterministic theory. With this interpretation, we can
also see why it is not necessarily a disaster for the theory if O1
fails sometimes. For example, event A could mark the launching of
two satellites into circular orbits from the same place on the Earth,
heading in opposite directions, and B could be their subsequent
collision on the opposite side of the planet. Although this violates
O1, it doesn’t violate determinism. Determinism only requires the
validity of O1 for events infinitesimally close together. Even for
randomly chosen events far apart, the probability that they will
violate O1 is zero.
1.5.3 Eötvös experiments
d / If the geodesics defined
by an airplane and a radio wave
differ from one another, then
it is not possible to treat both
problems exactly using the same
geometrical theory. In general
relativity, this would be analogous
to a violation of the equivalence
General relativity’s
validity as a purely geometrical
theory of gravity requires that the
equivalence principle be exactly
satisfied in all cases.
Einstein’s entire system breaks down if there is any violation, no
matter how small, of the proportionality between inertial and gravitational mass, and it therefore becomes very interesting to search
experimentally for such a violation. For example, we might wonder whether neutrons and protons had slightly different ratios of
gravitational and inertial mass, which in a Galileo-style experiment
would cause a small difference between the acceleration of a lead
weight, with a large neutron-to-proton ratio, and a wooden one,
which consists of light elements with nearly equal numbers of neutrons and protons. The first high-precision experiments of this type
were performed by Eötvös around the turn of the twentieth century,
and they verified the equivalence of inertial and gravitational mass
to within about one part in 108 . These are generically referred to
as Eötvös experiments.
Figure e shows a strategy for doing Eötvös experiments that allowed a test to about one part in 1012 . The top panel is a simplified
version. The platform is balanced, so the gravitational masses of
the two objects are observed to be equal. The objects are made
of different substances. If the equivalence of inertial and gravitational mass fails to hold for these two substances, then the force
of gravity on each mass will not be exact proportion to its inertia,
and the platform will experience a slight torque as the earth spins.
Chapter 1
For more justification of this statement, see ch. 9, problem 1, on page 344.
Regarding infinitesimals, see p. 93.
A geometrical theory of spacetime
The bottom panel shows a more realistic drawing of an experiment
by Braginskii and Panov.13 The whole thing was encased in a tall
vacuum tube, which was placed in a sealed basement whose temperature was controlled to within 0.02 ◦ C. The total mass of the
platinum and aluminum test masses, plus the tungsten wire and
the balance arms, was only 4.4 g. To detect tiny motions, a laser
beam was bounced off of a mirror attached to the wire. There was
so little friction that the balance would have taken on the order of
several years to calm down completely after being put in place; to
stop these vibrations, static electrical forces were applied through
the two circular plates to provide very gentle twists on the ellipsoidal
mass between them.
Equivalence of gravitational fields and accelerations
One consequence of the Eötvös experiments’ null results is that
it is not possible to tell the difference between an acceleration and
a gravitational field. At certain times during Betty’s field trip, she
feels herself pressed against her seat, and she interprets this as evidence that she’s in a space vessel that is undergoing violent accelerations and decelerations. But it’s equally possible that Tutorbot
has simply arranged for her capsule to be hung from a rope and
dangled into the gravitational field of a planet. Suppose that the
first explanation is correct. The capsule is initially at rest in outer
space, where there is no gravity. Betty can release a pencil and a
lead ball in the air inside the cabin, and they will stay in place. The
capsule then accelerates, and to Betty, who has adopted a frame
of reference tied to its deck, ceiling and walls, it appears that the
pencil and the ball fall to the deck. They are guaranteed to stay
side by side until they hit the deckplates, because in fact they aren’t
accelerating; they simply appear to accelerate, when in reality it’s
the deckplates that are coming up and hitting them. But now consider the second explanation, that the capsule has been dipped into
a gravitational field. The ball and the pencil will still fall side by
side to the floor, because they have the same ratio of gravitational
to inertial mass.
e / An
Top: simplified version. Bottom:
realistic version by Braginskii and
Panov. (Drawing after Braginskii
and Panov.)
1.5.4 The equivalence principle
This leads to one way of stating a central principle of relativity
known as the equivalence principle: Accelerations and gravitational
fields are equivalent. There is no experiment that can distinguish
one from the other.14
To see what a radical departure this is, we need to compare with
the completely different picture presented by Newtonian physics and
special relativity. Newton’s law of inertia states that “Every object
V.B. Braginskii and V.I. Panov, Soviet Physics JETP 34, 463 (1972).
This statement of the equivalence principle is summarized, along with some
other forms of it to be encountered later, in the back of the book on page 389.
Section 1.5
The equivalence principle
perseveres in its state of rest, or of uniform motion in a straight
line, unless it is compelled to change that state by forces impressed
thereon.”15 Newton’s intention here was to clearly state a contradiction of Aristotelian physics, in which objects were supposed to
naturally stop moving and come to rest in the absence of a force. For
Aristotle, “at rest” meant at rest relative to the Earth, which represented a special frame of reference. But if motion doesn’t naturally
stop of its own accord, then there is no longer any way to single out
one frame of reference, such as the one tied to the Earth, as being
special. An equally good frame of reference is a car driving in a
straight line down the interstate at constant speed. The earth and
the car both represent valid inertial frames of reference, in which
Newton’s law of inertia is valid. On the other hand, there are other,
noninertial frames of reference, in which the law of inertia is violated. For example, if the car decelerates suddenly, then it appears
to the people in the car as if their bodies are being jerked forward,
even though there is no physical object that could be exerting any
type of forward force on them. This distinction between inertial and
noninertial frames of reference was carried over by Einstein into his
theory of special relativity, published in 1905.
But by the time he published the general theory in 1915, Einstein
had realized that this distinction between inertial and noninertial
frames of reference was fundamentally suspect. How do we know
that a particular frame of reference is inertial? One way is to verify
that its motion relative to some other inertial frame, such as the
Earth’s, is in a straight line and at constant speed. But how does
the whole thing get started? We need to bootstrap the process
with at least one frame of reference to act as our standard. We
can look for a frame in which the law of inertia is valid, but now
we run into another difficulty. To verify that the law of inertia
holds, we have to check that an observer tied to that frame doesn’t
see objects accelerating for no reason. The trouble here is that by
the equivalence principle, there is no way to determine whether the
object is accelerating “for no reason” or because of a gravitational
force. Betty, for example, cannot tell by any local measurement
(i.e., any measurement carried out within the capsule) whether she
is in an inertial or a noninertial frame.
We could hope to resolve the ambiguity by making non-local
measurements instead. For example, if Betty had been allowed to
look out a porthole, she could have tried to tell whether her capsule
was accelerating relative to the stars. Even this possibility ends up
not being satisfactory. The stars in our galaxy are moving in circular
orbits around the galaxy. On an even larger scale, the universe is
expanding in the aftermath of the Big Bang. It spent about the
first half of its history decelerating due to gravitational attraction,
but the expansion is now observed to be accelerating, apparently
Chapter 1
paraphrased from a translation by Motte, 1729
A geometrical theory of spacetime
f / Wouldn’t it be nice if we could define the meaning of a Newtonian inertial frame of reference? Newton makes it sound easy: to define an inertial frame, just find some object that is not accelerating because it is
not being acted on by any external forces. But what object would we use? The earth? The “fixed stars?” Our
galaxy? Our supercluster of galaxies? All of these are accelerating — relative to something.
due to a poorly understood phenomenon referred to by the catch-all
term “dark energy.” In general, there is no distant background of
physical objects in the universe that is not accelerating.
Lorentz frames
The conclusion is that we need to abandon the entire distinction
between Newton-style inertial and noninertial frames of reference.
The best that we can do is to single out certain frames of reference
defined by the motion of objects that are not subject to any nongravitational forces. A falling rock defines such a frame of reference.
In this frame, the rock is at rest, and the ground is accelerating. The
rock’s world-line is a straight line of constant x = 0 and varying t.
Such a free-falling frame of reference is called a Lorentz frame. The
frame of reference defined by a rock sitting on a table is an inertial
frame of reference according to the Newtonian view, but it is not a
Lorentz frame.
In Newtonian physics, inertial frames are preferable because they
make motion simple: objects with no forces acting on them move
along straight world-lines. Similarly, Lorentz frames occupy a privileged position in general relativity because they make motion simple:
objects move along “straight” world-lines if they have no nongravitational forces acting on them.
The artificial horizon
Example: 1
The pilot of an airplane cannot always easily tell which way is up.
The horizon may not be level simply because the ground has an
actual slope, and in any case the horizon may not be visible if the
weather is foggy. One might imagine that the problem could be
solved simply by hanging a pendulum and observing which way
it pointed, but by the equivalence principle the pendulum cannot
tell the difference between a gravitational field and an acceleration of the aircraft relative to the ground — nor can any other
accelerometer, such as the pilot’s inner ear. For example, when
Section 1.5
g / An artificial horizon.
The equivalence principle
the plane is turning to the right, accelerometers will be tricked into
believing that “down” is down and to the left. To get around this
problem, airplanes use a device called an artificial horizon, which
is essentially a gyroscope. The gyroscope has to be initialized
when the plane is known to be oriented in a horizontal plane. No
gyroscope is perfect, so over time it will drift. For this reason the
instrument also contains an accelerometer, and the gyroscope is
automatically restored to agreement with the accelerometer, with
a time-constant of several minutes. If the plane is flown in circles for several minutes, the artificial horizon will be fooled into
indicating that the wrong direction is vertical.
No antigravity
Example: 2
This whole chain of reasoning was predicated on the null results
of Eötvös experiments. In the Rocky and Bullwinkle cartoons,
there is a non-Eötvösian substance called upsidasium, which falls
up instead of down. Its ratio of gravitational to inertial mass is
apparently negative. If such a substance could be found, it would
falsify the equivalence principle. Cf. example 10, p. 283.
Operational definition of a Lorentz frame
h / Bars of upsidasium are
kept in special warehouses,
bolted to the ground. Copyright
Jay Ward Productions, used
under U.S. fair use excpetion to
copyright law.
We can define a Lorentz frame in operational terms using an idealized variation (figure i) on a device actually built by Harold Waage
at Princeton as a lecture demonstration to be used by his partner
in crime John Wheeler. Build a sealed chamber whose contents are
isolated from all nongravitational forces. Of the four known forces
of nature, the three we need to exclude are the strong nuclear force,
the weak nuclear force, and the electromagnetic force. The strong
nuclear force has a range of only about 1 fm (10−15 m), so to exclude it we merely need to make the chamber thicker than that, and
also surround it with enough paraffin wax to keep out any neutrons
that happen to be flying by. The weak nuclear force also has a short
range, and although shielding against neutrinos is a practical impossibility, their influence on the apparatus inside will be negligible. To
shield against electromagnetic forces, we surround the chamber with
a Faraday cage and a solid sheet of mu-metal. Finally, we make sure
that the chamber is not being touched by any surrounding matter,
so that short-range residual electrical forces (sticky forces, chemical bonds, etc.) are excluded. That is, the chamber cannot be
supported; it is free-falling.
Crucially, the shielding does not exclude gravitational forces.
There is in fact no known way of shielding against gravitational effects such as the attraction of other masses (example 10, p. 283) or
the propagation of gravitational waves (ch. 9). Because the shielding is spherical, it exerts no gravitational force of its own on the
apparatus inside.
Inside, an observer carries out an initial calibration by firing
Chapter 1
A geometrical theory of spacetime
i / The spherical chamber, shown
in a cutaway view, has layers
of shielding to exclude all known
nongravitational forces.
the chamber has been calibrated
by marking the three dashed-line
trajectories under free-fall conditions, an observer inside the
chamber can always tell whether
she is in a Lorentz frame.
bullets along three Cartesian axes and tracing their paths, which
she defines to be linear.
We’ve gone to elaborate lengths to show that we can really determine, without reference to any external reference frame, that
the chamber is not being acted on by any nongravitational forces,
so that we know it is free-falling. In addition, we also want the
observer to be able to tell whether the chamber is rotating. She
could look out through a porthole at the stars, but that would be
missing the whole point, which is to show that without reference to
any other object, we can determine whether a particular frame is a
Lorentz frame. One way to do this would be to watch for precession
of a gyroscope. Or, without having to resort to additional apparatus, the observer can check whether the paths traced by the bullets
change when she changes the muzzle velocity. If they do, then she
infers that there are velocity-dependent Coriolis forces, so she must
be rotating. She can then use flywheels to get rid of the rotation,
and redo the calibration.
After the initial calibration, she can always tell whether or not
she is in a Lorentz frame. She simply has to fire the bullets, and see
whether or not they follow the precalibrated paths. For example,
she can detect that the frame has become non-Lorentzian if the
chamber is rotated, allowed to rest on the ground, or accelerated by
a rocket engine.
It may seem that the detailed construction of this elaborate
thought-experiment does nothing more than confirm something obvious. It is worth pointing out, then, that we don’t really know
whether it works or not. It works in general relativity, but there are
Section 1.5
The equivalence principle
other theories of gravity, such as Brans-Dicke gravity (p. 322), that
are also consistent with all known observations, but in which the
apparatus in figure i doesn’t work. Two of the assumptions made
above fail in this theory: gravitational shielding effects exist, and
Coriolis effects become undetectable if there is not enough other
matter nearby.
Locality of Lorentz frames
It would be convenient if we could define a single Lorentz frame
that would cover the entire universe, but we can’t. In figure j, two
girls simultaneously drop down from tree branches — one in Los Angeles and one in Mumbai. The girl free-falling in Los Angeles defines
a Lorentz frame, and in that frame, other objects falling nearby will
also have straight world-lines. But in the LA girl’s frame of reference, the girl falling in Mumbai does not have a straight world-line:
she is accelerating up toward the LA girl with an acceleration of
about 2g.
j / Two
A second way of stating the equivalence principle is that it is
always possible to define a local Lorentz frame in a particular neighborhood of spacetime.16 It is not possible to do so on a universal
The locality of Lorentz frames can be understood in the analogy of the string stretched across the globe. We don’t notice the
curvature of the Earth’s surface in everyday life because the radius
of curvature is thousands of kilometers. On a map of LA, we don’t
notice any curvature, nor do we detect it on a map of Mumbai, but
it is not possible to make a flat map that includes both LA and
Mumbai without seeing severe distortions.
The meanings of words evolve over time, and since relativity is
now a century old, there has been some confusing semantic drift
in its nomenclature. This applies both to “inertial frame” and to
“special relativity.”
k / One planet rotates about
its axis and the other does not.
As discussed in more detail on
p. 115, Einstein believed that
general relativity was even more
radically egalitarian about frames
of reference than it really is. He
thought that if the planets were
alone in an otherwise empty
universe, there would be no way
to tell which planet was really
rotating and which was not, so
that B’s tidal bulge would have
to disappear. There would be no
way to tell which planet’s surface
was a Lorentz frame.
Chapter 1
Early formulations of general relativity never refer to “inertial
frames,” “Lorentz frames,” or anything else of that flavor. The very
first topic in Einstein’s original systematic presentation of the theory17 is an example (figure k) involving two planets, the purpose
of which is to convince the reader that all frames of reference are
created equal, and that any attempt to make some of them into
second-class citizens is invidious. Other treatments of general relativity from the same era follow Einstein’s lead.18 The trouble is
This statement of the equivalence principle is summarized, along with some
other forms of it, in the back of the book on page 389.
Einstein, “The Foundation of the General Theory of Relativity,” 1916. An
excerpt is given on p. 360.
Two that I believe were relatively influential are Born’s 1920 Einstein’s The-
A geometrical theory of spacetime
that this example is more a statement of Einstein’s aspirations for
his theory than an accurate depiction of the physics that it actually implies. General relativity really does allow an unambiguous
distinction to be made between Lorentz frames and non-Lorentz
frames, as described on p. 26. Einstein’s statement should have
been weaker: the laws of physics (such as the Einstein field equation, p. 263) are the same in all frames (Lorentz or non-Lorentz).
This is different from the situation in Newtonian mechanics and special relativity, where the laws of physics take on their simplest form
only in Newton-inertial frames.
Because Einstein didn’t want to make distinctions between frames,
we ended up being saddled with inconvenient terminology for them.
The least verbally awkward choice is to hijack the term “inertial,”
redefining it from its Newtonian meaning. We then say that the
Earth’s surface is not an inertial frame, in the context of general
relativity, whereas in the Newtonian context it is an inertial frame
to a very good approximation. This usage is fairly standard,19 but
would have made Newton confused and Einstein unhappy. If we
follow this usage, then we may sometimes have to say “Newtonianinertial” or “Einstein-inertial.” A more awkward, but also more
precise, term is “Lorentz frame,” as used in this book; this seems to
be widely understood.20
The distinction between special and general relativity has undergone a similar shift over the decades. Einstein originally defined the
distinction in terms of the admissibility of accelerated frames of reference. This, however, puts us in the absurd position of saying that
special relativity, which is supposed to be a generalization of Newtonian mechanics, cannot handle accelerated frames of reference in the
same way that Newtonian mechanics can. In fact both Newtonian
mechanics and special relativity treat Newtonian-noninertial frames
of reference in the same way: by modifying the laws of physics so
that they do not take on their most simple form (e.g., violating Newton’s third law), while retaining the ability to change coordinates
back to a preferred frame in which the simpler laws apply. It was
realized fairly early on21 that the important distinction was between
special relativity as a theory of flat spacetime, and general relativity
as a theory that described gravity in terms of curved spacetime. All
relativists writing since about 1950 seem to be in agreement on this
more modern redefinition of the terms.22
ory of Relativity and Eddington’s 1924 The Mathematical Theory of Relativity.
Born follows Einstein’s “Foundation” paper slavishly. Eddington seems only to
mention inertial frames in a few places where the context is Newtonian.
Misner, Thorne, and Wheeler, Gravitation, 1973, p. 18
ibid, p. 19
Eddington, op. cit.
Misner, Thorne, and Wheeler, op. cit., pp.163-164. Penrose, The Road to
Reality, 2004, p. 422. Taylor and Wheeler, Spacetime Physics, 1992, p. 132.
Schutz, A First Course in General Relativity, 2009, pp. 3, 141. Hobson, General
Section 1.5
The equivalence principle
In an accelerating frame, the equivalence principle tells us that
measurements will come out the same as if there were a gravitational
field. But if the spacetime is flat, describing it in an accelerating
frame doesn’t make it curved. (Curvature is a physical property of
spacetime, and cannot be changed from zero to nonzero simply by
a choice of coordinates.) Thus relativity allows us to have gravitational fields in flat space — but only for certain special configurations like this one. Special relativity is capable of operating just fine
in this context. For example, Chung et al.23 did a high-precision
test of special relativity in 2009 using a matter interferometer in a
vertical plane, specifically in order to test whether there was any
violation of special relativity in a uniform gravitational field. Their
experiment is interpreted purely as a test of special relativity, not
general relativity.
Chiao’s paradox
The remainder of this subsection deals with the subtle question of whether and how the equivalence principle can be applied to
charged particles. You may wish to skip it on a first reading. The
short answer is that using the equivalence principle to make conclusions about charged particles is like the attempts by slaveholders
and abolitionists in the 19th century U.S. to support their positions
based on the Bible: you can probably prove whichever conclusion
was the one you set out to prove.
l / Chiao’s paradox: a charged
particle and a neutral particle are
in orbit around the earth. Will the
charged particle radiate, violating
the equivalence principle?
The equivalence principle is not a single, simple, mathematically well defined statement.24 As an example of an ambiguity that
is still somewhat controversial, 90 years after Einstein first proposed
the principle, consider the question of whether or not it applies to
charged particles. Raymond Chiao25 proposes the following thought
experiment, which I’ll refer to as Chiao’s paradox. Let a neutral particle and a charged particle be set, side by side, in orbit around the
earth. Assume (unrealistically) that the space around the earth has
no electric or magnetic field. If the equivalence principle applies
regardless of charge, then these two particles must go on orbiting
amicably, side by side. But then we have a violation of conservation
of energy, since the charged particle, which is accelerating, will radiate electromagnetic waves (with very low frequency and amplitude).
It seems as though the particle’s orbit must decay.
The resolution of the paradox, as demonstrated by hairy calculations26 is interesting because it exemplifies the local nature of
Relativity: An Introduction for Physicists, 2005, sec. 1.14.
A good recent discussion of this is “Theory of gravitation theories: a noprogress report,” Sotiriou, Faraoni, and Liberati,
The first detailed calculation appears to have been by Cécile and Bryce
DeWitt, “Falling Charges,” Physics 1 (1964) 3. This paper is unfortunately
Chapter 1
A geometrical theory of spacetime
the equivalence principle. When a charged particle moves through a
gravitational field, in general it is possible for the particle to experience a reaction from its own electromagnetic fields. This might seem
impossible, since an observer in a frame momentarily at rest with
respect to the particle sees the radiation fly off in all directions at
the speed of light. But there are in fact several different mechanisms
by which a charged particle can be reunited with its long-lost electromagnetic offspring. An example (not directly related to Chiao’s
scenario) is the following.
Bring a laser very close to a black hole, but not so close that it
has strayed inside the event horizon, which is the spherical point of
no return from within which nothing can escape. Example 15 on
page 65 gives a plausibility argument based on Newtonian physics
that the radius27 of the event horizon should be something like rH =
GM/c2 , and section 6.3.2 on page 224 derives the relativistically
correct factor of 2 in front, so that rH = 2GM/c2 . It turns out
that at r = (3/2)RH , a ray of light can have a circular orbit around
the black hole. Since this is greater than RH , we can, at least in
theory, hold the laser stationary at this value of r using a powerful
rocket engine. If we point the laser in the azimuthal direction, its
own beam will come back and hit it.
Since matter can experience a back-reaction from its own electromagnetic radiation, it becomes plausible how the paradox can be
resolved. The equivalence principle holds locally, i.e., within a small
patch of space and time. If Chiao’s charged and neutral particle are
released side by side, then they will obey the equivalence principle
for at least a certain amount of time — and “for at least a certain
amount of time” is all we should expect, since the principle is local.
But after a while, the charged particle will start to experience a
back-reaction from its own electromagnetic fields, and this causes
its orbit to decay, satisfying conservation of energy. Since Chiao’s
particles are orbiting the earth, and the earth is not a black hole, the
mechanism clearly can’t be as simple as the one described above,
but Grøn and Næss show that there are similar mechanisms that
can apply here, e.g., scattering of light waves by the nonuniform
gravitational field.
It is worth keeping in mind the DeWitts’ caution that “The questions answered by this investigation are of conceptual interest only,
since the forces involved are far too small to be detected experimenvery difficult to obtain now. A more recent treatment by Grøn and Næss is
accessible at A full exposition of the techniques
is given by Poisson, “The Motion of Point Particles in Curved Spacetime,” www.
Because relativity describes gravitational fields in terms of curvature of
spacetime, the Euclidean relationship between the radius and circumference of
a circle fails here. The r coordinate should be understood here not as the radius
measured from the center but as the circumference divided by 2π.
Section 1.5
The equivalence principle
tally” (see problem 8, p. 39).
1.5.5 Gravitational red-shifts
Starting on page 15, we saw experimental evidence that the rate
of flow of time changes with height in a gravitational field. We can
now see that this is required by the equivalence principle.
m / 1.
A photon is emitted
upward from the floor of the elevator. The elevator accelerates
upward. 2. By the time the
photon is detected at the ceiling,
the elevator has changed its
velocity, so the photon is detected
with a Doppler shift.
n / An electromagnetic wave
strikes an ohmic surface. The
wave’s electric field excites an
oscillating current density J. The
wave’s magnetic field then acts
on these currents, producing
a force in the direction of the
wave’s propagation. This is a
pre-relativistic argument that
light must possess inertia. The
first experimental confirmation of
this prediction is shown in figure
o. See Nichols and Hull, ”The
pressure due to radiation,” Phys.
Rev. (Series I) 17 (1903) 26.
By the equivalence principle, there is no way to tell the difference
between experimental results obtained in an accelerating laboratory
and those found in a laboratory immersed in a gravitational field.28
In a laboratory accelerating upward, a photon emitted from the
floor and would be Doppler-shifted toward lower frequencies when
observed at the ceiling, because of the change in the receiver’s velocity during the photon’s time of flight. The effect is given by
∆E/E = ∆f /f = ay/c2 , where a is the lab’s acceleration, y is the
height from floor to ceiling, and c is the speed of light.
Self-check: Verify this statement.
By the equivalence principle, we find that when such an experiment is done in a gravitational field g, there should be a gravitational
effect on the energy of a photon equal to ∆E/E = gy/c2 . Since the
quantity gy is the gravitational potential (gravitational energy per
unit mass), the photon’s fractional loss of energy is the same as the
(Newtonian) loss of energy experienced by a material object of mass
m and initial kinetic energy mc2 .
The interpretation is as follows. Classical electromagnetism requires that electromagnetic waves have inertia. For example, if a
plane wave strikes an ohmic surface, as in figure n, the wave’s electric field excites oscillating currents in the surface. These currents
then experience a magnetic force from the wave’s magnetic field,
and application of the right-hand rule shows that the resulting force
is in the direction of propagation of the wave. Thus the light wave
acts as if it has momentum. The equivalence principle says that
whatever has inertia must also participate in gravitational interactions. Therefore light waves must have weight, and must lose energy
when they rise through a gravitational field.
Self-check: Verify the application of the right-hand rule described above.
Further interpretation:
• The quantity mc2 is famous, even among people who don’t
know what m and c stand for. This is the first hint of where
it comes from. The full story is given in section 4.2.2.
• The relation p = E/c between the energy and momentum
of a light wave follows directly from Maxwell’s equations, by
Problem 4 on p. 38 verifies, in one specific example, that this way of stating
the equivalence principle is implied by the one on p. 21.
Chapter 1
A geometrical theory of spacetime
the argument above; however, we will see in section 4.2.2 that
according to relativity this relation must hold for any massless
• What we have found agrees with Niels Bohr’s correspondence
principle, which states that when a new physical theory, such
as relativity, replaces an older one, such as Newtonian physics,
the new theory must agree with the old one under the experimental conditions in which the old theory had been verified by
experiments. The gravitational mass of a beam of light with
energy E is E/c2 , and since c is a big number, it is not surprising that the weight of light rays had never been detected
before Einstein trying to detect it.
• This book describes one particular theory of gravity, Einstein’s
theory of general relativity. There are other theories of gravity, and some of these, such as the Brans-Dicke theory, do
just as well as general relativity in agreeing with the presently
available experimental data. Our prediction of gravitational
Doppler shifts of light only depended on the equivalence principle, which is one ingredient of general relativity. Experimental
tests of this prediction only test the equivalence principle; they
do not allow us to distinguish between one theory of gravity
and another if both theories incorporate the equivalence principle.
• If an object such as a radio transmitter or an atom in an excited state emits an electromagnetic wave with a frequency f ,
then the object can be considered to be a type of clock. We
can therefore interpret the gravitational red-shift as a gravitational time dilation: a difference in the rate at which time
itself flows, depending on the gravitational potential. This is
consistent with the empirical results presented in section 1.2.1,
p. 15.
o / A simplified drawing of
the 1903 experiment by Nichols
and Hull that verified the predicted momentum of light waves.
Two circular mirrors were hung
from a fine quartz fiber, inside
an evacuated bell jar. A 150
mW beam of light was shone
on one of the mirrors for 6 s,
producing a tiny rotation, which
was measurable by an optical
lever (not shown). The force was
within 0.6% of the theoretically
predicted value of 0.001 µN.
For comparison, a short clipping
of a single human hair weighs
∼ 1 µN.
Chiao’s paradox revisited
Example: 3
The equivalence principle says that electromagnetic waves have
gravitational mass as well as inertial mass, so it seems clear that
the same must hold for static fields. In Chiao’s paradox (p. 39), the
orbiting charged particle has an electric field that extends out to
infinity. When we measure the mass of a charged particle such as
an electron, there is no way to separate the mass of this field from
a more localized contribution. The electric field “falls” through the
gravitational field, and the equivalence principle, which is local,
cannot guarantee that all parts of the field rotate uniformly about
the earth, even in distant parts of the universe. The electric field
pattern becomes distorted, and this distortion causes a radiation
reaction which back-reacts on the particle, causing its orbit to decay.
Section 1.5
The equivalence principle
1.5.6 The Pound-Rebka experiment
The 1959 Pound-Rebka experiment at Harvard29 was the first
quantitative test of the equivalence principle to be carried out under
controlled conditions, and in this section we will discuss it in detail.
When y is on the order of magnitude of the height of a building,
the value of ∆E/E = gy/c2 is ∼ 10−14 , so an extremely highprecision experiment is necessary in order to detect a gravitational
red-shift. A number of other effects are big enough to obscure it entirely, and must somehow be eliminated or compensated for. These
are listed below, along with their orders of magnitude in the experimental design finally settled on by Pound and Rebka.
p / The
(1) Classical Doppler broadening due to temperature. Thermal motion causes Doppler shifts of
emitted photons, corresponding to the random
component of the emitting atom’s velocity vector along the direction of emission.
(2) The recoil Doppler shift. When an atom
emits a photon with energy E and momentum
p = E/c, conservation of momentum requires
that the atom recoil with momentum p = −E/c
and energy p2 /2m. This causes a downward
Doppler shift of the energy of the emitted photon. A similar effect occurs on absorption, doubling the problem.
(3) Natural line width. The Heisenberg uncertainty principle says that a state with a half-life
τ must have an uncertainty in its energy of at
least ∼ h/τ , where h is Planck’s constant.
(4) Special-relativistic Doppler shift due to temperature. Section 1.2 presented experimental evidence that time flows at a different rate depending on the motion of the observer. Therefore
the thermal motion of an atom emitting a photon has an effect on the frequency of the photon,
even if the atom’s motion is not along the line of
emission. The equations needed in order to calculate this effect will not be derived until section
2.2; a quantitative estimate is given in example
14 on page 62. For now, we only need to know
that this leads to a temperature-dependence in
the average frequency of emission, in addition
to the broadening of the bell curve described by
effect (1) above.
q / Emission of 14 keV gammarays by 57 Fe. The parent nucleus
Co absorbs an electron and
undergoes a weak-force decay
process that converts it into 57 Fe,
in an excited state. With 85%
probability, this state decays to a
state just above the ground state,
with an excitation energy of 14
keV and a half-life of 10−7 s. This
state finally decays, either by
gamma emission or emission of
an internal conversion electron,
to the ground state.
Chapter 1
∼ 10−12
∼ 10−12
∼ 10−14 per
degree C
The most straightforward way to mitigate effect (1) is to use
photons emitted from a solid. At first glance this would seem like
∼ 10−6
Phys. Rev. Lett. 4 (1960) 337
A geometrical theory of spacetime
a bad idea, since electrons in a solid emit a continuous spectrum of
light, not a discrete spectrum like the ones emitted by gases; this
is because we have N electrons, where N is on the order of Avogadro’s number, all interacting strongly with one another, so by the
correspondence principle the discrete quantum-mechanical behavior
must be averaged out. But the protons and neutrons within one
nucleus do not interact much at all with those in other nuclei, so
the photons emitted by a nucleus do have a discrete spectrum. The
energy scale of nuclear excitations is in the keV or MeV range, so
these photons are x-rays or gamma-rays. Furthermore, the timescale of the random vibrations of a nucleus in a solid are extremely
short. For a velocity on the order of 100 m/s, and vibrations with an
amplitude of ∼ 10−10 m, the time is about 10−12 s. In many cases,
this is much shorter than the half-life of the excited nuclear state
emitting the gamma-ray, and therefore the Doppler shift averages
out to nearly zero.
Effect (2) is still much bigger than the 10−14 size of the effect to
be measured. It can be avoided by exploiting the Mössbauer effect,
in which a nucleus in a solid substance at low temperature emits or
absorbs a gamma-ray photon, but with significant probability the
recoil is taken up not by the individual nucleus but by a vibration
of the atomic lattice as a whole. Since the recoil energy varies as
p2 /2m, the large mass of the lattice leads to a very small dissipation
of energy into the recoiling lattice. Thus if a photon is emitted and
absorbed by identical nuclei in a solid, and for both emission and
absorption the recoil momentum is taken up by the lattice as a
whole, then there is a negligible energy shift. One must pick an
isotope that emits photons with energies of about 10-100 keV. Xrays with energies lower than about 10 keV tend to be absorbed
strongly by matter and are difficult to detect, whereas for gammaray energies & 100 keV the Mössbauer effect is not sufficient to
eliminate the recoil effect completely enough.
If the Mössbauer effect is carried out in a horizontal plane, resonant absorption occurs. When the source and absorber are aligned
vertically, p, gravitational frequency shifts should cause a mismatch,
destroying the resonance. One can move the source at a small velocity (typically a few mm/s) in order to add a Doppler shift onto
the frequency; by determining the velocity that compensates for the
gravitational effect, one can determine how big the gravitational
effect is.
The typical half-life for deexcitation of a nucleus by emission
of a gamma-ray with energy E is in the nanosecond range. To
measure an gravitational effect at the 10−14 level, one would like to
have a natural line width, (3), with ∆E/E . 10−14 , which would
require a half-life of & 10 µs. In practice, Pound and Rebka found
that other effects, such as (4) and electron-nucleus interactions that
depended on the preparation of the sample, tended to put nuclei
Section 1.5
r / Top:
A graph of velocity
versus time for the source. The
velocity has both a constant
component and an oscillating one
with a frequency of 10-50 Hz.
The constant component vo was
used as a way of determining the
calibration of frequency shift as a
function of count rates. Data were
acquired during the quarter-cycle
periods of maximum oscillatory
velocity, 1 and 2. Bottom: Count
rates as a function of velocity,
for vo = 0 and v1 6= 0. The
dashed curve and black circles
represent the count rates that
would have been observed if
there were no gravitational effect.
The gravitational effect shifts
the resonance curve to one side
(solid curve), resulting in an
asymmetry of the count rates
(open circles). The shift, and the
resulting asymmetry, are greatly
exaggerated for readability; in
reality, the gravitational effect was
500 times smaller than the width
of the resonance curve.
The equivalence principle
in one sample “out of tune” with those in another sample at the
10−13 -10−12 level, so that resonance could not be achieved unless the
natural line width gave ∆E/E & 10−12 . As a result, they settled on
an experiment in which 14 keV gammas were emitted by 57 Fe nuclei
(figure q) at the top of a 22-meter tower, and absorbed by 57 Fe
nuclei at the bottom. The 100-ns half-life of the excited state leads
to ∆E/E ∼ 10−12 . This is 500 times greater than the gravitational
effect to be measured, so, as described in more detail below, the
experiment depended on high-precision measurements of small upand-down shifts of the bell-shaped resonance curve.
The absorbers were seven iron films isotopically enhanced in
applied directly to the faces of seven sodium-iodide scintillation detectors (bottom of figure p). When a gamma-ray impinges
on the absorbers, a number of different things can happen, of which
we can get away with considering only the following: (a) the gammaray is resonantly absorbed in one of the 57 Fe absorbers, after which
the excited nucleus decays by re-emission of another such photon (or
a conversion electron), in a random direction; (b) the gamma-ray
passes through the absorber and then produces ionization directly
in the sodium iodide crystal. In case b, the gamma-ray is detected.
In case a, there is a 50% probability that the re-emitted photon will
come out in the upward direction, so that it cannot be detected.
Thus when the conditions are right for resonance, a reduction in
count rate is expected. The Mössbauer effect never occurs with
100% probability; in this experiment, about a third of the gammas
incident on the absorbers were resonantly absorbed.
57 Fe,
The choice of y = 22 m was dictated mainly by systematic errors. The experiment was limited by the strength of the gamma-ray
source. For a source of a fixed strength, the count rate in the detector at a distance y would be proportional
to y −2 , leading to sta√
tistical errors proportional to 1/ count rate ∝ y. Since the effect
to be measured is also proportional to y, the signal-to-noise ratio
was independent of y. However, systematic effects such as (4) were
easier to monitor and account for when y was fairly large. A lab
building at Harvard happened to have a 22-meter tower, which was
used for the experiment. To reduce the absorption of the gammas
in the 22 meters of air, a long, cylindrical mylar bag full of helium
gas was placed in the shaft, p.
s / Pound and Rebka at the
top and bottom of the tower.
Chapter 1
The resonance was a bell-shaped curve with a minimum at the
natural frequency of emission. Since the curve was at a minimum,
where its derivative was zero, the sensitivity of the count rate to
the gravitational shift would have been nearly zero if the source had
been stationary. Therefore it was necessary to vibrate the source
up and down, so that the emitted photons would be Doppler shifted
onto the shoulders of the resonance curve, where the slope of the
curve was large. The resulting asymmetry in count rates is shown
in figure r. A further effort to cancel out possible systematic effects
A geometrical theory of spacetime
was made by frequently swapping the source and absorber between
the top and bottom of the tower.
For y = 22.6 m, the equivalence principle predicts a fractional
frequency shift due to gravity of 2.46 × 10−15 . Pound and Rebka
measured the shift to be (2.56 ± 0.25) × 10−15 . The results were in
statistical agreement with theory, and verified the predicted size of
the effect to a precision of 10%.
Section 1.5
The equivalence principle
In classical mechanics, one hears the term “the acceleration of
gravity,” which doesn’t literally make sense, since it is objects that
accelerate. Explain why this term’s usefulness is dependent on the
equivalence principle.
The New Horizons space probe communicates with the earth
using microwaves with a frequency of about 10 GHz. Estimate the
sizes of the following frequency shifts in this signal, when the probe
flies by Pluto in 2015, at a velocity of ∼ 10 A.U./year: (a) the
Doppler shift due to the probe’s velocity; (b) the Doppler shift due
to the Earth’s orbital velocity; (c) the gravitational Doppler shift.
Euclid’s axioms E1-E5 (p. 18) do not suffice to prove that
there are an infinite number of points in the plane, and therefore
they need to be supplemented by an extra axiom that states this
(unless one finds the nonstandard realizations with finitely many
points to be interesting enough to study for their own sake). Prove
that the axioms of ordered geometry O1-O4 on p. 19 do not have
this problem.
. Solution, p. 365
In the science fiction novel Have Spacesuit — Will Travel, by
Robert Heinlein, Kip, a high school student, answers a radio distress
call, encounters a flying saucer, and is knocked out and kidnapped
by aliens. When he wakes up, he finds himself in a locked cell
with a young girl named Peewee. Peewee claims they’re aboard an
accelerating spaceship. “If this was a spaceship,” Kip thinks. “The
floor felt as solid as concrete and motionless.”
The equivalence principle can be stated in a variety of ways. On
p. 21, I stated it as (1) gravitational and inertial mass are always
proportional to one another. An alternative formulation (p. 32)
is (2) that Kip has no way, by experiments or observarions inside
his sealed prison cell, to determine whether he’s in an accelerating
spaceship or on the surface of a planet, experiencing its gravitational
(a) Show that any violation of statement 1 also leads to a violation of
statement 2. (b) If we’d intended to construct a geometrical theory
of gravity roughly along the lines of axioms O1-O4 on p. 19, which
axiom is violated in this scenario?
. Solution, p. 365
Clock A sits on a desk. Clock B is tossed up in the air from
the same height as the desk and then comes back down. Compare
the elapsed times.
. Hint, p. 365 . Solution, p. 365
(a) Find the difference in rate between a clock at the center
of the earth and a clock at the south pole. (b) When an antenna
on earth receives a radio signal from a space probe that is in a
hyperbolic orbit in the outer solar system, the signal will show both
Chapter 1
A geometrical theory of spacetime
a kinematic red-shift and a gravitational blueshift. Compare the
orders of magnitude of these two effects.
. Solution, p. 365
Consider the following physical situations: (1) a charged
object lies on a desk on the planet earth; (2) a charged object orbits
the earth; (3) a charged object is released above the earth’s surface
and dropped straight down; (4) a charged object is subjected to a
constant acceleration by a rocket engine in outer space. In each case,
we want to know whether the charge radiates. Analyze the physics
in each case (a) based on conservation of energy; (b) by determining
whether the object’s motion is inertial in the sense intended by Isaac
Newton; (c) using the most straightforward interpretation of the
equivalence principle (i.e., not worrying about the issues discussed
on p. that surround the ambiguous definition of locality).
. Solution, p. 366
Consider the physical situation depicted in figure l, p. 30.
Let ag be the gravitational acceleration and ar the acceleration of
the charged particle due to radiation. Then ar /ag measures the violation of the equivalence principle. The goal of this problem is to
make an order-of-magnitude estimate of this ratio in the case of a
neutron and a proton in low earth orbit.
(a) Let m the mass of each particle, and q the charge of the charged
particle. Without doing a full calculation like the ones by the DeWitts and Grøn and Næss, use general ideas about the frequencyscaling of radiation (see section 9.2.5, p. 340) to find the proportionality that gives the dependence of ar /ag on q, m, and any convenient
parameters of the orbit.
(b) Based on considerations of units, insert the necessary universal
constants into your answer from part a.
(c) The result from part b will still be off by some unitless factor, but
we expect this to be of order unity. Under this assumption, make
an order-of-magnitude estimate of the violation of the equivalence
principle in the case of a neutron and a proton in low earth orbit.
. Solution, p. 366
Chapter 1
A geometrical theory of spacetime
Chapter 2
Geometry of flat spacetime
The geometrical treatment of space, time, and gravity only requires
as its basis the equivalence of inertial and gravitational mass. Given
this assumption, we can describe the trajectory of any free-falling
test particle as a geodesic. Equivalence of inertial and gravitational
mass holds for Newtonian gravity, so it is indeed possible to redo
Newtonian gravity as a theory of curved spacetime. This project was
carried out by the French mathematician Cartan. The geometry of
the local reference frames is very simple. The three space dimensions
have an approximately Euclidean geometry, and the time dimension
is entirely separate from them. This is referred to as a Euclidean
spacetime with 3+1 dimensions. Although the outlook is radically
different from Newton’s, all of the predictions of experimental results
are the same.
The experiments in section 1.2 show, however, that there are
real, experimentally verifiable violations of Newton’s laws. In Newtonian physics, time is supposed to flow at the same rate everywhere,
which we have found to be false. The flow of time is actually dependent on the observer’s state of motion through space, which shows
that the space and time dimensions are intertwined somehow. The
geometry of the local frames in relativity therefore must not be as
simple as Euclidean 3+1. Their actual geometry was implicit in
Einstein’s 1905 paper on special relativity, and had already been
developed mathematically, without the full physical interpretation,
by Hendrik Lorentz. Lorentz’s and Einstein’s work were explicitly
connected by Minkowski in 1907, so a Lorentz frame is often referred
to as a Minkowski frame.
To describe this Lorentz geometry, we need to add more structure on top of the axioms O1-O4 of ordered geometry, but it will not
be the additional Euclidean structure of E3-E4, it will be something
different. To see how to proceed, let’s start by thinking about what
bare minimum of geometrical machinery is needed in order to set
up frames of reference.
a / Hendrik
2.1 Affine properties of Lorentz geometry
2.1.1 Parallelism and measurement
We think of a frame of reference as a body of measurements
or possible measurements to be made by some observer. Ordered
geometry lacks measure. The following argument shows that merely
by adding a notion of parallelism to our geometry, we automatically
gain a system of measurement.
a / Objects are released at
rest at spacetime events P and
Q. They remain at rest, and their
world-lines define a notion of
b / There is no well-defined
angular measure in this geometry. In a different frame of
reference, the angles are not
right angles.
c / Simultaneity is not well
defined. The constant-time lines
PQ and RS from figure a are
not constant-time lines when
observed in a different frame of
Chapter 2
We only expect Lorentz frames to be local, but we do need them
to be big enough to cover at least some amount of spacetime. If
Betty does an Eötvös experiment by releasing a pencil and a lead
ball side by side, she is essentially trying to release them at the
same event A, so that she can observe them later and determine
whether their world-lines stay right on top of one another at point
B. That was all that was required for the Eötvös experiment, but
in order to set up a Lorentz frame we need to start dealing with
objects that are not right on top of one another. Suppose we release two lead balls in two different locations, at rest relative to one
another. This could be the first step toward adding measurement
to our geometry, since the balls mark two points in space that are
separated by a certain distance, like two marks on a ruler, or the
goals at the ends of a soccer field. Although the balls are separated
by some finite distance, they are still close enough together so that
if there is a gravitational field in the area, it is very nearly the same
in both locations, and we expect the distance defined by the gap
between them to stay the same. Since they are both subject only to
gravitational forces, their world-lines are by definition straight lines
(geodesics). The goal here is to end up with some kind of coordinate grid defining a (t, x) plane, and on such a grid, the two balls’
world-lines are vertical lines. If we release them at events P and
Q, then observe them again later at R and S, PQRS should form a
rectangle on such a plot. In the figure, the irregularly spaced tick
marks along the edges of the rectangle are meant to suggest that
although ordered geometry provides us with a well-defined ordering
along these lines, we have not yet constructed a complete system of
The depiction of PQSR as a rectangle, with right angles at its
vertices, might lead us to believe that our geometry would have
something like the concept of angular measure referred to in Euclid’s
E4, equality of right angles. But this is too naive even for the
Euclidean 3+1 spacetime of Newton and Galileo. Suppose we switch
to a frame that is moving relative to the first one, so that the balls
are not at rest. In the Euclidean spacetime, time is absolute, so
events P and Q would remain simultaneous, and so would R and
S; the top and bottom edges PQ and RS would remain horizontal
on the plot, but the balls’ world-lines PR and QS would become
slanted. The result would be a parallelogram. Since observers in
Geometry of flat spacetime
different states of motion do not agree on what constitutes a right
angle, the concept of angular measure is clearly not going to be
useful here. Similarly, if Euclid had observed that a right angle
drawn on a piece of paper no longer appeared to be a right angle
when the paper was turned around, he would never have decided
that angular measure was important enough to be enshrined in E4.
In the context of relativity, where time is not absolute, there is
not even any reason to believe that different observers must agree on
the simultaneity of PQ and RS. Our observation that time flows differently depending on the observer’s state of motion tells us specifically to expect this not to happen when we switch to a frame moving
to the relative one. Thus in general we expect that PQRS will be
distorted into a form like the one shown in figure c. We do expect,
however, that it will remain a parallelogram; a Lorentz frame is one
in which the gravitational field, if any, is constant, so the properties
of spacetime are uniform, and by symmetry the new frame should
still have PR=QS and PQ=RS.
With this motivation, we form the system of affine geometry by
adding the following axioms to set O1-O4.1 The notation [PQRS]
means that events P, Q, S, and R form a parallelogram, and is
defined as the statement that the lines determined by PQ and RS
never meet at a point, and similarly for PR and QS.
A1 Constructibility of parallelograms: Given any P, Q, and R,
there exists S such that [PQRS], and if P, Q, and R are distinct
then S is unique.
A2 Symmetric treatment of the sides of a parallelogram: If [PQRS],
then [QRSP], [QPSR], and [PRQS].
A3 Lines parallel to the same line are parallel to one another: If
[ABCD] and [ABEF], then [CDEF].
The following theorem is a stronger version of Playfair’s axiom
E5, the interpretation being that affine geometry describes a spacetime that is locally flat.
Theorem: Given any line ` and any point P not on the line, there
exists a unique line through P that is parallel to `.
This is stronger than E5, which only guarantees uniqueness, not
existence. Informally, the idea here is that A1 guarantees the existence of the parallel, and A3 makes it unique.2
The axioms are summarized for convenient reference in the back of the book
on page 388. This formulation is essentially the one given by Penrose, The Road
to Reality, in section 14.1.
Proof: Pick any two distinct points A and B on `, and construct the uniquely
determined parallelogram [ABPQ] (axiom A1). Points P and Q determine a line
(axiom O1), and this line is parallel to ` (definition of the parallelogram). To
Section 2.1
Affine properties of Lorentz geometry
d / Construction
Although these new axioms do nothing more than to introduce
the concept of parallelism lacking in ordered geometry, it turns out
that they also allow us to build up a concept of measurement. Let
` be a line, and suppose we want to define a number system on this
line that measures how far apart events are. Depending on the type
of line, this could be a measurement of time, of spatial distance, or
a mixture of the two. First we arbitrarily single out two distinct
points on ` and label them 0 and 1. Next, pick some auxiliary point
q0 not lying on `. By A1, construct the parallelogram 01q0 q1 . Next
construct q0 1q1 2. Continuing in this way, we have a scaffolding of
parallelograms adjacent to the line, determining an infinite lattice of
points 1, 2, 3, . . . on the line, which represent the positive integers.
Fractions can be defined in a similar way. For example, 21 is defined
as the point such that when the initial lattice segment 0 21 is extended
by the same construction, the next point on the lattice is 1.
The continuously varying variable constructed in this way is
called an affine parameter. The time measured by a free-falling
clock is an example of an affine parameter, as is the distance measured by the tick marks on a free-falling ruler. Since light rays travel
along geodesics, the wave crests on a light wave can even be used
analogously to the ruler’s tick marks.
Example: 1
The affine parameter can be used to define the centroid of a set
of points. In the simplest example, finding the centroid of two
points, we simply bisect the line segment as described above in
the construction of the number 12 . Similarly, the centroid of a triangle can be defined as the intersection of its three medians, the
lines joining each vertex to the midpoint of the opposite side.
e / Affine geometry gives a
well-defined centroid for the
Conservation of momentum
Example: 2
In nonrelativistic mechanics, the concept of the center of mass
is closely related to the law of conservation of momentum. For
example, a logically complete statement of the law is that if a system of particles is not subjected to any external force, and we
pick a frame in which its center of mass is initially at rest, then its
center of mass remains at rest in that frame. Since centroids are
well defined in affine geometry, and Lorentz frames have affine
properties, we have grounds to hope that it might be possible to
generalize the definition of momentum relativistically so that the
generalized version is conserved in a Lorentz frame. On the other
hand, we don’t expect to be able to define anything like a global
prove that this line is unique, we argue by contradiction. Suppose some other
parallel m to exist. If m crosses the infinite line BQ at some point Z, then both
[ABPQ] and [ABPZ], so by A1, Q=Z, so the ` and m are the same. The only
other possibility is that m is parallel to BQ, but then the following chain of
parallelisms holds: PQ k AB k m k BQ. By A3, lines parallel to another line are
parallel to each other, so PQ k BQ, but this is a contradiction, since they have
Q in common.
Chapter 2
Geometry of flat spacetime
Lorentz frame for the entire universe, so there is no such natural
expectation of being able to define a global principle of conservation of momentum. This is an example of a general fact about
relativity, which is that conservation laws are difficult or impossible
to formulate globally.
Although the affine parameter gives us a system of measurement
for free in a geometry whose axioms do not even explicitly mention
measurement, there are some restrictions:
The affine parameter is defined only along straight lines, i.e.,
geodesics. Alice’s clock defines an affine parameter, but Betty’s
does not, since it is subject to nongravitational forces.
We cannot compare distances along two arbitrarily chosen
lines, only along a single line or two parallel lines.
The affine parameter is arbitrary not only in the choice of its
origin 0 (which is to be expected in any case, since any frame
of reference requires such an arbitrary choice) but also in the
choice of scale. For example, there is no fundamental way of
deciding how fast to make a clock tick.
We will eventually want to lift some of these restrictions by
adding to our kit a tool called a metric, which allows us to define distances along arbitrary curves in space time, and to compare
distances in different directions. The affine parameter, however, will
not be entirely superseded. In particular, we’ll find that the metric
has a couple of properties that are not as nice as those of the affine
parameter. The square of a metric distance can be negative, and the
metric distance measured along a light ray is precisely zero, which
is not very useful.
Self-check: By the construction of the affine parameter above,
affine distances on the same line are comparable. By another construction, verify the claim made above that this can be extended to
distances measured along two different parallel lines.
Area and volume
Example: 3
It is possible to define area and volume in affine geometry. This
is a little surprising, since distances along different lines are not
even comparable. However, we are already accustomed to multiplying and dividing numbers that have different units (a concept
that would have given Euclid conniptions), and the situation in
affine geometry is really no different. To define area, we extend
the one-dimensional lattice to two dimensions. Any planar figure
can be superimposed on such a lattice, and dissected into parallelograms, each of which has a standard area.
Area on a graph of v versus t
Example: 4
If an object moves at a constant velocity v for time t, the distance
Section 2.1
f / Example 3.
The area of
the viola can be determined
by counting the parallelograms
formed by the lattice. The area
can be determined to any desired
precision, by dividing the parallelograms into fractional parts that
are as small as necessary.
g / Example 4.
Affine properties of Lorentz geometry
it travels can be represented by the area of a parallelogram in an
affine plane with sides having lengths v and t. These two lengths
are measured by affine parameters along two different directions,
so they are not comparable. For example, it is meaningless to
ask whether 1 m/s is greater than, less than, or equal to 1 s. If
we were graphing velocity as a function of time on a conventional
Cartesian graph, the v and t axes would be perpendicular, but
affine geometry has no notion of angular measure, so this is irrelevant here.
Self-check: If multiplication is defined in terms of affine area,
prove the commutative property ab = ba and the distributive rule
a(b + c) = ab + bc from axioms A1-A3.
2.1.2 Vectors
Vectors distinguished from scalars
We’ve been discussing subjects like the center of mass that in
freshman mechanics would be described in terms of vectors and
scalars, the distinction being that vectors have a direction in space
and scalars don’t. As we make the transition to relativity, we are
forced to refine this distinction. For example, we used to consider
time as a scalar, but the Hafele-Keating experiment shows that time
is different in different frames of reference, which isn’t something
that’s supposed to happen with scalars such as mass or temperature.
In affine geometry, it doesn’t make much sense to say that a vector
has a magnitude and direction, since non-parallel magnitudes aren’t
comparable, and there is no system of angular measurement in which
to describe a direction.
A better way of defining vectors and scalars is that scalars are
absolute, vectors relative. If I have three apples in a bowl, then all
observers in all frames of reference agree with me on the number
three. But if my terrier pup pulls on the leash with a certain force
vector, that vector has to be defined in relation to other things. It
might be three times the strength of some force that we define as
one newton, and in the same direction as the earth’s magnetic field.
In general, measurement means comparing one thing to another.
The number of apples in the bowl isn’t a measurement, it’s a count.
Affine measurement of vectors
Before even getting into the full system of affine geometry, let’s
consider the one-dimensional example of a line of time. We could
use the hourly emergence of a mechanical bird from a pendulumdriven cuckoo clock to measure the rate at which the earth spins,
but we could equally well take our planet’s rotation as the standard
and use it to measure the frequency with which the bird pops out of
the door. Once we have two things to compare against one another,
measurement is reduced to counting (figure d, p. 44). Schematically,
Chapter 2
Geometry of flat spacetime
let’s represent this measurement process with the following notation,
which is part of a system called called birdtracks:3
c e = 24
Here c represents the cuckoo clock and e the rotation of the earth.
Although the measurement relationship is nearly symmetric, the
arrow has a direction, because, for example, the measurement of
the earth’s rotational period in terms of the clock’s frequency is
c e = (24 hr)(1 hr−1 ) = 24, but the clock’s period in terms of the
earth’s frequency is e c = 1/24. We say that the relationship is
not symmetric but “dual.” By the way, it doesn’t matter how we
arrange these diagrams on the page. The notations c e and e c
mean exactly the same thing, and expressions like this can even be
drawn vertically.
Suppose that e is a displacement along some one-dimensional
line of time, and we want to think of it as the thing being measured.
Then we expect that the measurement process represented by c produces a real-valued result and is a linear function of e. Since the
relationship between c and e is dual, we expect that c also belongs
to some vector space. For example, vector spaces allow multiplication by a scalar: we could double the frequency of the cuckoo clock
by making the bird come out on the half hour as well as on the
hour, forming 2c. Measurement should be a linear function of both
vectors; we say it is “bilinear.”
The two vectors c and e have different units, hr−1 and hr, and
inhabit two different one-dimensional vector spaces. The “flavor” of
the vector is represented by whether the arrow goes into it or comes
out. Just as we used notation like →
v in freshman physics to tell
vectors apart from scalars, we can employ arrows in the birdtracks
notation as part of the notation for the vector, so that instead of
writing the two vectors as c and e, we can notate them as c
e . Performing a measurement is like plumbing. We join the two
“pipes” in c
e and simplify to c e .
A confusing and nonstandardized jungle of notation and terminology has grown up around these concepts. For now, let’s refer to a
vector such as e , with the arrow coming in, simply as a “vector,”
and the type like c
as a “dual vector.” In the one-dimensional
example of the earth and the cuckoo clock, the roles played by the
two vectors were completely equivalent, and it didn’t matter which
one we expressed as a vector and which as a dual vector. Example
5 shows that it is sometimes more natural to take one quantity as
The system used in this book follows the one defined by Cvitanović, which
was based closely on a graphical notation due to Penrose. For a more complete exposition, see the Wikipedia article “Penrose graphical notation” and
Cvitanović’s online book at
Section 2.1
Affine properties of Lorentz geometry
a vector and another as a dual vector. Example 6 shows that we
sometimes have no choice at all as to which is which.
In birdtracks notation, a scalar is a quantity that has no external
arrows at all. Since the expression c e = 24 has no external arrows,
only internal ones, it represents a scalar. This makes sense because
it’s a count, and a count is a scalar.
A convenient way of summarizing all of our categories of variablies is by their behavior when we convert units, i.e., when we
rescale our space. If we switch our time unit from hours to minutes,
the number of apples in a bowl is unchanged, the earth’s period
of rotation gets 60 times bigger, and the frequency of the cuckoo
clock changes by a factor of 1/60. In other words, a quantity u
under rescaling of coordinates by a factor α becomes αp u, where
the exponents −1, 0, and +1 correspond to dual vectors, scalars,
and vectors, respectively. We can therefore see that these distinctions are of interest even in one dimension, contrary to what one
would have expected from the freshman-physics concept of a vector
as something transforming in a certain way under rotations.
Geometrical visualization
h / 1.
A displacement vector.
2. A vector from the space dual
to the space of displacements.
Measurement is reduced
to counting. The cuckoo clock
chimes 24 times in one rotation
of the earth.
In two dimensions, there are natural ways of visualizing the different vector spaces inhabited by vectors and dual vectors. We’ve
already been describing a vector like e as a displacement. Its
vector space is the space of such displacements.4 A vector in the
dual space such as c can be visualized as a set of parallel, evenly
spaced lines on a topographic map, h/2, with an arrowhead to show
which way is “uphill.” The act of measurement consists of counting
how many of these lines are crossed by a certain vector, h/3.
Given a scalar field f , its gradient grad f at any given point
is a dual vector. In birdtracks notation, we have to indicate this
by writing it with an outward-pointing arrow, (grad f ) . Because
gradients occur so frequently, we have a special shorthand for them,
which is simply a circle:
In the context of spacetime with a metric and curvature, we’ll see
that the usual definition of the gradient in terms of partial derivatives should be modified with correction terms to form something
called a covariant derivative. When we get to that point on p. 178,
we’ll commandeer the circle notation for that operation.
i / Constant-temperature curves
for January in North America, at
intervals of 4 ◦ C. The temperature gradient at a given point is a
dual vector.
Chapter 2
Force is a dual vector
Example: 5
The dot product dW = F · dx for computing mechanical work
In terms of the primitive notions used in the axiomatization in section 2.1,
a displacement could be described as an equivalence class of segments such that
for any two segments in the class AB and CD, AB and CD form a parallelogram.
Geometry of flat spacetime
becomes, in birdtracks notation,
dW = F dx
This shows that force is more naturally considered to be a dual
vector rather than a vector. The symmetry between vectors and
dual vectors is broken by considering displacements like dx to
be vectors, and this asymmetry then spreads to other quantities
such as force.
The same result can be obtained from Newton’s second law; see
example 21 on p. 141.
Systems without a metric
Example: 6
The freshman-mechanics way of thinking about vectors and scalar
products contains the hidden assumption that we have, besides
affine measurement, an additional piece of measurement apparatus called the metric (section 3.5, p. 98). Without yet having to
formally define what we mean by a metric, we can say roughly
that it supplies the conveniences that we’re used to having in the
Euclidean plane, but that are not present in affine geometry. In
particular, it allows us to define the notion that one vector is perpendicular to another vector, or that one dual vector is perpendicular to another dual vector.
Let’s start with an example where the hidden assumption is valid,
and we do have a metric. Let a billiard ball of unit mass be constrained by a diagonal wall to have C ≤ 0, where C = y − x. The
Lagrangian formalism just leads to the expected Newtonian expressions for the momenta conjugate to x and y, px = ẋ, py = ẏ,
and these form a dual vector p . The force of constraint is
= dp /dt. Let w
= (grad C)
be the gradient of the con-
and w
both belong to the
straint function. The vectors F
space of dual vectors, and they are parallel to each other. Since
we do happen to have a metric in this example, it is also possible
to say, as most people would, that the force is perpendicular to
the wall.
j / There is no natural metric
on the space (θ, φ).
Now consider the example shown in figure j. The arm’s weight is
negligible compared to the unit mass of the gripped weight, and
both the upper and lower arm have unit length. Elbows don’t bend
backward, so we have a constraint C ≤ 0, where C = θ − φ, and
as before we can define define a dual vector w = (grad C)
that is parallel to the line of constraint in the (θ, φ) plane. The
conjugate momenta (which are actually angular momenta) turn
out to be pθ = θ̇+cos(φ−θ)φ̇ and a similar expression for pφ . As in
the example of the billiard ball, the force of constraint is parallel to
w . There is no metric that naturally applies to the (θ, φ) plane,
so we have no notion of perpendicularity, and it doesn’t make
sense to say that F
is perpendicular to the line of constraint.
Section 2.1
Affine properties of Lorentz geometry
Finally we remark that since four-dimensional Galilean spacetime
lacks a metric (see p. 100), the distinction between vectors and
dual vectors in Galilean relativity is a real and physically important
one. The only reason people were historically able to ignore this
distinction was that Galilean spacetime splits into independent
time and spatial parts, with the spatial part having a metric.
k / The
considers P and Q to be simultaneous.
No simultaneity without a metric
Example: 7
We’ll see in section 2.2 that one way of defining the distinction between Galilean and Lorentz geometry is that in Lorentzian spacetime, simultaneity is observer-dependent. Without a metric, there
can be no notion of simultaneity at all, not even a frame-dependent
one. In figure k, the fact that the observer considers events P
and Q to be simultaneous is represented by the fact that the
observer’s displacement vector o is perpendicular to the displacement s from P to Q. In affine geometry, we can’t express
Example: 8
Depictions of dual vectors as in figure h/2 on p. 48 evoke plane
waves. This makes sense, because the phase θ of a wave is
a scalar, and the gradient of a scalar is a dual vector. The dual
vector k = (grad θ) = (ω, kx , . . .) is called the frequency vector.
If o represents the displacement of an observer from one event
in spacetime to another, then k o is the number of radians of
the wave’s phase that wash over the observer’s position. The
frequency vector is discussed further in section 4.2.3 on p. 133.
l / Points on the graph satisfy
the dispersion relation C = 0 for
water waves. At a given point
on the graph, the dual vector
(grad C )
tells us the group
There is a dispersion relation between a wave’s frequency and
wavenumber. For example, surface waves in deep water obey
the constraint C = 0, where C = ω4 − α2 k 2 and α is a constant
with units of acceleration, relating to the acceleration of gravity.
(Since the water is infinitely deep, there is no other scale that
could enter into the constraint.)
When a wave is modulated, it can transport energy and momentum and transmit information, i.e., act as an agent of cause and
effect between events. How fast does it go? If a certain bump on
the envelope with which the wave is modulated visits spacetime
events P and Q, then whatever frequency and wavelength the
wave has near the bump are observed to be the same at P and Q.
In general, k and ω are constant along the spacetime displacement of any point on the envelope, so the spacetime displacement s from P to Q must satisfy the condition (grad C) s = 0.
The set of solutions to this equation is the world-line of the bump,
and the inverse slope of this world-line is called the group velocity.
In our example of water waves, the group velocity is α/2ω.
The quantity ω/k is referred to as the phase velocity. In our
example of the water waves, the phase velocity is ω/k = α/ω,
Chapter 2
Geometry of flat spacetime
which is twice the group velocity. The phase velocity lacks physical interest, because it is not the velocity at which any “stuff”
moves. The birdtracks notation expresses this fact by refusing
to let us use information about the phase θ to decide whether
events P and Q could have been linked by the wave’s motion.
The only quantity we can extract from the phase information is
the frequency vector k , and one can say loosely that this is the
phase velocity vector. But it’s not really a vector at all, it’s a dual
vector. The displacement s from P to Q is a vector, not a dual
vector. Since k
and s belong to different vector spaces, we
can’t say whether they are parallel.
Abstract index notation
Expressions in birdtracks notation such as
can be awkward to type on a computer, which is why we’ve already been occasionally resorting to more linear notations such as
(grad C) s. As we encounter more complicated birdtracks, the diagrams will sometimes look like complicated electrical schematics,
and the problem of generating them on a keyboard will get more
acute. There is in fact a systematic way of representing any such
expression using only ordinary subscripts and superscripts. This is
called abstract index notation, and was introduced by Roger Penrose at around the same time he invented birdtracks. For practical
reasons, it was the abstract index notation that caught on.
The idea is as follows. Suppose we wanted to describe a complicated birdtrack verbally, so that someone else could draw it. The
diagram would be made up of various smaller parts, a typical one
looking something like the scalar product u v. The verbal instructions might be: “We have an object u with an arrow coming out of
it. For reference, let’s label this arrow as a. Now remember that
other object v I had you draw before? There was an arrow coming
into that one, which we also labeled a. Now connect up the two
arrows labeled a.”
Shortening this lengthy description to its bare minimum, Penrose
renders it like this: ua v a . Subscripts depict arrows coming out of
a symbol (think of water flowing from a tank out through a pipe
below). Superscripts indicate arrows going in. When the same letter
is used as both a superscript and a subscript, the two arrows are to
be piped together.
Abstract index notation evolved out of an earlier one called
the Einstein summation convention, in which superscripts and subscripts referred to specific coordinates. For example, we might take
0 to be the time coordinate, 1 to be x, and so on. A symbol like uγ
would then indicate a component of the dual vector u, which could
Section 2.1
Affine properties of Lorentz geometry
be its x component if γ took on the value 1. Repeated indices were
summed over.
The advantage of the birdtrack and abstract index notations is
that they are coordinate-independent, so that an equation written
in them is valid regardless of the choice of coordinates. The Einstein
and Penrose notations look very similar, so for example if we want
to take a general result expressed in Penrose notation and apply it
in a specific coordinate system, there is essentially no translation
required. In fact, the two notations look so similar that we need an
explicit way to tell which is which, so that we can tell whether or
not a particular result is coordinate-independent. We therefore use
the convention that Latin indices represent abstract indices, whereas
Greek ones imply a specific coordinate system and can take on numerical values, e.g., γ = 1.
2.2 Relativistic properties of Lorentz geometry
We now want to pin down the properties of the Lorentz geometry
that are left unspecified by the affine treatment. We need some
further input from experiments in order to show us how to proceed.
We take the following as empirical facts about flat spacetime:5
L1 Spacetime is homogeneous and isotropic. No time or place
has special properties that make it distinguishable from other
points, nor is one direction in space distinguishable from another.6
L2 Inertial frames of reference exist. These are frames in which
particles move at constant velocity if not subject to any forces.
We can construct such a frame by using a particular particle,
which is not subject to any forces, as a reference point.
L3 Equivalence of inertial frames: If a frame is in constant-velocity
translational motion relative to an inertial frame, then it is also
an inertial frame. No experiment can distinguish one preferred
inertial frame from all the others.
L4 Causality: There exist events 1 and 2 such that t1 < t2 in all
L5 Relativity of time: There exist events 1 and 2 and frames of
reference (t, x) and (t0 , x0 ) such that t1 < t2 , but t01 > t02 .
L4 makes it possible to have an event 1 that causes an event 2,
with all observers agreeing on which caused which. L5 is supported
These facts are summarized for convenience on page 388 in the back of the
For the experimental evidence on isotropy,
see http://www.
Chapter 2
Geometry of flat spacetime
by the experimental evidence in section 1.2; if L5 were false, then
space and time could work as imagined by Galileo and Newton.
Define affine parameters t and x for time and position, and construct a (t, x) plane. Axiom L1 guarantees that spacetime is flat,
allowing us to do this; if spacetime had, for example, a curvature
like that of a sphere, then the axioms of affine geometry would fail,
and it would be impossible to lay out such a global grid of parallels. Although affine geometry treats all directions symmetrically,
we’re going beyond the affine aspects of the space, and t does play
a different role than x here, as shown, for example, by L4 and L5.
In the (t, x) plane, consider a rectangle with one corner at the
origin O. We can imagine its right and left edges as representing the
world-lines of two objects that are both initially at rest in this frame;
they remain at rest (L2), so the right and left edges are parallel.
How do we know that this is a rectangle and not some other kind
of parallelogram? In purely affine geometry, there is no notion of
perpendicularity, so this distinction is meaningless. But implicit in
the existence of inertial frames (L2) is the assumption that spacetime
has some additional structure that allows a particular observer to
decide what events he considers to be simultaneous (example 7,
p. 50). He then considers his own world-line, i.e., his t axis, to
be perpendicular to a proposed x axis if points on the x axis are
simultaneous to him.
We now define a second frame of reference such that the origins
of the two frames coincide, but they are in motion relative to one
another with velocity v. The transformation L from the first frame
to the second is referred to as a Lorentz boost with velocity v. L
depends on v. By equivalence of inertial frames (L3), an observer in
the new frame considers his own t axis to be perpendicular to his own
x, even though they don’t look that way in figure a. Thus, although
we assume some notion of perpendicularity, we do not assume that
it looks the same as the Euclidean one.
By homogeneity of spacetime (L1), L must be linear, so the
original rectangle will be transformed into a parallelogram in the
new frame; this is also consistent with L3, which requires that the
world-lines on the right and left edges remain parallel. The left edge
has inverse slope v. By L5 (no simultaneity), the top and bottom
edges are no longer horizontal.
a / Two objects at rest have
world-lines that define a rectangle. In a second frame of
reference in motion relative to the
first one, the rectangle becomes
a parallelogram.
For simplicity, let the original rectangle have unit area. Then
the area of the new parallelogram is still 1, by the following argument. Let the new area be A, which is a function of v. By isotropy
of spacetime (L1), A(v) = A(−v). Furthermore, the function A(v)
must have some universal form for all geometrical figures, not just
for a figure that is initially a particular rectangle; this follows because of our definition of affine area in terms of a dissection by
Section 2.2
Relativistic properties of Lorentz geometry
a two-dimensional lattice, which we can choose to be a lattice of
squares. Applying boosts +v and −v one after another results in a
transformation back into our original frame of reference, and since
A is universal for all shapes, it doesn’t matter that the second transformation starts from a parallelogram rather than a square. Scaling
the area once by A(v) and again by A(−v) must therefore give back
the original square with its original unit area, A(v)A(−v) = 1, and
since A(v) = A(−v), A(v) = ±1 for any value of v. Since A(0) = 1,
we must have A(v) = 1 for all v. The argument is independent of
the shape of the region, so we conclude that all areas are preserved
by Lorentz boosts. (See subsection 4.6.3 on p. 155 for further interpretation of A.)
If we consider a boost by an infinitesimal velocity dv, then the
vanishing change in area comes from the sum of the areas of the four
infinitesimally thin slivers where the rectangle lies either outside the
parallelogram (call this negative area) or inside it (positive). (We
don’t worry about what happens near the corners, because such
effects are of order dv 2 .) In other words, area flows around in the
x − t plane, and the flows in and out of the rectangle must cancel.
Let v be positive; the flow at the sides of the rectangle is then to the
right. The flows through the top and bottom cannot be in opposite
directions (one up, one down) while maintaining the parallelism of
the opposite sides, so we have the following three possible cases:
b / Flows of area: (I) a shear
that preserves simultaneity, (II) a
rotation, (III) upward flow at all
I There is no flow through the top and bottom. This case corresponds to Galilean relativity, in which the rectangle shears
horizontally under a boost, and simultaneity is preserved, violating L5.
II Area flows downward at both the top and the bottom. The
flow is clockwise at both the positive t axis and the positive
x axis. This makes it plausible that the flow is clockwise everywhere in the (t, x) plane, and the proof is straightforward.7
Proof: By linearity of L, the flow is clockwise at the negative axes as well.
Also by linearity, the handedness of the flow is the same at all points on a ray
extending out from the origin in the direction θ. If the flow were counterclockwise
somewhere, then it would have to switch handedness twice in that quadrant, at
θ1 and θ2 . But by writing out the vector cross product r × dr, where dr is the
displacement caused by L(dv), we find that it depends on sin(2θ +δ), which does
not oscillate rapidly enough to have two zeroes in the same quadrant.
Chapter 2
Geometry of flat spacetime
As v increases, a particular element of area flows continually
clockwise. This violates L4, because two events with a cause
and effect relationship could be time-reversed by a Lorentz
III Area flows upward at both the top and the bottom.
Only case III is possible, and given case III, there must be at least
one point P in the first quadrant where area flows neither clockwise
nor counterclockwise.8 The boost simply increases P’s distance from
the origin by some factor. By the linearity of the transformation,
the entire line running through O and P is simply rescaled. This
special line’s inverse slope, which has units of velocity, apparently
has some special significance, so we give it a name, c. We’ll see later
that c is the maximum speed of cause and effect whose existence
we inferred in section 1.3. Any world-line with a velocity equal to
c retains the same velocity as judged by moving observers, and by
isotropy the same must be true for −c.
For convenience, let’s adopt time and space units in which c = 1,
and let the original rectangle be a unit square. The upper right
tip of the parallelogram must slide along the line through the origin
with slope +1, and similarly the parallelogram’s other diagonal must
have a slope of −1. Since these diagonals bisected one another on
the original square, and since bisection is an affine property that
is preserved when we change frames of reference, the parallelogram
must be equilateral.
We can now determine the complete form of the Lorentz transformation. Let unit square PQRS, as described above, be transformed
to parallelogram P0 Q0 R0 S0 in the new coordinate system (x0 , t0 ). Let
the t0 coordinate of R0 be γ, interpreted as the ratio between the
time elapsed on a clock moving from P0 to R0 and the corresponding
time as measured by a clock that is at rest in the (x0 , t0 ) frame. By
the definition of v, R0 has coordinates (vγ, γ), and the other geometrical facts established above place Q0 symmetrically on the other
side of the diagonal, at (γ, vγ). Computing the cross product of vectors P0 R0 and P0 Q0 , we find the area of P0 Q0 R0 S0 to be γ 2 (1 − v 2 ),
and setting this equal to 1 gives
1 − v2
c / Unit square PQRS is Lorentzboosted to the parallelogram
P0 Q0 R0 S0 .
Self-check: Interpret the dependence of γ on the sign of v.
This follows from the fact that, as shown in the preceding footnote, the
handedness of the flow depends only on θ.
Section 2.2
Relativistic properties of Lorentz geometry
d / The behavior of the γ factor.
The result for the transformation L, a Lorentz boost along the
x axis with velocity v, is:
t0 = γt + vγx
x0 = vγt + γx
The symmetry of P0 Q0 R0 S0 with respect to reflection across the
diagonal indicates that the time and space dimensions are treated
symmetrically, although they are not entirely interchangeable as
they would have been in case II.
A measuring rod, unlike a clock, sweeps out a two-dimensional
strip on an x − t graph. As in Galilean relativity, the two observers
disagree on the positions of events at the two ends of their rods,
but in addition they disagree on the simultaneity of such events.
Calculation shows that a moving rod appears contracted by a factor
In summary, a clock runs fastest according to an observer who
is at rest relative to the clock, and a measuring rod likewise appears
longest in its own rest frame.
The lack of a universal notion of simultaneity has a similarly
symmetric interpretation. In prerelativistic physics, points in space
have no fixed identity. A brass plaque commemorating a Civil War
battle is not at the same location as the battle, according to an
observer who perceives the Earth has having been hurtling through
Chapter 2
Geometry of flat spacetime
space for the intervening centuries. By symmetry, points in time
have no fixed identity either.
In everyday life, we don’t notice relativistic effects like time dilation, so apparently γ ≈ 1, and v 1, i.e., the speed c must be
very large when expressed in meters per second. By setting c equal
to 1, we have chosen a the distance unit that is extremely long in
proportion to the time unit. This is an example of the correspondence principle, which states that when a new physical theory, such
as relativity, replaces an old one, such as Galilean relativity, it must
remain “backward-compatible” with all the experiments that verified the old theory; that is, it must agree with the old theory in
the appropriate limit. Despite my coyness, you probably know that
the speed of light is also equal to c. It is important to emphasize,
however, that light plays no special role in relativity, nor was it
necessary to assume the constancy of the speed of light in order to
derive the Lorentz transformation; we will in fact prove on page 68
that photons must travel at c, and on page ?? that this must be
true for any massless particle.
On the other hand, Einstein did originally develop relativity
based on a different set of assumptions than our L1-L5. His treatment, given in his 1905 paper “On the electrodynamics of moving
bodies,” is reproduced on p. 345. It starts from the following two
P1 The principle of relativity: “. . . the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest.”
P2 “. . . light is always propagated in empty space with a definite
velocity c which is independent of the state of motion of the
emitting body.”
Einstein’s P1 is essentially the same as our L3 (equivalence of inertial frames). He implicitly assumes something equivalent to our
L1 (homogeneity and isotropy of spacetime). In his system, our
L5 (relativity of time) is a theorem proved from the axioms P1-P2,
whereas in our system, his P2 is a theorem proved from the axioms
e / Example 9.
Flashes of
light travel along P0 T0 and Q0 T0 .
The observer in this frame of
reference judges them to have
been emitted at different times,
and to have traveled different
Section 2.2
Relativistic properties of Lorentz geometry
Example: 9
Let the intersection of the parallelogram’s two diagonals be T in
the original (rest) frame, and T0 in the Lorentz-boosted frame. An
observer at T in the original frame simultaneously detects the
passing by of the two flashes of light emitted at P and Q, and
since she is positioned at the midpoint of the diagram in space,
she infers that P and Q were simultaneous. Since the arrival of
both flashes of light at the same point in spacetime is a concrete
event, an observer in the Lorentz-boosted frame must agree on
their simultaneous arrival. (Simultaneity is well defined as long
as no spatial separation is involved.) But the distances traveled
by the two flashes in the boosted frame are unequal, and since
the speed of light is the same in all cases, the boosted observer
infers that they were not emitted simultaneously.
Example: 10
A different kind of symmetry is the symmetry between observers.
If observer A says observer B’s time is slow, shouldn’t B say that
A’s time is fast? This is what would happen if B took a pill that
slowed down all his thought processes: to him, the rest of the
world would seem faster than normal. But this can’t be correct
for Lorentz boosts, because it would introduce an asymmetry between observers. There is no preferred, “correct” frame corresponding to the observer who didn’t take a pill; either observer
can correctly consider himself to be the one who is at rest. It may
seem paradoxical that each observer could think that the other
was the slow one, but the paradox evaporates when we consider
the methods available to A and B for resolving the controversy.
They can either (1) send signals back and forth, or (2) get together and compare clocks in person. Signaling doesn’t establish one observer as correct and one as incorrect, because as
we’ll see in the following section, there is a limit to the speed of
propagation of signals; either observer ends up being able to explain the other observer’s observations by taking into account the
finite and changing time required for signals to propagate. Meeting in person requires one or both observers to accelerate, as in
the original story of Alice and Betty, and then we are no longer
dealing with pure Lorentz frames, which are described by nonaccelerating observers.
Einstein’s goof
Example: 11
Einstein’s original 1905 paper on special relativity, reproduced on
p. 345, contains a famous incorrect prediction, that “a spring-clock
at the equator must go more slowly, by a very small amount, than
a precisely similar clock situated at one of the poles under otherwise identical conditions” (p. 355). This was a reasonable prediction at the time, but we now know that it was incorrect because
it neglected gravitational time dilation. In the description of the
Hafele-Keating experiment using atomic clocks aboard airplanes
Chapter 2
Geometry of flat spacetime
(p. 15), we saw that both gravity and motion had effects on the
rate of flow of time. On p. 32 we found based on the equivalence
principle that the gravitational redshift of an electromagnetic wave
is ∆E/E = ∆Φ (where c = 1 and Φ is the gravitational potential
gy), and that this could also be interpreted as a gravitational time
dilation ∆t/t = ∆Φ.
The clock at the equator suffers a kinematic time dilation that
would tend to cause it to run more slowly than the one at the
pole. However, the earth is not a sphere, so the two clocks are
at different distances from the earth’s center, and the field they
inhabit is also not the simple field of a sphere. This suggests that
there may be an additional gravitational effect due to ∆Φ 6= 0.
Expanding the Lorentz gamma factor in a Taylor series, we find
that the kinematic effect amounts to ∆t/t = γ − 1 ≈ v 2 /2. The
mismatch in rates between the two clocks is
≈ v 2 − ∆Φ
where ∆Φ = Φequator − Φpole , and a factor of 1/c 2 on the right is
suppressed because c = 1. But this expression for ∆t/t vanishes,
because the surface of the earth’s oceans is in equilibrium, and
therefore a mass of water m can be brought from the north pole to
the equator, with the change in potential energy −m∆Φ being exactly sufficient to supply the necessary kinetic energy (1/2)mv 2 .
We therefore find that a change of latitude should have no effect
on the rate of a clock, provided that it remains at sea level.
This has been verified experimentally by Alley et al.9 Alley’s group
flew atomic clocks from Washington, DC to Thule, Greenland, left
them there for four days, and brought them back. The difference
between the clocks that went to Greenland and other clocks that
stayed in Washington was 38 ± 5 ns, which was consistent with
the 35±2 ns effect predicted purely based on kinematic and gravitational time dilation while the planes were in the air. If Einstein’s
1905 prediction had been correct, then there would have been an
additional difference of 224 ns due to the difference in latitude.
The perfect cancellation of kinematic and gravitational effects is
not as fortuitous as it might seem; this is discussed in example
17 on p. 113.
Example: 12
In the GPS system, as in example 11, both gravitational and kinematic time dilation must be considered. Let’s determine the directions and relative strengths of the two effects in the case of a GPS
C.O. Alley, et al., in NASA Goddard Space Flight Center, Proc. of the
13th Ann. Precise Time and Time Interval (PTTI) Appl. and Planning
Meeting, p. 687-724, 1981, available online at
Section 2.2
Relativistic properties of Lorentz geometry
A radio photon emitted by a GPS satellite gains energy as it falls
to the earth’s surface, so its energy and frequency are increased
by this effect. The observer on the ground, after accounting for
all non-relativistic effects such as Doppler shifts and the Sagnac
effect (p. 74), would interpret the frequency shift by saying that
time aboard the satellite was flowing more quickly than on the
However, the satellite is also moving at orbital speeds, so there is
a Lorentz time dilation effect. According to the observer on earth,
this causes time aboard the satellite to flow more slowly than on
the ground.
We can therefore see that the two effects are of opposite sign.
Which is stronger?
For a satellite in low earth orbit, we would have v 2 /r = g, where
r is only slightly greater than the radius of the earth. The relative
effect on the flow of time is γ − 1 ≈ v 2 /2 = gr /2. The gravitational effect, approximating g as a constant, is −gy, where y
is the satellite’s altitude above the earth. For such a satellite, the
gravitational effect is down by a factor of 2y /r , so the Lorentz time
dilation dominates.
GPS satellites, however, are not in low earth orbit. They orbit
at an altitude of about 20,200 km, which is quite a bit greater
than the radius of the earth. We therefore expect the gravitational
effect to dominate. To confirm this, we need to generalize the
equation ∆t/t = ∆Φ (with c = 1) from example 11 to the case
where g is not a constant. Integrating the equation dt/t = dΦ,
we find that the time dilation factor is equal to e∆Φ . When ∆Φ is
small, e∆Φ ≈ 1 + ∆Φ, and we have a relative effect equal to ∆Φ.
The total effect for a GPS satellite is thus (inserting factors of c for
calculation with SI units, and using positive signs for blueshifts)
+∆Φ −
= 5.2 × 10−10 − 0.9 × 10−10
where the first term is gravitational and the second kinematic. A
more detailed analysis includes various time-varying effects, but
this is the constant part. For this reason, the atomic clocks aboard
the satellites are set to a frequency of 10.22999999543 MHz before launching them into orbit; on the average, this is perceived
on the ground as 10.23 MHz. A more complete analysis of the
general relativity involved in the GPS system can be found in the
review article by Ashby.10
Self-check: Suppose that positioning a clock at a certain distance from a certain planet produces a fractional change δ in the
N. Ashby, “Relativity in the Global Positioning System,” http://www.
Chapter 2
Geometry of flat spacetime
rate at which time flows. In other words, the time dilation factor
is 1 + δ. Now suppose that a second, identical planet is brought
into the picture, at an equal distance from the clock. The clock is
positioned on the line joining the two planets’ centers, so that the
gravitational field it experiences is zero. Is the fractional time dilation now approximately 0, or approximately 2δ? Why is this only
an approximation?
f / Apparatus used for the test of
relativistic time dilation described
in example 13.
The prominent black and white blocks are
large magnets surrounding a circular pipe with a vacuum inside.
(c) 1974 by CERN.
Section 2.2
Relativistic properties of Lorentz geometry
Large time dilation
Example: 13
The time dilation effect in the Hafele-Keating experiment was very
small. If we want to see a large time dilation effect, we can’t do
it with something the size of the atomic clocks they used; the
kinetic energy would be greater than the total megatonnage of
all the world’s nuclear arsenals. We can, however, accelerate
subatomic particles to speeds at which γ is large. An early, lowprecision experiment of this kind was performed by Rossi and Hall
in 1941, using naturally occurring cosmic rays. Figure f shows a
1974 experiment11 of a similar type which verified the time dilation predicted by relativity to a precision of about one part per
g / Muons accelerated to nearly c
undergo radioactive decay much
more slowly than they would
according to an observer at rest
with respect to the muons. The
first two data-points (unfilled
circles) were subject to large
systematic errors.
h / The change in the frequency
of x-ray photons emitted by 57 Fe
as a function of temperature,
drawn after Pound And Rebka
(1960). Dots are experimental
measurements. The solid curve
is Pound and Rebka’s theoretical
calculation using the Debye theory of the lattice vibrations with
a Debye temperature of 420 degrees C. The dashed line is one
with the slope calculated in the
text using a simplified treatment
of the thermodynamics. There is
an arbitrary vertical offset in the
experimental data, as well as the
theoretical curves.
Muons were produced by an accelerator at CERN, near Geneva.
A muon is essentially a heavier version of the electron. Muons undergo radioactive decay, lasting an average of only 2.197 µs before they evaporate into an electron and two neutrinos. The 1974
experiment was actually built in order to measure the magnetic
properties of muons, but it produced a high-precision test of time
dilation as a byproduct. Because muons have the same electric
charge as electrons, they can be trapped using magnetic fields.
Muons were injected into the ring shown in figure f, circling around
it until they underwent radioactive decay. At the speed at which
these muons were traveling, they had γ = 29.33, so on the average they lasted 29.33 times longer than the normal lifetime. In
other words, they were like tiny alarm clocks that self-destructed
at a randomly selected time. Figure g shows the number of radioactive decays counted, as a function of the time elapsed after a given stream of muons was injected into the storage ring.
The two dashed lines show the rates of decay predicted with and
without relativity. The relativistic line is the one that agrees with
Time dilation in the Pound-Rebka experiment
Example: 14
In the description of the Pound-Rebka experiment on page 34, I
postponed the quantitative estimation of the frequency shift due
to temperature. Classically, one expects only a broadening of
the line, since the Doppler shift is proportional to vk /c, where
vk , the component of the emitting atom’s velocity along the line
of sight, averages to zero. But relativity tells us to expect that if
the emitting atom is moving, its time will flow more slowly, so the
frequency of the light it emits will also be systematically shifted
downward. This frequency shift should increase with temperature. In other words, the Pound-Rebka experiment was designed
as a test of general relativity (the equivalence principle), but this
special-relativistic effect is just as strong as the relativistic one,
and needed to be accounted for carefully.
Chapter 2
Bailey at al., Nucl. Phys. B150(1979) 1
Geometry of flat spacetime
In Pound and Rebka’s paper describing their experiment,12 they
refer to a preliminary measurement13 in which they carefully measured this effect, showed that it was consistent with theory, and
pointed out that a previous claim by Cranshaw et al. of having
measured the gravitational frequency shift was vitiated by their
failure to control for the temperature dependence.
It turns out that the full Debye treatment of the lattice vibrations is
not really necessary near room temperature, so we’ll simplify the
thermodynamics. At absolute temperature T , the mean translational kinetic energy of each iron nucleus is (3/2)k T . The velocity
is much less than c(= 1), so we can use the nonrelativistic expression for kinetic energy, K = (1/2)mv 2 , which gives a mean value
for v 2 of 3k T /m. In the limit of v 1, time dilation produces a
change in frequency by a factor of 1/γ, which differs from unity
by approximately −v 2 /2. The relative time dilation is therefore
−3kT /2m, or, in metric units, −3k T /2mc 2 . The vertical scale
in figure h contains an arbitrary offset, since Pound and Rebka’s
measurements were the best absolute measurements to date of
the frequency. The predicted slope of −3k/2mc 2 , however, is
not arbitrary. Plugging in 57 atomic mass units for m, we find
the slope to be 2.4 × 10−15 , which, as shown in the figure is an
excellent approximation (off by only 10%) near room temperature.
Phys. Rev. Lett. 4 (1960) 337
Phys. Rev. Lett. 4 (1960) 274
Section 2.2
Relativistic properties of Lorentz geometry
2.3 The light cone
a / The light cone in 2+1 dimensions.
Given an event P, we can now classify all the causal relationships in
which P can participate. In Newtonian physics, these relationships
fell into two classes: P could potentially cause any event that lay in
its future, and could have been caused by any event in its past. In a
Lorentz spacetime, we have a trichotomy rather than a dichotomy.
There is a third class of events that are too far away from P in space,
and too close in time, to allow any cause and effect relationship, since
causality’s maximum velocity is c. Since we’re working in units in
which c = 1, the boundary of this set is formed by the lines with
slope ±1 on a (t, x) plot. This is referred to as the light cone, and in
the generalization from 1+1 to 3+1 dimensions, it literally becomes
a (four-dimensional) cone. The terminology comes from the fact
that light happens to travel at c, the maximum speed of cause and
effect. If we make a cut through the cone defined by a surface of constant time in P’s future, the resulting section is a sphere (analogous
to the circle formed by cutting a three-dimensional cone), and this
sphere is interpreted as the set of events on which P could have had
a causal effect by radiating a light pulse outward in all directions.
Events lying inside one another’s light cones are said to have
a timelike relationship. Events outside each other’s light cones are
spacelike in relation to one another, and in the case where they lie
on the surfaces of each other’s light cones the term is lightlike.
b / The circle plays a privileged role in Euclidean geometry.
When rotated, it stays the same.
The pie slice is not invariant as
the circle is. A similar privileged
place is occupied by the light
cone in Lorentz geometry. Under
a Lorentz boost, the spacetime
parallelograms change, but the
light cone doesn’t.
The light cone plays the same role in the Lorentz geometry that
the circle plays in Euclidean geometry. The truth or falsehood of
propositions in Euclidean geometry remains the same regardless of
how we rotate the figures, and this is expressed by Euclid’s E3 asserting the existence of circles, which remain invariant under rotation. Similarly, Lorentz boosts preserve light cones and truth of
propositions in a Lorentz frame.
Self-check: Under what circumstances is the time-ordering of
events P and Q preserved under a Lorentz boost?
In a uniform Lorentz spacetime, all the light cones line up like
soldiers with their axes parallel with one another. When gravity is
present, however, this uniformity is disturbed in the vicinity of the
masses that constitute the sources. The light cones lying near the
sources tip toward the sources. Superimposed on top of this gravitational tipping together, recent observations have demonstrated a
systematic tipping-apart effect which becomes significant on cosmological distance scales. The parameter Λ that sets the strength of
this effect is known as the cosmological constant. The cosmological constant is not related to the presence of any sources (such as
negative masses), and can be interpreted instead as a tendency for
space to expand over time on its own initiative. In the present era,
the cosmological constant has overpowered the gravitation of the
universe’s mass, causing the expansion of the universe to accelerate.
Chapter 2
Geometry of flat spacetime
Self-check: In the bottom panel of figure c, can an observer look
at the properties of the spacetime in her immediate vicinity and
tell how much her light cones are tipping, and in which direction?
Compare with figure j on page 28.
A Newtonian black hole
Example: 15
In the case of a black hole, the light cone tips over so far that
the entire future timelike region lies within the black hole. If an
observer is present at such an event, then that observer’s entire potential future lies within the black hole, not outside it. By
expanding on the logical consequences of this statement, we arrive at an example of relativity’s proper interpretation as a theory
of causality, not a theory of objects exerting forces on one another as in Newton’s vision of action at a distance, or Lorentz’s
original ether-drag interpretation of the factor γ, in which length
contraction arose from a physical strain imposed on the atoms
composing a physical body.
Imagine a black hole from a Newtonian point of view, as proposed
in 1783 by geologist John Michell. Setting the escape velocity
equal to the speed of light, we find that this will occur for any gravitating spherical body compact enough to have M/r > c 2 /2G.
(A fully relativistic argument, as given in section 6.2, agrees on
M/r ∝ c 2 /G, which is fixed by units. The correct unitless factor
depends on the definition of r , which is flexible in general relativity.) A flash of light emitted from the surface of such a Newtonian
black hole would fall back down like water from a fountain, but
it would nevertheless be possible for physical objects to escape,
e.g., if they were lifted out in a bucket dangling from a cable. If the
cable is to support its own weight, it must have a tensile strength
per unit density of at least c 2 /2, which is about ten orders of magnitude greater than that of carbon nanotube fibers. (The factor of
1/2 is not to be taken seriously, since it comes from a nonrelativistic calculation.)
The cause-and-effect interpretation of relativity tells us that this
Newtonian picture is incorrect. A physical object that approaches
to within a distance r of a concentration of mass M, with M/r
sufficiently large, has no causal future lying at larger values of r .
The conclusion is that there is a limit on the tensile strength of
any substance, imposed purely by general relativity, and we can
state this limit without having to know anything about the physical
nature of the interatomic forces. A more complete treatment of
the tension in the rope is given in example 5 on p. 273. Cf. also
homework problem 4 and section 3.5.4, as well as some references given in the remark following problem 4.
Section 2.3
c / Light cones tip over for
two reasons in general relativity:
because of the presence of
masses, which have gravitational fields, and because of
the cosmological constant. The
time and distance scales in the
bottom figure are many orders of
magnitude greater than those in
the top.
d / Example 15.
Matter is
lifted out of a Newtonian black
hole with a bucket. The dashed
line represents the point at which
the escape velocity equals the
speed of light.
The light cone
2.3.1 Velocity addition
In classical physics, velocities add in relative motion. For example, if a boat moves relative to a river, and the river moves relative
to the land, then the boat’s velocity relative to the land is found
by vector addition. This linear behavior cannot hold relativistically.
For example, if a spaceship is moving at 0.60c relative to the earth,
and it launches a probe at 0.60c relative to itself, we can’t have the
probe moving at 1.20c relative to the earth, because this would be
greater than the maximum speed of cause and effect, c. To see how
to add velocities relativistically, we start by rewriting the Lorentz
transformation as the matrix
cosh η sinh η
sinh η cosh η
e / The rapidity, η = tanh−1 v , as
a function of v .
where η = tanh−1 v is called the rapidity. We are guaranteed that
the matrix can be written in this form, because its area-preserving
property says that the determinant equals 1, and cosh2 η − sinh2 η =
1 is an identity of the hyperbolic trig functions. It is now straightforward to verify that multiplication of two matrices of this form
gives a third matrix that is also of this form, with η = η1 + η2 . In
other words, rapidities add linearly; velocities don’t. In the example
of the spaceship and the probe, the rapidities add as tanh−1 .60 +
tanh−1 .60 = .693 + .693 = 1.386, giving the probe a velocity of
tanh 1.386 = 0.88 relative to the earth. Any number of velocities
can be added in this way, η1 + η2 + . . . + ηn .
Self-check: Interpret the asymptotes of the graph in figure e.
Bell’s spaceship paradox
Example: 16
A difficult philosophical question is whether the time dilation and
length contractions predicted by relativity are “real.” This depends, of course, on what one means by “real.” They are framedependent, i.e., observers in different frames of reference disagree about them. But this doesn’t tell us much about their reality,
since velocities are frame-dependent in Newtonian mechanics,
but nobody worries about whether velocities are real. John Bell
(1928-1990) proposed the following thought experiment to physicists in the CERN cafeteria, and found that nearly all of them got
it wrong. He took this as evidence that their intuitions had been
misguided by the standard way of approaching this question of
the reality of Lorentz contractions.
f / Example 16.
Chapter 2
Let spaceships A and B accelerate as shown in figure f along a
straight line. Observer C does not accelerate. The accelerations,
as judged by C, are constant, and equal for the two ships. Each
ship is equipped with a yard-arm, and a thread is tied between
the two arms. Does the thread break, due to Lorentz contraction?
(We assume that the acceleration is gentle enough that the thread
does not break simply because of its own inertia.)
Geometry of flat spacetime
The popular answer in the CERN cafeteria was that the thread
would not break, the reasoning being that Lorentz contraction is
a frame-dependent effect, and no such contraction would be observed in A and B’s frame. The ships maintain a constant distance from one another, so C merely disagrees with A and B
about the length of the thread, as well as other lengths like the
lengths of the spaceships.
The error in this reasoning is that the accelerations of A and B
were specified to be equal and constant in C’s frame, not in A and
B’s. Bell’s interpretation is that the frame-dependence is a distraction, that Lorentz contraction is in some sense a real effect, and
that it is therefore immediately clear that the thread must break,
without even having to bother going into any other frame. To convince his peers in the cafeteria, however, Bell presumably needed
to satisfy them as to the specific errors in their reasoning, and this
requires that we consider the frame-dependence explicitly.
We can first see that it is impossible, in general, for different observers to agree about what is meant by constant acceleration.
Suppose that A and B agree with C about the constancy of their
acceleration. Then A and B experience a voyage in which the
rapidities of the stars around them (and of observer C) increase
linearly with time. As the rapidity approaches infinity, both C and
the stars approach the speed of light. But since A and C agree on
the magnitude of their velocity relative to one another, this means
that A’s velocity as measured by C must approach c, and this
contradicts the premise that C observes constant acceleration for
both ships. Therefore A and B do not consider their own accelerations to be constant.
A and B do not agree with C about simultaneity, and since they
also do not agree that their accelerations are constant, they do
not consider their own accelerations to be equal at a given moment of time. Therefore the string changes its length, and this
is consistent with Bell’s original, simple answer, which did not require comparing different frames of reference. To establish that
the string comes under tension, rather than going slack, we can
apply the equivalence principle. By the equivalence principle, any
experiments done by A and B give the same results as if they
were immersed in a gravitational field. The leading ship B sees A
as experiencing a gravitational time dilation. According to B, the
slowpoke A isn’t accelerating as rapidly as it should, causing the
string to break.
These ideas are closely related to the fact that general relativity
does not admit any spacetime that can be interpreted as a uniform gravitational field (see problem 6, p. 199).
Section 2.3
The light cone
2.3.2 Logic
The trichotomous classification of causal relationships has interesting logical implications. In classical Aristotelian logic, every
proposition is either true or false, but not both, and given propositions p and q, we can form propositions such as p ∧ q (both p and
q) or p ∨ q (either p or q). Propositions about physical phenomena
can only be verified by observation. Let p be the statement that
a certain observation carried out at event P gives a certain result,
and similarly for q at Q. If PQ is spacelike, then the truth or falsehood of p ∧ q cannot be checked by physically traveling to P and
Q, because no observer would be able to attend both events. The
truth-value of p ∧ q is unknown to any observer in the universe until
a certain time, at which the relevant information has been able to
propagate back and forth. What if P and Q lie inside two different
black holes? Then the truth-value of p ∧ q can never be determined
by any observer. Another example is the case in which P and Q
are separated by such a great distance that, due to the accelerating
expansion of the universe, their future light cones do not overlap.
We conclude that Aristotelian logic cannot be appropriately applied to relativistic observation in this way. Some workers attempting to construct a quantum-mechanical theory of gravity have suggested an even more radically observer-dependent logic, in which
different observers may contradict one another on the truth-value of
a single proposition p1 , unless they agree in advance on the list p2 ,
p3 , . . . of all the other propositions that they intend to test as well.
We’ll return to these questions on page 232.
2.4 Experimental tests of Lorentz geometry
We’ve already seen, in section 1.2, a variety of evidence for the nonclassical behavior of spacetime. We’re now in a position to discuss
tests of relativity more quantitatively. An up-to-date review of such
tests is given by Mattingly.14
One such test is that relativity requires the speed of light to
be the same in all frames of reference, for the following reasons.
Compare with the speed of sound in air. The speed of sound is not
the same in all frames of reference, because the wave propagates
at a fixed speed relative to the air. An observer at who is moving
relative to the air will measure a different speed of sound. Light, on
the other hand, isn’t a vibration of any physical medium. Maxwell’s
equations predict a definite value for the speed of light, regardless
of the motion of the source. This speed also can’t be relative to
any medium. If the speed of light isn’t fixed relative to the source,
and isn’t fixed relative to a medium, then it must be fixed relative
to anything at all. The only speed in relativity that is equal in all
Chapter 2
Geometry of flat spacetime
frames of reference is c, so light must propagate at c. We will see
on page ?? that there is a deeper reason for this; relativity requires
that any massless particle propagate at c. The requirement of v = c
for massless particles is so intimately hard-wired into the structure
of relativity that any violation of it, no matter how tiny, would be of
great interest. Essentially, such a violation would disprove Lorentz
invariance, i.e., the invariance of the laws of physics under Lorentz
transformations. There are two types of tests we could do: (1)
test whether photons of all energies travel at the same speed, i.e.,
whether the vacuum is dispersive; (2) test whether observers in all
frames of reference measure the same speed of light.
2.4.1 Dispersion of the vacuum
Some candidate quantum-mechanical theories of gravity, such
as loop quantum gravity,
p predict a granular structure for spacetime
at the Planck scale, ~G/c3 = 10−35 m, which one could imagine
might lead to deviations from v = 1 that would become more and
more significant for photons with wavelengths getting closer and
closer to that scale. Lorentz-invariance would then be an approximation valid only at large scales. It turns out that the state of the
art in loop quantum gravity is not yet sufficient to say whether or
not such an effect should exist.
Presently the best experimental tests of the invariance of the
speed of light with respect to wavelength come from astronomical
observations of gamma-ray bursts, which are sudden outpourings of
high-energy photons, believed to originate from a supernova explosion in another galaxy. One such observation, in 2009,15 collected
photons from such a burst, with a duration of 2 seconds, indicating
that the propagation time of all the photons differed by no more
than 2 seconds out of a total time in flight on the order of ten billion years, or about one part in 1017 ! A single superlative photon in
the burst had an energy of 31 GeV, and its arrival within the same
2-second time window demonstrates Lorentz invariance over a vast
range of photon energies, contradicting heuristic estimates that had
been made by some researchers in loop quantum gravity.
a / An artist’s conception of a
gamma-ray burst, resulting from
a supernova explosion.
2.4.2 Observer-independence of c
The constancy of the speed of light for observers in all frames of
reference was originally detected in 1887 when Michelson and Morley
set up a clever apparatus to measure any difference in the speed of
light beams traveling east-west and north-south. The motion of
the earth around the sun at 110,000 km/hour (about 0.01% of the
speed of light) is to our west during the day. Michelson and Morley
believed that light was a vibration of a physical medium, the ether,
so they expected that the speed of light would be a fixed value
relative to the ether. As the earth moved through the ether, they
Section 2.4
Experimental tests of Lorentz geometry
thought they would observe an effect on the velocity of light along
an east-west line. For instance, if they released a beam of light in
a westward direction during the day, they expected that it would
move away from them at less than the normal speed because the
earth was chasing it through the ether. They were surprised when
they found that the expected 0.01% change in the speed of light did
not occur.
b / The Michelson-Morley experiment, shown in photographs, and
drawings from the original 1887
paper. 1. A simplified drawing of the apparatus. A beam of
light from the source, s, is partially reflected and partially transmitted by the half-silvered mirror
h1 . The two half-intensity parts of
the beam are reflected by the mirrors at a and b, reunited, and observed in the telescope, t. If the
earth’s surface was supposed to
be moving through the ether, then
the times taken by the two light
waves to pass through the moving ether would be unequal, and
the resulting time lag would be
detectable by observing the interference between the waves when
they were reunited. 2. In the real
apparatus, the light beams were
reflected multiple times. The effective length of each arm was
increased to 11 meters, which
greatly improved its sensitivity to
the small expected difference in
the speed of light. 3. In an
earlier version of the experiment,
they had run into problems with
its “extreme sensitiveness to vibration,” which was “so great that
it was impossible to see the interference fringes except at brief
intervals . . . even at two o’clock
in the morning.” They therefore
mounted the whole thing on a
massive stone floating in a pool of
mercury, which also made it possible to rotate it easily. 4. A photo
of the apparatus. Note that it is
underground, in a room with solid
brick walls.
Although the Michelson-Morley experiment was nearly two decades in the past by the time Einstein published his first paper on
relativity in 1905, and Einstein did know about it,16 it’s unclear how
much it influenced him. Michelson and Morley themselves were uncertain about whether the result was to be trusted, or whether systematic and random errors were masking a real effect from the ether.
There were a variety of competing theories, each of which could
claim some support from the shaky data. Some physicists believed
that the ether could be dragged along by matter moving through it,
which inspired variations on the experiment that were conducted on
mountaintops in thin-walled buildings, (figure), or with one arm of
the apparatus out in the open, and the other surrounded by massive
lead walls. In the standard sanitized textbook version of the history
of science, every scientist does his experiments without any preconceived notions about the truth, and any disagreement is quickly
settled by a definitive experiment. In reality, this period of confu16
Chapter 2
J. van Dongen,
Geometry of flat spacetime
sion about the Michelson-Morley experiment lasted for four decades,
and a few reputable skeptics, including Miller, continued to believe
that Einstein was wrong, and kept trying different variations of the
experiment as late as the 1920’s. Most of the remaining doubters
were convinced by an extremely precise version of the experiment
performed by Joos in 1930, although you can still find kooks on the
internet who insist that Miller was right, and that there was a vast
conspiracy to cover up his results.
c / Dayton Miller thought that the result of the Michelson-Morley experiment could be explained because the ether had been pulled along by
the dirt, and the walls of the laboratory. This motivated him to carry out a
series of experiments at the top of Mount Wilson, in a building with thin
Before Einstein, some physicists who did believe the negative
result of the Michelson-Morley experiment came up with explanations that preserved the ether. In the period from 1889 to 1895, both
Lorentz and George FitzGerald suggested that the negative result
of the Michelson-Morley experiment could be explained if the earth,
and every physical object on its surface, was contracted slightly by
the strain of the earth’s motion through the ether. Thus although
Lorentz developed all the mathematics of Lorentz frames, and got
them named after himself, he got the interpretation wrong.
2.4.3 Lorentz violation by gravitational forces
The tests described in sections 2.4.1 and 2.4.2 both involve the
behavior of light, i.e., they test whether or not electromagnetism
really has the exact Lorentz-invariant behavior contained implicitly
in Maxwell’s equations. In the jargon of the field, they test Lorentz
invariance in the “photon sector.” Since relativity is a theory of
gravity, it is natural to ask whether the Lorentz invariance holds
for gravitational forces as well as electromagnetic ones. If Lorentz
invariance is violated by gravity, then the strength of gravitational
forces might depend on the observer’s motion through space, relative to some fixed reference frame analogous to that of the ether.
Historically, gravitational Lorentz violations have been much more
difficult to test, since gravitational forces are so weak, and the first
high-precision data were obtained by Nordtvedt and Will in 1957,
70 years after Michelson and Morley. Nordtvedt and Will measured
Section 2.4
Experimental tests of Lorentz geometry
d / The results of the measurement of g by Chung et al., section 2.4.3. The experiment was
done on the Stanford University
campus, surrounded by the Pacific ocean and San Francisco
Bay, so it was subject to varying gravitational from both astronomical bodies and the rising and
falling ocean tides. Once both of
these effects are subtracted out
of the data, there is no Lorentzviolating variation in g due to
the earth’s motion through space.
Note that the data are broken up
into three periods, with gaps of
three months and four years separating them. (c) APS, used under the U.S. fair use exception to
the strength of the earth’s gravitational field as a function of time,
and found that it did not vary on a 24-hour cycle with the earth’s
rotation, once tidal effects had been accounted for. Further constraints come from data on the moon’s orbit obtained by reflecting
laser beams from a mirror left behind by the Apollo astronauts.
e / The matter interferometer
used by Chung et al. Each atom’s
wavefunction is split into two
parts, which travel along two
different paths (solid and dashed
A recent high-precision laboratory experiment was done in 2009
by Chung et al.17 They constructed an interferometer in a vertical plane that is conceptually similar to a Michelson interferometer,
except that it uses cesium atoms rather than photons. That is,
the light waves of the Michelson-Morley experiment are replaced by
quantum-mechanical matter waves. The roles of the half-silvered
and fully silvered mirrors are filled by lasers, which kick the atoms
electromagnetically. Each atom’s wavefunction is split into two
parts, which travel by two different paths through spacetime, eventually reuniting and interfering. The result is a measurement of g
to about one part per billion. The results, shown in figure d, put a
strict limit on violations of Lorentz geometry by gravity.
2.5 Three spatial dimensions
New and nontrivial phenomena arise when we generalize from 1+1
dimensions to 3+1.
2.5.1 Lorentz boosts in three dimensions
How does a Lorentz boost along one axis, say x, affect the other
two spatial coordinates y and z?
First, we can rule out the possibility that such a transformation
could have various terms such as t0 = . . . + (. . .)y + . . .. For example,
Chapter 2
Geometry of flat spacetime
if the t coefficient was positive for v > 0, then the laws of physics
would be different from the laws that applied in a universe where
the y or t axis was inverted, but this would violate parity or timereversal symmetry. This establishes that observers in the two frames
agree on the directions of the y and z axes and on simultaneity along
those axes when they coincide.
Now suppose that two observers, in motion relative to one another along the x axis, each carry a stick, represented by line segments AB and CD, oriented along the y axis, such that the bases
of the sticks A and C coincide at some time. Due to the vanishing
of the types of terms in the transformation referred to above, they
agree that B and D are collinear with A (and C) at this time. Then
by O3 and O4, either B lies between A and D, D lies between A and
B, or B=D. That is, they must agree whether the sticks are equal
in length or, if not, then on whose is longer. This would violate L1,
isotropy of space, since it would distinguish +x from −x.
Another simple way to obtain this result is as follows. We have
already proved that area in the (t, x) plane is preserved. The same
proof applies to volume in the spaces (t, x, y) and (t, x, z), hence
lengths in the y and z directions are preserved. (The proof does not
apply to volume in, e.g., (x, y, z) space, because the x transformation
depends on t, and therefore if we are given a region in (x, y, z), we
do not have enough information to say how it will change under a
Lorentz boost.)
The complete form of the transformation L(vx̂), a Lorentz boost
along the x axis with velocity v, is therefore:
t0 = γt + vγx
x0 = vγt + γx
a / A boost along x followed
by a boost along y results in
tangling up of the x and y coordinates, so the result is not just a
boost but a boost plus a rotation.
y0 = y
z0 = z
Based on the trivial nature of this generalization, it might seem
as though no qualitatively new considerations would arise in 3+1
dimensions as compared with 1+1. To see that this is not the case,
consider figure a. A boost along the x axis tangles up the x and
t coordinates. A y-boost mingles y and t. Therefore consecutive
boosts along x and y can cause x and y to mix. The result, as
we’ll see in more detail below, is that two consecutive boosts along
non-collinear axes are not equivalent to a single boost; they are
equivalent to a boost plus a spatial rotation. The remainder of this
section discusses this effect, known as Thomas precession, in more
detail; it can be omitted on a first reading.
Self-check: Apply similar reasoning to a Galilean boost.
Section 2.5
Three spatial dimensions
2.5.2 Gyroscopes and the equivalence principle
To see how this mathematical fact would play out as a physical
effect, we need to consider how to make a physical manifestation of
the concept of a direction in space.
b / Inertial devices for maintaining a direction in space: 1.
A ring laser. 2. The photon in
a perfectly reflective spherical
cavity. 3. A gyroscope.
In two space dimensions, we can construct a ring laser, b/1,
which in its simplest incarnation is a closed loop of optical fiber
with a bidirectional laser inserted in one place. Coherent light traverses the loop simultaneously in both directions, interfering in a
beat pattern, which can be observed by sampling the light at some
point along the loop’s circumference. If the loop is rotated in its
own plane, the interference pattern is altered, because the beamsampling device is in a different place, and the path lengths traveled
by the two beams has been altered. This phase shift is called the
Sagnac effect, after M. Georges Sagnac, who observed the effect in
1913 and interpreted it, incorrectly, as evidence for the existence of
the aether.18 The loop senses its own angular velocity relative to
an inertial reference frame. If we transport the loop while always
carefully adjusting its orientation so as to prevent phase shifts, then
its orientation has been preserved. The atomic clocks used in the
Hafele-Keating atomic-clock experiment described on page 15 were
sensitive to Sagnac effects, and it was not practical to maintain their
orientations while they were strapped into seats on a passenger jet,
so this orientational effect had to be subtracted out of the data at
the end of the experiment.
In three spatial dimensions, we could build a spherical cavity
with a reflective inner surface, and release a photon inside, b/2.
In reality, the photon-in-a-cavity is not very practical. The photon would eventually be absorbed or scattered, and it would also be
difficult to accurately initialize the device and read it out later. A
more practical tool is a gyroscope. For example, one of the classic
tests of general relativity is the 2007 Gravity Probe B experiment
(discussed in detail on pages 170 and 212), in which four gyroscopes aboard a satellite were observed to precess due to specialand general-relativistic effects.
c / A ring laser gyroscope
built for use in inertial guidance of
The gyroscope, however, is not so obviously a literal implementation of our basic concept of a direction. How, then, can we be sure
that its behavior is equivalent to that of the photon-in-a-cavity? We
could, for example, carry out a complete mathematical development
of the angular momentum vector in relativity.19 The equivalence
principle, however, allows us to bypass such technical details. Suppose that we seal the two devices inside black boxes, with identical
external control panels for initializing them and reading them out.
We initialize them identically, and then transport them along side18
Comptes rendus de l’Académie des science 157 (1913) 708
This is done, for example, in Misner, Thorne, and Wheeler, Gravitation, pp.
Chapter 2
Geometry of flat spacetime
by-side world-lines. Classically, both the mechanical gyroscope and
the photon-gyroscope would maintain absolute, fixed directions in
space. Relativistically, they will not necessarily maintain their orientations. For example, we’ve already seen in section 2.5.1 that
there are reasons to expect that their orientations will change if
they are subjected to accelerations that are not all along the same
line. Because relativity is a geometrical theory of spacetime, this
difference between the classical and relativistic behavior must be
determinable from purely geometrical considerations, such as the
shape of the world-line. If it depended on something else, then we
could conceivably see a disagreement in the outputs of the two instruments, but this would violate the equivalence principle.
Suppose there were such a discrepancy. That discrepancy would
be a physically measurable property of the spacetime region through
which the two gyroscopes had been transported. The effect would
have a certain magnitude and direction, so by collecting enough data
we could map it out as vector field covering that region of spacetime.
This field evidently causes material particles to accelerate, since it
has an effect on the mechanical gyroscope. Roughly speaking (the
reasoning will be filled in more rigorously on page 142), the fact
that this field acts differently on the two gyroscopes is like getting a
non-null result from an Eötvös experiment, and it therefore violates
the equivalence principle. We conclude that gyroscopes b/2 and
b/3 are equivalent. In other words, there can only be one uniquely
defined notion of direction, and the details of how it is implemented
are irrelevant.
2.5.3 Boosts causing rotations
As a quantitative example, consider the following thought experiment. Put a gyroscope in a box, and send the box around the
square path shown in figure d at constant speed. The gyroscope defines a local coordinate system, which according to classical physics
would maintain its orientation. At each corner of the square, the
box has its velocity vector changed abruptly, as represented by the
hammer. We assume that the hits with the hammer are transmitted
to the gyroscope at its center of mass, so that they do not result in
any torque. Classically, if the set of gyroscopes travels once around
the square, it should end up at the same place and in the same orientation, so that the coordinate system it defines is identical with
the original one.
For notation, let L(vx̂) indicate the boost along the x axis described by the transformation on page 72. This is a transformation
that changes to a frame of reference moving in the negative x direction compared to the original frame. A particle considered to be at
rest in the original frame is described in the new frame as moving
in the positive x direction. Applying such an L to a vector p, we
calculate Lp, which gives the coordinates of the event as measured
Section 2.5
d / Classically, the gyroscope
should not rotate as long as the
forces from the hammer are all
transmitted to it at its center of
Three spatial dimensions
in the new frame. An expression like M Lp is equivalent by associativity to M (Lp), i.e., M L represents applying L first, and then
In this notation, the hammer strikes can be represented by a
series of four Lorentz boosts,
T = L(vx̂) L(vŷ) L(−vx̂) L(−vŷ)
where we assume that the square has negligible size, so that all four
Lorentz boosts act in a way that preserves the origin of the coordinate systems. (We have no convenient way in our notation L(. . .) to
describe a transformation that does not preserve the origin.) The
first transformation, L(−vŷ), changes coordinates measured by the
original gyroscope-defined frame to new coordinates measured by
the new gyroscope-defined frame, after the box has been accelerated in the positive y direction.
e / A page from one of Einstein’s
The calculation of T is messy, and to be honest, I made a series
of mistakes when I tried to crank it out by hand. Calculations in
relativity have a reputation for being like this. Figure e shows a page
from one of Einstein’s notebooks, written in fountain pen around
1913. At the bottom of the page, he wrote “zu umstaendlich,”
meaning “too involved.” Luckily we live in an era in which this sort
Chapter 2
Geometry of flat spacetime
of thing can be handled by computers. Starting at this point in the
book, I will take appropriate opportunities to demonstrate how to
use the free and open-source computer algebra system Maxima to
keep complicated calculations manageable. The following Maxima
program calculates a particular element of the matrix T .
/* For convenience, define gamma in terms of v: */
/* Define Lx as L(x-hat), Lmx as L(-x-hat), etc.: */
gamma*v, 0],
[gamma*v, gamma,
[gamma*v, 0,
Lmx:matrix([gamma, -gamma*v, 0],
[-gamma*v, gamma,
Lmy:matrix([gamma, 0,
[-gamma*v, 0,
/* Calculate the product of the four matrices: */
/* Define a column vector along the x axis: */
/* Find the result of T acting on this vector,
expressed as a Taylor series to second order in v: */
Statements are terminated by semicolons, and comments are written like /* ... */ On line 2, we see a symbolic definition of the
symbol gamma in terms of the symbol v. The colon means “is defined as.” Line 2 does not mean, as it would in most programming
languages, to take a stored numerical value of v and use it to calculate a numerical value of γ. In fact, v does not have a numerical
value defined at this point, nor will it ever have a numerical value
defined for it throughout this program. Line 2 simply means that
whenever Maxima encounters the symbol gamma, it should take it as
an abbreviation for the symbol 1/sqrt(1-v*v). Lines 5-16 define
some 3 × 3 matrices that represent the L transformations. The basis
is t̂, x̂, ŷ. Line 18 calculates the product of the four matrices; the
dots represent matrix multiplication. Line 23 defines a vector along
the x axis, expressed as a column matrix (three rows of one column
each) so that Maxima will know how to operate on it using matrix
multiplication by T .
Finally line 26 outputs20 the result of T acting on P:
I’ve omitted some output generated automatically from the earlier steps in
Section 2.5
Three spatial dimensions
[ 0 + . . .
[ 1 + . . .
[ - v + . . .
In other words,
 
T  1  =  1  + ...
−v 2
where . . . represents higher-order terms in v. Suppose that we use
the initial frame of reference, before T is applied, to determine that
a particular reference point, such as a distant star, is along the x
axis. Applying T , we get a new vector T P, which we find has a nonvanishing y component approximately equal to −v 2 . This result is
entirely unexpected classically. It tells us that the gyroscope, rather
than maintaining its original orientation as it would have done classically, has rotated slightly. It has precessed in the counterclockwise
direction in the x−y plane, so that the direction to the star, as measured in the coordinate system defined by the gyroscope, appears
to have rotated clockwise. As the box moved clockwise around the
square, the gyroscope has apparently rotated by a counterclockwise
angle χ ≈ v 2 about the z axis. We can see that this is a purely
relativistic effect, since for v 1 the effect is small. For historical
reasons discussed in section 2.5.4, this phenomenon is referred to as
the Thomas precession.
The particular features of this square geometry are not necessary.
I chose them so that (1) the boosts would be along the Cartesian
axes, so that we would be able to write them down easily; (2) it is
clear that the effect doesn’t arise from any asymmetric treatment
of the spatial axes; and (3) the change in the orientation of the
gyroscope can be measured at the same point in space, e.g., by
comparing it with a twin gyroscope that stays at home. In general:
A gyroscope transported around a closed loop in flat spacetime changes its orientation compared with one that is not
This is a purely relativistic effect, since a classical gyroscope
does not change its axis of rotation unless subjected to a
torque; if the boosts are accomplished by forces that act at
the gyroscope’s center of mass, then there is no classical explanation for the effect.
the computation. The (%o9) indicates that this is Maxima’s output from the
ninth and final step.
Chapter 2
Geometry of flat spacetime
The effect can occur in the absence of any gravitational fields.
That is, this is a phenomenon of special relativity.
The composition of two or more Lorentz boosts along different
axes is not equivalent to a single boost; it is equivalent to a
boost plus a spatial rotation.
Lorentz boosts do not commute, i.e., it makes a difference what
order we perform them in. Even if there is almost no time lag
between the first boost and the second, the order of the boosts
matters. If we had applied the boosts in the opposite order,
the handedness of the effect would have been reversed.
Self-check: If Lorentz boosts did commute, what would be the
consequences for the expression L(vx̂) L(vŷ) L(−vx̂) L(−vŷ)?
The velocity disk
Figure f shows a useful way of visualizing the combined effects
of boosts and rotations in 2+1 dimensions. The disk depicts all
possible states of motion relative to some arbitrarily chosen frame
of reference. Lack of motion is represented by the point at the
center. A point at distance v from the center represents motion at
velocity v in a particular direction in the x − y plane. By drawing
little axes at a particular point, we can represent a particular frame
of reference: the frame is in motion at some velocity, with its own
x and y axes are oriented in a particular way.
It turns out to be easier to understand the qualitative behavior
of our mysterious rotations if we switch from the low-velocity limit
to the contrary limit of ultrarelativistic velocities. Suppose we have
a rocket-ship with an inertial navigation system consisting of two
gyroscopes at right angles to one another. We first accelerate the
ship in the y direction, and the acceleration is steady in the sense
that it feels constant to observers aboard the ship. Since it is rapidities, not velocities, that add linearly, this means that as an observer
aboard the ship reads clock times τ1 , τ2 , . . . , all separated by equal
intervals ∆τ , the ship’s rapidity changes at a constant rate, η1 , η2 ,
. . . . This results in a series of frames of reference that appear closer
and closer together on the diagram as the ship approaches the speed
of light, at the edge of the disk. We can start over from the center
again and repeat the whole process along the x axis, resulting in
a similar succession of frames. In both cases, the boosts are being
applied along a single line, so that there is no rotation of the x and
y axes.
f / The velocity disk.
g / Two excursions in a rocketship: one along the y axis and
one along x .
Now suppose that the ship were to accelerate along a route like
the one shown in figure h. It first accelerates along the y axis at a
constant rate (again, as judged by its own sensors), until its velocity
is very close to the speed of light, A. It then accelerates, again at
a self-perceived constant rate and with thrust in a fixed direction
Section 2.5
Three spatial dimensions
as judged by its own gyroscopes, until it is moving at the same
ultrarelativistic speed in the x direction, B. Finally, it decelerates
in the x direction until it is again at rest, O. This motion traces out
a clockwise loop on the velocity disk. The motion in space is also
h / A round-trip involving ultrarelativistic velocities. All three
legs are at constant acceleration.
i / In the limit where A and B
are ultrarelativistic velocities, leg
AB is perpendicular to the edge
of the velocity disk. The result is
that the x − y frame determined
by the ship’s gyroscopes has
rotated by 90 degrees by the time
it gets home.
We might naively think that the middle leg of the trip, from A
to B, would be a straight line on the velocity disk, but this can’t be
the case. First, we know that non-collinear boosts cause rotations.
Traveling around a clockwise path causes counterclockwise rotation,
and vice-versa. Therefore an observer in the rest frame O sees the
ship (and its gyroscopes) as rotating as it moves from A to B. The
ship’s trajectory through space is clockwise, so according to O the
ship rotates counterclockwise as it goes A to B. The ship is always
firing its engines in a fixed direction as judged by its gyroscopes, but
according to O the ship is rotating counterclockwise, its thrust is
progressively rotating counterclockwise, and therefore its trajectory
turns counterclockwise. We conclude that leg AB on the velocity
disk is concave, rather than being a straight-line hypotenuse of a
triangle OAB.
We can also determine, by the following argument, that leg AB
is perpendicular to the edge of the disk where it touches the edge of
the disk. In the transformation from frame A to frame O, y coordinates are dilated by a factor of γ, which approaches infinity in the
limit we’re presently considering. Observers aboard the rocket-ship,
occupying frame A, believe that their task is to fire the rocket’s
engines at an angle of 45 degrees with respect to the y axis, so as
to eliminate their velocity with respect to the origin, and simultaneously add an equal amount of velocity in the x direction. This
45-degree angle in frame A, however, is not a 45-degree angle in
frame O. From the stern of the ship to its bow we have displacements ∆x and ∆y, and in the transformation from A to O, ∆y
is magnified almost infinitely. As perceived in frame O, the ship’s
orientation is almost exactly antiparallel to the y axis.21
As the ship travels from A to B, its orientation (as judged in
frame O) changes from −ŷ to x̂. This establishes, in a much more
direct fashion, the direction of the Thomas precession: its handedness is contrary to the handedness of the direction of motion. We
can now also see something new about the fundamental reason for
the effect. It has to do with the fact that observers in different
states of motion disagree on spatial angles. Similarly, imagine that
Although we will not need any more than this for the purposes of our present
analysis, a longer and more detailed discussion by Rhodes and Semon, www., Am. J. Phys. 72(7)2004, shows
that this type of inertially guided, constant-thrust motion is always represented
on the velocity disk by an arc of a circle that is perpendicular to the disk at its
edge. (We consider a diameter of the disk to be the limiting case of a circle with
infinite radius.)
Chapter 2
Geometry of flat spacetime
you are a two-dimensional being who was told about the existence
of a new, third, spatial dimension. You have always believed that
the cosine of the angle between two unit vectors u and v is given by
the vector dot product ux vx + uy vy . If you were allowed to explore
a two-dimensional projection of a three-dimensional scene, e.g., on
the flat screen of a television, it would seem to you as if all the
angles had been distorted. You would have no way to interpret the
visual conventions of perspective. But once you had learned about
the existence of a z axis, you would realize that these angular distortions were happening because of rotations out of the x − y plane.
Such rotations really conserve the quantity ux vx + uy vy + uz vz ; only
because you were ignoring the uz vz term did it seem that angles
were not being preserved. Similarly, the generalization from three
Euclidean spatial dimensions to 3+1-dimensional spacetime means
that three-dimensional dot products are no longer conserved.
The general low-v limit
Let’s find the low-v limit of the Thomas precession in general,
not just in the highly artificial special case of χ ≈ v 2 for the example
involving the four hammer hits. To generalize to the case of smooth
acceleration, we first note that the rate of precession dχ/dt must
have the following properties.
It is odd under a reversal of the direction of motion, v → −v.
(This corresponds to sending the gyroscope around the square
in the opposite direction.)
It is odd under a reversal of the acceleration due to the second
boost, a → −a.
It is a rotation about the spatial axis perpendicular to the
plane of the v and a vectors, in the opposite direction compared to the handedness of the curving trajectory.
It is approximately linear in v and a, for small v and a.
The only rotationally invariant mathematical operation that has
these symmetry properties is the vector cross product, so the rate
of precession must be ka × v, where k > 0 is nearly independent of
v and a for small v and a.
To pin down the value of k, we need to find a connection between
our two results: χ ≈ v 2 for the four hammer hits, and dχ/dt ≈ ka×v
for smooth acceleration. We can do this by considering the physical
significance of areas on the velocity disk. As shown in figure j, the
rotation χ due to carrying the velocity around the boundary of a
region is additive when adjacent regions are joined together. We
can therefore find χ for any region by breaking the region down into
elements of area dA and integrating their contributions dχ. What is
the relationship between dA and dχ? The velocity disk’s structure
Section 2.5
Three spatial dimensions
is nonuniform, in the sense that near the edge of the disk, it takes a
larger boost to move a small distance. But we’re investigating the
low-velocity limit, and in the low-velocity region near the center of
the disk, the disk’s structure is approximately uniform. We therefore expect that there is an approximately constant proportionality
factor relating dA and dχ at low velocities. The example of the
hammer corresponds geometrically to a square with area v 2 , so we
find that this proportionality factor is unity, dA ≈ dχ.
j / If the crack between the
two areas is squashed flat, the
two pieces of the path on the
interior coincide, and their contributions to the precession cancel
out (v → −v, but a → +a, so
a × v → −a × v). Therefore the
precession χ obtained by going
around the outside is equal to the
sum χ1 + χ2 of the precessions
that would have been obtained by
going around the two parts.
To relate this to smooth acceleration, consider a particle per2
forming circular motion with period T , which has |a ×
R v| = 2πv /T .
Over one full period of the motion, we have χ = k|a × v|dt =
2πkv 2 , and the particle’s velocity vector traces a circle of area A =
πv 2 on the velocity disk. Equating A and χ, we find k = 1/2. The
result is that in the limit of low velocities, the rate of rotation is
Ω≈ a×v
where Ω is the angular velocity vector of the rotation. In the special
case of circular motion, this can be written as Ω = (1/2)v 2 ω, where
ω = 2π/T is the angular frequency of the motion.
2.5.4 An experimental test: Thomas precession in hydrogen
If we want to see this precession effect in real life, we should look
for a system in which both v and a are large. An atom is such a
k / States in hydrogen are labeled with their ` and s quantum
representing their
orbital and spin angular momenta
in units of ~. The state with
s = +1/2 has its spin angular
momentum aligned with its orbital
angular momentum, while the
s = −1/2 state has the two
angular momenta in opposite
The direction and
order of magnitude of the splitting
between the two ` = 1 states
is successfully explained by
magnetic interactions with the
proton, but the calculated effect
is too big by a factor of 2. The
relativistic Thomas precession
cancels out half of the effect.
Chapter 2
The Bohr model, introduced in 1913, marked the first quantitatively successful, if conceptually muddled, description of the atomic
energy levels of hydrogen. Continuing to take c = 1, the over-all
scale of the energies was calculated to be proportional to mα2 , where
m is the mass of the electron, and α = ke2 /~ ≈ 1/137, known as the
fine structure constant, is essentially just a unitless way of expressing the coupling constant for electrical forces. At higher resolution,
each excited energy level is found to be split into several sub-levels.
The transitions among these close-lying states are in the millimeter region of the microwave spectrum. The energy scale of this fine
structure is ∼ mα4 . This is down by a factor of α2 compared to the
visible-light transitions, hence the name of the constant. Uhlenbeck
and Goudsmit showed in 1926 that a splitting on this order of magnitude was to be expected due to the magnetic interaction between
the proton and the electron’s magnetic moment, oriented along its
spin. The effect they calculated, however, was too big by a factor
of two.
The explanation of the mysterious factor of two had in fact been
implicit in a 1916 calculation by Willem de Sitter, one of the first
applications of general relativity. De Sitter treated the earth-moon
system as a gyroscope, and found the precession of its axis of rotation, which was partly due to the curvature of spacetime and partly
Geometry of flat spacetime
due to the type of rotation described earlier in this section. The
effect on the motion of the moon was noncumulative, and was only
about one meter, which was much too small to be measured at the
time. In 1927, however, Llewellyn Thomas applied similar reasoning to the hydrogen atom, with the electron’s spin vector playing
the role of gyroscope. Since gravity is negligible here, the effect has
nothing to do with curvature of spacetime, and Thomas’s effect corresponds purely to the special-relativistic part of de Sitter’s result.
It is simply the rotation described above, with Ω = (1/2)v 2 ω. Although Thomas was not the first to calculate it, the effect is known
as Thomas precession. Since the electron’s spin is ~/2, the energy
splitting is ±(~/2)Ω, depending on whether the electron’s spin is in
the same direction as its orbital motion, or in the opposite direction. This is less than the atom’s gross energy scale ~ω by a factor
of v 2 /2, which is ∼ α2 . The Thomas precession cancels out half of
the magnetic effect, bringing theory in agreement with experiment.
Uhlenbeck later recalled: “...when I first heard about [the Thomas
precession], it seemed unbelievable that a relativistic effect could
give a factor of 2 instead of something of order v/c... Even the
cognoscenti of relativity theory (Einstein included!) were quite surprised.”
Section 2.5
Three spatial dimensions
Suppose that we don’t yet know the exact form of the Lorentz
transformation, but we know based on the Michelson-Morley experiment that the speed of light is the same in all inertial frames, and
we’ve already determined, e.g., by arguments like those on p. 72,
that there can be no length contraction in the direction perpendicular to the motion. We construct a “light clock,” consisting simply
of two mirrors facing each other, with a light pulse bouncing back
and forth between them.
(a) Suppose this light clock is moving at a constant velocity v in the
direction perpendicular to its own optical arm, which is of length L.
Use the Pythagorean theorem√to prove that the clock experiences a
time dilation given by γ = 1/ 1 − v 2 , thereby fixing the time-time
portion of the Lorentz transformation.
(b) Why is it significant for the interpretation of special relativity
that the result from part a is independent of L?
(c) Carry out a similar calculation in the case where the clock moves
with constant acceleration a as measured in some inertial frame. Although the result depends on L, prove that in the limit of small L,
we recover the earlier constant-velocity result, with no explicit dependence on a.
Remark: Some authors state a “clock postulate” for special relativity, which
says that for a clock that is sufficiently small, the rate at which it runs depends only on v, not a (except in the trivial sense that v and a are related
by calculus). The result of part c shows that the clock “postulate” is really a
theorem, not a statement that is logically independent of the other postulates
of special relativity. Although this argument only applies to a particular family of light clocks of various sizes, one can also make any small clock into an
acceleration-insensitive clock, by attaching an accelerometer to it and applying an appropriate correction to compensate for the clock’s observed sensitivity
to acceleration. (It’s still necessary for the clock to be small, since otherwise
the lack of simultaneity in relativity makes it impossible to describe the whole
clock as having a certain acceleration at a certain instant.) Farley at al.22 have
verified the “clock postulate” to within 2% for the radioactive decay of muons
with γ ∼ 12 being accelerated by magnetic fields at 5 × 1018 m/s2 . Some people get confused by this acceleration-independent property of small clocks and
think that it contradicts the equivalence principle. For a good explanation, see
. Solution, p. 367
Chapter 2
Nuovo Cimento 45 (1966) 281
Geometry of flat spacetime
Some of the most conceptually direct tests of relativistic time
dilation were carried out by comparing the rates of twin atomic
clocks, one left on a mountaintop for a certain amount of time, the
other in a nearby valley below.23 Unlike the clocks in the HafeleKeating experiment, these are stationary for almost the entire duration of the experiment, so any time dilation is purely gravitational,
not kinematic. One could object, however, that the clocks are not
really at rest relative to one another, due to the earth’s rotation.
This is an example of how the distinction between gravitational
and kinematic time dilations is frame-dependent, since the effect is
purely gravitational in the rotating frame, where the gravitational
field is reduced by the fictitious centrifugal force. Show that, in the
non-rotating frame, the ratio of the kinematic effect to the gravitational one comes out to be 2.8 × 10−3 at the latitude of Tokyo.
This small value indicates that the experiment can be interpreted
as a very pure test of the gravitational time dilation effect. To calculate the effect, you will need to use the fact that, as discussed
on p. 33, gravitational redshifts can be interpreted as gravitational
time dilations.
. Solution, p. 367
(a) On p. 82 (see figure j), we showed that the Thomas precession is proportional to area on the velocity disk. Use a similar
argument to show that the Sagnac effect (p. 74) is proportional to
the area enclosed by the loop.
(b) Verify this more directly in the special case of a circular loop.
(c) Show that a light clock of the type described in problem 1 is
insensitive to rotation with constant angular velocity.
(d) Connect these results to the commutativity and transitivity assumptions in the Einstein clock synchronization procedure described
on p. 347.
. Solution, p. 367
A graph from the paper by
Iijima, showing the time difference between the two clocks.
One clock was kept at Mitaka
Observatory, at 58 m above sea
The other was moved
back and forth between a second
observatory, Norikura Corona
Station, and the peak of the
Norikura volcano, 2876 m above
sea level. The plateaus on the
graph are data from the periods
when the clocks were compared
side by side at Mitaka.
difference between one plateau
and the next is the gravitational
time dilation accumulated during
the period when the mobile clock
was at the top of Norikura.
Example 15 on page 65 discusses relativistic bounds on the
properties of matter, using the example of pulling a bucket out of a
black hole. Derive a similar bound by considering the possibility of
sending signals out of the black hole using longitudinal vibrations of
a cable, as in the child’s telephone made of two tin cans connected
by a piece of string.
Remark: Surprisingly subtle issues can arise in such calculations; see A.Y.
Shiekh, Can. J. Phys. 70, 458 (1992). For a quantitative treatment of a dangling
rope in relativity, see Greg Egan, “The Rindler Horizon,” http://gregegan.
The Maxima program on page 77 demonstrates how to multiply matrices and find Taylor series. Apply this technique to the
following problem. For successive Lorentz boosts along the same
L. Briatore and S. Leschiutta, “Evidence for the earth gravitational shift by
direct atomic-time-scale comparison,” Il Nuovo Cimento B, 37B (2): 219 (1977).
Iijima et al., “An experiment for the potential blue shift at the Norikura Corona
Station,” Annals of the Tokyo Astronomical Observatory, Second Series, Vol.
XVII, 2 (1978) 68.
axis with rapidities η1 and η2 , find the matrix representing the combined Lorentz transformation, in a Taylor series up to the first nonclassical terms in each matrix element. A mixed Taylor series in
two variables can be obtained simply by nesting taylor functions.
The taylor function will happily work on matrices, not just scalars.
. Solution, p. 367
Chapter 2
Geometry of flat spacetime
Chapter 3
Differential geometry
General relativity is described mathematically in the language of
differential geometry. Let’s take those two terms in reverse order.
The geometry of spacetime is non-Euclidean, not just in the
sense that the 3+1-dimensional geometry of Lorentz frames is different than that of 4 interchangeable Euclidean dimensions, but also
in the sense that parallels do not behave in the way described by
E5 or A1-A3. In a Lorentz frame, which describes space without
any gravitational fields, particles whose world-lines are initially parallel will continue along their parallel world-lines forever. But in
the presence of gravitational fields, initially parallel world-lines of
free-falling particles will in general diverge, approach, or even cross.
Thus, neither the existence nor the uniqueness of parallels can be
assumed. We can’t describe this lack of parallelism as arising from
the curvature of the world-lines, because we’re using the world-lines
of free-falling particles as our definition of a “straight” line. Instead,
we describe the effect as coming from the curvature of spacetime itself. The Lorentzian geometry is a description of the case in which
this curvature is negligible.
What about the word differential ? The equivalence principle
states that even in the presence of gravitational fields, local Lorentz
frames exist. How local is “local?” If we use a microscope to zoom in
on smaller and smaller regions of spacetime, the Lorentzian approximation becomes better and better. Suppose we want to do experiments in a laboratory, and we want to ensure that when we compare
some physically observable quantity against predictions made based
on the Lorentz geometry, the resulting discrepancy will not be too
large. If the acceptable error is , then we should be able to get the
error down that low if we’re willing to make the size of our laboratory no bigger than δ. This is clearly very similar to the Weierstrass
style of defining limits and derivatives in calculus. In calculus, the
idea expressed by differentiation is that every smooth curve can be
approximated locally by a line; in general relativity, the equivalence
principle tells us that curved spacetime can be approximated locally
by flat spacetime. But consider that no practitioner of calculus habitually solves problems by filling sheets of scratch paper with epsilons and deltas. Instead, she uses the Leibniz notation, in which dy
and dx are interpreted as infinitesimally small numbers. You may
be inclined, based on your previous training, to dismiss infinitesi-
mals as neither rigorous nor necessary. In 1966, Abraham Robinson
demonstrated that concerns about rigor had been unfounded; we’ll
come back to this point in section 3.3. Although it is true that any
calculation written using infinitesimals can also be carried out using
limits, the following example shows how much more well suited the
infinitesimal language is to differential geometry.
Areas on a sphere
Example: 1
TheR area of a region S in the Cartesian plane can be calculated
as S dA, where dA = dxdy is the area of an infinitesimal rectangle of width dx and height dy. A curved surface such as a sphere
does not admit a global Cartesian coordinate system in which the
constant coordinate curves are both uniformly spaced and perpendicular to one another. For example, lines of longitude on the
earth’s surface grow closer together as one moves away from the
equator. Letting θ be the angle with respect to the pole, and φ the
azimuthal angle, the approximately rectangular patch bounded by
θ, θ + dθ, φ, and φ + dφ has width r sin θdθ and height r dφ, giving dA = r 2 sin θdθdφ. If you look at the corresponding derivation
in an elementary calculus textbook that strictly eschews infinitesimals, the technique is to start from scratch with Riemann sums.
This is extremely laborious, and moreover must be carried out
again for every new case. In differential geometry, the curvature
of the space varies from one point to the next, and clearly we
don’t want to reinvent the wheel with Riemann sums an infinite
number of times, once at each point in space.
3.1 Tangent vectors
a / A vector can be thought
of as lying in the plane tangent to
a certain point.
Chapter 3
It’s not immediately clear what a vector means in the context of
curved spacetime. The freshman physics notion of a vector carries
all kinds of baggage, including ideas like rotation of vectors and a
magnitude that is positive for nonzero vectors. We also used to
assume the ability to represent vectors as arrows, i.e., geometrical
figures of finite size that could be transported to other places —
but in a curved geometry, it is not in general possible to transport
a figure to another location without distorting its shape, so there is
no notion of congruence. For this reason, it’s better to visualize vectors as tangents to the underlying space, as in figure a. Intuitively,
we want to think of these vectors as arrows that are infinitesimally
small, so that they fit on the curved surface without having to be
bent. In the pictures, we simply scale them up to make them visible without an infinitely powerful microscope, and this scaling only
makes them appear to rise out of the space in which they live.
3.2 Affine notions and parallel transport
Differential geometry
3.2.1 The affine parameter in curved spacetime
An important example of the differential, i.e., local, nature of our
geometry is the generalization of the affine parameter to a context
broader than affine geometry.
Our construction of the affine parameter with a scaffolding of
parallelograms depended on the existence and uniqueness of parallels expressed by A1, so we might imagine that there was no point in
trying to generalize the construction to curved spacetime. But the
equivalence principle tells us that spacetime is locally affine to some
approximation. Concretely, clock-time is one example of an affine
parameter, and the curvature of spacetime clearly can’t prevent us
from building a clock and releasing it on a free-fall trajectory. To
generalize the recipe for the construction (figure a), the first obstacle is the ambiguity of the instruction to construct parallelogram
01q0 q1 , which requires us to draw 1q1 parallel to 0q0 . Suppose we
construe this as an instruction to make the two segments initially
parallel, i.e., parallel as they depart the line at 0 and 1. By the time
they get to q0 and q1 , they may be converging or diverging.
Because parallelism is only approximate here, there will be a
certain amount of error in the construction of the affine parameter.
One way of detecting such an error is that lattices constructed with
different initial distances will get out of step with one another. For
example, we can define 21 as before by requiring that the lattice
constructed with initial segment 0 12 line up with the original lattice
at 1. We will find, however, that they do not quite line up at
other points, such as 2. Let’s use this discrepancy = 2 − 20 as
a numerical measure of the error. It will depend on both δ1 , the
distance 01, and on δ2 , the distance between 0 and q0 . Since vanishes for either δ1 = 0 or δ2 = 0, and since the equivalence
principle guarantees smooth behavior on small scales, the leading
term in the error will in general be proportional to the product
δ1 δ2 . In the language of infinitesimals, we can replace δ1 and δ2
with infinitesimally short distances, which for simplicity we assume
to be equal, and which
we call dλ. Then the affine parameter λ
is defined as λ = dλ, where the error of order dλ2 is, as usual,
interpreted as the negligible discrepancy between the integral and
its approximation as a Riemann sum.
a / Construction of an affine
parameter in curved spacetime.
3.2.2 Parallel transport
If you were alert, you may have realized that I cheated you at
a crucial point in this construction. We were to make 1q1 and 0q0
“initially parallel” as they left 01. How should we even define this
idea of “initially parallel?” We could try to do it by making angles
q0 01 and q1 12 equal, but this doesn’t quite work, because it doesn’t
specify whether the angle is to the left or the right on the twodimensional plane of the page. In three or more dimensions, the
issue becomes even more serious. The construction workers building
Section 3.2
Affine notions and parallel transport
the lattice need to keep it all in one plane, but how do they do that
in curved spacetime?
A mathematician’s answer would be that our geometry lacks
some additional structure called a connection, which is a rule that
specifies how one locally flat neighborhood is to be joined seamlessly
onto another locally flat neighborhood nearby. If you’ve ever bought
two maps and tried to tape them together to make a big map, you’ve
formed a connection. If the maps were on a large enough scale,
you also probably noticed that this was impossible to do perfectly,
because of the curvature of the earth.
Physically, the idea is that in flat spacetime, it is possible to
construct inertial guidance systems like the ones discussed on page
74. Since they are possible in flat spacetime, they are also possible
in locally flat neighborhoods of spacetime, and they can then be
carried from one neighborhood to another.
b / Parallel transport is pathdependent. On the surface of
this sphere, parallel-transporting
a vector along ABC gives a
different answer than transporting
it along AC.
In three space dimensions, a gyroscope’s angular momentum vector maintains its direction, and we can orient other vectors, such as
1q1 , relative to it. Suppose for concreteness that the construction
of the affine parameter above is being carried out in three space dimensions. We place a gyroscope at 0, orient its axis along 0q0 , slide
it along the line to 1, and then construct 1q1 along that axis.
In 3+1 dimensions, a gyroscope only does part of the job. We
now have to maintain the direction of a four-dimensional vector.
Four-vectors will not be discussed in detail until section 4.2, but
similar devices can be used to maintain their orientations in spacetime. These physical devices are ways of defining a mathematical
notion known as parallel transport, which allows us to take a vector
from one point to another in space. In general, specifying a notion
of parallel transport is equivalent to specifying a connection.
Parallel transport is path-dependent, as shown in figure b.
c / Bad things happen if we
try to construct an affine parameter along a curve that isn’t a
geodesic. This curve is similar
to path ABC in figure b. Parallel transport doesn’t preserve
the vectors’ angle relative to
the curve, as it would with a
The errors in the
construction blow up in a way
that wouldn’t happen if the curve
had been a geodesic. The fourth
dashed parallel flies off wildly
around the back of the sphere,
wrapping around and meeting
the curve at a point, 4, that is
essentially random.
Chapter 3
Affine parameters defined only along geodesics
In the context of flat spacetime, the affine parameter was defined
only along lines, not arbitrary curves, and could not be compared
between lines running in different directions. In curved spacetime,
the same limitation is present, but with “along lines” replaced by
“along geodesics.” Figure c shows what goes wrong if we try to
apply the construction to a world-line that isn’t a geodesic. One
definition of a geodesic is that it’s the course we’ll end up following
if we navigate by keeping a fixed bearing relative to an inertial guidance device such as gyroscope; that is, the tangent to a geodesic,
when parallel-transported farther along the geodesic, is still tangent.
A non-geodesic curve lacks this property, and the effect on the construction of the affine parameter is that the segments nqn drift more
and more out of alignment with the curve.
Differential geometry
3.3 Models
A typical first reaction to the phrase “curved spacetime” — or even
“curved space,” for that matter — is that it sounds like nonsense.
How can featureless, empty space itself be curved or distorted? The
concept of a distortion would seem to imply taking all the points
and shoving them around in various directions as in a Picasso painting, so that distances between points are altered. But if space has
no identifiable dents or scratches, it would seem impossible to determine which old points had been sent to which new points, and the
distortion would have no observable effect at all. Why should we
expect to be able to build differential geometry on such a logically
dubious foundation? Indeed, historically, various mathematicians
have had strong doubts about the logical self-consistency of both
non-Euclidean geometry and infinitesimals. And even if an authoritative source assures you that the resulting system is self-consistent,
its mysterious and abstract nature would seem to make it difficult
for you to develop any working picture of the theory that could
play the role that mental sketches of graphs play in organizing your
knowledge of calculus.
Models provide a way of dealing with both the logical issues and
the conceptual ones. Figure a on page 89 “pops” off of the page,
presenting a strong psychological impression of a curved surface rendered in perspective. This suggests finding an actual mathematical
object, such as a curved surface, that satisfies all the axioms of a
certain logical system, such as non-Euclidean geometry. Note that
the model may contain extrinsic elements, such as the existence of
a third dimension, that are not connected to the system being modeled.
Let’s focus first on consistency. In general, what can we say
about the self-consistency of a mathematical system? To start with,
we can never prove anything about the consistency or lack of consistency of something that is not a well-defined formal system, e.g., the
Bible. Even Euclid’s Elements, which was a model of formal rigor for
thousands of years, is loose enough to allow considerable ambiguity.
If you’re inclined to scoff at the silly Renaissance mathematicians
who kept trying to prove the parallel postulate E5 from postulates
E1-E4, consider the following argument. Suppose that we replace
E5 with E50 , which states that parallels don’t exist: given a line and
a point not on the line, no line can ever be drawn through the point
and parallel to the given line. In the new system of plane geometry
E0 consisting of E1-E4 plus E50 , we can prove a variety of theorems,
and one of them is that there is an upper limit on the area of any figure. This imposes a limit on the size of circles, and that appears to
contradict E3, which says we can construct a circle with any radius.
We therefore conclude that E0 lacks self-consistency. Oops! As your
high school geometry text undoubtedly mentioned in passing, E0 is
a / Tullio
(18731941) worked on models of
number systems possessing
infinitesimals and on differential
He invented the
tensor notation, which Einstein
learned from his textbook. He
was appointed to prestigious
endowed chairs at Padua and the
University of Rome, but was fired
in 1938 because he was a Jew
and an anti-fascist.
Section 3.3
a perfectly respectable system called elliptic geometry. So what’s
wrong with this supposed proof of its lack of self-consistency? The
issue is the exact statement of E3. E3 does not say that we can
construct a circle given any real number as its radius. Euclid could
not have intended any such interpretation, since he had no notion of
real numbers. To Euclid, geometry was primary, and numbers were
geometrically constructed objects, being represented as lengths, angles, areas, and volumes. A literal translation of Euclid’s statement
of the axiom is “To describe a circle with any center and distance.”1
“Distance” means a line segment. There is therefore no contradiction in E0 , because E0 has a limit on the lengths of line segments.
Now suppose that such ambiguities have been eliminated from
the system’s basic definitions and axioms. In general, we expect
it to be easier to prove an inconsistent system’s inconsistency than
to demonstrate the consistency of a consistent one. In the former
case, we can start cranking out theorems, and if we can find a way
to prove both proposition P and its negation ¬P, then obviously
something is wrong with the system. One might wonder whether
such a contradiction could remain contained within one corner of
the system, like nuclear waste. It can’t. Aristotelian logic allows
proof by contradiction: if we prove both P and ¬P based on certain
assumptions, then our assumptions must have been wrong. If we
can prove both P and ¬P without making any assumptions, then
proof by contradiction allows us to establish the truth of any randomly chosen proposition. Thus a single contradiction is sufficient,
in Aristotelian logic, to invalidate the entire system. This goes by
the Latin rubric ex falso quodlibet, meaning “from a falsehood, whatever you please.” Thus any contradiction proves the inconsistency
of the entire system.
Proving consistency is harder. If you’re mathematically sophisticated, you may be tempted to leap directly to Gödel’s theorem, and
state that nobody can ever prove the self-consistency of a mathematical system. This would be a misapplication of Gödel. Gödel’s theorem only applies to mathematical systems that meet certain technical criteria, and some of the interesting systems we’re dealing with
don’t meet those criteria; in particular, Gödel’s theorem doesn’t
apply to Euclidean geometry, and Euclidean geometry was proved
self-consistent by Tarski and his students around 1950. Furthermore, we usually don’t require an absolute proof of self-consistency.
Usually we’re satisfied if we can prove that a certain system, such
as elliptic geometry, is at least as self-consistent as another system,
such as Euclidean geometry. This is called equiconsistency. The
general technique for proving equiconsistency of two theories is to
show that a model of one can be constructed within the other.
Suppose, for example, that we construct a geometry in which the
Chapter 3
Heath, pp. 195-202
Differential geometry
space of points is the surface of a sphere, and lines are understood
to be the geodesics, i.e., the great circles whose centers coincide at
the sphere’s center. This geometry, called spherical geometry, is
useful in cartography and navigation. It is non-Euclidean, as we
can demonstrate by exhibiting at least one proposition that is false
in Euclidean geometry. For example, construct a triangle on the
earth’s surface with one corner at the north pole, and the other
two at the equator, separated by 90 degrees of longitude. The sum
of its interior angles is 270 degrees, contradicting Euclid, book I,
proposition 32. Spherical geometry must therefore violate at least
one of the axioms E1-E5, and indeed it violates both E1 (because
no unique line is determined by two antipodal points such as the
north and south poles) and E5 (because parallels don’t exist at all).
A closely related construction gives a model of elliptic geometry,
in which E1 holds, and only E5 is thrown overboard. To accomplish
this, we model a point using a diameter of the sphere,2 and a line as
the set of all diameters lying in a certain plane. This has the effect
of identifying antipodal points, so that there is now no violation of
E1. Roughly speaking, this is like lopping off half of the sphere, but
making the edges wrap around. Since this model of elliptic geometry
is embedded within a Euclidean space, all the axioms of elliptic
geometry can now be proved as theorems in Euclidean geometry. If a
contradiction arose from them, it would imply a contradiction in the
axioms of Euclidean geometry. We conclude that elliptic geometry
is equiconsistent with Euclidean geometry. This was known long
before Tarski’s 1950 proof of Euclidean geometry’s self-consistency,
but since nobody was losing any sleep over hidden contradictions
in Euclidean geometry, mathematicians stopped wasting their time
looking for contradictions in elliptic geometry.
Example: 2
Consider the following axiomatically defined system of numbers:
1. It is a field, i.e., it has addition, subtraction, multiplication, and
division with the usual properties.
2. It is an ordered geometry in the sense of O1-O4 on p. 19, and
the ordering relates to addition and multiplication in the usual
3. Existence of infinitesimals: There exists a positive number d
such that d < 1, d < 1/2, d < 1/3, . . .
A model of this system can be constructed within the real number
system by defining d as the identity function d(x) = x and forming
the set of functions of the form f (d) = P(d)/Q(d), where P and Q
are polynomials with real coefficients. The ordering of functions f
The term “elliptic” may be somewhat misleading here. The model is still
constructed from a sphere, not an ellipsoid.
Section 3.3
and g is defined according to the sign of limx→0+ f (x) − g(x). Axioms 1-3 can all be proved from the real-number axioms. Therefore this system, which includes infinitesimals, is equiconsistent
with the reals. More elaborate constructions can extend this to
systems that have more of the properties of the reals, and a
browser-based calculator that implements such a system is available at Abraham Robinson extended this in 1966 to all of analysis, and thus there is nothing intrinsically nonrigorous about doing analysis in the style of Gauss
and Euler, with symbols like dx representing infinitesimally small
Besides proving consistency, these models give us insight into
what’s going on. The model of elliptic geometry suggests an insight into the reason that there is an upper limit on lengths and
areas: it is because the space wraps around on itself. The model of
infinitesimals suggests a fact that is not immediately obvious from
the axioms: the infinitesimal quantities compose a hierarchy, so that
for example 7d is in finite proportion to d, while d2 is like a “lesser
flea” in Swift’s doggerel: “Big fleas have little fleas/ On their backs
to ride ’em,/ and little fleas have lesser fleas,/And so, ad infinitum.”
Spherical and elliptic geometry are not valid models of a generalrelativistic spacetime, since they are locally Euclidean rather than
Lorentzian, but they still provide us with enough conceptual guidance to come up with some ideas that might never have occurred to
us otherwise:
b / An Einstein’s ring is formed
when there is a chance alignment
of a distant source with a closer
gravitating body. Here, a quasar,
MG1131+0456, is seen as a
ring due to focusing of light by
an unknown object, possibly a
supermassive black hole. Because the entire arrangement
lacks perfect axial symmetry, the
ring is nonuniform; most of its
brightness is concentrated in two
lumps on opposite sides. This
type of gravitational lensing is
direct evidence for the curvature
of space predicted by gravitational lensing. The two geodesics
form a lune, which is a figure
that cannot exist in Euclidean
Chapter 3
• In spherical geometry, we can have a two-sided polygon called
a lune that encloses a nonzero area. In general relativity, a
lune formed by the world-lines of two particles represents motion in which the particles separate but are later reunited,
presumably because of some mass between them that created
a gravitational field. An example is gravitational lensing.
• Both spherical models wraps around on themselves, so that
they are not topologically equivalent to infinite planes. We
therefore form a conjecture there may be a link between curvature, which is a local property, and topology, which is global.
Such a connection is indeed observed in relativity. For example, cosmological solutions of the equations of general relativity come in two flavors. One type has enough matter in it to
produce more than a certain critical amount of curvature, and
this type is topologically closed. It describes a universe that
has finite spatial volume, and that will only exist for a finite
time before it recontracts in a Big Crunch. The other type,
More on this topic is available in, for example, Keisler’s Elementary Calculus: An Infinitesimal Approach, Stroyan’s A Brief Introduction to Infinitesimal
Calculus, or my own Calculus, all of which are available for free online.
Differential geometry
corresponding to the universe we actually inhabit, has infinite
spatial volume, will exist for infinite time, and is topologically
• There is a distance scale set by the size of the sphere, with its
inverse being a measure of curvature. In general relativity,
we expect there to be a similar way to measure curvature
numerically, although the curvature may vary from point to
Self-check: Prove from the axioms E0 that elliptic geometry, unlike spherical geometry, cannot have a lune with two distinct vertices. Convince yourself nevertheless, using the spherical model of
E0 , that it is possible in elliptic geometry for two lines to enclose a
region of space, in the sense that from any point P in the region,
a ray emitted in any direction must intersect one of the two lines.
Summarize these observations with a characterization of lunes in
elliptic geometry versus lunes in spherical geometry.
3.4 Intrinsic quantities
Models can be dangerous, because they can tempt us to impute
physical reality to features that are purely extrinsic, i.e., that are
only present in that particular model. This is as opposed to intrinsic
features, which are present in all models, and which are therefore
logically implied by the axioms of the system itself. The existence
of lunes is clearly an intrinsic feature of non-Euclidean geometries,
because intersection of lines was defined before any model has even
been proposed.
Curvature in elliptic geometry
Example: 3
What about curvature? In the spherical model of elliptic geometry, the size of the sphere is an inverse measure of curvature.
Is this a valid intrinsic quantity, or is it extrinsic? It seems suspect, because it is a feature of the model. If we try to define
“size” as the radius R of the sphere, there is clearly reason for
concern, because this seems to refer to the center of the sphere,
but existence of a three-dimensional Euclidean space inside and
outside the surface is clearly an extrinsic feature of the model.
There is, however, a way in which a creature confined to the surface can determine R, by constructing geodesic and an affine
parameter along that geodesic, and measuring the distance λ accumulated until the geodesic returns to the initial point. Since
antipodal points are identified, λ equals half the circumference of
the sphere, not its whole circumference, so R = λ/π, by wholly
intrinsic methods.
Extrinsic curvature
Example: 4
Euclid’s axioms E1-E5 refer to explicit constructions. If a twodimensional being can physically verify them all as descriptions of
Section 3.4
Intrinsic quantities
the two-dimensional space she inhabits, then she knows that her
space is Euclidean, and that propositions such as the Pythagorean
theorem are physically valid in her universe. But the diagram in
a/1 illustrating illustrating the proof of the Pythagorean theorem in
Euclid’s Elements (proposition I.47) is equally valid if the page is
rolled onto a cylinder, 2, or formed into a wavy corrugated shape,
3. These types of curvature, which can be achieved without tearing or crumpling the surface, are extrinsic rather than intrinsic. Of
the curved surfaces in figure a, only the sphere, 4, has intrinsic
curvature; the diagram can’t be plastered onto the sphere without
folding or cutting and pasting.
a / Example 4.
Self-check: How would the ideas of example 4 apply to a cone?
Example 4 shows that it can be difficult to sniff out bogus extrinsic features that seem intrinsic, and example 3 suggests the desirability of developing methods of calculation that never refer to
any extrinsic quantities, so that we never have to worry whether a
symbol like R staring up at us from a piece of paper is intrinsic.
This is why it is unlikely to be helpful to a student of general relativity to pick up a book on differential geometry that was written
without general relativity specifically in mind. Such books have a
tendency to casually mix together intrinsic and extrinsic notation.
For example, a vector cross product a × b refers to a vector poking
out of the plane occupied by a and b, and the space outside the
plane may be extrinsic; it is not obvious how to generalize this operation to the 3+1 dimensions of relativity (since the cross product
is a three-dimensional beast), and even if it were, we could not be
assured that it would have any intrinsically well defined meaning.
3.4.1 Coordinate independence
To see how to proceed in creating a manifestly intrinsic notation,
consider the two types of intrinsic observations that are available in
general relativity:
Chapter 3
Differential geometry
• 1. We can tell whether events and world-lines are incident:
whether or not two lines intersect, two events coincide, or an
event lies on a certain line.
Incidence measurements, for example detection of gravitational lensing, are global, but they are the only global observations we can
do.4 If we were limited entirely to incidence, spacetime would be
described by the austere system of projective geometry, a geometry
without parallels or measurement. In projective geometry, all propositions are essentially statements about combinatorics, e.g., that it
is impossible to plant seven trees so that they form seven lines of
three trees each.
• 2. We can also do measurements in local Lorentz frames.
This gives us more power, but not as much as we might expect.
Suppose we define a coordinate such as t or x. In Newtonian mechanics, these coordinates would form a predefined background, a
preexisting stage for the actors. In relativity, on the other hand,
consider a completely arbitrary change of coordinates of the form
x → x0 = f (x), where f is a smooth one-to-one function. For example, we could have x → x + px3 + q sin(rx) (with p and q chosen
small enough so that the mapping is always one-to-one). Since the
mapping is one-to-one, the new coordinate system preserves all the
incidence relations. Since the mapping is smooth, the new coordinate system is still compatible with the existence of local Lorentz
frames. The difference between the two coordinate systems is therefore entirely extrinsic, and we conclude that a manifestly intrinsic
notation should avoid any explicit reference to a coordinate system.
That is, if we write a calculation in which a symbol such as x appears, we need to make sure that nowhere in the notation is there
any hidden assumption that x comes from any particular coordinate
system. For example, the equation should still be valid if the generic
symbol x is later taken to represent the distance r from some center
of symmetry. This coordinate-independence property is also known
as general covariance, and this type of smooth change of coordinates
is also called a diffeomorphism.
The Dehn twist
Example: 5
As an exotic example of a change of coordinates, take a torus
and label it with coordinates (θ, φ), where θ + 2π is taken to be the
same as θ, and similarly for φ. Now subject it to the coordinate
transformation T defined by θ → θ + φ, which is like opening the
torus, twisting it by a full circle, and then joining the ends back
together. T is known as the “Dehn twist,” and it is different from
Einstein referred to incidence measurements as “determinations of spacetime coincidences.” For his presentation of this idea, see p. 364.
Section 3.4
Intrinsic quantities
most of the coordinate transformations we do in relativity because
it can’t be done smoothly, i.e., there is no continuous function f (x)
on 0 ≤ x ≤ 1 such that every value of f is a smooth coordinate
transformation, f (0) is the identity transformation, and f (1) = T .
Frames moving at c?
A good application of these ideas is to the question of what the
world would look like in a frame of reference moving at the speed
of light. This question has a long and honorable history. As a
young student, Einstein tried to imagine what an electromagnetic
wave would look like from the point of view of a motorcyclist riding
alongside it. We now know, thanks to Einstein himself, that it really
doesn’t make sense to talk about such observers.
The most straightforward argument is based on the positivist
idea that concepts only mean something if you can define how to
measure them operationally. If we accept this philosophical stance
(which is by no means compatible with every concept we ever discuss
in physics), then we need to be able to physically realize this frame
in terms of an observer and measuring devices. But we can’t. It
would take an infinite amount of energy to accelerate Einstein and
his motorcycle to the speed of light.
b / A series of Lorentz boosts
acts on a square.
Since arguments from positivism can often kill off perfectly interesting and reasonable concepts, we might ask whether there are
other reasons not to allow such frames. There are. Recall that
we placed two technical conditions on coordinate transformations:
they are supposed to be smooth and one-to-one. The smoothness
condition is related to the inability to boost Einstein’s motorcycle
into the speed-of-light frame by any continuous, classical process.
(Relativity is a classical theory.) But independent of that, we have
a problem with the one-to-one requirement. Figure b shows what
happens if we do a series of Lorentz boosts to higher and higher
velocities. It should be clear that if we could do a boost up to a velocity of c, we would have effected a coordinate transformation that
was not one-to-one. Every point in the plane would be mapped onto
a single lightlike line.
3.5 The metric
Consider a coordinate x defined along a certain curve, which is not
necessarily a geodesic. For concreteness, imagine this curve to exist
in two spacelike dimensions, which we can visualize as the surface
of a sphere embedded in Euclidean 3-space. These concrete features
are not strictly necessary, but they drive home the point that we
should not expect to be able to define x so that it varies at a steady
rate with elapsed distance; for example, we know that it will not be
possible to define a two-dimensional Cartesian grid on the surface
of a sphere. In the figure, the tick marks are therefore not evenly
Chapter 3
Differential geometry
spaced. This is perfectly all right, given the coordinate invariance of
general relativity. Since the incremental changes in x are equal, I’ve
represented them below the curve as little vectors of equal length.
They are the wrong length to represent distances along the curve,
but this wrongness is an inevitable fact of life in relativity.
Now suppose we want to integrate the arc length of a segment
of this curve. The little vectors are infinitesimal. In the integrated
length, each little vector should contribute some amount, which is
a scalar.
√ This scalar is not simply the magnitude of the vector,
ds 6= dx · dx, since the vectors are the wrong length. Figure a
is clearly reminiscent of the geometrical picture of vectors and dual
vectors developed on p. 48. But the purely affine notion of vectors
and their duals is not enough to define the length of a vector in
general; it is only sufficient to define a length relative to other lengths
along the same geodesic. When vectors lie along different geodesics,
we need to be able to specify the additional conversion factor that
allows us to compare one to the other. The piece of machinery that
allows us to do this is called a metric.
a / The tick marks on the line
define a coordinate measured
along the line. It is not possible to
set up such a coordinate system
globally so that the coordinate
is uniform everywhere.
arrows represent changes in the
value of the coordinate; since the
changes in the coordinate are
all equal, the arrows are all the
same length.
Fixing a metric allows us to define the proper scaling of the tick
marks relative to the arrows at a given point, i.e., in the birdtracks
notation it gives us a natural way of taking a displacement vector
such as s , with the arrow pointing into the symbol, and making
a corresponding dual vector s , with the arrow coming out. This
is a little like cloning a person but making the clone be of the opposite sex. Hooking them up like s s then tells us the squared
magnitude of the vector. For example, if dx is an infinitesimal
timelike displacement, then dx dx is the squared time interval dx2
measured by a clock traveling along that displacement in spacetime.
(Note that in the notation dx2 , it’s clear that dx is a scalar, because
unlike dx and dx
it doesn’t have any arrow coming in or out
of it.) Figure b shows the resulting picture.
b / The vectors dx and dx
are duals of each other.
In the abstract index notation introduced on p. 51, the vectors
dx and dx are written dxa and dxa . When a specific coordinate
system has been fixed, we write these with concrete, Greek indices,
dxµ and dxµ . In an older and conceptually incompatible notation
and terminology due to Sylvester (1853), one refers to dxµ as a contravariant vector, and dxµ as covariant. The confusing terminology
Section 3.5
The metric
is summarized on p. 389.
The assumption that a metric exists is nontrivial. There is no
metric in Galilean spacetime, for example, since in the limit c → ∞
the units used to measure timelike and spacelike displacements are
not comparable. Assuming the existence of a metric is equivalent to
assuming that the universe holds at least one physically manipulable
clock or ruler that can be moved over long distances and accelerated
as desired. In the distant future, large and causally isolated regions
of the cosmos may contain only massless particles such as photons,
which cannot be used to build clocks (or, equivalently, rulers); the
physics of these regions will be fully describable without a metric.
If, on the other hand, our world contains not just zero or one but
two or more clocks, then the metric hypothesis requires that these
clocks maintain a consistent relative rate when accelerated along
the same world-line. This consistency is what allows us to think
of relativity as a theory of space and time rather than a theory of
clocks and rulers. There are other relativistic theories of gravity
besides general relativity, and some of these violate this hypothesis.
Given a dxµ , how do we find its dual dxµ , and vice versa? In
one dimension, we simply need to introduce a real number g as a
correction factor. If one of the vectors is shorter than it should be
in a certain region, the correction factor serves to compensate by
making its dual proportionately longer. The two possible mappings
(covariant to contravariant and contravariant to covariant) are accomplished with factors of g and 1/g. The number g is the metric,
and it encodes all the information about distances. For example, if
φ represents longitude measured at the arctic circle, then the metric
is the only source for the datum that a displacement dφ corresponds
to 2540 km per radian.
Now let’s generalize to more than one dimension. Because globally Cartesian coordinate systems can’t be imposed on a curved
space, the constant-coordinate lines will in general be neither evenly
spaced nor perpendicular to one another. If we construct a local
set of basis vectors lying along the intersections of the constantcoordinate surfaces, they will not form an orthonormal set. We
would like to have an expression of the form ds2 = Σdxµ dxµ for the
squared arc length, and using the Einstein summation notation this
ds2 = dxµ dxµ
3.5.1 The Euclidean metric
For Cartesian coordinates in a Euclidean plane, where one doesn’t
normally bother with the distinction between covariant and contravariant vectors, this expression for ds2 is simply the Pythagorean
theorem, summed over two values of µ for the two coordinates:
ds2 = dxµ dxµ = dx2 + dy 2
Chapter 3
Differential geometry
The symbols dx, ds0 , dx0 , and dx0 are all synonyms, and likewise
for dy, ds1 , dx1 , and dx1 . (Because notations such as ds1 force the
reader to keep track of which digits have been assigned to which
letters, it is better practice to use notation such as dy or dsy ; the
latter notation could in principle be confused with one in which y
was a variable taking on values such as 0 or 1, but in reality we
understand it from context, just as we understand that the d’s in
dy/dx are not referring to some variable d that stands for a number.)
In the non-Euclidean case, the Pythagorean theorem is false; dxµ
and dxµ are no longer synonyms, so their product is no longer simply
the square of a distance. To see this more explicitly, let’s write the
expression so that only the covariant quantities occur. By local
flatness, the relationship between the covariant and contravariant
vectors is linear, and the most general relationship of this kind is
given by making the metric a symmetric matrix gµν . Substituting
dxµ = gµν xν , we have
ds2 = gµν dxµ dxν
where there are now implied sums over both µ and ν. Notice how
implied sums occur only when the repeated index occurs once as
a superscript and once as a subscript; other combinations are ungrammatical.
Self-check: Why does it make sense to demand that the metric
be symmetric?
On p. 46 we encountered the distinction among scalars, vectors,
and dual vectors. These are specific examples of tensors, which can
be expressed in the birdtracks notation as objects with m arrows
coming in and n coming out, or. In index notation, we have m
superscripts and n subscripts. A scalar has m = n = 0. A dual
vector has (m, n) = (0, 1), a vector (1, 0), and the metric (0, 2). We
refer to the number of indices as the rank of the tensor. Tensors are
discussed in more detail, and defined more rigorously, in chapter 4.
For our present purposes, it is important to note that just because
we write a symbol with subscripts or superscripts, that doesn’t mean
it deserves to be called a tensor. This point can be understood in
the more elementary context of Newtonian scalars and vectors. For
example, we can define a Euclidean “vector” u = (m, T , e), where
m is the mass of the moon, T is the temperature in Chicago, and
e is the charge of the electron. This creature u doesn’t deserve
to be called a vector, because it doesn’t behave as a vector under
rotation. The general philosophy is that a tensor is something that
has certain properties under changes of coordinates. For example,
we’ve already seen on p. 48 the different scaling behavior of tensors
with ranks (1, 0), (0, 0), and (0, 1).
When discussing the symmetry of rank-2 tensors, it is convenient
Section 3.5
The metric
to introduce the following notation:
(Tab + Tba )
= (Tab − Tba )
T(ab) =
Any Tab can be split into symmetric and antisymmetric parts. This
is similar to writing an arbitrary function as a sum of and odd
function and an even function. The metric has only a symmetric
part: g(ab) = gab , and g[ab] = 0. This notation is generalized to
ranks greater than 2 on page 184.
Self-check: Characterize an antisymmetric rank-2 tensor in two
A change of scale
Example: 6
. How is the effect of a uniform rescaling of coordinates represented in g?
. If we change our units of measurement so that x µ → αx µ , while
demanding that ds2 come out the same, then we need gµν →
α−2 gµν .
Comparing with p. 48, we deduce the general rule that a tensor
of rank (m, n) transforms under scaling by picking up a factor of
αm−n .
Polar coordinates
Example: 7
Consider polar coordinates (r , θ) in a Euclidean plane. The constant-coordinate curves happen to be orthogonal everywhere, so
the off-diagonal elements of the metric gr θ and gθr vanish. Infinitesimal coordinate changes dr and dθ correspond to infinitesimal displacements dr and r dθ in orthogonal directions, so by the
Pythagorean theorem, ds2 = dr 2 + r 2 dθ2 , and we read off the
elements of the metric gr r = 1 and gθθ = r 2 .
Notice how in example 7 we started from the generally valid
relation ds2 = gµν dxµ dxν , but soon began writing down facts like
gθθ = r2 that were only valid in this particular coordinate system.
To make it clear when this is happening, we maintain the distinction
between abtract Latin indices and concrete Greek indices introduced
on p. 51. For example, we can write the general expression for
squared differential arc length with Latin indices,
ds2 = gij dxi dxj
because it holds regardless of the coordinate system, whereas the
vanishing of the off-diagonal elements of the metric in Euclidean
polar coordinates has to be written as gµν = 0 for µ 6= ν, since it
would in general be false if we used a different coordinate system to
describe the same Euclidean plane.
Chapter 3
Differential geometry
Oblique Cartesian coordinates
Example: 8
. Oblique Cartesian coordinates are like normal Cartesian coordinates in the plane, but their axes are at at an angle φ 6= π/2 to
one another. Find the metric in these coordinates. The space is
globally Euclidean.
. Since the coordinates differ from Cartesian coordinates only in
the angle between the axes, not in their scales, a displacement
dx i along either axis, i = 1 or 2, must give ds = dx, so for the diagonal elements we have g11 = g22 = 1. The metric is always symmetric, so g12 = g21 . To fix these off-diagonal elements, consider
a displacement by ds in the direction perpendicular to axis 1. This
changes the coordinates by dx 1 = −ds cot φ and dx 2 = ds csc φ.
We then have
ds2 = gij dx i dx j
= ds2 (cot2 φ + csc2 φ − 2g12 cos φ csc φ)
g12 = cos φ
c / Example 8.
Example: 9
In one dimension, g is a single number, and lengths are given
by ds = gdx. The square root can also be understood through
example 6 on page 102, in which we saw that a uniform rescaling
x → αx is reflected in gµν → α−2 gµν .
In two-dimensional Cartesian coordinates, multiplication of the
width and height of a rectangle gives the element of area dA =
g11 g22 dx 1 dx 2 . Because the coordinates are orthogonal, g is di√
agonal, and the factor of gp
11 g22 is identified as the square root
of its determinant, so dA = |g|dx 1 dx 2 . Note that the scales on
the two axes are not necessarily the same, g11 6= g22 .
The same expression for the element of area holds even if the coordinates
pare not orthogonal. In example 8, for instance, we have
|g| = 1 − cos2 φ = sin φ, which is the right correction factor
corresponding to the fact that dx 1 and dx 2 form a parallelepiped
rather than a rectangle.
Area of a sphere
Example: 10
For coordinates (θ, φ) on the surface of a sphere of radius r , we
have, by an argument similar to that of example 7 on page 102,
gθθ = r 2 , gφφ = r 2 sin2 θ, gθφ = 0. The area of the sphere is
A = dA
Z Z p
= r2
sin θdθdφ
= 4πr 2
Section 3.5
The metric
Inverse of the metric
. Relate g ij to gij .
Example: 11
. The notation is intended to treat covariant and contravariant
vectors completely symmetrically. The metric with lower indices
gij can be interpreted as a change-of-basis transformation from a
contravariant basis to a covariant one, and if the symmetry of the
notation is to be maintained, g ij must be the corresponding inverse matrix, which changes from the covariant basis to the contravariant one. The metric must always be invertible.
In the one-dimensional case, p. 99, the metric at any given
point was simply some number g, and we used factors of g and
1/g to convert back and forth between covariant and contravariant
vectors. Example 11 makes it clear how to generalize this to more
xa = gab xb
xa = g ab xb
This is referred to as raising and lowering indices. There is no need
to memorize the positions of the indices in these rules; they are
the only ones possible based on the grammatical rules, which are
that summation only occurs over top-bottom pairs, and upper and
lower indices have to match on both sides of the equals sign. This
whole system, introduced by Einstein, is called “index-gymnastics”
Raising and lowering indices on a rank-two tensor Example: 12
In physics we encounter various examples of matrices, such as
the moment of inertia tensor from classical mechanics. These
have two indices, not just one like a vector. Again, the rules for
raising and lowering indices follow directly from grammar. For
Aab = g ac Acb
Aab = gac gbd Acd
A matrix operating on a vector
Example: 13
The row and column vectors from linear algebra are the covariant and contravariant vectors in our present terminology. (The
convention is that covariant vectors are row vectors and contravariant ones column vectors, but I don’t find this worth memorizing.) What about matrices? A matrix acting on a column vector
gives another column vector, q = Up. Translating this into indexgymnastics notation, we have
q a = U ...... pb
Chapter 3
Differential geometry
where we want to figure out the correct placement of the indices
on U. Grammatically, the only possible placement is
q a = U ab pb
This shows that the natural way to represent a column-vector-tocolumn-vector linear operator is as a rank-2 tensor with one upper
index and one lower index.
In birdtracks notation, a rank-2 tensor is something that has two
arrows connected to it. Our example becomes → q =→ U → p.
That the result is itself an upper-index vector is shown by the fact
that the right-hand-side taken as a whole has a single external
arrow coming into it.
The distinction between vectors and their duals may seem irrelevant if we can always raise and lower indices at will. We can’t
always do that, however, because in many perfectly ordinary situations there is no metric. See example 6, p. 49.
3.5.2 The Lorentz metric
In a locally Euclidean space, the Pythagorean theorem allows us
to express the metric in local Cartesian coordinates in the simple
form gµµ = +1, gµν = 0, i.e., g = diag(+1, +1, . . . , +1). This is
not the appropriate metric for a locally Lorentz space. The axioms
of Euclidean geometry E3 (existence of circles) and E4 (equality of
right angles) describe the theory’s invariance under rotations, and
the Pythagorean theorem is consistent with this, because it gives the
same answer for the length of a vector even if its components are
reexpressed in a new basis that is rotated with respect to the original
one. In a Lorentzian geometry, however, we care about invariance
under Lorentz boosts, which do not preserve the quantity t2 + x2 .
It is not circles in the (t, x) plane that are invariant, but light cones,
and this is described by giving gtt and gxx opposite signs and equal
absolute values. A lightlike vector (t, x), with t = x, therefore has
a magnitude of exactly zero,
s2 = gtt t2 + gxx x2 = 0
and this remains true after the Lorentz boost (t, x) → (γt, γx). It
is a matter of convention which element of the metric to make positive and which to make negative. In this book, I’ll use gtt = +1
and gxx = −1, so that g = diag(+1, −1). This has the advantage that any line segment representing the timelike world-line of a
physical object has a positive squared magnitude; the forward flow
of time is represented as a positive number, in keeping with the
philosophy that relativity is basically a theory of how causal relationships work. With this sign convention, spacelike vectors have
positive squared magnitudes, timelike ones negative. The same convention is followed, for example, by Penrose. The opposite version,
Section 3.5
The metric
with g = diag(−1, +1) is used by authors such as Wald and Misner,
Thorne, and Wheeler.
Our universe does not have just one spatial dimension, it has
three, so the full metric in a Lorentz frame is given by
g = diag(+1, −1, −1, −1).
Mixed covariant-contravariant form of the metric
Example: 14
In example 12 on p. 104, we saw how to raise and lower indices
on a rank-two tensor, and example 13 showed that it is sometimes
natural to consider the form in which one index is raised and one
lowered. The metric itself is a rank-two tensor, so let’s see what
happens when we compute the mixed form g ab from the lowerindex form. In general, we have
Aab = g ac Acb
and substituting g for A gives
g ab = g ac gcb
But we already know that g ... is simply the inverse matrix of g...
(example 11, p. 104), which means that g ab is simply the identity
matrix. That is, whereas a quantity like gab or g ab carries all the
information about our system of measurement at a given point,
g ab carries no information at all. Where gab or g ab can have both
positive and negative elements, elements that have units, and
off-diagonal elements, g ab is just a generic symbol carrying no
information other than the dimensionality of the space.
The metric tensor is so commonly used that it is simply left out of
birdtrack diagrams. Consistency is maintained because because
g ab is the identity matrix, so → g → is the same as →→.
3.5.3 Isometry, inner products, and the Erlangen program
In Euclidean geometry, the dot product of vectors a and b is
given by gxx ax bx + gyy ay by + gzz az bz = ax bx + ay by + az bz , and in
the special case where a = b we have the squared magnitude. In
the tensor notation, aµ bν = a1 b1 + a2 b2 + a3 b3 . Like magnitudes,
dot products are invariant under rotations. This is because knowing the dot product of vectors a and b entails knowing the value
of a · b = |a||a| cos θab , and Euclid’s E4 (equality of right angles)
implies that the angle θab is invariant. the same axioms also entail
invariance of dot products under translation; Euclid waits only until
the second proposition of the Elements to prove that line segments
can be copied from one location to another. This seeming triviality is
actually false as a description of physical space, because it amounts
to a statement that space has the same properties everywhere.
The set of all transformations that can be built out of successive translations, rotations, and reflections is called the group of
Chapter 3
Differential geometry
isometries. It can also be defined as the group5 that preserves dot
products, or the group that preserves congruence of triangles.
In Lorentzian geometry, we usually avoid the Euclidean term
dot product and refer to the corresponding operation by the more
general term inner product. In a specific coordinate system we have
aµ bν = a0 b0 −a1 b1 −a2 b2 −a3 b3 . The inner product is invariant under
Lorentz boosts, and also under the Euclidean isometries. The group
found by making all possible combinations of continuous transformations6 from these two sets is called the Poincaré group. The
Poincaré group is not the symmetry group of all of spacetime, since
curved spacetime has different properties in different locations. The
equivalence principle tells us, however, that space can be approximated locally as being flat, so the Poincaré group is locally valid,
just as the Euclidean isometries are locally valid as a description of
geometry on the Earth’s curved surface.
The triangle inequality
Example: 15
In Euclidean geometry, the triangle inequality |b + c| < |b| + |c|
follows from
(|b| + |c|)2 − (b + c) · (b + c) = 2(|b||c| − b · c) ≥ 0
The reason this quantity always comes out positive is that for two
vectors of fixed magnitude, the greatest dot product is always
achieved in the case where they lie along the same direction.
In Lorentzian geometry, the situation is different. Let b and c be
timelike vectors, so that they represent possible world-lines. Then
the relation a = b+c suggests the existence of two observers who
take two different paths from one event to another. A goes by a
direct route while B takes a detour. The magnitude of each timelike vector represents the time elapsed on a clock carried by the
observer moving along that vector. The triangle equality is now
reversed, becoming |b + c| > |b| + |c|. The difference from the
Euclidean case arises because inner products are no longer necessarily maximized if vectors are in the same direction. E.g., for
two lightlike vectors, bi cj vanishes entirely if b and c are parallel. For timelike vectors, parallelism actually minimizes the inner
product rather than maximizing it.7
In mathematics, a group is defined as a binary operation that has an identity,
inverses, and associativity. For example, addition of integers is a group. In the
present context, the members of the group are not numbers but the transformations applied to the Euclidean plane. The group operation on transformations
T1 and T2 consists of finding the transformation that results from doing one and
then the other, i.e., composition of functions.
The discontinuous transformations of spatial reflection and time reversal are
not included in the definition of the Poincaré group, although they do preserve
inner products. General relativity has symmetry under spatial reflection (called
P for parity), time reversal (T), and charge inversion (C), but the standard
model of particle physics is only invariant under the composition of all three,
CPT, not under any of these symmetries individually.
Proof: Let b and c be parallel and timelike, and directed forward in time.
Section 3.5
The metric
In his 1872 inaugural address at the University of Erlangen, Felix
Klein used the idea of groups of transformations to lay out a general classification scheme, known as the Erlangen program, for all
the different types of geometry. Each geometry is described by the
group of transformations, called the principal group, that preserves
the truth of geometrical statements. Euclidean geometry’s principal
group consists of the isometries combined with arbitrary changes of
scale, since there is nothing in Euclid’s axioms that singles out a
particular distance as a unit of measurement. In other words, the
principal group consists of the transformations that preserve similarity, not just those that preserve congruence. Affine geometry’s
principal group is the transformations that preserve parallelism; it
includes shear transformations, and there is therefore no invariant
notion of angular measure or congruence. Unlike Euclidean and
affine geometry, elliptic geometry does not have scale invariance.
This is because there is a particular unit of distance that has special
status; as we saw in example 3 on page 95, a being living in an elliptic plane can determine, by entirely intrinsic methods, a distance
scale R, which we can interpret in the hemispherical model as the
radius of the sphere. General relativity breaks this symmetry even
more severely. Not only is there a scale associated with curvature,
but the scale is different from one point in space to another.
3.5.4 Einstein’s carousel
Non-Euclidean geometry observed in the rotating frame
d / Observer A, rotating with
the carousel, measures an
azimuthal distance with a ruler.
The following example was historically important, because Einstein used it to convince himself that general relativity should be
described by non-Euclidean geometry.8 Its interpretation is also
fairly subtle, and the early relativists had some trouble with it.
Suppose that observer A is on a spinning carousel while observer
B stands on the ground. B says that A is accelerating, but by the
equivalence principle A can say that she is at rest in a gravitational
field, while B is free-falling out from under her. B measures the
radius and circumference of the carousel, and finds that their ratio
is 2π. A carries out similar measurements, but when she puts her
meter-stick in the azimuthal direction it becomes Lorentz-contracted
by the factor γ = (1−ω 2 r2 )−1/2 , so she finds that the ratio is greater
than 2π. In A’s coordinates, the spatial geometry is non-Euclidean,
Adopt a frame of reference in which every spatial component of each vector
vanishes. This entails no loss of generality, since inner products are invariant
under such a transformation. Since the time-ordering is also preserved under
transformations in the Poincaré group, each is still directed forward in time, not
backward. Now let b and c be pulled away from parallelism, like opening a pair
of scissors in the x − t plane. This reduces bt ct , while causing bx cx to become
negative. Both effects increase the inner product.
The example is described in Einstein’s paper “The Foundation of the General
Theory of Relativity.” An excerpt, which includes the example, is given on
p. 360.
Chapter 3
Differential geometry
and the metric differs from the Euclidean one found in example 7
on page 102.
Observer A feels a force that B considers to be fictitious, but
that, by the equivalence principle, A can say is a perfectly real
gravitational force. According to A, an observer like B is free-falling
away from the center of the disk under the influence of this gravitational field. A also observes that the spatial geometry of the carousel
is non-Euclidean. Therefore it seems reasonable to conjecture that
gravity can be described by non-Euclidean geometry, rather than as
a physical force in the Newtonian sense.
At this point, you know as much about this example as Einstein
did in 1912, when he began using it as the seed from which general
relativity sprouted, collaborating with his old schoolmate, mathematician Marcel Grossmann, who knew about differential geometry.
The remainder of subsection 3.5.4, which you may want to skip on a
first reading, goes into more detail on the interpretation and mathematical description of the rotating frame of reference. Even more
detailed treatments are given by Grøn9 and Dieks.10 .
Ehrenfest’s paradox
Ehrenfest11 described the following paradox. Suppose that observer B, in the lab frame, measures the radius of the disk to be r
when the disk is at rest, and r0 when the disk is spinning. B can
also measure the corresponding circumferences C and C 0 . Because
B is in an inertial frame, the spatial geometry does not appear nonEuclidean according to measurements carried out with his meter
sticks, and therefore the Euclidean relations C = 2πr and C 0 = 2πr0
both hold. The radial lines are perpendicular to their own motion,
and they therefore have no length contraction, r = r0 , implying
C = C 0 . The outer edge of the disk, however, is everywhere tangent to its own direction of motion, so it is Lorentz contracted, and
therefore C 0 < C. The resolution of the paradox is that it rests on
the incorrect assumption that a rigid disk can be made to rotate.
If a perfectly rigid disk was initially not rotating, one would have
to distort it in order to set it into rotation, because once it was
rotating its outer edge would no longer have a length equal to 2π
times its radius. Therefore if the disk is perfectly rigid, it can never
be rotated. As discussed on page 65, relativity does not allow the
existence of infinitely rigid or infinitely strong materials. If it did,
then one could violate causality. If a perfectly rigid disk existed, vibrations in the disk would propagate at infinite velocity, so tapping
the disk with a hammer in one place would result in the transmis9
Relativistic description of a rotating disk, Am. J. Phys. 43 (1975) 869
Space, Time, and Coordinates in a Rotating World, http://www.phys.uu.
P. Ehrenfest, Gleichförmige Rotation starrer Körper und Relativitätstheorie,
Z. Phys. 10 (1909) 918, available in English translation at
Section 3.5
The metric
sion of information at v > c to other parts of the disk, and then
there would exist frames of reference in which the information was
received before it was transmitted. The same applies if the hammer
tap is used to impart rotational motion to the disk.
Self-check: What if we build the disk by assembling the building
materials so that they are already rotating properly before they are
joined together?
e / Einstein
The metric in the rotating frame
What if we try to get around these problems by applying torque
uniformly all over the disk, so that the rotation starts smoothly and
simultaneously everywhere? We then run into issues identical to the
ones raised by Bell’s spaceship paradox (p. 66). In fact, Ehrenfest’s
paradox is nothing more than Bell’s paradox wrapped around into
a circle. The same question of time synchronization comes up.
To spell this out mathematically, let’s find the metric according
to observer A by applying the change of coordinates θ0 = θ − ωt.
First we take the Euclidean metric of example 7 on page 102 and
rewrite it as a (globally) Lorentzian metric in spacetime for observer
ds2 = dt2 − dr2 − r2 dθ2
Applying the transformation into A’s coordinates, we find
ds2 = (1 − ω 2 r2 )dt2 − dr2 − r2 dθ02 − 2ωr2 dθ0 dt
Recognizing ωr as the velocity of one frame relative to another, and
(1−ω 2 r2 )−1/2 as γ, we see that we do have a relativistic time dilation
effect in the dt2 term. But the dr2 and dθ02 terms look Euclidean.
Why don’t we see any Lorentz contraction of the length scale in the
azimuthal direction?
The answer is that coordinates in general relativity are arbitrary, and just because we can write down a certain set of coordinates, that doesn’t mean they have any special physical interpretation. The coordinates (t, r, θ0 ) do not correspond physically to the
quantities that A would measure with clocks and meter-sticks. The
tip-off is the dθ0 dt cross-term. Suppose that A sends two cars driving around the circumference of the carousel, one clockwise and one
counterclockwise, from the same point. If (t, r, θ0 ) coordinates corresponded to clock and meter-stick measurements, then we would
expect that when the cars met up again on the far side of the disk,
their dashboards would show equal values of the arc length rθ0 on
their odometers and equal proper times ds on their clocks. But this
is not the case, because the sign of the dθ0 dt term is opposite for the
two world-lines. The same effect occurs if we send beams of light
in both directions around the disk, and this is the Sagnac effect (p.
Chapter 3
Differential geometry
This is a symptom of the fact that the coordinate t is not properly synchronized between different places on the disk. We already
know that we should not expect to be able to find a universal time
coordinate that will match up with every clock, regardless of the
clock’s state of motion. Suppose we set ourselves a more modest
goal. Can we find a universal time coordinate that will match up
with every clock, provided that the clock is at rest relative to the
rotating disk?
The spatial metric and synchronization of clocks
A trick for improving the situation is to eliminate the dθ0 dt crossterm by completing the square in the metric [2]. The result is
2 2
ds = (1 − ω r ) dt +
1 − ω2 r2
1 − ω2 r2
The interpretation of the quantity in square brackets is as follows.
Suppose that two observers situate themselves on the edge of the
disk, separated by an infinitesimal angle dθ0 . They then synchronize
their clocks by exchanging light pulses. The time of flight, measured
in the lab frame, for each light pulse is the solution of the equation
ds2 = 0, and the only difference between the clockwise result dt1
and the counterclockwise one dt2 arises from the sign of dθ0 . The
quantity in square brackets is the same in both cases, so the amount
by which the clocks must be adjusted is dt = (dt2 − dt1 )/2, or
dt =
dθ0 .
1 − ω2 r2
Substituting this into the metric, we are left with the purely spatial
ds2 = −dr2 −
1 − ω2 r2
The factor of (1 − ω 2 r2 )−1 = γ 2 in the dθ02 term is simply the
expected Lorentz-contraction factor. In other words, the circumference is, as expected, greater than 2πr by a factor of γ.
Does the metric [3] represent the same non-Euclidean spatial
geometry that A, rotating with the disk, would determine by meterstick measurements? Yes and no. It can be interpreted as the
one that A would determine by radar measurements. That is, if
A measures a round-trip travel time dt for a light signal between
points separated by coordinate distances dr and dθ0 , then A can say
that the spatial separation is dt/2, and such measurements will be
described correctly by [3]. Physical meter-sticks, however, present
some problems. Meter-sticks rotating with the disk are subject to
Coriolis and centrifugal forces, and this problem can’t be avoided
simply by making the meter-sticks infinitely rigid, because infinitely
rigid objects are forbidden by relativity. In fact, these forces will inevitably be strong enough to destroy any meter stick that is brought
Section 3.5
The metric
out to r = 1/ω, where the speed of the disk becomes equal to the
speed of light.
It might appear that we could now define a global coordinate
T =t+
1 − ω2 r2
interpreted as a time coordinate that was synchronized in a consistent way for all points on the disk. The trouble with this interpretation becomes evident when we imagine driving a car around
the circumference of the disk, at a speed slow enough so that there
is negligible time dilation of the car’s dashboard clock relative to
the clocks tied to the disk. Once the car gets back to its original
position, θ0 has increased by 2π, so it is no longer possible for the
car’s clock to be synchronized with the clocks tied to the disk. We
conclude that it is not possible to synchronize clocks in a rotating
frame of reference; if we try to do it, we will inevitably have to have
a discontinuity somewhere. This problem is present even locally, as
demonstrated by the possibility of measuring the Sagnac effect with
apparatus that is small compared to the disk. The only reason we
were able to get away with time synchronization in order to establish
the metric [3] is that all the physical manifestations of the impossibility of synchronization, e.g., the Sagnac effect, are proportional to
the area of the region in which synchronization is attempted. Since
we were only synchronizing two nearby points, the area enclosed by
the light rays was zero.
Example: 16
As a practical example, the GPS system is designed mainly to
allow people to find their positions relative to the rotating surface
of the earth (although it can also be used by space vehicles). That
is, they are interested in their (r , θ0 , φ) coordinates. The frame of
reference defined by these coordinates is referred to as ECEF, for
Earth-Centered, Earth-Fixed.
The system requires synchronization of the atomic clocks carried
aboard the satellites, and this synchronization also needs to be
extended to the (less accurate) clocks built into the receiver units.
It is impossible to carry out such a synchronization globally in the
rotating frame in order to create coordinates (T , r , θ0 , φ). If we
tried, it would result in discontinuities (see problem 8, p. 120).
Instead, the GPS system handles clock synchronization in coordinates (t, r , θ0 , φ), as in equation [2]. These are known as the
Earth-Centered Inertial (ECI) coordinates. The t coordinate in
this system is not the one that users at neighboring points on
the earth’s surface would establish if they carried out clock synchronization using electromagnetic signals. It is simply the time
coordinate of the nonrotating frame of reference tied to the earth’s
center. Conceptually, we can imagine this time coordinate as one
that is established by sending out an electromagnetic “tick-tock”
Chapter 3
Differential geometry
signal from the earth’s center, with each satellite correcting the
phase of the signal based on the propagation time inferred from
its own r . In reality, this is accomplished by communication with a
master control station in Colorado Springs, which communicates
with the satellites via relays at Kwajalein, Ascension Island, Diego
Garcia, and Cape Canaveral.
Einstein’s goof, in the rotating frame
Example: 17
Example 11 on p. 58 recounted Einstein’s famous mistake in predicting that a clock at the pole would experience a time dilation
relative to a clock at the equator, and the empirical test of this fact
by Alley et al. using atomic clocks. The perfect cancellation of
gravitational and kinematic time dilations might seem fortuitous,
but it fact it isn’t. When we transform into the frame rotating along
with the earth, there is no longer any kinematic effect at all, because neither clock is moving. In this frame, the surface of the
earth’s oceans is an equipotential, so the gravitational time dilation vanishes as well, assuming both clocks are at sea level.
In the transformation to the rotating frame, the metric picks up a
dθ0 dt term, but since both clocks are fixed to the earth’s surface,
they have dθ0 = 0, and there is no Sagnac effect.
Impossibility of rigid rotation, even with external forces
The determination of the spatial metric with rulers at rest relative to the disk is appealing because of its conceptual simplicity
compared to complicated procedures involving radar, and this was
presumably why Einstein presented the concept using ruler measurements in his 1916 paper laying out the general theory of relativity.12
In an effort to recover this simplicity, we could propose using external forces to compensate for the centrifugal and Coriolis forces to
which the rulers would be subjected, causing them to stay straight
and maintain their correct lengths. Something of this kind is carried out with the large mirrors of some telescopes, which have active
systems that compensate for gravitational deflections and other effects. The first issue to worry about is that one would need some
way to monitor a ruler’s length and straightness. The monitoring
system would presumably be based on measurements with beams
of light, in which case the physical rulers themselves would become
In addition, we would need to be able to manipulate the rulers in
order to place them where we wanted them, and these manipulations
would include angular accelerations. If such a thing was possible,
then it would also amount to a loophole in the resolution of the
Ehrenfest paradox. Could Ehrenfest’s rotating disk be accelerated
and decelerated with help from external forces, which would keep it
from contorting into a potato chip? The problem we run into with
The paper is reproduced in the back of the book, and the relevant part is
on p. 362.
Section 3.5
The metric
such a strategy is one of clock synchronization. When it was time to
impart an angular acceleration to the disk, all of the control systems
would have to be activated simultaneously. But we have already
seen that global clock synchronization cannot be realized for an
object with finite area, and therefore there is a logical contradiction
in this proposal. This makes it impossible to apply rigid angular
acceleration to the disk, but not necessarily the rulers, which could
in theory be one-dimensional.
3.6 The metric in general relativity
So far we’ve considered a variety of examples in which the metric
is predetermined. This is not the case in general relativity. For
example, Einstein published general relativity in 1915, but it was
not until 1916 that Schwarzschild found the metric for a spherical,
gravitating body such as the sun or the earth.
When masses are present, finding the metric is analogous to
finding the electric field made by charges, but the interpretation is
more difficult. In the electromagnetic case, the field is found on
a preexisting background of space and time. In general relativity,
there is no preexisting geometry of spacetime. The metric tells us
how to find distances in terms of our coordinates, but the coordinates themselves are completely arbitrary. So what does the metric
even mean? This was an issue that caused Einstein great distress
and confusion, and at one point, in 1914, it even led him to publish an incorrect, dead-end theory of gravity in which he abandoned
With the benefit of hindsight, we can consider these issues in
terms of the general description of measurements in relativity given
on page 96:
1. We can tell whether events and world-lines are incident.
2. We can do measurements in local Lorentz frames.
3.6.1 The hole argument
The main factor that led Einstein to his false start is known as
the hole argument. Suppose that we know about the distribution of
matter throughout all of spacetime, including a particular region of
finite size — the “hole” — which contains no matter. By analogy
with other classical field theories, such as electromagnetism, we expect that the metric will be a solution to some kind of differential
equation, in which matter acts as the source term. We find a metric
g(x) that solves the field equations for this set of sources, where x is
some set of coordinates. Now if the field equations are coordinateindependent, we can introduce a new set of coordinates x0 , which is
identical to x outside the hole, but differs from it on the inside. If
Chapter 3
Differential geometry
we reexpress the metric in terms of these new coordinates as g 0 (x0 ),
then we are guaranteed that g 0 (x0 ) is also a solution. But furthermore, we can substitute x for x0 , and g 0 (x) will still be a solution.
For outside the hole there is no difference between the primed and
unprimed quantities, and inside the hole there is no mass distribution that has to match the metric’s behavior on a point-by-point
We conclude that in any coordinate-invariant theory, it is impossible to uniquely determine the metric inside such a hole. Einstein
initially decided that this was unacceptable, because it showed a
lack of determinism; in a classical theory such as general relativity,
we ought to be able to predict the evolution of the fields, and it
would seem that there is no way to predict the metric inside the
hole. He eventually realized that this was an incorrect interpretation. The only type of global observation that general relativity lets
us do is measurements of the incidence of world-lines. Relabeling all
the points inside the hole doesn’t change any of the incidence relations. For example, if two test particles sent into the region collide
at a point x inside the hole, then changing the point’s name to x0
doesn’t change the observable fact that they collided.
a / Einstein’s
3.6.2 A Machian paradox
Another type of argument that made Einstein suffer is also resolved by a correct understanding of measurements, this time the
use of measurements in local Lorentz frames. The earth is in hydrostatic equilibrium, and its equator bulges due to its rotation.
Suppose that the universe was empty except for two planets, each
rotating about the line connecting their centers.13 Since there are
no stars or other external points of reference, the inhabitants of each
planet have no external reference points against which to judge their
rotation or lack of rotation. They can only determine their rotation,
Einstein said, relative to the other planet. Now suppose that one
planet has an equatorial bulge and the other doesn’t. This seems to
violate determinism, since there is no cause that could produce the
differing effect. The people on either planet can consider themselves
as rotating and the other planet as stationary, or they can describe
the situation the other way around. Einstein believed that this argument proved that there could be no difference between the sizes
of the two planets’ equatorial bulges.
The flaw in Einstein’s argument was that measurements in local
Lorentz frames do allow one to make a distinction between rotation
and a lack of rotation. For example, suppose that scientists on
planet A notice that their world has no equatorial bulge, while planet
B has one. They send a space probe with a clock to B, let it stay
The example is described in Einstein’s paper “The Foundation of the General
Theory of Relativity.” An excerpt, which includes the example, is given on
p. 360.
Section 3.6
b / A paradox?
Planet A has
no equatorial bulge, but B does.
What cause produces this effect?
Einstein reasoned that the cause
couldn’t be B’s rotation, because
each planet rotates relative to the
The metric in general relativity
on B’s surface for a few years, and then order it to return. When
the clock is back in the lab, they compare it with another clock that
stayed in the lab on planet A, and they find that less time has elapsed
according to the one that spent time on B’s surface. They conclude
that planet B is rotating more quickly than planet A, and that the
motion of B’s surface was the cause of the observed time dilation.
This resolution of the apparent paradox depends specifically on the
Lorentzian form of the local geometry of spacetime; it is not available
in, e.g., Cartan’s curved-spacetime description of Newtonian gravity
(see page 41).
Einstein’s original, incorrect use of this example sprang from his
interest in the ideas of the physicist and philosopher Ernst Mach.
Mach had a somewhat ill-defined idea that since motion is only a
well-defined notion when we speak of one object moving relative
to another object, the inertia of an object must be caused by the
influence of all the other matter in the universe. Einstein referred
to this as Mach’s principle. Einstein’s false starts in constructing
general relativity were frequently related to his attempts to make his
theory too “Machian.” Section 8.3 on p. 322 discusses an alternative,
more Machian theory of gravity proposed by Brans and Dicke in
3.7 Interpretation of coordinate independence
This section discusses some of the issues that arise in the interpretation of coordinate independence. It can be skipped on a first
3.7.1 Is coordinate independence obvious?
One often hears statements like the following from relativists:
“Coordinate independence isn’t really a physical principle. It’s
merely an obvious statement about the relationship between mathematics and the physical universe. Obviously the universe doesn’t
come equipped with coordinates. We impose those coordinates on
it, and the way in which we do so can never be dictated by nature.”
The impressionable reader who is tempted to say, “Ah, yes, that is
obvious,” should consider that it was far from obvious to Newton
(“Absolute, true and mathematical time, of itself, and from its own
nature flows equably without regard to anything external . . . ”), nor
was it obvious to Einstein. Levi-Civita nudged Einstein in the direction of coordinate independence in 1912. Einstein tried hard to
make a coordinate-independent theory, but for reasons described in
section 3.6.1 (p. 114), he convinced himself that that was a dead
end. In 1914-15 he published theories that were not coordinateindependent, which you will hear relativists describe as “obvious”
dead ends because they lack any geometrical interpretation. It seems
to me that it takes a highly refined intuition to regard as intuitively
Chapter 3
Differential geometry
“obvious” an issue that Einstein struggled with like Jacob wrestling
with Elohim.
3.7.2 Is coordinate independence trivial?
It has also been alleged that coordinate independence is trivial.
To gauge the justice of this complaint, let’s distinguish between two
reasons for caring about coordinate independence:
1. Coordinate independence tells us that when we solve problems,
we should avoid writing down any equations in notation that
isn’t manifestly intrinsic, and avoid interpreting those equations as if the coordinates had intrinsic meaning. Violating
this advice doesn’t guarantee that you’ve made a mistake, but
it makes it much harder to tell whether or not you have.
2. Coordinate independence can be used as a criterion for judging
whether a particular theory is likely to be successful.
Nobody questions the first justification. The second is a little trickier. Laying out the general theory systematically in a 1916 paper,14
Einstein wrote “The general laws of nature are to be expressed by
equations which hold good for all the systems of coordinates, that is,
are covariant with respect to any substitutions whatever (generally
covariant).” In other words, he was explaining why, with hindsight,
his 1914-1915 coordinate-dependent theory had to be a dead end.
The only trouble with this is that Einstein’s way of posing the
criterion didn’t quite hit the nail on the head mathematically. As
Hilbert famously remarked, “Every boy in the streets of Göttingen
understands more about four-dimensional geometry than Einstein.
Yet, in spite of that, Einstein did the work and not the mathematicians.” What Einstein had in mind was that a theory like Newtonian
mechanics not only lacks coordinate independence, but would also
be impossible to put into a coordinate-independent form without
making it look hopelessly complicated and ugly, like putting lipstick
on a pig. But Kretschmann showed in 1917 that any theory could
be put in coordinate independent form, and Cartan demonstrated in
1923 that this could be done for Newtonian mechanics in a way that
didn’t come out particularly ugly. Physicists today are more apt to
pose the distinction in terms of “background independence” (meaning that a theory should not be phrased in terms of an assumed geometrical background) or lack of a “prior geometry” (meaning that
the curvature of spacetime should come from the solution of field
equations rather than being imposed by fiat). But these concepts as
well have resisted precise mathematical formulation.15 My feeling
is that this general idea of coordinate independence or background
see p. 364
Giulini, “Some remarks on the notions of general covariance and background
Section 3.7
Interpretation of coordinate independence
independence is like the equivalence principle: a crucial conceptual
principle that doesn’t lose its importance just because we can’t put
it in a mathematical box with a ribbon and a bow. For example,
string theorists take it as a serious criticism of their theory that it is
not manifestly background independent, and one of their goals is to
show that it has a background independence that just isn’t obvious
on the surface.
3.7.3 Coordinate independence as a choice of gauge
a / Since magnetic field lines
can never intersect, a magnetic
field pattern contains coordinateindependent information in the
form of the knotting of the lines.
This figure shows the magnetic field pattern of the star
SU Aurigae, as measured by
Zeeman-Doppler imaging (Petit
at al.). White lines represent
magnetic field lines that close
upon themselves in the immediate vicinity of the star; blue lines
are those that extend out into the
interstellar medium.
It is instructive to consider coordinate independence from the
point of view of a field theory. Newtonian gravity can be described
in three equivalent ways: as a gravitational field g, as a gravitational
potential φ, or as a set of gravitational field lines. The field lines are
never incident on one another, and locally the field satisfies Poisson’s
The electromagnetic field has polarization properties different
from those of the gravitational field, so we describe it using either
the two fields (E, B), a pair of potentials,16 or two sets of field
lines. There are similar incidence conditions and local field equations
(Maxwell’s equations).
Gravitational fields in relativity have polarization properties unknown to Newton, but the situation is qualitatively similar to the
two foregoing cases. Now consider the analogy between electromagnetism and relativity. In electromagnetism, it is the fields that are
directly observable, so we expect the potentials to have some extrinsic properties. We can, for example, redefine our electrical ground,
Φ → Φ + C, without any observable consequences. As discussed in
more detail in section 5.6.1 on page 173, it is even possible to modify
the electromagnetic potentials in an entirely arbitrary and nonlinear
way that changes from point to point in spacetime. This is called a
gauge transformation. In relativity, the gauge transformations are
the smooth coordinate transformations. These gauge transformations distort the field lines without making them cut through one
There is the familiar electrical potential φ, measured in volts, but also a
vector potential A, which you may or may not have encountered. Briefly, the
electric field is given not by −∇φ but by −∇φ − ∂A/∂t, while the magnetic field
is the curl of A. This is introduced at greater length in section 4.2.5 on page
Chapter 3
Differential geometry
Consider a spacetime that is locally exactly like the standard Lorentzian spacetime described in ch. 2, but that has a global
structure differing in the following way from the one we have implicitly assumed. This spacetime has global property G: Let two
material particles have world-lines that coincide at event A, with
some nonzero relative velocity; then there may be some event B in
the future light-cone of A at which the particles’ world-lines coincide
again. This sounds like a description of something that we would
expect to happen in curved spacetime, but let’s see whether that
is necessary. We want to know whether this violates the flat-space
properties L1-L5 on page 388, if those properties are taken as local.
(a) Demonstrate that it does not violate them, by using a model in
which space “wraps around” like a cylinder.
(b) Now consider the possibility of interpreting L1-L5 as global statements. Do spacetimes with property G always violate L3 if L3 is
taken globally?
. Solution, p. 368
Usually in relativity we pick units in which c = 1. Suppose,
however, that we want to use SI units. The convention is that coordinates are written with upper indices, so that, fixing the usual
Cartesian coordinates in 1+1 dimensions of spacetime, an infinitesimal displacement between two events is notated (dst , dsx ). In SI
units, the two components of this vector have different units, which
may seem strange but is perfectly legal. Describe the form of the
metric, including the units of its elements. Describe the lower-index
vector dsa .
. Solution, p. 368
(a) Explain why the following expressions ain’t got good
grammar: Uaa , xa y a , pa −qa . (Recall our notational convention that
Latin indices represent abstract indices, so that it would not make
sense, for example, to interpret Uaa as U ’s ath diagonal element
rather than as an implied sum.)
(b) Which of these could also be nonsense in terms of units?
. Solution, p. 369
Suppose that a mountaineer describes her location using coordinates (θ, φ, h), representing colatitude, longitude, and altitude.
Infer the units of the components of dsa and of the elements of gab
and g ab . Given that the units of mechanical work should be newtonmeters (cf 5, p. 48), infer the components of a force vector Fa and
its upper-index version F a .
. Solution, p. 369
Generalize figure h/2 on p. 48 to three dimensions.
. Solution, p. 369
Suppose you have a collection of pencils, some of which
have been sharpened more times than others so that they they’re
shorter. You toss them all on the floor in random orientations,
and you’re then allowed to slide them around but not to rotate
them. Someone asks you to make up a definition of whether or
not a given set of three pencils “cancels.” If all pencils are treated
equally (i.e., order doesn’t matter), and if we respect the rotational
invariance of Euclidean geometry, then you will be forced to reinvent
vector addition and define cancellation of pencils p, q, and r as
p + q + r = 0. Do something similar with “pencil” replaced by “an
oriented pairs of lines as in figure h/2 on p. 48.
Describe the quantity g aa . (Note the repeated index.)
. Solution, p. 369
Example 16 on page 112 discusses the discontinuity that
would result if one attempted to define a time coordinate for the
GPS system that was synchronized globally according to observers
in the rotating frame, in the sense that neighboring observers could
verify the synchronization by exchanging electromagnetic signals.
Calculate this discontinuity at the equator, and estimate the resulting error in position that would be experienced by GPS users.
. Solution, p. 369
Resolve the following paradox.
Equation [3] on page 111 claims to give the metric obtained by an observer on the surface of a rotating disk. This metric is shown to lead
to a non-Euclidean value for the ratio of the circumference of a circle
to its radius, so the metric is clearly non-Euclidean. Therefore a local observer should be able to detect violations of the Pythagorean
And yet this metric was originally derived by a series of changes
of coordinates, starting from the Euclidean metric in polar coordinates, as derived in example 7 on page 102. Section 3.4 (p. 95)
argued that the intrinsic measurements available in relativity are
not capable of detecting an arbitrary smooth, one-to-one change of
coordinates. This contradicts our earlier conclusion that there are
locally detectable violations of the Pythagorean theorem.
. Solution, p. 369
This problem deals with properties of the metric [3] on page
111. (a) A pulse of collimated light is emitted from the center of
the disk in a certain direction. Does the spatial track of the pulse
form a geodesic of this metric? (b) Characterize the behavior of the
geodesics near r = 1/ω. (c) An observer at rest with respect to the
surface of the disk proposes to verify the non-Euclidean nature of
the metric by doing local tests in which right triangles are formed
out of laser beams, and violations of the Pythagorean theorem are
detected. Will this work?
. Solution, p. 370
Chapter 3
Differential geometry
In the early decades of relativity, many physicists were in the
habit of speaking as if the Lorentz transformation described what an
observer would actually “see” optically, e.g., with an eye or a camera.
This is not the case, because there is an additional effect due to optical aberration: observers in different states of motion disagree about
the direction from which a light ray originated. This is analogous
to the situation in which a person driving in a convertible observes
raindrops falling from the sky at an angle, even if an observer on the
sidewalk sees them as falling vertically. In 1959, Terrell and Penrose
independently provided correct analyses,17 showing that in reality
an object may appear contracted, expanded, or rotated, depending
on whether it is approaching the observer, passing by, or receding.
The case of a sphere is especially interesting. Consider the following
four cases:
A The sphere is not rotating. The sphere’s center is at rest. The
observer is moving in a straight line.
B The sphere is not rotating, but its center is moving in a straight
line. The observer is at rest.
C The sphere is at rest and not rotating. The observer moves
around it in a circle whose center coincides with that of the
D The sphere is rotating, with its center at rest. The observer is
at rest.
Penrose showed that in case A, the outline of the sphere is still
seen to be a circle, although regions on the sphere’s surface appear
What can we say about the generalization to cases B, C, and D?
. Solution, p. 370
This problem involves a relativistic particle of mass m which
is also a wave, as described by quantum mechanics. Let c = 1 and
~ = 1 throughout. Starting from the de Broglie relations E = ω
and p = k, where k is the wavenumber, find the dispersion relation
connecting ω to k. Calculate the group velocity, and verify that it is
consistent with the usual relations p = mγv and E = mγ for m > 0.
What goes wrong if you instead try to associate v with the phase
. Solution, p. 370
James Terrell, “Invisibility of the Lorentz Contraction,” Physical Review 116
(1959) 1045. Roger Penrose, “The Apparent Shape of a Relativistically Moving
Sphere,” Proceedings of the Cambridge Philosophical Society 55 (1959) 139.
Chapter 3
Differential geometry
Chapter 4
We now have enough machinery to be able to calculate quite a bit of
interesting physics, and to be sure that the results are actually meaningful in a relativistic context. The strategy is to identify relativistic
quantities that behave as Lorentz scalars and Lorentz vectors, and
then combine them in various ways. The notion of a tensor has been
introduced on page 101. A Lorentz scalar is a tensor of rank 0, and
a Lorentz vector is a rank-1 tensor.
4.1 Lorentz scalars
A Lorentz scalar is a quantity that remains invariant under both spatial rotations and Lorentz boosts. Mass is a Lorentz scalar.1 Electric charge is also a Lorentz scalar, as demonstrated to extremely
high precision by experiments measuring the electrical neutrality of
atoms and molecules to a relative precision of better than 10−20 ; the
electron in a hydrogen atom has typically velocities of about 1/100,
and those in heavier elements such as uranium are highly relativistic, so any violation of Lorentz invariance would give the atoms a
nonvanishing net electric charge.
The time measured by a clock traveling along a particular worldline from one event to another is something that all observers will
agree upon; they will simply note the mismatch with their own
clocks. It is therefore a Lorentz scalar. This clock-time as measured
by a clock attached to the moving body in question is often referred
to as proper time, “proper” being used here in the somewhat archaic
sense of “own” or “self,” as in “The Vatican does not lie within Italy
proper.” Proper time, which we notate τ , can only be defined for
timelike world-lines, since a lightlike or spacelike world-line isn’t
possible for a material clock.
More generally, when we express a metric as ds2 = . . ., the
quantity ds is a Lorentz scalar. In the special case of a timelike
world-line, ds and dτ are the same thing. (In books that use a
− + ++ metric, one has ds = −dτ .)
Even more generally, affine parameters, which exist independent
of any metric at all, are scalars. As a trivial example, if τ is a
particular object’s proper time, then τ is a valid affine parameter,
Some older books define mass as transforming according to m → γm, which
can be made to give a self-consistent theory, but is ugly.
but so is 2τ +7. Less trivially, a photon’s proper time is always zero,
but one can still define an affine parameter along its trajectory. We
will need such an affine parameter, for example, in section 6.2.7,
page 220, when we calculate the deflection of light rays by the sun,
one of the early classic experimental tests of general relativity.
Another example of a Lorentz scalar is the pressure of a perfect
fluid, which is often assumed as a description of matter in cosmological models.
Infinitesimals and the clock “postulate”
Example: 1
At the beginning of chapter 3, I motivated the use of infinitesimals
as useful tools for doing differential geometry in curved spacetime. Even in the context of special relativity, however, infinitesimals can be useful. One way of expressing the proper time accumulated on a moving clock is
s = ds
Z q
gij dx i dx j
2 2 2
which only contains an explicit dependence on the clock’s velocity, not its acceleration. This is an example of the clock “postulate”
referred to in the remark at the end of homework problem 1 on
page 84. Note that the clock postulate only applies in the limit of
a small clock. This is represented in the above equation by the
use of infinitesimal quantities like dx.
4.2 Four-vectors
4.2.1 The velocity and acceleration four-vectors
Our basic Lorentz vector is the spacetime displacement dxi . Any
other quantity that has the same behavior as dxi under rotations
and boosts is also a valid Lorentz vector. Consider a particle moving
through space, as described in a Lorentz frame. Since the particle
may be subject to nongravitational forces, the Lorentz frame cannot be made to coincide (except perhaps momentarily) with the
particle’s rest frame. If dxi is not lightlike, then the corresponding
infinitesimal proper time interval dτ is nonzero. As with Newtonian
three-vectors, dividing a four-vector by a Lorentz scalar produces
another quantity that transforms as a four-vector, so dividing the
infinitesimal displacement by a nonzero infinitesimal proper time
interval, we have the four-velocity vector v i = dxi /dτ , whose components in a Lorentz coordinate system are (γ, γu1 , γu2 , γu3 ), where
(u1 , u2 , u3 ) is the ordinary three-component velocity vector as defined in classical mechanics. The four-velocity’s squared magnitude
Chapter 4
v i vi is always exactly 1, even though the particle is not moving at
the speed of light. (If it were moving at the speed of light, we would
have dτ = 0, and v would be undefined.)
When we hear something referred to as a “vector,” we usually
take this is a statement that it not only transforms as a vector, but
also that it adds as a vector. But we have already seen in section
2.3.1 on page 66 that even collinear velocities in relativity do not
add linearly; therefore they clearly cannot add linearly when dressed
in the clothing of four-vectors. We’ve also seen in section 2.5.3 that
the combination of non-collinear boosts is noncommutative, and is
generally equivalent to a boost plus a spatial rotation; this is also
not consistent with linear addition of four-vectors. At the risk of
beating a dead horse, a four-velocity’s squared magnitude is always
1, and this is not consistent with being able to add four-velocity
A zero velocity vector?
Example: 2
. Suppose an object has a certain four-velocity v i in a certain
frame of reference. Can we transform into a different frame in
which the object is at rest, and its four-velocity is zero?
. No. In general, the Lorentz transformation preserves the magnitude of vectors, so it can never transform a vector with a zero
magnitude into one with zero magnitude. Since this is a material
object (not a ray oflight) we can transform into a frame in which
the object is at rest, but an object at rest does not have a vanishing four-velocity. It has a four-velocity of (1, 0, 0, 0).
Example 2 suggests a nice way of thinking about velocity vectors, which is that every timelike velocity vector represents a potential observer. An observer is a material object, and therefore has
a timelike velocity vector. This observer writes her own velocity
vector as (1, 0, 0, 0), i.e., as the unit vector in the timelike direction. Often when we see an expression involving a velocity vector,
we can interpret it as describing a measurement taken by a specific
Orthogonality as simultaneity
Example: 3
In a space where the inner product can be negative, orthogonality
doesn’t mean what our euclidean intuition thinks it means. For example, a lightlike vector can be orthogonal to itself — a situation
that never occurs in a euclidean space.
Suppose we have a timelike vector t and a spacelike one x. What
would it mean for t and x to be orthogonal, with t·x = 0? Since t is
timelike, we can make a unit vector t̂ = t/|t| out of it, and interpret
t̂ as the velocity vector of some hypothetical observer. We then
know that in that observer’s frame, t̂ is simply a unit vector along
the time axis. It now becomes clear that x must be parallel to the
x axis, i.e., it represents a displacement between two events that
Section 4.2
this observer considers to be simultaneous.
This is an example of the idea that expressions involving velocity
vectors can be interpreted as measurements taken by a certain
observer. The expression t · x = 0 can be interpreted as meaning
that according to an observer whose world-line is tangent to t, x
represents a relationship of simultaneity.
The four-acceleration is found by taking a second derivative with
respect to proper time. Its squared magnitude is only approximately
equal to minus the squared magnitude of the classical acceleration
three-vector, in the limit of small velocities.
Constant acceleration
Example: 4
. Suppose a spaceship moves so that the acceleration is judged
to be the constant value a by an observer on board. Find the
motion x(t) as measured by an observer in an inertial frame.
. Let τ stand for the ship’s proper time, and let dots indicate
derivatives with respect to τ. The ship’s velocity has magnitude
1, so
ṫ 2 − ẋ 2 = 1
An observer who is instantaneously at rest with respect to the
ship judges is to have a four-acceleration (0, a, 0, 0) (because the
low-velocity limit applies). The observer in the (t, x) frame agrees
on the magnitude of this vector, so
ẗ 2 − ẍ 2 = −a2
The solution of these differential equations is t =
x = a1 cosh aτ, and eliminating τ gives
1 + a2 t 2
sinh aτ,
As t approaches infinity, dx/dt approaches the speed of light.
4.2.2 The momentum four-vector
Definition for a material particle
If we hope to find something that plays the role of momentum
in relativity, then the momentum three-vector probably needs to
be generalized to some kind of four-vector. If so, then the law of
conservation of momentum will be valid regardless of one’s frame of
reference, which is necessary.2
If we are to satisfy the correspondence principle then the relativistic definition of momentum should probably look as much as
possible like the nonrelativistic one. In subsection 4.2.1, we defined
We are not guaranteed that this is the right way to proceed, since the converse is not true: some three-vectors such as the electric and magnetic fields are
embedded in rank-2 tensors in more complicated ways than this. See section
4.2.4, p. 136.
Chapter 4
the velocity four-vector in the case of a particle whose dxi is not
lightlike. Let’s assume for the moment that it makes sense to think
of mass as a scalar. As with Newtonian three-vectors, multiplying
a Lorentz scalar by a four-vector vector produces another quantity
that transforms as a four-vector. We therefore conjecture that the
four-momentum of a material particle can be defined as pi = mv i ,
which in Lorentz coordinates is (mγ, mγv 1 , mγv 2 , mγv 3 ). There is
no a priori guarantee that this is right, but it’s the most reasonable
thing to guess. It needs to be checked against experiment, and also
for consistency with the other parts of our theory.
The spacelike components look like the classical momentum vector multiplied by a factor of γ, the interpretation being that to an
observer in this frame, the moving particle’s inertia is increased relative to its value in the particle’s rest frame. Such an effect is indeed
observed experimentally. This is why particle accelerators are so big
and expensive. As the particle approaches the speed of light, γ diverges, so greater and greater forces are needed in order to produce
the same acceleration. In relativistic scattering processes with material particles, we find empirically that the four-momentum we’ve
defined is conserved, which confirms that our conjectures above are
valid, and in particular that the quantity we’re calling m can be
treated as a Lorentz scalar, and this is what all physicists do today.
The reader is cautioned, however, that up until about 1950, it was
common to use the word “mass” for the combination mγ (which
is what occurs in the Lorentz-coordinate form of the momentum
vector), while referring to m as the “rest mass.” This archaic terminology is only used today in some popular-level books and low-level
school textbooks.
Equivalence of mass and energy
The momentum four-vector has locked within it the reason for
Einstein’s famous E = mc2 , which in our relativistic units becomes
simply E = m. To see why, consider the experimentally measured
inertia of a physical object made out of atoms. The subatomic
particles are all moving, and many of the velocities, e.g., the velocities of the electrons, are quite relativistic. This has the effect
of increasing the experimentally determined inertial mass of the
whole object, by a factor of γ averaged over all the particles — even
though the masses of the individual particles are invariant Lorentz
scalars. (This same increase must also be observed for the gravitational mass, based on the equivalence principle as verified by Eötvös
Now if the object is heated, the velocities will increase on the
average, resulting in a further increase in its mass. Thus, a certain
amount of heat energy is equivalent to a certain amount of mass.
But if heat energy contributes to mass, then the same must be true
for other forms of energy. For example, suppose that heating leads to
Section 4.2
a chemical reaction, which converts some heat into electromagnetic
binding energy. If one joule of binding energy did not convert to
the same amount of mass as one joule of heat, then this would allow
the object to spontaneously change its own mass, and then by conservation of momentum it would have to spontaneously change its
own velocity, which would clearly violate the principle of relativity.
We conclude that mass and energy are equivalent, both inertially
and gravitationally. In relativity, neither is separately conserved;
the conserved quantity is their sum, referred to as the mass-energy,
E. An alternative derivation, by Einstein, is given in example 16 on
page 135.
Energy is the timelike component of the four-momentum
The Lorentz transformation of a zero vector is always zero. This
means that the momentum four-vector of a material object can’t
equal zero in the object’s rest frame, since then it would be zero
in all other frames as well. So for an object of mass m, let its
momentum four-vector in its rest frame be (f (m), 0, 0, 0), where f
is some function that we need to determine, and f can depend only
on m since there is no other property of the object that can be
dynamically relevant here. Since conservation laws are additive, f
has to be f (m) = km for some universal constant k. In where c = 1,
k is unitless. Since we want to recover the appropriate Newtonian
limit for massive bodies, and since vt = 1 in that limit, we need k =
1. Transforming the momentum four-vector from the particle’s rest
frame into some other frame, we find that the timelike component
is no longer m. We interpret this as the relativistic mass-energy, E.
Since the momentum four-vector was obtained from the magnitude-1 velocity four-vector through multiplication by m, its squared
magnitude pi pi is equal to the square of the particle’s mass. Writing
p for the magnitude of the momentum three-vector, and E for the
mass-energy, we find the useful relation m2 = E 2 − p2 . We take this
to be the relativistic definition the mass of any particle, including
one whose dxi is lightlike.
Particles traveling at c
The definition of four-momentum as pi = mv i only works for
particles that move at less than c. For those that move at c, the
four-velocity is undefined. As we’ll see in example 6 on p. 129, this
class of particles is exactly those that are massless. As shown on
p. 32, the three-momentum of a light wave is given by p = E. The
fact that this momentum is nonzero implies that for light pi = mv i
represents an indeterminate form. The fact that this momentum
equals E is consistent with our definition of mass as m2 = E 2 − p2 .
Mass is not additive
Since the momentum four-vector pa is additive, and our definition of mass as pa pa depends on the vector in a nonlinear way, it
Chapter 4
follows that mass is not additive (even for particles that are not
interacting but are simply considered collectively).
Mass of two light waves
Example: 5
Let the momentum of a certain light wave be (pt , px ) = (E, E),
and let another such wave have momentum (E, −E). The total
momentum is (2E, 0). Thus this pair of massless particles has a
collective mass of 2E.
Massless particles travel at c
Example: 6
We demonstrate this by showing that if we suppose the opposite,
then there are two different consequences, either of which would
be physically unacceptable.
When a particle does have a nonvanishing mass, we have
|v |
E/m→∞ E
Thus if we had a massless particle with |v | 6= 1, its behavior
would be different from the limiting behavior of massive particles.
But this is physically unacceptable because then we would have
a magic method for detecting arbitrarily small masses such as
10−10000000000 kg. We don’t actually know that the photon, for
example, is exactly massless; see example 13 on p. 131.
Furthermore, suppose that a massless particle had |v | < 1 in
the frame of some observer. Then some other observer could
be at rest relative to the particle. In such a frame, the particle’s
three-momentum p is zero by symmetry, since there is no preferred direction for it. Then E 2 = p2 + m2 is zero as well, so
the particle’s entire energy-momentum four-vector is zero. But
a four-vector that vanishes in one frame also vanishes in every
other frame. That means we’re talking about a particle that can’t
undergo scattering, emission, or absorption, and is therefore undetectable by any experiment. This is physically unacceptable
because we don’t consider phenomena (e.g., invisible fairies) to
be of physical interest if they are undetectable even in principle.
Gravitational redshifts
Example: 7
Since a photon’s energy E is equivalent to a certain gravitational
mass m, photons that rise or fall in a gravitational field must
lose or gain energy, and this should be observed as a redshift
or blueshift in the frequency. We expect the change in gravitational potential energy to be E∆φ, giving a corresponding opposite change in the photon’s energy, so that ∆E/E = ∆φ. In
metric units, this becomes ∆E/E = ∆φ/c 2 , and in the field near
the Earth’s surface we have ∆E/E = gh/c 2 . This is the same
result that was found in section 1.5.5 based only on the equivalence principle, and verified experimentally by Pound and Rebka
as described in section 1.5.6.
Section 4.2
Constraints on polarization
Example: 8
We observe that electromagnetic waves are always polarized
transversely, never longitudinally. Such a constraint can only apply to a wave that propagates at c. If it applied to a wave that
propagated at less than c, we could move into a frame of reference in which the wave was at rest. In this frame, all directions in
space would be equivalent, and there would be no way to decide
which directions of polarization should be permitted. For a wave
that propagates at c, there is no frame in which the wave is at rest
(see p. 98).
Relativistic work-energy theorem
Example: 9
In Einstein’s original 1905 paper on relativity, he assumed without
providing any justification that the Newtonian work-energy relation W = F d was valid relativistically. One way of justifying this is
that we can construct a simple machine with a mechanical advantage A and a reduction of motion by 1/A, with these ratios being
exact relativistically.3 One can then calculate, as Einstein did,
dp dx
W =
dx =
dv = m(γ − 1)
dv dt
which is consistent with our result for E as a function of γ if we
equate it to E(γ) − E(1).
The Dirac sea
Example: 10
A great deal of physics can be derived from the T.H. White’s
principle that “whatever is not forbidden in compulsory” — originally intended for ants but applied to particles by Gell-Mann.
In quantum mechanics, any process that is not forbidden by a
law is supposed to occur. The relativistic relation
E = ± p + m2 has two roots, a positive one and a negative one.
The positive-energy and negative-energy states are separated by
a no-man’s land of width 2m, so no continuous classical process
can lead from one side to the other. p
But quantum-mechanically, if
an electron exists with energy E = + p2 + m2 ,p
it should be able to
make a quantum leap into a state with E = − p2 + m2 , emitting
the energy difference of 2E in the form of photons. Why doesn’t
this happen? One explanation is that the states with E < 0 are all
already occupied. This is the “Dirac sea,” which we now interpret
as being full of electrons. A vacancy in the sea manifests itself as
an antielectron.
Massive neutrinos
Example: 11
Neutrinos were long thought to be massless, but are now believed
to have masses in the eV range. If they had been massless, they
would always have had to propagate at the speed of light. Although they are now thought to have mass, that mass is six orders of magnitude less than the MeV energy scale of the nuclear
Chapter 4
For an explicit example, see
reactions in which they are produced, so all neutrinos observed
in experiments are moving at velocities very close to the speed of
No radioactive decay of massless particles
Example: 12
A photon cannot decay into an electron and a positron, γ → e+ +
e− , in the absence of a charged particle to interact with. To see
this, consider the process in the frame of reference in which the
electron-positron pair has zero total momentum. In this frame, the
photon must have had zero (three-)momentum, but a photon with
zero momentum must have zero energy as well. This means that
conservation of relativistic four -momentum has been violated: the
timelike component of the four-momentum is the mass-energy,
and it has increased from 0 in the initial state to at least 2mc 2 in
the final state.
To demonstrate the consistency of the theory, we can arrive at the
same conclusion by a different method. Whenever a particle has
a small mass (small compared to its energy, say), it must travel
at close to c. It must therefore have a very large time dilation,
and will take a very long time to undergo radioactive decay. In
the limit as the mass approaches zero, the time required for the
decay approaches infinity. Another way of saying this is that the
rate of radioactive decay must be fixed in terms of proper time,
but there is no such thing as proper time for a massless particle.
Thus it is not only this specific process that is forbidden, but any
radioactive decay process involving a massless particle.
There are various loopholes in this argument. The question is
investigated more thoroughly by Fiore and Modanese.4
Massive photons
Example: 13
Continuing in the same vein as example 11, we can consider the
possibility that the photon has some nonvanishing mass. A 2003
experiment by Luo et al.5 has placed a limit of about 10−54 kg
on this mass. This is incredibly small, but suppose that future experimental work using improved techniques shows that the mass
is less than this, but actually nonzero. A naive reaction to this
scenario is that it would shake relativity to its core, since relativity
is based upon the assumption that the speed of light is a constant, whereas for a massive particle it need not be constant. But
this is a misinterpretation of the role of c in relativity. As should
be clear from the approach taken in section 2.2, c is primarily a
geometrical property of spacetime, not a property of light.
Luo et al., “New Experimental Limit on the Photon Rest Mass with a Rotating Torsion Balance,” Phys. Rev. Lett. 90 (2003) 081801. The interpretation
of such experiments is difficult, and this paper attracted a series of comments. A
weaker but more universally accepted bound is 8 × 10−52 kg, Davis, Goldhaber,
and Nieto, Phys. Rev. Lett. 35 (1975) 1402.
Section 4.2
In reality, such a discovery would be more of a problem for particle physicists than for relativists, as we can see by the following
sketch of an argument. Imagine two charged particles, at rest,
interacting via an electrical attraction. Quantum mechanics describes this as an exchange of photons. Since the particles are
at rest, there is no source of energy, so where do we get the
energy to make the photons? The Heisenberg uncertainty principle, ∆E∆t & h, allows us to steal this energy, provided that we
give it back within a time ∆t. This time limit imposes a limit on
the distance the photons can travel, but by using photons of low
enough energy, we can make this distance limit as large as we
like, and there is therefore no limit on the range of the force. But
suppose that the photon has a mass. Then there is a minimum
mass-energy mc 2 required in order to create a photon, the maximum time is h/mc 2 , and the maximum range is h/mc. Refining
these crude arguments a little, one finds that exchange of zeromass particles gives a force that goes like 1/r 2 , while a nonzero
mass results in e−µr /r 2 , where µ−1 = ~/mc. For the photon,
the best current mass limit corresponds to µ−1 & 1011 m, so the
deviation from 1/r 2 would be difficult to measure in earthbound
Now Gauss’s law is a specific characteristic of 1/r 2 fields. It would
be violated slightly if photons had mass. We would have to modify
Maxwell’s equations, and it turns out6 that the necessary change
to Gauss’s law would be of the form ∇ · E = (. . .)ρ − (. . .)µ2 Φ,
where Φ is the electrical potential, and (. . . ) indicates factors
that depend on the choice of units. This tells us that Φ, which
in classical electromagnetism can only be measured in terms of
differences between different points in space, can now be measured in absolute terms. Gauge symmetry has been broken. But
gauge symmetry is indispensible in creating well-behaved relativistic field theories, and this is the reason that, in general, particle physicists have a hard time with forces arising from the exchange of massive particles. The hypothetical Higgs particle,
which may be observed at the Large Hadron Collider in the near
future, is essentially a mechanism for wriggling out of this difficulty
in the case of the massive W and Z particles that are responsible
for the weak nuclear force; the mechanism cannot, however, be
extended to allow a massive photon.
Dust and radiation in cosmological models
Example: 14
In cosmological models, one needs an equation of state that relates the pressure P to the mass-energy density ρ. The pressure
is a Lorentz scalar. The mass-energy density is not (since massenergy is just the timelike component of a particular vector), but
in a coordinate system without any net flow of mass, we can ap6
Goldhaber and Nieto, ”Terrestrial and Extraterrestrial Limits on The Photon Mass,” Rev. Mod. Phys. 43 (1971) 277
Chapter 4
proximate it as one.
The early universe was dominated by radiation. A photon in a
box contributes a pressure on each wall that is proportional to
|pµ |, where µ is a spacelike index. In thermal equilibrium, each of
these three degrees of freedom carries an equal amount of energy, and since momentum and energy are equal for a massless
particle, the average momentum along each axis is equal to 31 E.
The resulting equation of state is P = 13 ρ. As the universe expanded, the wavelengths of the photons expanded in proportion
to the stretching of the space they occupied, resulting in λ ∝ a−1 ,
where a is a distance scale describing the universe’s intrinsic curvature at a fixed time. Since the number density of photons is
diluted in proportion to a−3 , and the mass per photon varies as
a−1 , both ρ and P vary as a−4 .
Cosmologists refer to noninteracting, nonrelativistic materials as
“dust,” which could mean many things, including hydrogen gas,
actual dust, stars, galaxies, and some forms of dark matter. For
dust, the momentum is negligible compared to the mass-energy,
so the equation of state is P = 0, regardless of ρ. The massenergy density is dominated simply by the mass of the dust, so
there is no red-shift scaling of the a−1 type. The mass-energy
density scales as a−3 . Since this is a less steep dependence on
a than the a−4 , there was a point, about a thousand years after
the Big Bang, when matter began to dominate over radiation. At
this point, the rate of expansion of the universe made a transition
to a qualitatively different behavior resulting from the change in
the equation of state.
In the present era, the universe’s equation of state is dominated
by neither dust nor radiation but by the cosmological constant
(see page 285). Figure a shows the evolution of the size of the
universe for the three different regimes. Some of the simpler
cases are derived in sections 8.2.7 and 8.2.8, starting on page
a / Example 14.
4.2.3 The frequency vector and the relativistic Doppler shift
The frequency vector was introduced in example 8 on p. 50. In
the spirit of index-gymnastics notation, frequency is to time as the
wavenumber k = 1/λ is to space, so when treating waves relativistically it is natural to conjecture that there is a four-frequency fa
made by assembling (f , k), which behaves as a Lorentz vector. This
is correct, since we already know that ∂a transforms as a covariant
vector, and for a scalar wave of the form A = Ao exp [2πifa xa ] the
partial derivative operator is identical to multiplication by 2πfa .
As an application, consider the relativistic Doppler shift of a light
wave. For simpicity, let’s restrict ourselves to one spatial dimension.
For a light wave, f = k, so the frequency vector in 1+1 dimensions
Section 4.2
is simply (f , f ). Putting this through a Lorentz transformation, we
f = (1 + v)γf =
where the second form displays more clearly the symmetic form
of the relativistic relationship, such that interchanging the roles of
source and observer is equivalent to flipping the sign of v. That is,
the relativistic version only depends on the relative motion of the
source and the observer, whereas the Newtonian one also depends
on the source’s motion relative to the medium (i.e., relative to the
preferred frame in which the waves have the “right” velocity). In
Newtonian mechanics, we have f 0 = (1 + v)f for a moving observer.
Relativistically, there is also a time dilation of the oscillation of the
source, providing an additional factor of γ.
This analysis is extended to 3+1 dimensions in problem 11.
Ives-Stilwell experiments
Example: 15
The relativistic Doppler shift differs from the nonrelativistic one by
the time-dilation factor γ, so that there is still a shift even when
the relative motion of the source and the observer is perpendicular to the direction of propagation. This is called the transverse Doppler shift. Einstein suggested this early on as a test
of relativity. However, such experiments are difficult to carry out
with high precision, because they are sensitive to any error in
the alignment of the 90-degree angle. Such experiments were
eventually performed, with results that confirmed relativity,7 but
one-dimensional measurements provided both the earliest tests
of the relativistic Doppler shift and the most precise ones to date.
The first such test was done by Ives and Stilwell inp1938, using the
following trick. The relativistic expression Sv = (1 + v )/(1 − v )
for the Doppler shift has the property that Sv S−v = 1, which differs
from the nonrelativistic result of (1 + v )(1 − v ) = 1 − v 2 . One can
therefore accelerate an ion up to a relativistic speed, measure
both the forward Doppler
p shifted frequency ff and the backward
one fb , and compute ff fb . According to relativity, this should
exactly equal the frequency fo measured in the ion’s rest frame.
In a particularly exquisite modern version of the Ives-Stilwell idea,8
Saathoff et al. circulated Li+ ions at v = .064 in a storage ring.
An electron-cooler technique was used in order to reduce the
variation in velocity among ions in the beam. Since the identity
Sv S−v = 1 is independent of v , it was not necessary to measure v to the same incredible precision as the frequencies; it was
See, e.g., Hasselkamp, Mondry, and Scharmann, Zeitschrift für Physik A:
Hadrons and Nuclei 289 (1979) 151.
G. Saathoff et al., “Improved Test of Time Dilation in Relativity,” Phys.
Rev. Lett. 91 (2003) 190403. A publicly available description of the experiment
is given in Saathoff’s PhD thesis,
Chapter 4
only necessary that it be stable and well-defined. The natural line
width was 7 MHz, and other experimental effects broadened it further to 11 MHz. By curve-fitting the line, it was possible to achieve
results good to a few tenths of a MHz. The resulting frequencies,
in units of MHz, were:
p fb
ff fb
= 582490203.44 ± .09
= 512671442.9 ± 0.5
= 546466918.6 ± 0.3
= 546466918.8 ± 0.4 (from previous experimental work)
The spectacular agreement with theory has made this experiment
a lightning rod for anti-relativity kooks.
If one is searching for small deviations from the predictions of
special relativity, a natural place to look is at high velocities. IvesStilwell experiments have been performed at velocities as high as
0.84, and they confirm special relativity.9
Einstein’s derivation of E = mc2
Example: 16
On page 126, we showed that the celebrated E = mc 2 follows directly from the form of the Lorentz transformation. An alternative
derivation was given by Einstein in one of his classic 1905 papers laying out the theory of special relativity; the paper is short,
and is reproduced in English translation on page 358 of this book.
Having laid the groundwork of four-vectors and relativistic Doppler
shifts, we can give an even shorter version of Einstein’s argument.
The discussion is also streamlined by restricting the discussion to
1+1 dimensions and by invoking photons.
Suppose that a lantern, at rest in the lab frame, is floating weightlessly in outer space, and simultaneously emits two pulses of
light in opposite directions, each with energy E/2 and frequency
f . By symmetry, the momentum of the pulses cancels, and the
lantern remains at rest. An observer in motion at velocity v relative to the lab sees the frequencies of the beams shifted to
f 0 = (1 ± v )γf . The effect on the energies of the beams can
be found purely classically, by transforming the electric and magnetic fields to the moving frame, but as a shortcut we can apply the quantum-mechanical relation Eph = hf for the energies of
the photons making up the beams. The result is that the moving observer finds the total energy of the beams to be not E but
(E/2)(1 + v )γ + (E/2)(1 − v )γ = Eγ.
Both observers agree that the lantern had to use up some of the
energy stored in its fuel in order to make the two pulses. But
the moving observer says that in addition to this energy E, there
was a further energy E(γ − 1). Where could this energy have
come from? It must have come from the kinetic energy of the
lantern. The lantern’s velocity remained constant throughout the
MacArthur et al., Phys. Rev. Lett. 56 (1986) 282 (1986)
Section 4.2
experiment, so this decrease in kinetic energy seen by the moving
observer must have come from a decrease in the lantern’s inertial
mass — hence the title of Einstein’s paper, “Does the inertia of a
body depend upon its energy content?”
To figure out how much mass the lantern has lost, we have to
decide how we can even define mass in this new context. In
Newtonian mechanics, we had K = (1/2)mv 2 , and by the correspondence principle this must still hold in the low-velocity limit.
Expanding E(γ − 1) in a Taylor series, we find that it equals
E(v 2 /2) + . . ., and in the low-velocity limit this must be the same
as ∆K = (1/2)∆mv 2 , so ∆m = E. Reinserting factors of c to get
back to nonrelativistic units, we have E = ∆mc 2 .
4.2.4 A non-example: electric and magnetic fields
It is fairly easy to see that the electric and magnetic fields cannot
be the spacelike parts of two four-vectors. Consider the arrangement
shown in figure b/1. We have two infinite trains of moving charges
superimposed on the same line, and a single charge alongside the
line. Even though the line charges formed by the two trains are
moving in opposite directions, their currents don’t cancel. A negative charge moving to the left makes a current that goes to the right,
so in frame 1, the total current is twice that contributed by either
line charge.
b / Magnetism is a purely relativistic effect.
In frame 1 the charge densities of the two line charges cancel out,
and the electric field experienced by the lone charge is therefore zero.
Frame 2 shows what we’d see if we were observing all this from a
frame of reference moving along with the lone charge. Both line
charges are in motion in both frames of reference, but in frame
1, the line charges were moving at equal speeds, so their Lorentz
contractions were equal, and their charge densities canceled out. In
frame 2, however, their speeds are unequal. The positive charges
are moving more slowly than in frame 1, so in frame 2 they are
less contracted. The negative charges are moving more quickly, so
their contraction is greater now. Since the charge densities don’t
cancel, there is an electric field in frame 2, which points into the
wire, attracting the lone charge.
We appear to have a logical contradiction here, because an observer in frame 2 predicts that the charge will collide with the wire,
whereas in frame 1 it looks as though it should move with constant
velocity parallel to the wire. Experiments show that the charge does
collide with the wire, so to maintain the Lorentz-invariance of electromagnetism, we are forced to invent a new kind of interaction, one
between moving charges and other moving charges, which causes the
acceleration in frame 2. This is the magnetic interaction, and if we
hadn’t known about it already, we would have been forced to invent
it. That is, magnetism is a purely relativistic effect. The reason a
relativistic effect can be strong enough to stick a magnet to a re-
Chapter 4
frigerator is that it breaks the delicate cancellation of the extremely
large electrical interactions between electrically neutral objects.
Although the example shows that the electric and magnetic fields
do transform when we change from one frame to another, it is easy
to show that they do not transform as the spacelike parts of a relativistic four-vector. This is because transformation between frames
1 and 2 is along the axis parallel to the wire, but it affects the components of the fields perpendicular to the wire. The electromagnetic
field actually transforms as a rank-2 tensor.
4.2.5 The electromagnetic potential four-vector
An electromagnetic quantity that does transform as a four-vector
is the potential. On page 118, I mentioned the fact, which may or
may not already be familiar to you, that whereas the Newtonian
gravitational field’s polarization properties allow it to be described
using a single scalar potential φ or a single vector field g = −∇φ,
the pair of electromagnetic fields (E, B) needs a pair of potentials,
Φ and A. It’s easy to see that Φ can’t be a Lorentz scalar. Electric charge q is a scalar, so if Φ were a scalar as well, then the
product qΦ would be a scalar. But this is equal to the energy of
the charged particle, which is only the timelike component of the
energy-momentum four-vector, and therefore not a Lorentz scaler
itself. This is a contradiction, so Φ is not a scalar.
c / The charged particle follows
a trajectory that extremizes
fb dx b compared to other
nearby trajectories.
Relativistically, the trajectory should be
understood as a world-line in
3+1-dimensional spacetime.
To see how to fit Φ into relativity, consider the nonrelativistic
quantum mechanical relation qΦ = hf for a charged particle in a
potential Φ. Since f is the timelike component of a four-vector in
relativity, we need Φ to be the timelike component of some four
vector, Ab . For the spacelike part of this four-vector, let’s write A,
so that Ab = (Φ, A). We can see by the following argument that
this mysterious A must have something to do with the magnetic
Consider the example of figure c from a quantum-mechanical
point of view. The charged particle q has wave properties, but let’s
say that it can be well approximated in this example as following a
specific trajectory. This is like the ray approximation to wave optics.
A light ray in classical optics follows Fermat’s principle, also known
as the principle of least time, which states that the ray’s path from
point A to point B is one that extremizes the optical path length
(essentially the number of oscillations). The reason for this is that
the ray approximation is only an approximation. The ray actually
has some width, which we can visualize as a bundle of neighboring
trajectories. Only if the trajectory follows Fermat’s principle will
the interference among the neighboring paths be constructive. The
classical optical path length is found by integrating k · ds, where k
is the wavenumber. To make this relativistic, we need to use the
frequency four-vector to form fb dxb , which can also be expressed
as fb v b dτ = γ(f − k · v)dτ . If the charge is at rest and there
Section 4.2
d / The magnetic field (top)
and vector potential (bottom) of
a solenoid. The lower diagram is
in the plane cutting through the
waist of the solenoid, as indicated
by the dashed line in the upper
diagram. For an infinite solenoid,
the magnetic field is uniform
on the inside and zero on the
outside, while the vector potential
is proportional to r on the inside
and to 1/r on the outside.
are no magnetic fields, then the quantity in parentheses is f =
E/h = (q/h)Φ. The correct relativistic generalization is clearly
fb = (q/h)Ab .
Since Ab ’s spacelike part, A, results in the velocity-dependent
effects, we conclude that A is a kind of potential that relates to the
magnetic field, in the same way that the potential Φ relates to the
electric field. A is known as the vector potential, and the relation
between the potentials and the fields is
E = −∇Φ −
B = ∇A
An excellent discussion of the vector potential from a purely classical
point of view is given in the classic Feynman Lectures.10 Figure d
shows an example.
4.3 The tensor transformation laws
We may wish to represent a vector in more than one coordinate
system, and to convert back and forth between the two representations. In general relativity, the transformation of the coordinates
need not be linear, as in the Lorentz transformations; it can be any
smooth, one-to-one function. For simplicity, however, we start by
considering the one-dimensional case, and by assuming the coordinates are related in an affine manner, x0µ = axµ + b. The addition
of the constant b is merely a change in the choice of origin, so it
has no effect on the components of the vector, but the dilation by
the factor a gives a change in scale, which results in v 0µ = av µ for a
contravariant vector. In the special case where v is an infinitesimal
displacement, this is consistent with the result found by implicit differentiation of the coordinate transformation. For a contravariant
vector, vµ0 = a1 vµ . Generalizing to more than one dimension, and to
a possibly nonlinear transformation, we have
vµ0 = vκ 0µ
v 0µ = v κ
Note the inversion of the partial derivative in one equation compared
to the other. Because these equations describe a change from one
coordinate system to another, they clearly depend on the coordinate
system, so we use Greek indices rather than the Latin ones that
would indicate a coordinate-independent equation. Note that the
letter µ in these equations always appears as an index referring to
the new coordinates, κ to the old ones. For this reason, we can get
The Feynman Lectures on Physics, Feynman, Leighton, and Sands, Addison
Wesley Longman, 1970
Chapter 4
away with dropping the primes and writing, e.g., v µ = v κ ∂x0µ /∂xκ
rather than v 0 , counting on context to show that v µ is the vector
expressed in the new coordinates, v κ in the old ones. This becomes
especially natural if we start working in a specific coordinate system
where the coordinates have names. For example, if we transform
from coordinates (t, x, y, z) to (a, b, c, d), then it is clear that v t is
expressed in one system and v c in the other.
Self-check: Recall that the gauge transformations allowed in general relativity are not just any coordinate transformations; they
must be (1) smooth and (2) one-to-one. Relate both of these requirements to the features of the vector transformation laws above.
In equation [2], µ appears as a subscript on the left side of the
equation, but as a superscript on the right. This would appear
to violate our rules of notation, but the interpretation here is that
in expressions of the form ∂/∂xi and ∂/∂xi , the superscripts and
subscripts should be understood as being turned upside-down. Similarly, [1] appears to have the implied sum over κ written ungrammatically, with both κ’s appearing as superscripts. Normally we
only have implied sums in which the index appears once as a superscript and once as a subscript. With our new rule for interpreting
indices on the bottom of derivatives, the implied sum is seen to be
written correctly. This rule is similar to the one for analyzing the
units of derivatives written in Leibniz notation, with, e.g., d2 x/dt2
having units of meters per second squared. That is, the flipping of
the indices like this is required for consistency so that everything
will work out properly when we change our units of measurement,
causing all our vector components to be rescaled.
A quantity v that transforms according to [1] or [2] is referred
to as a rank-1 tensor, which is the same thing as a vector.
The identity transformation
Example: 17
In the case of the identity transformation x = x , equation [1]
clearly gives v 0 = v , since all the mixed partial derivatives ∂x 0µ /∂x κ
with µ 6= κ are zero, and all the derivatives for κ = µ equal 1.
In equation [2], it is tempting to write
∂x κ
= ∂x 0µ
∂x κ
but this would give infinite results for the mixed terms! Only in the
case of functions of a single variable is it possible to flip derivatives in this way; it doesn’t work for partial derivatives. To evaluate these partial derivatives, we have to invert the transformation
(which in this example is trivial to accomplish) and then take the
partial derivatives.
The metric is a rank-2 tensor, and transforms analogously:
gµν = gκλ
∂xκ ∂xλ
∂x0µ ∂x0ν
Section 4.3
The tensor transformation laws
(writing g rather than g 0 on the left, because context makes the
distinction clear).
Self-check: Write the similar expressions for g µν , gνµ , and gµν ,
which are entirely determined by the grammatical rules for writing
superscripts and subscripts. Interpret the case of a rank-0 tensor.
An accelerated coordinate system?
Example: 18
Let’s see the effect on Lorentzian metric g of the transformation
t0 = t
x 0 = x + at 2
The inverse transformation is
t = t0
x = x 0 − at 02
The tensor transformation law gives
gt00 t 0 = 1 − (at 0 )2
gx0 0 x 0 = −1
gx0 0 t 0 = −at 0
Clearly something bad happens at at 0 = ±1, when the relative
velocity surpasses the speed of light: the t 0 t 0 component of the
metric vanishes and then reverses its sign. This would be physically unreasonable if we viewed this as a transformation from observer A’s Lorentzian frame into the accelerating reference frame
of observer B aboard a spaceship who feels a constant acceleration. Several things prevent such an interpretation: (1) B cannot
exceed the speed of light. (2) Even before B gets to the speed
of light, the coordinate t 0 cannot correspond to B’s proper time,
which is dilated. (3) Due to time dilation, A and B do not agree
on the rate at which B is accelerating. If B measures her own
acceleration to be a0 , A will judge it to be a < a0 , with a → 0 as B
approaches the speed of light. There is nothing invalid about the
coordinate system (t 0 , x 0 ), but neither does it have any physically
interesting interpretation.
Physically meaningful constant acceleration
Example: 19
To make a more physically meaningful version of example 18, we
need to use the result of example 4 on page 126. The somewhat messy derivation of the coordinate transformation is given
by Semay.11 The result is
t = x+
sinh at
x = x+
cosh at
Chapter 4
Applying the tensor transformation law gives (problem 7, page
gt00 t 0 = (1 + ax 0 )2
gx0 0 x 0 = −1
Unlike the result of example 18, this one never misbehaves.
The closely related topic of a uniform gravitational field in general
relativity is considered in problem 6 on page 199.
Accurate timing signals
Example: 20
The relation between the potential A and the fields E and B
given on page 138 can be written in manifestly covariant form as
Fij = ∂[i Aj] , where F , called the electromagnetic tensor, is an antisymmetric rank-two tensor whose six independent components
correspond in a certain way with the components of the E and
B three-vectors. If F vanishes completely at a certain point in
spacetime, then the linear form of the tensor transformation laws
guarantees that it will vanish in all coordinate systems, not just
one. The GPS system takes advantage of this fact in the transmission of timing signals from the satellites to the users. The
electromagnetic wave is modulated so that the bits it transmits
are represented by phase reversals of the wave. At these phase
reversals, F vanishes, and this vanishing holds true regardless of
the motion of the user’s unit or its position in the earth’s gravitational field. Cf. problem 17 on p. 157.
Momentum wants a lower index
Example: 21
In example 5 on p. 48, we saw that once we arbitrarily chose to
write ruler measurements in Euclidean three-space as ∆x a rather
than ∆xa , it became natural to think of the Newtonian force threevector as “wanting” to be notated with a lower index. We can
do something similar with the momentum 3- or 4-vector. The
Lagrangian is a relativistic scalar, and in Lagrangian mechanics
momentum is defined by pa = ∂L/∂v a . The upper index in the
denominator on the right becomes a lower index on the left by
the same reasoning as was employed in the notation of the tensor transformation laws. Newton’s second law shows that this is
consistent with the result of example 5 on p. 48.
Section 4.3
The tensor transformation laws
4.4 Experimental tests
4.4.1 Universality of tensor behavior
The techniques developed in this chapter allow us to make a variety of new predictions that can be tested by experiment. In general,
the mathematical treatment of all observables in relativity as tensors means that all observables must obey the same transformation
laws. This is an extremely strict statement, because it requires that
a wide variety of physical systems show identical behavior. For example, we already mentioned on page 74 the 2007 Gravity Probe
B experiment (discussed in detail on pages 170 and 212), in which
four gyroscopes aboard a satellite were observed to precess due to
special- and general-relativistic effects. The gyroscopes were complicated electromechanical systems, but the predicted precession was
entirely independent of these complications. We argued that if two
different types of gyroscopes displayed different behaviors, then the
resulting discrepancy would allow us to map out some mysterious
vector field. This field would be a built-in characteristic of spacetime (not produced by any physical objects nearby), and since all
observables in general relativity are supposed to be tensors, the field
would have to transform as a tensor. Let’s say that this tensor was
of rank 1. Since the tensor transformation law is linear, a nonzero
tensor can never be transformed into a vanishing tensor in another
coordinate system. But by the equivalence principle, any special,
local property of spacetime can be made to vanish by transforming
into a free-falling frame of reference, in which the spacetime is has a
generic Lorentzian geometry. The mysterious new field should therefore vanish in such a frame. This is a contradiction, so we conclude
that different types of gyroscopes cannot differ in their behavior.
This is an example of a new way of stating the equivalence principle: there is no way to associate a preferred tensor field with spacetime.12
4.4.2 Speed of light differing from c
In a Lorentz invariant theory, we interpret c as a property of
the underlying spacetime, not of the particles that inhabit it. One
way in which Lorentz invariance could be violated would be if different types of particles had different maximum velocities. In 1997,
Coleman and Glashow suggested a sensitive test for such an effect.13
Assuming Lorentz invariance, a photon cannot decay into an
electron and a positron, γ → e+ + e− (example 12, page 131).
Suppose, however, that material particles have a maximum speed
cm = 1, while photons have a maximum speed cp > 1. Then the photon’s momentum four-vector, (E, E/cp ) is timelike, so a frame does
This statement of the equivalence principle, along with the others we have
encountered, is summarized in the back of the book on page 389.
Chapter 4
exist in which its three-momentum is zero. The detection of cosmicray gammas from distant sources with energies on the order of 10
TeV puts an upper limit on the decay rate, implying cp − 1 . 10−15 .
An even more stringent limit can be put on the possibility of
cp < 1. When a charged particle moves through a medium at a speed
higher than the speed of light in the medium, Cerenkov radiation
results. If cp is less than 1, then Cerenkov radiation could be emitted
by high-energy charged particles in a vacuum, and the particles
would rapidly lose energy. The observation of cosmic-ray protons
with energies ∼ 108 TeV requires cp − 1 & −10−23 .
4.4.3 Degenerate matter
The straightforward properties of the momentum four-vector
have surprisingly far-reaching implications for matter subject to extreme pressure, as in a star that uses up all its fuel for nuclear fusion
and collapses. These implications were initially considered too exotic to be taken seriously by astronomers. For historical perspective,
consider that in 1916, when Einstein published the theory of general relativity, the Milky Way was believed to constitute the entire
universe; the “spiral nebulae” were believed to be inside it, rather
than being similar objects exterior to it. The only types of stars
whose structure was understood even vaguely were those that were
roughly analogous to our own sun. (It was not known that nuclear
fusion was their source of energy.) The term “white dwarf” had not
been invented, and neutron stars were unknown.
An ordinary, smallish star such as our own sun has enough hydrogen to sustain fusion reactions for billions of years, maintaining
an equilibrium between its gravity and the pressure of its gases.
When the hydrogen is used up, it has to begin fusing heavier elements. This leads to a period of relatively rapid fluctuations in
structure. Nuclear fusion proceeds up until the formation of elements as heavy as oxygen (Z = 8), but the temperatures are not
high enough to overcome the strong electrical repulsion of these nuclei to create even heavier ones. Some matter is blown off, but finally
nuclear reactions cease and the star collapses under the pull of its
own gravity.
To understand what happens in such a collapse, we have to understand the behavior of gases under very high pressures. In general, a surface area A within a gas is subject to collisions in a time t
from the n particles occupying the volume V = Avt, where v is the
typical velocity of the particles. The resulting pressure is given by
P ∼ npv/V , where p is the typical momentum.
Nondegenerate gas: In an ordinary gas such as air, the particles are nonrelativistic, so v = p/m, and the thermal energy
per particle is p2 /2m ∼ kT , so the pressure is P ∼ nkT /V .
Section 4.4
Experimental tests
Nonrelativistic, degenerate gas: When a fermionic gas is subject to extreme pressure, the dominant effects creating pressure are quantum-mechanical. Because of the Pauli exclusion principle, the volume available to each particle is ∼ V /n,
so its wavelength is no more than ∼ (V /n)1/3 , leading to
p = h/λ ∼ h(n/V )1/3 . If the speeds of the particles are still
nonrelativistic, then v = p/m still holds, so the pressure becomes P ∼ (h2 /m)(n/V )5/3 .
Relativistic, degenerate gas: If the compression is strong enough
to cause highly relativistic motion for the particles, then v ≈ c,
and the result is P ∼ hc(n/V )4/3 .
As a star with the mass of our sun collapses, it reaches a point
at which the electrons begin to behave as a degenerate gas, and
the collapse stops. The resulting object is called a white dwarf. A
white dwarf should be an extremely compact body, about the size
of the Earth. Because of its small surface area, it should emit very
little light. In 1910, before the theoretical predictions had been
made, Russell, Pickering, and Fleming discovered that 40 Eridani B
had these characteristics. Russell recalled: “I knew enough about
it, even in these paleozoic days, to realize at once that there was
an extreme inconsistency between what we would then have called
‘possible’ values of the surface brightness and density. I must have
shown that I was not only puzzled but crestfallen, at this exception
to what looked like a very pretty rule of stellar characteristics; but
Pickering smiled upon me, and said: ‘It is just these exceptions
that lead to an advance in our knowledge,’ and so the white dwarfs
entered the realm of study!”
S. Chandrasekhar showed in that 1930’s that there was an upper
limit to the mass of a white dwarf. We will recapitulate his calculation briefly in condensed order-of-magnitude form. The pressure
at the core of the star is P ∼ ρgr ∼ GM 2 /r4 , where M is the total
mass of the star. The star contains roughly equal numbers of neutrons, protons, and electrons, so M = Knm, where m is the mass of
the electron, n is the number of electrons, and K ≈ 4000. For stars
near the limit, the electrons are relativistic. Setting the pressure at
the core equal to the degeneracy pressure of a relativistic gas, we
find that the Chandrasekhar limit is ∼ (hc/G)3/2 (Km)−2 = 6M .
A less sloppy calculation gives something more like 1.4M . The selfconsistency of this solution is investigated in homework problem 15
on page 157.
a / Subrahmanyan
drasekhar (1910-1995)
Chapter 4
What happens to a star whose mass is above the Chandrasekhar
limit? As nuclear fusion reactions flicker out, the core of the star becomes a white dwarf, but once fusion ceases completely this cannot
be an equilibrium state. Now consider the nuclear reactions
n → p + e− + ν̄
p + e− → n + ν
which happen due to the weak nuclear force. The first of these releases 0.8 MeV, and has a half-life of 14 minutes. This explains
why free neutrons are not observed in significant numbers in our
universe, e.g., in cosmic rays. The second reaction requires an input
of 0.8 MeV of energy, so a free hydrogen atom is stable. The white
dwarf contains fairly heavy nuclei, not individual protons, but similar considerations would seem to apply. A nucleus can absorb an
electron and convert a proton into a neutron, and in this context the
process is called electron capture. Ordinarily this process will only
occur if the nucleus is neutron-deficient; once it reaches a neutronto-proton ratio that optimizes its binding energy, neutron capture
cannot proceed without a source of energy to make the reaction go.
In the environment of a white dwarf, however, there is such a source.
The annihilation of an electron opens up a hole in the “Fermi sea.”
There is now an state into which another electron is allowed to drop
without violating the exclusion principle, and the effect cascades
upward. In a star with a mass above the Chandrasekhar limit, this
process runs to completion, with every proton being converted into a
neutron. The result is a neutron star, which is essentially an atomic
nucleus (with Z = 0) with the mass of a star!
Observational evidence for the existence of neutron stars came
in 1967 with the detection by Bell and Hewish at Cambridge of a
mysterious radio signal with a period of 1.3373011 seconds. The signal’s observability was synchronized with the rotation of the earth
relative to the stars, rather than with legal clock time or the earth’s
rotation relative to the sun. This led to the conclusion that its origin
was in space rather than on earth, and Bell and Hewish originally
dubbed it LGM-1 for “little green men.” The discovery of a second
signal, from a different direction in the sky, convinced them that it
was not actually an artificial signal being generated by aliens. Bell
published the observation as an appendix to her PhD thesis, and
it was soon interpreted as a signal from a neutron star. Neutron
stars can be highly magnetized, and because of this magnetization
they may emit a directional beam of electromagnetic radiation that
sweeps across the sky once per rotational period — the “lighthouse
effect.” If the earth lies in the plane of the beam, a periodic signal
can be detected, and the star is referred to as a pulsar. It is fairly
easy to see that the short period of rotation makes it difficult to
explain a pulsar as any kind of less exotic rotating object. In the
approximation of Newtonian mechanics,
a spherical body of density
ρ, rotating with a period T = 3π/Gρ, has zero apparent gravity
at its equator, since gravity is just strong enough to accelerate an
object so that it follows a circular trajectory above a fixed point on
Section 4.4
Experimental tests
the surface (problem 14). In reality, astronomical bodies of planetary size and√greater are held together by their own gravity, so we
have T & 1/ Gρ for any body that does not fly apart spontaneously
due to its own rotation. In the case of the Bell-Hewish pulsar, this
implies ρ & 1010 kg/m3 , which is far larger than the density of normal matter, and also 10-100 times greater than the typical density
of a white dwarf near the Chandrasekhar limit.
An upper limit on the mass of a neutron star can be found in a
manner entirely analogous to the calculation of the Chandrasekhar
limit. The only difference is that the mass of a neutron is much
greater than the mass of an electron, and the neutrons are the only
particles present, so there is no factor of K. Assuming the more
precise result of 1.4M for the Chandrasekhar limit rather than
our sloppy one, and ignoring the interaction of the neutrons via the
strong nuclear force, we can infer an upper limit on the mass of a
neutron star:
Kme 2
≈ 5M
The theoretical uncertainties in such an estimate are fairly large.
Tolman, Oppenheimer, and Volkoff originally estimated it in 1939
as 0.7M , whereas modern estimates are more in the range of 1.5
to 3M . These are significantly lower than our crude estimate of
5M , mainly because the attractive nature of the strong nuclear
force tends to pull the star toward collapse. Unambiguous results
are presently impossible because of uncertainties in extrapolating
the behavior of the strong force from the regime of ordinary nuclei,
where it has been relatively well parametrized, into the exotic environment of a neutron star, where the density is significantly different
and no protons are present. There are a variety of effects that may
be difficult to anticipate or to calculate. For example, Brown and
Bethe found in 199414 that it might be possible for the mass limit to
be drastically revised because of the process e− → K − +νe , which is
impossible in free space due to conservation of energy, but might be
possible in a neutron star. Observationally, nearly all neutron stars
seem to lie in a surprisingly small range of mass, between 1.3 and
1.45M , but in 2010 a neutron star with a mass of 1.97 ± .04 M
was discovered, ruling out most neutron-star models that included
exotic matter.15
For stars with masses above the Tolman-Oppenheimer-Volkoff
limit, theoretical predictions become even more speculative. A variety of bizarre objects has been proposed, including black stars,
gravastars, quark stars, boson stars, Q-balls, and electroweak stars.
H.A. Bethe and G.E. Brown, “Observational constraints on the maximum
neutron star mass,” Astrophys. J. 445 (1995) L129. G.E. Brown and H.A.
Bethe, “A Scenario for a Large Number of Low-Mass Black Holes in the Galaxy,”
Astrophys. J. 423 (1994) 659. Both papers are available at
Demorest et al.,
Chapter 4
It seems likely, however, both on theoretical and observational grounds,
that objects with masses of about 3 to 20 solar masses end up as
black holes; see section 6.3.3.
Section 4.4
Experimental tests
4.5 Conservation laws
4.5.1 No general conservation laws
a / Two
Some of the first tensors we discussed were mass and charge, both
rank-0 tensors, and the rank-1 momentum tensor, which contains
both the classical energy and the classical momentum. Physicists
originally decided that mass, charge, energy, and momentum were
interesting because these things were found to be conserved. This
makes it natural to ask how conservation laws can be formulated
in relativity. We’re used to stating conservation laws casually in
terms of the amount of something in the whole universe, e.g., that
classically the total amount of mass in the universe stays constant.
Relativity does allow us to make physical models of the universe as
a whole, so it seems as though we ought to be able to talk about
conservation laws in relativity.
We can’t.
First, how do we define “stays constant?” Simultaneity isn’t
well-defined, so we can’t just take two snapshots, call them initial
and final, and compare the total amount of, say, electric charge in
each snapshot. This difficulty isn’t insurmountable. As in figure
a, we can arbitrarily pick out three-dimensional spacelike surfaces
— one initial and one final — and integrate the charge over each
one. A law of conservation of charge would say that no matter what
spacelike surface we picked, the total charge on each would be the
b / We define a boundary
around a region whose charge
we want to measure.
c / This boundary cuts
sphere into equal parts.
Next there’s the issue that the integral might diverge, especially
if the universe was spatially infinite. For now, let’s assume a spatially finite universe. For simplicity, let’s assume that it has the
topology of a three-sphere (see section 8.2 for reassurance that this
isn’t physically unreasonable), and we can visualize it as a twosphere.
In the case of the momentum four-vector, what coordinate system would we express it in? In general, we do not even expect to
be able to define a smooth, well-behaved coordinate system that
covers the entire universe, and even if we did, it would not make
sense to add a vector expressed in that coordinate system at point
A to another vector from point B; the best we could do would be
to parallel-transport the vectors to one point and then add them,
but parallel transport is path-dependent. (Similar issues occur with
angular momentum.) For this reason, let’s restrict ourselves to the
easier case of a scalar, such as electric charge.
But now we’re in real trouble. How would we go about actually
measuring the total electric charge of the universe? The only way to
do it is to measure electric fields, and then apply Gauss’s law. This
requires us to single out some surface that we can integrate the flux
over, as in b. This would really be a two-dimensional surface on the
Chapter 4
three-sphere, but we can visualize it as a one-dimensional surface —
a closed curve — on the two-sphere. But now suppose this curve is
a great circle, c. If we measure a nonvanishing total flux across it,
how do we know where the charge is? It could be on either side.
The conclusion is that conservation laws only make sense in relativity under very special circumstances.16 We do not have anything
like over-arching, global principles of conservation. As an example
of the appropriate special circumstances, section 6.2.6, p. 216 shows
how to define conserved quantities, which behave like energy and
momentum, for the motion of a test particle in a particular metric
that has a certain symmetry. This is generalized on p. 246 to a
general, global conservation law corresponding to every continuous
symmetry of a spacetime.
4.5.2 Conservation of angular momentum and frame
Another special case where conservation laws work is that if
the spacetime we’re studying gets very flat at large distances from a
small system we’re studying, then we can define a far-away boundary
that surrounds the system, measure the flux through that boundary,
and find the system’s charge. For such asymptotically flat spacetimes, we can also get around the problems that crop up with conserved vectors, such as momentum. If the spacetime far away is
nearly flat, then parallel transport loses its path-dependence, so we
can unambiguously define a notion of parallel-transporting all the
contributions to the flux to one arbitrarily chosen point P and then
adding them. Asymptotic flatness also allows us to define an approximate notion of a global Lorentz frame, so that the choice of P
doesn’t matter.
As an example, figure d shows a jet of matter being ejected from
the galaxy M87 at ultrarelativistic fields. The blue color of the jet in
the visible-light image comes from synchrotron radiation, which is
the electromagnetic radiation emitted by relativistic charged particles accelerated by a magnetic field. The jet is believed to be coming
from a supermassive black hole at the center of M87. The emission
of the jet in a particular direction suggests that the black hole is not
spherically symmetric. It seems to have a particular axis associated
with it. How can this be? Our sun’s spherical symmetry is broken
by the existence of externally observable features such as sunspots
and the equatorial bulge, but the only information we can get about
a black hole comes from its external gravitational (and possibly electromagnetic) fields. It appears that something about the spacetime
metric surrounding this black hole breaks spherical symmetry, but
preserves symmetry about some preferred axis. What aspect of the
initial conditions in the formation of the hole could have determined
d / A relativistic jet.
For another argument leading to the same conclusion, see subsection 7.4.1,
p. 258.
Section 4.5
Conservation laws
such an axis? The most likely candidate is the angular momentum.
We are thus led to suspect that black holes can possess angular momentum, that angular momentum preserves information about their
formation, and that angular momentum is externally detectable via
its effect on the spacetime metric.
What would the form of such a metric be? Spherical coordinates
in flat spacetime give a metric like this:
ds2 = dt2 − dr2 − r2 dθ2 − r2 sin2 θdφ2
We’ll see in chapter 6 that for a non-rotating black hole, the metric
is of the form
ds2 = (. . .)dt2 − (. . .)dr2 − r2 dθ2 − r2 sin2 θdφ2
where (. . .) represents functions of r. In fact, there is nothing special about the metric of a black hole, at least far away; the same
external metric applies to any spherically symmetric, non-rotating
body, such as the moon. Now what about the metric of a rotating
body? We expect it to have the following properties:
1. It has terms that are odd under time-reversal, corresponding
to reversal of the body’s angular momentum.
2. Similarly, it has terms that are odd under reversal of the differential dφ of the azimuthal coordinate.
3. The metric should have axial symmetry, i.e., it should be independent of φ.
Restricting our attention to the equatorial plane θ = π/2, the simplest modification that has these three properties is to add a term
of the form
f (. . .)L dφ dt
where (. . .) again gives the r-dependence and L is a constant, interpreted as the angular momentum. A detailed treatment is beyond
the scope of this book, but solutions of this form to the relativistic
field equations were found by New Zealand-born physicist Roy Kerr
in 1963 at the University of Texas at Austin.
The astrophysical modeling of observations like figure d is complicated, but we can see in a simplified thought experiment that if
we want to determine the angular momentum of a rotating body
via its gravitational field, it will be difficult unless we use a measuring process that takes advantage of the asymptotic flatness of the
space. For example, suppose we send two beams of light past the
earth, in its equatorial plane, one on each side, and measure their
deflections, e. The deflections will be different, because the sign of
dφdt will be opposite for the two beams. But the entire notion of a
Chapter 4
“deflection” only makes sense if we have an asymptotically flat background, as indicated by the dashed tangent lines. Also, if spacetime
were not asymptotically flat in this example, then there might be
no unambiguous way to determine whether the asymmetry was due
to the earth’s rotation, to some external factor, or to some kind of
interaction between the earth and other bodies nearby.
It also turns out that a gyroscope in such a gravitational field
precesses. This effect, called frame dragging, was predicted by Lense
and Thirring in 1918, and was finally verified experimentally in 2008
by analysis of data from the Gravity Probe B experiment, to a precision of about 15%. The experiment was arranged so that the relatively strong geodetic effect (6.6 arc-seconds per year) and the much
weaker Lense-Thirring effect (.041 arc-sec/yr) produced precessions
in perpendicular directions. Again, the presence of an asymptotically flat background was involved, because the probe measured the
orientations of its gyroscopes relative to the guide star IM Pegasi.
4.6 Things that aren’t quite tensors
e / Two light rays travel in the
earth’s equatorial plane from A to
B. Due to frame-dragging, the ray
moving with the earth’s rotation
is deflected by a greater amount
than the one moving contrary to
it. As a result, the figure has an
asymmetric banana shape. Both
the deflection and its asymmetry
are greatly exaggerated.
This section can be skipped on a first reading.
4.6.1 Area, volume, and tensor densities
We’ve embarked on a program of redefining every possible physical quantity as a tensor, but so far we haven’t tackled area and
volume. Is there, for example, an area tensor in a locally Euclidean
plane? We are encouraged to hope that there is such a thing, because on p. 45 we saw that we could cook up a measure of area
with no other ingredients than the axioms of affine geometry. What
kind of tensor would it be? The notions of vector and scalar from
freshman mechanics are distinguished from one another by the fact
that one has a direction in space and the other does not. Therefore
we expect that area would be a scalar, i.e., a rank-0 tensor. But
this can’t be right, for the following reason. Under a rescaling of
coordinates by a factor k, area should change by a factor of k 2 . But
by the tensor transformation laws, a rank-0 tensor is supposed to
be invariant under a change of coordinates. We therefore conclude
that quantities like area and volume are not tensors.
In the language of ordinary vectors and scalars in Euclidean
three-space, one way to express area and volume is by using dot and
cross products. The area of the parallelogram spanned by u and v
is measured by the area vector u × v, and similarly the volume of
the parallelepiped formed by u, v, and w can be computed as the
scalar triple product u·(v ×w). Both of these quantities are defined
such that interchanging two of the inputs negates the output. In
differential geometry, we do have a scalar product, which is defined
by contracting the indices of two vectors, as in ua va . If we also had a
a tensorial cross product, we would be able to define area and volume
Section 4.6
f / Gravity Probe B verified
the existence of frame-dragging.
The rotational axis of the gyroscope precesses in two
perpendicular planes due to the
two separate effects: geodetic
and frame-dragging.
Things that aren’t quite tensors
tensors, so we conclude that there is no tensorial cross product, i.e.,
an operation that would multiply two rank-1 tensors to produce a
rank-1 tensor. Since one of the most important physical applications
of the cross product is to calculate the angular momentum L = r×p,
we find that angular momentum in relativity is either not a tensor
or not a rank-1 tensor.
When someone tells you that it’s impossible to do a seemingly
straightforward thing, the typical response is to look for a way to get
around the supposed limitation. In the case of a locally Euclidean
plane, what is to stop us from making a small, standard square, and
then sliding the square around to any desired location? If we have
some figure whose area we wish to measure, we can then dissect it
into squares of that size and count the number of squares.
a / A Möbius strip is not an
orientable surface.
There are two problems with this plan, neither of which is completely insurmountable. First, the area vector u×v is a vector, with
its orientation specified by the direction of the normal to the surface.
We need this
R orientation, for example, when we calculate the electric flux as E · dA. Figure a shows that we cannot always define
such an orientation in a consistent way. When the x − y coordinate
system is slid around the Möbius strip, it ends up with the opposite orientation. In general relativity, there is not any guarantee of
orientability in space — or even in time! But the vast majority of
spacetimes of physical interest are in fact orientable in every desired
way, and even for those that aren’t, orientability still holds in any
sufficiently small neighborhood.
The other problem is that area has the wrong scaling properties
to be a rank-0 tensor. We can get around this problem by being willing to discuss quantities that don’t transform exactly like tensors.
Often we only care about transformations, such as rotations and
translations, that don’t involve any scaling. We saw in section 2.2 on
p. 52 that Lorentz boosts also have the special property of preserving area in a space-time plane containing the boost. We therefore
define a tensor density as a quantity that transforms like a tensor
under rotations, translations, and boosts, but that rescales and possibly flips its sign under other types of coordinate transformations.
In general, the additional factor comes from the determinant d of
the matrix consisting of the partial derivatives ∂x0µ /∂xν (called the
Jacobian matrix). This determinant is raised to a power W , known
as the weight of the tensor density. Weight zero corresponds to the
case of a real tensor.
Area as a tensor density
Example: 22
In a Euclidean plane, making our rulers longer by a factor of k
causes the area measured in the new coordinates to decrease
by a factor of 1/k 2 . The rescaling is represented by a matrix of
partial derivatives that is simply k I, where I is the identity matrix. The determinant is k 2 . Therefore area is a tensor density of
Chapter 4
weight −1.
Mass density
Example: 23
A piece of aluminum foil as a certain number of milligrams per
square centimeter. Stretching rulers by k causes this number to
increase by k 2 , so this mass density has W = +1.
Length in the Euclidean plane
Example: 24
Let a line segment of unit length be parallel to the x axis in the Euclidean plane. Then the transformation (x, y) → (x/k, y) changes
its length to 1/k , which would lead us to imagine that length was
a tensor density with W = −1/2. But (x, y ) → (x, y/k) doesn’t
change the length at all, suggesting W = 0 (a pure tensor). The
result is that length is not a tensor or tensor density of any kind.
Generalizing to more than two dimensions, an m-dimensional volume embedded in an n-dimensional space is a tensor density
with W = −1 if and only if m = n; for m < n, the m-volume isn’t a
tensor density at all.
In Weyl’s apt characterization,17 tensors represent intensities,
while tensor densities measure quantity.
4.6.2 The Levi-Civita symbol
Although there is no tensorial vector cross product, we can define
a similar operation whose output is a tensor density. This is most
easily expressed in terms of the Levi-Civita symbol . (See p. 91 for
biographical information about Levi-Civita.)
In n dimensions, the Levi-Civita symbol has n indices. It is
defined so as to be totally asymmetric, in the sense that if any two of
the indices are interchanged, its sign flips. This is sufficient to define
the symbol completely except for an over-all scaling, which is fixed
by arbitrarily taking one of the nonvanishing elements and setting
it to +1. To see that this is enough to define completely, first note
that it must vanish when any index is repeated. For example, in
three dimensions labeled by κ, λ, and µ, κλλ is unchanged under
an interchange of the second and third indices, but it must also flip
its sign under this operation, which means that it must be zero. If
we arbitrarily fix κλµ = +1, then interchange of the second and
third indices gives κµλ = −1, and a further interchange of the
first and second yields µκλ = +1. Any permutation of the three
distinct indices can be reached from any other by a series of such
pairwise swaps, and the number of swaps is uniquely odd or even.18
In Cartesian coordinates in three dimensions, it is conventional to
choose xyz = +1 when x, y, and z form a right-handed spatial
coordinate system. In four dimensions, we take txyz = +1 when t
is future-timelike and (x, y, z) are right-handed.
Hermann Weyl, “Space-Time-Matter,” 1922, p. 109, available online at
For a proof, see the Wikipedia article “Parity of a permutation.”
Section 4.6
Things that aren’t quite tensors
In Euclidean three-space, in coordinates such that g = diag(1, 1, 1),
the vector cross product A = u × v, where we have in mind the interpretation of A as area, can be expressed as Aµ = µκλ uκ v λ .
Self-check: Check that this matches up with the more familiar
definition of the vector cross product.
Now suppose that we want to generalize to curved spaces, where
g cannot be constant. There are two ways to proceed.
Tensorial One is to let have the values 0 and ±1 at some arbitrarily
chosen point, in some arbitrarily chosen coordinate system, but to
let it transform like a tensor. Then Aµ = µκλ uκ v λ needs to be
modified, since the right-hand side is a tensor, and that would make
A a tensor, but if A is an area we don’t want it to transform like
a 1-tensor. We therefore need to revise the definition of area to
be Aµ = g −1/2 µκλ uκ v λ , where g is the determinant of the lowerindex form of the metric. The following two examples justify this
procedure in a locally Euclidean three-space.
Scaling coordinates with tensorial Example: 25
Then scaling of coordinates by k scales all the elements of the
metric by k −2 , g by k −6 , g −1/2 by k 3 , µκλ by k −3 , and u κ v λ by
k 2 . The result is to scale Aµ by k +3−3+2 = k 2 , which makes sense
if A is an area.
Oblique coordinates with tensorial Example: 26
In oblique coordinates (example 8, p. 103), the two basis vectors
have unit length but are at an angle φ 6= π/2 to one another. The
determinant of the metric is g = sin2 φ, so g = sin φ, which is
exactly the correction factor needed in order to get the right area
when u and v are the two basis vectors.
This procedure works more generally, the sole modification being
that in a space such as a locally Lorentzian one where g < 0 we need
to use −g as the correction factor rather than g.
Tensor-density The other option is to let have the same 0 and ±1 values at
all points. Then is clearly not a tensor, because it doesn’t scale
by a factor of k n when the coordinates are scaled by k; is a tensor
density with weight +1 for the upper-index version and −1 for the
lower-index one. The relation Aµ = µκλ uκ v λ gives an area that is
a tensor density, not a tensor, because A is not written in terms of
purely tensorial quantities. Scaling the coordinates by k leaves µκλ
unchanged, scales up uκ v λ by k 2 , and scales up the area by k 2 , as
Unfortunately, there is no consistency in the literature as to
whether should be a tensor or a tensor density. Some authors
Chapter 4
define both a tensor and a nontensor version, with notations like
and ˜, or19 0123 and [0123]. Others avoid writing the letter completely.20 The tensor-density version is convenient because we
always know that its value is 0 or ±1. The tensor version has the
advantage that it transforms as a tensor.
4.6.3 Spacetime volume
We saw on p. 54 that area in the 1 + 1-dimensional plane of flat
spacetime is preserved by a Lorentz boost. This makes sense because
when we express the area spanned by a parallelogram with edges p
and q as ab pa sb , all the indices have been contracted, leaving a
rank-0 tensor density. In 3 + 1 dimensions, we have the spacetime
volume V = abcd pa qb rc sd spanned by the paralellepiped with edges
p, q, r, and s. A typical situation in which this volume is nonzero
would be that in which one of the vectors is timelike and the other
three spacelike. Let the timelike one be p. Assume |p| = 1, since
an example with |p| 6= 1 can be reduced to this by scaling. Then
p can be interpreted as the velocity vector of some observer, and
V as the spatial volume that the observer says is spanned by the
3-paralellepiped with edges q, r, and s.
4.6.4 Angular momentum
As discussed above, angular momentum cannot be a rank-1 tensor. One approach is to define a rank-2 angular momentum tensor
Lab = ra pb .
In a frame whose origin is instantaneously moving along with
a certain system’s center of mass at a certain time, the time-space
components of L vanish, and the components Lyz , Lzx , and Lxy
coincide in the nonrelativistic limit with the x, y, and z components
of the Newtonian angular momentum vector. We can also define
a three-dimensional object La = abc Lbc (with three-dimensional
tensor-density in the spatial dimensions) that doesn’t transform
like a tensor.
Misner, Thorne, and Wheeler
Hawking and Ellis
Section 4.6
Things that aren’t quite tensors
Describe the four-velocity of a photon.
. Solution, p. 370
The Large Hadron Collider is designed to accelerate protons
to energies of 7 TeV. Find 1 − v for such a proton.
. Solution, p. 370
Prove that a photon in a vacuum cannot absorb a photon.
(This is the reason that the ability of materials to absorb gammarays is strongly dependent on atomic number Z. The case of Z = 0
corresponds to the vacuum.)
(a) For an object moving in a circle at constant speed, the
dot product of the classical three-vectors v and a is zero. Give
an interpretation in terms of the work-kinetic energy theorem. (b)
In the case of relativistic four-vectors, v i ai = 0 for any world-line.
Give a similar interpretation. Hint: find the rate of change of the
four-velocity’s squared magnitude.
Starting from coordinates (t, x) having a Lorentzian metric
g, transform the metric tensor into reflected coordinates (t0 , x0 ) =
(t, −x), and verify that g 0 is the same as g.
Starting from coordinates (t, x) having a Lorentzian metric g,
transform the metric tensor into Lorentz-boosted coordinates (t0 , x0 ),
and verify that g 0 is the same as g.
Verify the transformation of the metric given in example 19
on page 140.
A skeptic claims that the Hafele-Keating experiment can only
be explained correctly by relativity in a frame in which the earth’s
axis is at rest. Prove mathematically that this is incorrect. Does it
matter whether the frame is inertial?
. Solution, p. 370
Assume the metric g = diag(+1, +1, +1). Which of the following correctly expresses the noncommutative property of ordinary
matrix multiplication?
Aij Bjk 6= Bjk Aij
Aij Bjk 6= Bi j Ajk
Example 10 on page 130 introduced the Dirac sea, whose
is implied by the two roots of the relativistic relation E =
± p2 + m2 . Prove that a Lorentz boost will never transform a
positive-energy state into a negative-energy state.
. Solution, p. 371
On page 133, we found the relativistic Doppler shift in 1+1
dimensions. Extend this to 3+1 dimensions, and check your result
against the one given by Einstein on page 358.
Chapter 4
. Solution, p. 371
Estimate the energy contained in the electric field of an
electron, if the electron’s radius is r. Classically (i.e., assuming
relativity but no quantum mechanics), this energy contributes to the
electron’s rest mass, so it must be less than the rest mass. Estimate
the resulting lower limit on r, which is known as the classical electron
. Solution, p. 371
For gamma-rays in the MeV range, the most frequent mode of
interaction with matter is Compton scattering, in which the photon
is scattered by an electron without being absorbed. Only part of
the gamma’s energy is deposited, and the amount is related to the
angle of scattering. Use conservation of four-momentum to show
that in the case of scattering at 180 degrees, the scattered photon
has energy E 0 = E/(1+2E/m), where m is the mass of the electron.
Derive the equation T = 3π/Gρ given on page 146 for the
period of a rotating, spherical object that results in zero apparent
gravity at its surface.
Section 4.4.3 presented an estimate of the upper limit on the
mass of a white dwarf. Check the self-consistency of the solution
in the following respects: (1) Why is it valid to ignore the contribution of the nuclei to the degeneracy pressure? (2) Although the
electrons are ultrarelativistic, spacetime is approximated as being
flat. As suggested in example 15 on page 65, a reasonable order-ofmagnitude check on this result is that we should have M/r c2 /G.
The laws of physics in our universe imply that for bodies with
a certain range of masses, a neutron star is the unique equilibrium
state. Suppose we knew of the existence of neutron stars, but didn’t
know the mass of the neutron. Infer upper and lower bounds on the
mass of the neutron.
Example 20 on p. 141 briefly introduced the electromagnetic
potential four-vector Fij , and this implicitly defines the transformation properties of the electric and magnetic fields under a Lorentz
boost v. To lowest order in v, this transformation is given by
E0 ≈ E + v × B
B ≈B−v×E
I’m not a historian of science, but apparently ca. 1905 people like
Hertz believed that these were the exact transformations of the
field.21 Show that this can’t be the case, because performing two
such transformations in a row does not in general result in a transformation of the same form.
. Solution, p. 371
Montigny and Rousseaux,
We know of massive particles, whose velocity vectors always
lie inside the future light cone, and massless particles, whose velocities lie on it. In principle, we could have a third class of particles,
called tachyons, with spacelike velocity vectors. Tachyons would
have m2 < 0, i.e., their masses would have to be imaginary. Show
that it is possible to pick momentum four-vectors p1 and p2 for
a pair of tachyons such that p1 + p2 = 0. This implies that the
vacuum would be unstable with respect to spontaneous creation of
tachyon-antitachyon pairs.
Chapter 4
Chapter 5
General relativity describes gravitation as a curvature of spacetime,
with matter acting as the source of the curvature in the same way
that electric charge acts as the source of electric fields. Our goal is
to arrive at Einstein’s field equations, which relate the local intrinsic curvature to the locally ambient matter in the same way that
Gauss’s law relates the local divergence of the electric field to the
charge density. The locality of the equations is necessary because
relativity has no action at a distance; cause and effect propagate at
a maximum velocity of c(= 1).
The hard part is arriving at the right way of defining curvature.
We’ve already seen that it can be tricky to distinguish intrinsic
curvature, which is real, from extrinsic curvature, which can never
produce observable effects. E.g., example 4 on page 95 showed that
spheres have intrinsic curvature, while cylinders do not. The manifestly intrinsic tensor notation protects us from being misled in this
respect. If we can formulate a definition of curvature expressed using
only tensors that are expressed without reference to any preordained
coordinate system, then we know it is physically observable, and not
just a superficial feature of a particular model.
As an example, drop two rocks side by side, b. Their trajectories
are vertical, but on a (t, x) coordinate plot rendered in the Earth’s
frame of reference, they appear as parallel parabolas. The curvature of these parabolas is extrinsic. The Earth-fixed frame of reference is defined by an observer who is subject to non-gravitational
forces, and is therefore not a valid Lorentz frame. In a free-falling
Lorentz frame (t0 , x0 ), the two rocks are either motionless or moving
at constant velocity in straight lines. We can therefore see that the
curvature of world-lines in a particular coordinate system is not an
intrinsic measure of curvature; it can arise simply from the choice
of the coordinate system. What would indicate intrinsic curvature
would be, for example, if geodesics that were initially parallel were
to converge or diverge.
Nor is the metric a measure of intrinsic curvature. In example
19 on page 140, we found the metric for an accelerated observer to
gt0 0 t0 = (1 + ax0 )2
gx0 x0 = −1
where the primes indicate the accelerated observer’s frame. The fact
that the timelike element is not equal to −1 is not an indication of
a / The expected structure of
the field equations in general
b / Two rocks are dropped
side by side. The curvatures of
their world-lines are not intrinsic.
In a free-falling frame, both would
appear straight. If initially parallel
world-lines became non-parallel,
that would be evidence of intrinsic
intrinsic curvature. It arises only from the choice of the coordinates
(t0 , x0 ) defined by a frame tied to the accelerating rocket ship.
The fact that the above metric has nonvanishing derivatives, unlike a constant Lorentz metric, does indicate the presence of a gravitational field. However, a gravitational field is not the same thing
as intrinsic curvature. The gravitational field seen by an observer
aboard the ship is, by the equivalence principle, indistinguishable
from an acceleration, and indeed the Lorentzian observer in the
earth’s frame does describe it as arising from the ship’s acceleration, not from a gravitational field permeating all of space. Both
observers must agree that “I got plenty of nothin’ ” — that the
region of the universe to which they have access lacks any stars,
neutrinos, or clouds of dust. The observer aboard the ship must describe the gravitational field he detects as arising from some source
very far away, perhaps a hypothetical vast sheet of lead lying billions
of light-years aft of the ship’s deckplates. Such a hypothesis is fine,
but it is unrelated to the structure of our hoped-for field equation,
which is to be local in nature.
Not only does the metric tensor not represent the gravitational
field, but no tensor can represent it. By the equivalence principle, any gravitational field seen by observer A can be eliminated by
switching to the frame of a free-falling observer B who is instantaneously at rest with respect to A at a certain time. The structure of
the tensor transformation law guarantees that A and B will agree on
whether a given tensor is zero at the point in spacetime where they
pass by one another. Since they agree on all tensors, and disagree
on the gravitational field, the gravitational field cannot be a tensor.
We therefore conclude that a nonzero intrinsic curvature of the
type that is to be included in the Einstein field equations is not
encoded in any simple way in the metric or its first derivatives.
Since neither the metric nor its first derivatives indicate curvature,
we can reasonably conjecture that the curvature might be encoded
in its second derivatives.
5.1 Tidal curvature versus curvature caused
by local sources
a / Tidal forces disrupt comet
Chapter 5
A further complication is the need to distinguish tidal curvature from curvature caused by local sources. Figure a shows Comet
Shoemaker-Levy, broken up into a string of fragments by Jupiter’s
tidal forces shortly before its spectacular impact with the planet in
1994. Immediately after each fracture, the newly separated chunks
had almost zero velocity relative to one another, so once the comet
finished breaking up, the fragments’ world-lines were a sheaf of
nearly parallel lines separated by spatial distances of only 1 km.
These initially parallel geodesics then diverged, eventually fanning
out to span millions of kilometers.
If initially parallel lines lose their parallelism, that is clearly an
indication of intrinsic curvature. We call it a measure of sectional
curvature, because the loss of parallelism occurs within a particular
plane, in this case the (t, x) plane represented by figure b.
But this curvature was not caused by a local source lurking in
among the fragments. It was caused by a distant source: Jupiter.
We therefore see that the mere presence of sectional curvature is not
enough to demonstrate the existence of local sources. Even the sign
of the sectional curvature is not a reliable indication. Although this
example showed a divergence of initially parallel geodesics, referred
to as a negative curvature, it is also possible for tidal forces exerted
by distant masses to create positive curvature. For example, the
ocean tides on earth oscillate both above and below mean sea level,
As an example that really would indicate the presence of a local
source, we could release a cloud of test masses at rest in a spherical shell around the earth, and allow them to drop, d. We would
then have positive and equal sectional curvature in the t − x, t − y,
and t − z planes. Such an observation cannot be due to a distant
mass. It demonstrates an over-all contraction of the volume of an
initially parallel sheaf of geodesics, which can never be induced by
tidal forces. The earth’s oceans, for example, do not change their
total volume due to the tides, and this would be true even if the
oceans were a gas rather than an incompressible fluid. It is a unique
property of 1/r2 forces such as gravity that they conserve volume
in this way; this is essentially a restatement of Gauss’s law in a
b / Tidal forces cause the initially parallel world-lines of the
fragments to diverge. The spacetime occupied by the comet has
intrinsic curvature, but it is not
caused by any local mass; it is
caused by the distant mass of
c / The
field causes the Earth’s oceans to
be distorted into an ellipsoid. The
sign of the sectional curvature is
negative in the x − t plane, but
positive in the y − t plane.
5.2 The stress-energy tensor
In general, the curvature of spacetime will contain contributions
from both tidal forces and local sources, superimposed on one another. To develop the right formulation for the Einstein field equations, we need to eliminate the tidal part. Roughly speaking, we
will do this by averaging the sectional curvature over all three of the
planes t−x, t−y, and t−z, giving a measure of curvature called the
Ricci curvature. The “roughly speaking” is because such a prescription would treat the time and space coordinates in an extremely
asymmetric manner, which would violate local Lorentz invariance.
To get an idea of how this would work, let’s compare with the
Newtonian case, where there really is an asymmetry between the
treatment of time and space. In the Cartan curved-spacetime theory of Newtonian gravity (page 41), the field equation has a kind of
scalar Ricci curvature on one side, and on the other side is the density of mass, which is also a scalar. In relativity, however, the source
Section 5.2
d / A cloud of test masses is
released at rest in a spherical
shell around the earth, shown
here as a circle because the z
axis is omitted. The volume of
the shell contracts over time,
which demonstrates that the
local curvature of spacetime is
generated by a local source —
the earth — rather than some
distant one.
The stress-energy tensor
a / This curve has no intrinsic curvature.
b / A surveyor on a mountaintop
uses a heliotrope.
term in the equation clearly cannot be the scalar mass density. We
know that mass and energy are equivalent in relativity, so for example the curvature of spacetime around the earth depends not just
on the mass of its atoms but also on all the other forms of energy it
contains, such as thermal energy and electromagnetic and nuclear
binding energy. Can the source term in the Einstein field equations
therefore be the mass-energy E? No, because E is merely the timelike component of a particle’s momentum four-vector. To single it
out would violate Lorentz invariance just as much as an asymmetric
treatment of time and space in constructing a Ricci measure of curvature. To get a properly Lorentz invariant theory, we need to find a
way to formulate everything in terms of tensor equations that make
no explicit reference to coordinates. The proper generalization of
the Newtonian mass density in relativity is the stress-energy tensor
T ij , whose 16 elements measure the local density of mass-energy
and momentum, and also the rate of transport of these quantities
in various directions. If we happen to be able to find a frame of
reference in which the local matter is all at rest, then T tt represents
the mass density. The reason for the word “stress” in the name is
that, for example, the flux of x-momentum in the x direction is a
measure of pressure.
For the purposes of the present discussion, it’s not necessary to
introduce the explicit definition of T ; the point is merely that we
should expect the Einstein field equations to be tensor equations,
which tells us that the definition of curvature we’re seeking clearly
has to be a rank-2 tensor, not a scalar. The implications in fourdimensional spacetime are fairly complex. We’ll end up with a rank4 tensor that measures the sectional curvature, and a rank-2 Ricci
tensor derived from it that averages away the tidal effects. The
Einstein field equations then relate the Ricci tensor to the energymomentum tensor in a certain way. The stress-energy tensor is
discussed further in section 8.1.2 on page 263.
5.3 Curvature in two spacelike dimensions
c / A map of a triangulation
survey such as the one Gauss
carried out. By measuring the
interior angles of the triangles,
one can determine not just the
two-dimensional projection of
the grid but its complete threedimensional form, including both
the curvature of the earth (note
the curvature of the lines of latitude) and the height of features
above and below sea level.
Chapter 5
Since the curvature tensors in 3+1 dimensions are complicated, let’s
start by considering lower dimensions. In one dimension, a, there
is no such thing as intrinsic curvature. This is because curvature
describes the failure of parallelism to behave as in E5, but there is
no notion of parallelism in one dimension.
The lowest interesting dimension is therefore two, and this case
was studied by Carl Friedrich Gauss in the early nineteenth century.
Gauss ran a geodesic survey of the state of Hanover, inventing an
optical surveying instrument called a heliotrope that in effect was
used to cover the Earth’s surface with a triangular mesh of light
rays. If one of the mesh points lies, for example, at the peak of a
mountain, then the sum Σθ of the angles of the vertices meeting at
that point will be less than 2π, in contradiction to Euclid. Although
the light rays do travel through the air above the dirt, we can think
of them as approximations to geodesics painted directly on the dirt,
which would be intrinsic rather than extrinsic. The angular defect
around a vertex now vanishes, because the space is locally Euclidean,
but we now pick up a different kind of angular defect, which is that
the interior angles of a triangle no longer add up to the Euclidean
value of π.
A polygonal survey of a soccer ball
Example: 1
Figure d applies similar ideas to a soccer ball, the only difference
being the use of pentagons and hexagons rather than triangles.
In d/1, the survey is extrinsic, because the lines pass below the
surface of the sphere. The curvature is detectable because the
angles at each vertex add up to 120 + 120 + 110 = 350 degrees,
giving an angular defect of 10 degrees.
In d/2, the lines have been projected to form arcs of great circles
on the surface of the sphere. Because the space is locally Euclidean, the sum of the angles at a vertex has its Euclidean value
of 360 degrees. The curvature can be detected, however, because the sum of the internal angles of a polygon is greater than
the Euclidean value. For example, each spherical hexagon gives
a sum of 6 × 124.31 degrees, rather than the Euclidean 6 × 120.
The angular defect of 6 × 4.31 degrees is an intrinsic measure of
d / Example 1.
Angular defect on the earth’s surface
Example: 2
Divide the Earth’s northern hemisphere into four octants, with
their boundaries running through the north pole. These octants
have sides that are geodesics, so they are equilateral triangles.
Assuming Euclidean geometry, the interior angles of an equilateral triangle are each equal to 60 degrees, and, as with any triangle, they add up to 180 degrees. The octant-triangle in figure
e has angles that are each 90 degrees, and the sum is 270. This
shows that the Earth’s surface has intrinsic curvature.
This example suggests another way of measuring intrinsic curvature, in terms of the ratio C/r of the circumference of a circle to
its radius. In Euclidean geometry, this ratio equals 2π. Let ρ be
the radius of the Earth, and consider the equator to be a circle
centered on the north pole, so that its radius is the length of one
of the sides of the triangle in figure e, r = (π/2)ρ. (Don’t confuse
r , which is intrinsic, with ρ, the radius of the sphere, which is extrinsic and not equal to r .) Then the ratio C/r is equal to 4, which
is smaller than the Euclidean value of 2π.
e / Example 2.
Let = Σθ − π be the angular defect of a triangle, and for
concreteness let the triangle be in a space with an elliptic geometry,
so that it has constant curvature and can be modeled as a sphere of
Section 5.3
Curvature in two spacelike dimensions
radius ρ, with antipodal points identified.
Self-check: In elliptic geometry, what is the minimum possible
value of the quantity C/r discussed in example 2? How does this
differ from the case of spherical geometry?
We want a measure of curvature that is local, but if our space
is locally flat, we must have → 0 as the size of the triangles approaches zero. This is why Euclidean geometry is a good approximation for small-scale maps of the earth. The discrete nature of
the triangular mesh is just an artifact of the definition, so we want
a measure of curvature that, unlike , approaches some finite limit
as the scale of the triangles approaches zero. Should we expect this
scaling to go as ∝ ρ? ρ2 ? Let’s determine the scaling. First
we prove a classic lemma by Gauss, concerning a slightly different
version of the angular defect, for a single triangle.
f / Proof that the angular defect
of a triangle in elliptic geometry
is proportional to its area. Each
white circle represents the entire elliptic plane. The dashed
line at the edge is not really a
boundary; lines that go off the
edge simply wrap back around.
In the spherical model, the white
circle corresponds to one hemisphere, which is identified with
the opposite hemisphere.
Theorem: In elliptic geometry, the angular defect = α+β+γ−π
of a triangle is proportional to its area A.
Proof: By axiom E2, extend each side of the triangle to form a line,
figure f/1. Each pair of lines crosses at only one point (E1) and
divides the plane into two lunes with their four vertices touching at
this point, figure f/2. Of the six lunes, we focus on the three shaded
ones, which overlap the triangle. In each of these, the two interior
angles at the vertex are the same (Euclid I.15). The area of a lune
is proportional to its interior angle, as follows from dissection into
narrower lunes; since a lune with an interior angle of π covers the
entire area P of the plane, the constant of proportionality is P/π.
The sum of the areas of the three lunes is (P/π)(α + β + γ), but
these three areas also cover the entire plane, overlapping three times
on the given triangle, and therefore their sum also equals P + 2A.
Equating the two expressions leads to the desired result.
This calculation was purely intrinsic, because it made no use of
any model or coordinates. We can therefore construct a measure
of curvature that we can be assured is intrinsic, K = /A. This is
called the Gaussian curvature, and in elliptic geometry it is constant
rather than varying from point to point. In the model on a sphere
of radius ρ, we have K = 1/ρ2 .
Self-check: Verify the equation K = 1/ρ2 by considering a triangle covering one octant of the sphere, as in example 2.
g / Gaussian
nates on a sphere.
Chapter 5
It is useful to introduce normal or Gaussian normal coordinates, defined as follows. Through point O, construct perpendicular
geodesics, and define affine coordinates x and y along these. For
any point P off the axis, define coordinates by constructing the lines
through P that cross the axes perpendicularly. For P in a sufficiently small neighborhood of O, these lines exist and are uniquely
determined. Gaussian polar coordinates can be defined in a similar
Here are two useful interpretations of K.
1. The Gaussian curvature measures the failure of parallelism in
the following sense. Let line ` be constructed so that it crosses the
normal y axis at (0, dy) at an angle that differs from perpendicular
by the infinitesimal amount dα (figure h). Construct the line x0 =
dx, and let dα0 be the angle its perpendicular forms with `. Then1
the Gaussian curvature at O is
d2 α
h / 1.
Gaussian curvature
can be interpreted as the failure
of parallelism represented by
d2 α/dx dy .
where d2 α = dα0 − dα.
2. From a point P, emit a fan of rays at angles filling a certain
range θ of angles in Gaussian polar coordinates (figure i). Let the
arc length of this fan at r be L, which may not be equal to its
Euclidean value LE = rθ. Then2
K = −3 2
Let’s now generalize beyond elliptic geometry. Consider a space
modeled by a surface embedded in three dimensions, with geodesics
defined as curves of extremal length, i.e., the curves made by a piece
of string stretched taut across the surface. At a particular point
P, we can always pick a coordinate system (x, y, z) such that the
surface z = 12 k1 x2 + 12 k2 y 2 locally approximates the surface to the
level of precision needed in order to discuss curvature. The surface
is either paraboloidal or hyperboloidal (a saddle), depending on the
signs of k1 and k2 . We might naively think that k1 and k2 could be
independently determined by intrinsic measurements, but as we’ve
seen in example 4 on page 95, a cylinder is locally indistinguishable
from a Euclidean plane, so if one k is zero, the other k clearly cannot
be determined. In fact all that can be measured is the Gaussian
curvature, which equals the product k1 k2 . To see why this should
be true, first consider that any measure of curvature has units of
inverse distance squared, and the k’s have units of inverse distance.
The only possible intrinsic measures of curvature based on the k’s
are therefore k12 + k22 and k1 k2 . (We can’t have, for example, just k12 ,
because that would change under an extrinsic rotation about the z
axis.) Only k1 k2 vanishes on a cylinder, so it is the only possible
intrinsic curvature.
i / 2.
Gaussian curvature as
L 6= r θ.
Proof: Since any two lines cross in elliptic geometry, ` crosses the x axis. The
corollary then follows by application of the definition of the Gaussian curvature
to the right triangles formed by `, the x axis, and the lines at x = 0 and x = dx,
so that K = d/dA = d2 α/dxdy, where third powers of infinitesimals have been
In the spherical model, L = ρθ sin u, where u is the angle subtended at the
center of the sphere by an arc of length r. We then have L/LE = sin u/u, whose
second derivative with respect to u is −1/3. Since r = ρu, the second derivative
of the same quantity with respect to r equals −1/3ρ2 = −K/3.
Section 5.3
Curvature in two spacelike dimensions
Eating pizza
Example: 3
When people eat pizza by folding the slice lengthwise, they are
taking advantage of the intrinsic nature of the Gaussian curvature. Once k1 is fixed to a nonzero value, k2 can’t change without
varying K , so the slice can’t droop.
j / A triangle in a space with
negative curvature has angles
that add to less than π.
Elliptic and hyperbolic geometry
Example: 4
We’ve seen that figures behaving according to the axioms of elliptic geometry can be modeled on part of a sphere, which is a
surface of constant K > 0. The model can be made into global
one satisfying all the axioms if the appropriate topological properties are ensured by identifying antipodal points. A paraboloidal
surface z = k1 x 2 + k2 y 2 can be a good local approximation to
a sphere, but for points far from its apex, K varies significantly.
Elliptic geometry has no parallels; all lines meet if extended far
A space of constant negative curvature has a geometry called hyperbolic, and is of some interest because it appears to be the one
that describes the spatial dimensions of our universe on a cosmological scale. A hyperboloidal surface works locally as a model,
but its curvature is only approximately constant; the surface of
constant curvature is a horn-shaped one created by revolving a
mountain-shaped curve called a tractrix about its axis. The tractrix of revolution is not as satisfactory a model as the sphere is
for elliptic geometry, because lines are cut off at the cusp of the
horn. Hyperbolic geometry is richer in parallels than Euclidean
geometry; given a line ` and a point P not on `, there are infinitely
many lines through P that do not pass through `.
A flea on a football
Example: 5
We might imagine that a flea on the surface of an American football could determine by intrinsic, local measurements which direction to go in order to get to the nearest tip. This is impossible,
because the flea would have to determine a vector, and curvature
cannot be a vector, since z = 12 k1 x 2 + 21 k2 y 2 is invariant under the
parity inversion x → −x, y → −y . For similar reasons, a measure
of curvature can never have odd rank.
Without violating reflection symmetry, it is still conceivable that the
flea could determine the orientation of the tip-to-tip line running
through his position. Surprisingly, even this is impossible. The
flea can only measure the single number K , which carries no
information about directions in space.
k / A flea on the football cannot orient himself by intrinsic,
local measurements.
Chapter 5
The lightning rod
Example: 6
Suppose you have a pear-shaped conductor like the one in figure
l/1. Since the pear is a conductor, there are free charges everywhere inside it. Panels 1 and 2 of the figure show a computer simulation with 100 identical electric charges. In 1, the charges are
released at random positions inside the pear. Repulsion causes
them all to fly outward onto the surface and then settle down into
an orderly but nonuniform pattern.
We might not have been able to guess the pattern in advance, but
we can verify that some of its features make sense. For example,
charge A has more neighbors on the right than on the left, which
would tend to make it accelerate off to the left. But when we
look at the picture as a whole, it appears reasonable that this is
prevented by the larger number of more distant charges on its left
than on its right.
There also seems to be a pattern to the nonuniformity: the charges
collect more densely in areas like B, where the Gaussian curvature is large, and less densely in areas like C, where K is nearly
zero (slightly negative).
To understand the reason for this pattern, consider l/3. It’s straightforward to show that the density of charge σ on each sphere is
inversely proportional to its radius, or proportional to K 1/2 . Lord
Kelvin proved that on a conducting ellipsoid, the density of charge
is proportional to the distance from the center to the tangent
plane, which is equivalent3 to σ ∝ K 1/4 ; this result looks similar except for the different exponent. McAllister showed in 19904
that this K 1/4 behavior applies to a certain class of examples, but
it clearly can’t apply in all cases, since, for example, K could be
negative, or we could have a deep concavity, which would form
a Faraday cage. Problem 1 on p. 199 discusses the case of a
l / Example 6.
In 1 and 2,
charges that are visible on the
front surface of the conductor are
shown as solid dots; the others
would have to be seen through
the conductor, which we imagine
is semi-transparent.
Similar reasoning shows why Benjamin Franklin used a sharp tip
when he invented the lightning rod. The charged stormclouds induce positive and negative charges to move to opposite ends of
the rod. At the pointed upper end of the rod, the charge tends
to concentrate at the point, and this charge attracts the lightning.
The same effect can sometimes be seen when a scrap of aluminum foil is inadvertently put in a microwave oven. Modern experiments5 show that although a sharp tip is best at starting a
spark, a more moderate curve, like the right-hand tip of the pear
in this example, is better at successfully sustaining the spark for
long enough to connect a discharge to the clouds.
I W McAllister 1990 J. Phys. D: Appl. Phys. 23 359
Moore et al., Journal of Applied Meteorology 39 (1999) 593
Section 5.3
Curvature in two spacelike dimensions
5.4 Curvature tensors
The example of the flea suggests that if we want to express curvature
as a tensor, it should have even rank. Also, in a coordinate system
in which the coordinates have units of distance (they are not angles,
for instance, as in spherical coordinates), we expect that the units of
curvature will always be inverse distance squared. More elegantly,
we expect that under a uniform rescaling of coordinates by a factor
of µ, a curvature tensor should scale down by µ−2 .
Combining these two facts, we find that a curvature tensor should
have one of the forms Rab , Rabcd , . . . , i.e., the number of lower indices should be two greater than the number of upper indices. The
following definition has this property, and is equivalent to the earlier
definitions of the Gaussian curvature that were not written in tensor
Definition of the Riemann curvature tensor: Let dpc and dq d
be two infinitesimal vectors, and use them to form a quadrilateral
that is a good approximation to a parallelogram.6 Parallel-transport
vector v b all the way around the parallelogram. When it comes back
to its starting place, it has a new value v b → v b + dv b . Then the
Riemann curvature tensor is defined as the tensor that computes dv a
according to dv a = Rabcd v b dpc dq d . (There is no standardization in
the literature of the order of the indices.)
a / The definition of the Riemann
tensor. The vector v b changes
by dv b when parallel-transported
around the approximate parallelogram. (v b is drawn on a scale
that makes its length comparable
to the infinitesimals dpc , dq d , and
dv b ; in reality, its size would be
greater than theirs by an infinite
A symmetry of the Riemann tensor
Example: 7
If vectors dpc and dq d lie along the same line, then dv a must vanish, and interchanging dpc and dq d simply reverses the direction
of the circuit around the quadrilateral, giving dv a → −dv a . This
shows that R abcd must be antisymmetric under interchange of the
indices c and d, R abcd = −R abdc .
In local normal coordinates, the interpretation of the Riemann
tensor becomes particularly transparent. The constant-coordinate
lines are geodesics, so when the vector v b is transported along them,
it maintains a constant angle with respect to them. Any rotation
of the vector after it is brought around the perimeter of the quadrilateral can therefore be attributed to something that happens at
the vertices. In other words, it is simply a measure of the angular
defect. We can therefore see that the Riemann tensor is really just
a tensorial way of writing the Gaussian curvature K = d/dA.
In normal coordinates, the local geometry is nearly Cartesian,
and when we take the product of two vectors in an antisymmetric
manner, we are essentially measuring the area of the parallelogram
they span, as in the three-dimensional vector cross product. We can
therefore see that the Riemann tensor tells us something about the
amount of curvature contained within the infinitesimal area spanned
by dpc and dq d . A finite two-dimensional region can be broken
Chapter 5
Section 5.8 discusses the sense in which this approximation is good enough.
down into infinitesimal elements of area, and the Riemann tensor
integrated over them. The result is equal to the finite change ∆v b
in a vector transported around the whole boundary of the region.
Curvature tensors on a sphere
Example: 8
Let’s find the curvature tensors on a sphere of radius ρ.
Construct normal coordinates (x, y) with origin O, and let vectors dpc and dq d represent infinitesimal displacements along x
and y , forming a quadrilateral as described above. Then R xy xy
represents the change in the x direction that occurs in a vector
that is initially in the y direction. If the vector has unit magnitude, then R xy xy equals the angular deficit of the quadrilateral.
Comparing with the definition of the Gaussian curvature, we find
R xy xy = K = 1/ρ2 . Interchanging x and y , we find the same result
for R xy x . Thus although the Riemann tensor in two dimensions
has sixteen components, only these two are nonzero, and they
are equal to each other.
This result represents the defect in parallel transport around a
closed loop per unit area. Suppose we parallel-transport a vector
around an octant, as shown in figure b. The area of the octant
is (π/2)ρ2 , and multiplying it by the Riemann tensor, we find that
the defect in parallel transport is π/2, i.e., a right angle, as is also
evident from the figure.
The above treatment may be somewhat misleading in that it may
lead you to believe that there is a single coordinate system in
which the Riemann tensor is always constant. This is not the
case, since the calculation of the Riemann tensor was only valid
near the origin O of the normal coordinates. The character of
these coordinates becomes quite complicated far from O; we end
up with all our constant-x lines converging at north and south
poles of the sphere, and all the constant-y lines at east and west
b / The change in the vector
due to parallel transport around
the octant equals the integral
of the Riemann tensor over the
Angular coordinates (φ, θ) are more suitable as a large-scale description of the sphere. We can use the tensor transformation law
to find the Riemann tensor in these coordinates. If O, the origin
of the (x, y) coordinates, is at coordinates (φ, θ), then dx/dφ =
ρ sin θ and dy /dθ = ρ. The result is R φθφθ = R xy xy (dy /dθ)2 = 1
and R θφθφ = R xy x (dx/dφ)2 = sin2 θ. The variation in R θφθφ is
not due to any variation in the sphere’s intrinsic curvature; it represents the behavior of the coordinate system.
The Riemann tensor only measures curvature within a particular
plane, the one defined by dpc and dq d , so it is a kind of sectional curvature. Since we’re currently working in two dimensions, however,
there is only one plane, and no real distinction between sectional
curvature and Ricci curvature, which is the average of the sectional
curvature over all planes that include dq d : Rcd = Racad . The Ricci
Section 5.4
Curvature tensors
curvature in two spacelike dimensions, expressed in normal coordinates, is simply the diagonal matrix diag(K, K).
5.5 Some order-of-magnitude estimates
As a general proposition, calculating an order-of-magnitude estimate
of a physical effect requires an understanding of 50% of the physics,
while an exact calculation requires about 75%.7 We’ve reached
the point where it’s reasonable to attempt a variety of order-ofmagnitude estimates.
5.5.1 The geodetic effect
How could we confirm experimentally that parallel transport
around a closed path can cause a vector to rotate? The rotation
is related to the amount of spacetime curvature contained within
the path, so it would make sense to choose a loop going around
a gravitating body. The rotation is a purely relativistic effect, so
we expect it to be small. To make it easier to detect, we should
go around the loop many times, causing the effect to accumulate.
This is essentially a description of a body orbiting another body. A
gyroscope aboard the orbiting body is expected to precess. This is
known as the geodetic effect. In 1916, shortly after Einstein published the general theory of relativity, Willem de Sitter calculated
the effect on the earth-moon system. The effect was not directly
verified until the 1980’s, and the first high-precision measurement
was in 2007, from analysis of the results collected by the Gravity
Probe B satellite experiment. The probe carried four gyroscopes
made of quartz, which were the most perfect spheres ever manufactured, varying from sphericity by no more than about 40 atoms.
Let’s estimate the size of the effect. The first derivative of the
metric is, roughly, the gravitational field, whereas the second derivative has to do with curvature. The curvature of spacetime around
the earth should therefore vary as GM r−3 , where M is the earth’s
mass and G is the gravitational constant. The area enclosed by a
circular orbit is proportional to r2 , so we expect the geodetic effect
to vary as nGM/r, where n is the number of orbits. The angle of
precession is unitless, and the only way to make this result unitless
is to put in a factor of 1/c2 . In units with c = 1, this factor is unnecessary. In ordinary metric units, the 1/c2 makes sense, because
it causes the purely relativistic effect to come out to be small. The
result, up to unitless factors that we didn’t pretend to find, is
∆θ ∼
a / The geodetic effect as
measured by Gravity Probe B.
Chapter 5
c2 r
This statement is itself only a rough estimate. Anyone who has taught
physics knows that students will often calculate an effect exactly while not understanding the underlying physics at all.
We might also expect a Thomas precession. Like the spacetime
curvature effect, it would be proportional to nGM/c2 r. Since we’re
not worrying about unitless factors, we can just lump the Thomas
precession together with the effect already calculated.
The data for Gravity Probe B are r = re +(650 km) and n ≈ 5000
(orbiting once every 90 minutes for the 353-day duration of the
experiment), giving ∆θ ∼ 3 × 10−6 radians. Figure b shows the
actual results8 the four gyroscopes aboard the probe. The precession
was about 6 arc-seconds, or 3 × 10−5 radians. Our crude estimate
was on the right order of magnitude. The missing unitless factor on
the right-hand side of the equation above is 3π, which brings the two
results into fairly close quantitative agreement. The full derivation,
including the factor of 3π, is given on page 212.
b / Precession angle as a function of time as measured by the four gyroscopes aboard Gravity Probe B.
5.5.2 Deflection of light rays
In the discussion of the momentum four vector in section 4.2.2,
we saw that due to the equivalence principle, light must be affected
by gravity. There are two ways in which such an effect could occur.
Light can gain and lose momentum as it travels up and down in
a gravitational field, or its momentum vector can be deflected by
a transverse gravitational field. As an example of the latter, a ray
of starlight can be deflected by the sun’s gravity, causing the star’s
apparent position in the sky to be shifted. The detection of this
effect was one of the first experimental tests of general relativity.
Ordinarily the bright light from the sun would make it impossible
to accurately measure a star’s location on the celestial sphere, but
this problem was sidestepped by Arthur Eddington during an eclipse
of the sun in 1919.
Let’s estimate the size of this effect. We’ve already seen that
Section 5.5
Some order-of-magnitude estimates
c / One of the photos from Eddington’s observations of the
1919 eclipse. This is a photographic negative, so the circle that appears bright is actually
the dark face of the moon, and
the dark area is really the bright
corona of the sun. The stars,
marked by lines above and below them, appeared at positions
slightly different than their normal ones, indicating that their light
had been bent by the sun’s gravity
on its way to our planet.
the Riemann tensor is essentially just a tensorial way of writing
the Gaussian curvature K = d/dA. Suppose, for the sake of this
rough estimate, that the sun, earth, and star form a non-Euclidean
triangle with a right angle at the sun. Then the angular deflection
is the same as the angular defect of this triangle, and equals the
integral of the curvature over the interior of the triangle. Ignoring
unitless constants, this ends up being exactly the same calculation
as in section 5.5.1, and the result is ∼ GM/c2 r, where r is the
light ray’s distance of closest approach to the sun. The value of r
can’t be less than the radius of the sun, so the maximum size of the
effect is on the order of GM/c2 r, where M is the sun’s mass, and r
is its radius. We find ∼ 10−5 radians, or about a second of arc. To
measure a star’s position to within an arc second was well within
the state of the art in 1919, under good conditions in a comfortable
observatory. This observation, however, required that Eddington’s
team travel to the island of Principe, off the coast of West Africa.
The weather was cloudy, and only during the last 10 seconds of the
seven-minute eclipse did the sky clear enough to allow photographic
plates to be taken of the Hyades star cluster against the background
of the eclipse-darkened sky. The observed deflection was 1.6 seconds
of arc, in agreement with the relativistic prediction. The relativistic
prediction is derived on page 220.
5.6 The covariant derivative
In the preceding section we were able to estimate a nontrivial general
relativistic effect, the geodetic precession of the gyroscopes aboard
Gravity Probe B, up to a unitless constant 3π. Let’s think about
Chapter 5
what additional machinery would be needed in order to carry out
the calculation in detail, including the 3π.
First we would need to know the Einstein field equation, but in a
vacuum this is fairly straightforward: Rab = 0. Einstein posited this
equation based essentially on the considerations laid out in section
But just knowing that a certain tensor vanishes identically in the
space surrounding the earth clearly doesn’t tell us anything explicit
about the structure of the spacetime in that region. We want to
know the metric. As suggested at the beginning of the chapter, we
expect that the first derivatives of the metric will give a quantity
analogous to the gravitational field of Newtonian mechanics, but this
quantity will not be directly observable, and will not be a tensor.
The second derivatives of the metric are the ones that we expect to
relate to the Ricci tensor Rab .
5.6.1 The covariant derivative in electromagnetism
We’re talking blithely about derivatives, but it’s not obvious how
to define a derivative in the context of general relativity in such a
way that taking a derivative results in well-behaved tensor.
To see how this issue arises, let’s retreat to the more familiar
terrain of electromagnetism. In quantum mechanics, the phase of a
charged particle’s wavefunction is unobservable, so that for example
the transformation Ψ → −Ψ does not change the results of experiments. As a less trivial example, we can redefine the ground of our
electrical potential, Φ → Φ + δΦ, and this will add a constant onto
the energy of every electron in the universe, causing their phases to
oscillate at a greater rate due to the quantum-mechanical relation
E = hf . There are no observable consequences, however, because
what is observable is the phase of one electron relative to another,
as in a double-slit interference experiment. Since every electron has
been made to oscillate faster, the effect is simply like letting the conductor of an orchestra wave her baton more quickly; every musician
is still in step with every other musician. The rate of change of the
wavefunction, i.e., its derivative, has some built-in ambiguity.
For simplicity, let’s now restrict ourselves to spin-zero particles, since details of electrons’ polarization clearly won’t tell us
anything useful when we make the analogy with relativity. For a
spin-zero particle, the wavefunction is simply a complex number,
and there are no observable consequences arising from the transformation Ψ → Ψ0 = eiα Ψ, where α is a constant. The transformation
Φ → Φ − δΦ is also allowed, and it gives α(t) = (qδΦ/~)t, so that
the phase factor eiα(t) is a function of time t. Now from the point
of view of electromagnetism in the age of Maxwell, with the electric and magnetic fields imagined as playing their roles against a
background of Euclidean space and absolute time, the form of this
Section 5.6
with electrons. If we add an
arbitrary constant to the potential,
no observable changes result.
The wavelength is shortened, but
the relative phase of the two parts
of the waves stays the same.
b / Two
constant wavelengths, and a
third with a varying wavelength.
None of these are physically
distinguishable, provided that the
same variation in wavelength is
applied to all electrons in the
universe at any given point in
spacetime. There is not even
any unambiguous way to pick out
the third one as the one with a
varying wavelength. We could
choose a different gauge in which
the third wave was the only one
with a constant wavelength.
The covariant derivative
time-dependent phase factor is very special and symmetrical; it depends only on the absolute time variable. But to a relativist, there is
nothing very nice about this function at all, because there is nothing
special about a time coordinate. If we’re going to allow a function
of this form, then based on the coordinate-invariance of relativity, it
seems that we should probably allow α to be any function at all of
the spacetime coordinates. The proper generalization of Φ → Φ−δΦ
is now Ab → Ab − ∂b α, where Ab is the electromagnetic potential
four-vector (section 4.2.5, page 137).
Self-check: Suppose we said we would allow α to be a function
of t, but forbid it to depend on the spatial coordinates. Prove that
this would violate Lorentz invariance.
The transformation has no effect on the electromagnetic fields,
which are the direct observables. We can also verify that the change
of gauge will have no effect on observable behavior of charged particles. This is because the phase of a wavefunction can only be
determined relative to the phase of another particle’s wavefunction,
when they occupy the same point in space and, for example, interfere. Since the phase shift depends only on the location in spacetime,
there is no change in the relative phase.
But bad things will happen if we don’t make a corresponding
adjustment to the derivatives appearing in the Schrödinger equation.
These derivatives are essentially the momentum operators, and they
give different results when applied to Ψ0 than when applied to Ψ:
∂b Ψ → ∂b eiα Ψ
= eiα ∂b Ψ + i∂b α eiα Ψ
= ∂b + A0b − Ab Ψ0
To avoid getting incorrect results, we have to do the substitution
∂b → ∂b + ieAb , where the correction term compensates for the
change of gauge. We call the operator ∇ defined as
∇b = ∂b + ieAb
the covariant derivative. It gives the right answer regardless of a
change of gauge.
5.6.2 The covariant derivative in general relativity
Now consider how all of this plays out in the context of general relativity. The gauge transformations of general relativity are
arbitrary smooth changes of coordinates. One of the most basic
properties we could require of a derivative operator is that it must
give zero on a constant function. A constant scalar function remains
constant when expressed in a new coordinate system, but the same
is not true for a constant vector function, or for any tensor of higher
rank. This is because the change of coordinates changes the units
Chapter 5
in which the vector is measured, and if the change of coordinates is
nonlinear, the units vary from point to point.
Consider the one-dimensional case, in which a vector v a has only
one component, and the metric is also a single number, so that we
can omit the indices and simply write v and g. (We just have to
remember that v is really a covariant vector, even though we’re
leaving out the upper index.) If v is constant, its derivative dv/dx,
computed in the ordinary way without any correction term, is zero.
If we further assume that the coordinate x is a normal coordinate, so
that the metric is simply the constant g = 1, then zero is not just the
answer but the right answer. (The existence of a preferred, global
set of normal coordinates is a special feature of a one-dimensional
space, because there is no curvature in one dimension. In more than
one dimension, there will typically be no possible set of coordinates
in which the metric is constant, and normal coordinates only give a
metric that is approximately constant in the neighborhood around
a certain point. See figure g pn page 164 for an example of normal
coordinates on a sphere, which do not have a constant metric.)
Now suppose we transform into a new coordinate system X,
which is not normal. The metric G, expressed in this coordinate
system, is not constant. Applying the tensor transformation law,
we have V = v dX/dx, and differentiation with respect to X will
not give zero, because the factor dX/dx isn’t constant. This is the
wrong answer: V isn’t really varying, it just appears to vary because
G does.
c / These three rulers represent
three choices of coordinates. As
in figure b on page 173, switching
from one set of coordinates to
another has no effect on any
experimental observables. It is
merely a choice of gauge.
We want to add a correction term onto the derivative operator
d/dX, forming a covariant derivative operator ∇X that gives the
right answer. This correction term is easy to find if we consider
what the result ought to be when differentiating the metric itself.
In general, if a tensor appears to vary, it could vary either because
it really does vary or because the metric varies. If the metric itself
varies, it could be either because the metric really does vary or
. . . because the metric varies. In other words, there is no sensible
way to assign a nonzero covariant derivative to the metric itself, so
we must have ∇X G = 0. The required correction therefore consists
of replacing d/dX with
− G−1
Applying this to G gives zero. G is a second-rank contravariant
tensor. If we apply the same correction to the derivatives of other
second-rank contravariant tensors, we will get nonzero results, and
they will be the right nonzero results. For example, the covariant
derivative of the stress-energy tensor T (assuming such a thing could
have some physical significance in one dimension!) will be ∇X T =
dT /dX − G−1 (dG/dX)T .
∇X =
Physically, the correction term is a derivative of the metric, and
Section 5.6
The covariant derivative
we’ve already seen that the derivatives of the metric (1) are the closest thing we get in general relativity to the gravitational field, and
(2) are not tensors. In 1+1 dimensions, suppose we observe that a
free-falling rock has dV /dT = 9.8 m/s2 . This acceleration cannot be
a tensor, because we could make it vanish by changing from Earthfixed coordinates X to free-falling (normal, locally Lorentzian) coordinates x, and a tensor cannot be made to vanish by a change of
coordinates. According to a free-falling observer, the vector v isn’t
changing at all; it is only the variation in the Earth-fixed observer’s
metric G that makes it appear to change.
Mathematically, the form of the derivative is (1/y)dy/dx, which
is known as a logarithmic derivative, since it equals d(ln y)/dx. It
measures the multiplicative rate of change of y. For example, if
y scales up by a factor of k when x increases by 1 unit, then the
logarithmic derivative of y is ln k. The logarithmic derivative of
ecx is c. The logarithmic nature of the correction term to ∇X is a
good thing, because it lets us take changes of scale, which are multiplicative changes, and convert them to additive corrections to the
derivative operator. The additivity of the corrections is necessary if
the result of a covariant derivative is to be a tensor, since tensors
are additive creatures.
What about quantities that are not second-rank covariant tensors? Under a rescaling of contravariant coordinates by a factor of
k, covariant vectors scale by k −1 , and second-rank covariant tensors
by k −2 . The correction term should therefore be half as much for
covariant vectors,
∇X =
− G−1
and should have an opposite sign for contravariant vectors.
Generalizing the correction term to derivatives of vectors in more
than one dimension, we should have something of this form:
∇a v b = ∂a v b + Γbac v c
∇a vb = ∂a vb − Γcba vc
where Γbac , called the Christoffel symbol, does not transform like
a tensor, and involves derivatives of the metric. (“Christoffel” is
pronounced “Krist-AWful,” with the accent on the middle syllable.)
The explicit computation of the Christoffel symbols from the metric
is deferred until section 5.9, but the intervening sections 5.7 and 5.8
can be omitted on a first reading without loss of continuity.
An important gotcha is that when we evaluate a particular component of a covariant derivative such as ∇2 v 3 , it is possible for the
result to be nonzero even if the component v 3 vanishes identically.
This can be seen in example 5 on p. 273.
Chapter 5
Christoffel symbols on the globe
Example: 9
As a qualitative example, consider the geodesic airplane trajectory shown in figure d, from London to Mexico City. In physics
it is customary to work with the colatitude, θ, measured down
from the north pole, rather then the latitude, measured from the
equator. At P, over the North Atlantic, the plane’s colatitude has
a minimum. (We can see, without having to take it on faith from
the figure, that such a minimum must occur. The easiest way to
convince oneself of this is to consider a path that goes directly
over the pole, at θ = 0.)
At P, the plane’s velocity vector points directly west. At Q, over
New England, its velocity has a large component to the south.
Since the path is a geodesic and the plane has constant speed,
the velocity vector is simply being parallel-transported; the vector’s covariant derivative is zero. Since we have vθ = 0 at P, the
only way to explain the nonzero and positive value of ∂φ v θ is that
we have a nonzero and negative value of Γ θφφ .
By symmetry, we can infer that Γ θφφ must have a positive value
in the southern hemisphere, and must vanish at the equator.
Γ θφφ is computed in example 10 on page 188.
Symmetry also requires that this Christoffel symbol be independent of φ, and it must also be independent of the radius of the
d / Example 9.
Example 9 is in two spatial dimensions. In spacetime, Γ is essentially the gravitational field (see problem 6, p. 199), and early
papers in relativity essentially refer to it that way.9 This may feel
like a joyous reunion with our old friend from freshman mechanics,
g = 9.8 m/s. But our old friend has changed. In Newtonian mechanics, accelerations like g are frame-invariant (considering only
inertial frames, which are the only legitimate ones in that theory).
In general relativity they are frame-dependent, and as we saw on
page 176, the acceleration of gravity can be made to equal anything
we like, based on our choice of a frame of reference.
To compute the covariant derivative of a higher-rank tensor, we
just add more correction terms, e.g.,
∇a Ubc = ∂a Ubc − Γdba Udc − Γdca Ubd
∇a Ubc = ∂a Ubc − Γdba Udc + Γcad Ubd
With the partial derivative ∂µ , it does not make sense to use the
“On the gravitational field of a point mass according to Einstein’s theory,” Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften
1 (1916) 189, translated in
Section 5.6
The covariant derivative
metric to raise the index and form ∂ µ . It does make sense to do so
with covariant derivatives, so ∇a = g ab ∇b is a correct identity.
Comma, semicolon, and birdtracks notation
e / Birdtracks notation for the
covariant derivative.
Some authors use superscripts with commas and semicolons to
indicate partial and covariant derivatives. The following equations
give equivalent notations for the same derivatives:
∂µ Xν = Xν,µ
∇a Xb = Xb;a
∇a Xb = Xb ;a
Figure e shows two examples of the corresponding birdtracks notation. Because birdtracks are meant to be manifestly coordinateindependent, they do not have a way of expressing non-covariant
derivatives. We no longer want to use the circle as a notation for
a non-covariant gradient as we did when we first introduced it on
p. 48.
5.7 The geodesic equation
In this section, which can be skipped at a first reading, we show how
the Christoffel symbols can be used to find differential equations that
describe geodesics.
5.7.1 Characterization of the geodesic
A geodesic can be defined as a world-line that preserves tangency
under parallel transport, a. This is essentially a mathematical way
of expressing the notion that we have previously expressed more
informally in terms of “staying on course” or moving “inertially.”
a / The geodesic, 1, preserves
tangency under parallel transport. The non-geodesic curve,
2, doesn’t have this property;
a vector initially tangent to the
curve is no longer tangent to it
when parallel-transported along
Chapter 5
A curve can be specified by giving functions xi (λ) for its coordinates, where λ is a real parameter. A vector lying tangent to the
curve can then be calculated using partial derivatives, T i = ∂xi /∂λ.
There are three ways in which a vector function of λ could change:
(1) it could change for the trivial reason that the metric is changing,
so that its components changed when expressed in the new metric;
(2) it could change its components perpendicular to the curve; or
(3) it could change its component parallel to the curve. Possibility
1 should not really be considered a change at all, and the definition
of the covariant derivative is specifically designed to be insensitive
to this kind of thing. 2 cannot apply to T i , which is tangent by
construction. It would therefore be convenient if T i happened to
be always the same length. If so, then 3 would not happen either,
and we could reexpress the definition of a geodesic by saying that
the covariant derivative of T i was zero. For this reason, we will
assume for the remainder of this section that the parametrization
of the curve has this property. In a Newtonian context, we could
imagine the xi to be purely spatial coordinates, and λ to be a uni-
versal time coordinate. We would then interpret T i as the velocity,
and the restriction would be to a parametrization describing motion
with constant speed. In relativity, the restriction is that λ must be
an affine parameter. For example, it could be the proper time of a
particle, if the curve in question is timelike.
5.7.2 Covariant derivative with respect to a parameter
The notation of section 5.6 is not quite adapted to our present
purposes, since it allows us to express a covariant derivative with
respect to one of the coordinates, but not with respect to a parameter such as λ. We would like to notate the covariant derivative of
T i with respect to λ as ∇λ T i , even though λ isn’t a coordinate. To
connect the two types of derivatives, we can use a total derivative.
To make the idea clear, here is how we calculate a total derivative
for a scalar function f (x, y), without tensor notation:
∂f ∂x ∂f ∂y
∂x ∂λ ∂y ∂λ
This is just the generalization of the chain rule to a function of two
variables. For example, if λ represents time and f temperature,
then this would tell us the rate of change of the temperature as
a thermometer was carried through space. Applying this to the
present problem, we express the total covariant derivative as
∇λ T i = (∇b T i )
= ∂b T i + Γibc T c
5.7.3 The geodesic equation
Recognizing ∂b T i dxb /dλ as a total non-covariant derivative, we
dT i
∇λ T i =
+ Γibc T c
Substituting ∂xi /∂λ for T i , and setting the covariant derivative
equal to zero, we obtain
d2 xi
i dx dx
= 0.
dλ dλ
This is known as the geodesic equation.
If this differential equation is satisfied for one affine parameter
λ, then it is also satisfied for any other affine parameter λ0 = aλ + b,
where a and b are constants (problem 4). Recall that affine parameters are only defined along geodesics, not along arbitrary curves.
We can’t start by defining an affine parameter and then use it to
find geodesics using this equation, because we can’t define an affine
parameter without first specifying a geodesic. Likewise, we can’t
Section 5.7
The geodesic equation
do the geodesic first and then the affine parameter, because if we
already had a geodesic in hand, we wouldn’t need the differential
equation in order to find a geodesic. The solution to this chickenand-egg conundrum is to write down the differential equations and
try to find a solution, without trying to specify either the affine parameter or the geodesic in advance. We will seldom have occasion
to resort to this technique, an exception being example 16 on page
5.7.4 Uniqueness
The geodesic equation is useful in establishing one of the necessary theoretical foundations of relativity, which is the uniqueness of
geodesics for a given set of initial conditions. This is related to axiom O1 of ordered geometry, that two points determine a line, and
is necessary physically for the reasons discussed on page 22; briefly,
if the geodesic were not uniquely determined, then particles would
have no way of deciding how to move. The form of the geodesic
equation guarantees uniqueness. To see this, consider the following
algorithm for determining a numerical approximation to a geodesic:
1. Initialize λ, the xi and their derivatives dxi /dλ. Also, set a
small step-size ∆λ by which to increment λ at each step below.
2. For each i, calculate d2 xi /dλ2 using the geodesic equation.
3. Add (d2 xi /dλ2 )∆λ to the currently stored value of dxi /dλ.
4. Add (dxi /dλ)∆λ to xi .
5. Add ∆λ to λ.
6. Repeat steps 2-5 until the geodesic has been extended to the
desired affine distance.
Since the result of the calculation depends only on the inputs at
step 1, we find that the geodesic is uniquely determined.
To see that this is really a valid way of proving uniqueness, it
may be helpful to consider how the proof could have failed. Omitting
some of the details of the tensors and the multidimensionality of the
space, the form of the geodesic equation is essentially ẍ + f ẋ2 = 0,
where dots indicate derivatives with respect to λ. Suppose that it
had instead had the form ẍ2 + f ẋ = 0. Then at step 2 we would
have had to pick either a positive or a negative square root for ẍ.
Although continuity would usually suffice to maintain a consistent
sign from one iteration to the next, that would not work if we ever
came to a point where ẍ vanished momentarily. An equation of this
form therefore would not have a unique solution for a given set of
initial conditions.
The practical use of this algorithm to compute geodesics numerically is demonstrated in section 5.9.2 on page 188.
Chapter 5
5.8 Torsion
This section describes the concept of gravitational torsion. It can
be skipped without loss of continuity, provided that you accept the
symmetry property Γa[bc] = 0 without worrying about what it means
physically or what empirical evidence supports it.
Self-check: Interpret the mathematical meaning of the equation
Γa[bc] = 0, which is expressed in the notation introduced on page
5.8.1 Are scalars path-dependent?
It seems clear that something like the covariant derivative is
needed for vectors, since they have a direction in spacetime, and
thus their measures vary when the measure of spacetime itself varies.
Since scalars don’t have a direction in spacetime, the same reasoning
doesn’t apply to them, and this is reflected in our rules for covariant
derivatives. The covariant derivative has one Γ term for every index
of the tensor being differentiated, so for a scalar there should be no
Γ terms at all, i.e., ∇a is the same as ∂a .
But just because derivatives of scalars don’t require special treatment for this particular reason, that doesn’t mean they are guaranteed to behave as we intuitively expect, in the strange world of
coordinate-invariant relativity.
One possible way for scalars to behave counterintuitively would
be by analogy with parallel transport of vectors. If we stick a vector
in a box (as with, e.g., the gyroscopes aboard Gravity Probe B) and
carry it around a closed loop, it changes. Could the same happen
with a scalar? This is extremely counterintuitive, since there is no
reason to imagine such an effect in any of the models we’ve constructed of curved spaces. In fact, it is not just counterintuitive but
mathematically impossible, according to the following argument.
The only reason we can interpret the vector-in-a-box effect as arising from the geometry of spacetime is that it applies equally to all
vectors. If, for example, it only applied to the magnetic polarization vectors of ferromagnetic substances, then we would interpret
it as a magnetic field living in spacetime, not a property of spacetime itself. If the value of a scalar-in-a-box was path-dependent,
and this path-dependence was a geometric property of spacetime,
then it would have to apply to all scalars, including, say, masses
and charges of particles. Thus if an electron’s mass increased by 1%
when transported in a box along a certain path, its charge would
have to increase by 1% as well. But then its charge-to-mass ratio would remain invariant, and this is a contradiction, since the
charge-to-mass ratio is also a scalar, and should have felt the same
1% effect. Since the varying scalar-in-a-box idea leads to a contradiction, it wasn’t a coincidence that we couldn’t find a model that
produced such an effect; a theory that lacks self-consistency doesn’t
Section 5.8
have any models.
Self-check: Explain why parallel transporting a vector can only
rotate it, not change its magnitude.
a / Measuring
a scalar T .
∂ 2 T /∂ x ∂ y
There is, however, a different way in which scalars could behave
counterintuitively, and this one is mathematically self-consistent.
Suppose that Helen lives in two spatial dimensions and owns a thermometer. She wants to measure the spatial variation of temperature, in particular its mixed second derivative ∂ 2 T /∂x∂y. At home
in the morning at point A, she prepares by calibrating her gyrocompass to point north and measuring the temperature. Then she
travels ` = 1 km east along a geodesic to B, consults her gyrocompass, and turns north. She continues one kilometer north to C,
samples the change in temperature ∆T1 relative to her home, and
then retraces her steps to come home for lunch. In the afternoon,
she checks her work by carrying out the same process, but this time
she interchanges the roles of north and east, traveling along ADE.
If she were living in a flat space, this would form the other two sides
of a square, and her afternoon temperature sample ∆T2 would be
at the same point in space C as her morning sample. She actually
doesn’t recognize the landscape, so the sample points C and E are
different, but this just confirms what she already knew: the space
isn’t flat.10
None of this seems surprising yet, but there are now two qualitatively different ways that her analysis of her data could turn out,
indicating qualitatively different things about the laws of physics
in her universe. The definition of the derivative as a limit requires
that she repeat the experiment at smaller scales. As ` → 0, the
result for ∂ 2 T /∂x∂y should approach a definite limit, and the error should diminish in proportion to `. In particular the difference
between the results inferred from ∆T1 and ∆T2 indicate an error,
and the discrepancy between the second derivatives inferred from
them should shrink appropriately as ` shrinks. Suppose this doesn’t
happen. Since partial derivatives commute, we conclude that her
measuring procedure is not the same as a partial derivative. Let’s
call her measuring procedure ∇, so that she is observing a discrepancy between ∇x ∇y and ∇y ∇x . The fact that the commutator
∇x ∇y − ∇y ∇x doesn’t vanish cannot be explained by the Christoffel symbols, because what she’s differentiating is a scalar. Since the
discrepancy arises entirely from the failure of ∆T1 − ∆T2 to scale
down appropriately, the conclusion is that the distance δ between
the two sampling points is not scaling down as quickly as we expect. In our familiar models of two-dimensional spaces as surfaces
embedded in three-space, we always have δ ∼ `3 for small `, but she
has found that it only shrinks as quickly as `2 .
This point was mentioned on page 168, in connection with the definition of
the Riemann tensor.
Chapter 5
For a clue as to what is going on, note that the commutator
∇x ∇y − ∇y ∇x has a particular handedness to it. For example,
it flips its sign under a reflection across the line y = x. When we
“parallel”-transport vectors, they aren’t actually staying parallel. In
this hypothetical universe, a vector in a box transported by a small
distance ` rotates by an angle proportional to `. This effect is called
torsion. Although no torsion effect shows up in our familiar models,
that is not because torsion lacks self-consistency. Models of spaces
with torsion do exist. In particular, we can see that torsion doesn’t
lead to the same kind of logical contradiction as the varying-scalarin-a-box idea. Since all vectors twist by the same amount when
transported, inner products are preserved, so it is not possible to
put two vectors in one box and get the scalar-in-a-box paradox by
watching their inner product change when the box is transported.
Note that the elbows ABC and ADE are not right angles. If
Helen had brought a pair of gyrocompasses with her, one for x and
one for y, she would have found that the right angle between the
gyrocompasses was preserved under parallel transport, but that a
gyrocompass initially tangent to a geodesic did not remain so. There
are in fact two inequivalent definitions of a geodesic in a space with
torsion. The shortest path between two points is not necessarily
the same as the straightest possible path, i.e., the one that paralleltransports its own tangent vector.
5.8.2 The torsion tensor
b / The gyroscopes both rotate when transported from A
to B, causing Helen to navigate
along BC, which does not form
a right angle with AB. The angle
between the two gyroscopes’
axes is always the same, so the
rotation is not locally observable,
but it does produce an observable
gap between C and E.
Since torsion is odd under parity, it must be represented by an
odd-rank tensor, which we call τ cab and define according to
(∇a ∇b − ∇b ∇a )f = −τ cab ∇c f
where f is any scalar field, such as the temperature in the preceding section. There are two different ways in which a space can be
non-Euclidean: it can have curvature, or it can have torsion. For
a full discussion of how to handle the mathematics of a spacetime
with both curvature and torsion, see the article by Steuard Jensen at
pdf. For our present purposes, the main mathematical fact worth
noting is that vanishing torsion is equivalent to the symmetry Γabc =
Γacb of the Christoffel symbols. Using the notation introduced on
page 102, Γa[bc] = 0 if τ = 0.
Self-check: Use an argument similar to the one in example 5
on page 166 to prove that no model of a two-space embedded in a
three-space can have torsion.
Generalizing to more dimensions, the torsion tensor is odd under
the full spacetime reflection xa → −xa , i.e., a parity inversion plus
a time-reversal, PT.
In the story above, we had a torsion that didn’t preserve tangent vectors. In three or more dimensions, however, it is possible
c / Three gyroscopes are initially aligned with the x , y , and
z axes. After parallel transport
along the geodesic x axis, the
x gyro is still aligned with the x
axis, but the y and z gyros have
Section 5.8
to have torsion that does preserve tangent vectors. For example,
transporting a vector along the x axis could cause only a rotation in
the y-z plane. This relates to the symmetries of the torsion tensor,
which for convenience we’ll write in an x-y-z coordinate system and
in the fully covariant form τλµν . The definition of the torsion tensor
implies τλ(µν) = 0, i.e., that the torsion tensor is antisymmetric in
its two final indices. Torsion that does not preserve tangent vectors
will have nonvanishing elements such as τxxy , meaning that paralleltransporting a vector along the x axis can change its x component.
Torsion that preserves tangent vectors will have vanishing τλµν unless λ, µ, and ν are all distinct. This is an example of the type
of antisymmetry that is familiar from the vector cross product, in
which the cross products of the basis vectors behave as x × y = z,
y × z = x, y × z = x. Generalizing the notation for symmetrization
and antisymmetrization of tensors from page 102, we have
= Σabc Tabc
T(abc) =
where the sums are over all permutations of the indices, and in the
second line we have used the Levi-Civita symbol. In this notation,
a totally antisymmetric torsion tensor is one with τλµν= τ[λµν] , and
torsion of this type preserves tangent vectors under translation.
In two dimensions, there are no totally antisymmetric objects
with three indices, because we can’t write three indices without
repeating one. In three dimensions, an antisymmetric object with
three indices is simply a multiple of the Levi-Civita tensor, so a
totally antisymmetric torsion, if it exists, is represented by a single
number; under translation, vectors rotate like either right-handed
or left-handed screws, and this number tells us the rate of rotation.
In four dimensions, we have four independently variable quantities,
τxyz , τtyz , τtxz , and τtxy . In other words, an antisymmetric torsion of
3+1 spacetime can be represented by a four-vector, τ a = abcd τbcd .
5.8.3 Experimental searches for torsion
One way of stating the equivalence principle (see p. 142) is that
it forbids spacetime from coming equipped with a vector field that
could be measured by free-falling observers, i.e., observers in local
Lorentz frames. A variety of high-precision tests of the equivalence
principle have been carried out. From the point of view of an experimenter doing this kind of test, it is important to distinguish
between fields that are “built in” to spacetime and those that live
in spacetime. For example, the existence of the earth’s magnetic
field does not violate the equivalence principle, but if an experiment was sensitive to the earth’s field, and the experimenter didn’t
know about it, there would appear to be a violation. Antisymmetric torsion in four dimensions acts like a vector. If it constitutes
Chapter 5
a universal background effect built into spacetime, then it violates
the equivalence principle. If it instead arises from specific material
sources, then it may still show up as a measurable effect in experimental tests designed to detect Lorentz-invariance. Let’s consider
the latter possibility.
Since curvature in general relativity comes from mass and energy, as represented by the stress-energy tensor Tab , we could ask
what would be the sources of torsion, if it exists in our universe.
The source can’t be the rank-2 stress-energy tensor. It would have
to be an odd-rank tensor, i.e., a quantity that is odd under PT, and
in theories that include torsion it is commonly assumed that the
source is the quantum-mechanical angular momentum of subatomic
particles. If this is the case, then torsion effects are expected to be
proportional to ~G, the product of Planck’s constant and the gravitational constant, and they should therefore be extremely small and
hard to measure. String theory, for example, includes torsion, but
nobody has found a way to test string theory empirically because it
makes predictions about phenomena at the Planck scale,
~G/c ∼ 10−35 m, where both gravity and quantum mechanics
are strong effects.
There are, however, some high-precision experiments that have
a reasonable chance of detecting whether our universe has torsion.
Torsion violates the equivalence principle, and by the turn of the
century tests of the equivalence principle had reached a level of
precision sufficient to rule out some models that include torsion.
Figure d shows a torsion pendulum used in an experiment by the
Eöt-Wash group at the University of Washington.11 If torsion exists,
then the intrinsic spin σ of an electron should have an energy σ · τ ,
where τ is the spacelike part of the torsion vector. The torsion
could be generated by the earth, the sun, or some other object at a
greater distance. The interaction σ · τ will modify the behavior of a
torsion pendulum if the spins of the electrons in the pendulum are
polarized nonrandomly, as in a magnetic material. The pendulum
will tend to precess around the axis defined by τ .
This type of experiment is extremely difficult, because the pendulum tends to act as an ultra-sensitive magnetic compass, resulting
in a measurement of the ambient magnetic field rather than the hypothetical torsion field τ . To eliminate this source of systematic
error, the UW group first eliminated the ambient magnetic field
as well as possible, using mu-metal shielding and Helmholtz coils.
They also constructed the pendulum out of a combination of two
magnetic materials, Alnico 5 and SmCo5 , in such a way that the
magnetic dipole moment vanished, but the spin dipole moment did
not; Alnico 5’s magnetic field is due almost entirely to electron spin,
whereas the magnetic field of SmCo5 contains significant contribu11
d / The University of Washington torsion pendulum used to
search for torsion. The light gray
wedges are Alnico, the darker
ones SmCo5 . The arrows with
the filled heads represent the
directions of the electron spins,
with denser arrows indicating
higher polarization. The arrows
with the open heads show the
direction of the B field.
Section 5.8
tions from orbital motion. The result was a nonmagnetic object
whose spins were polarized. After four years of data collection, they
found |τ | . 10−21 eV. Models that include torsion typically predict
such effects to be of the order of m2ep
/mP ∼ 10−17 eV, where me is
the mass of the electron and mP = ~c/G ≈ 1019 GeV ≈ 20 µg is
the Planck mass. A wide class of these models is therefore ruled out
by these experiments.
Since there appears to be no experimental evidence for the existence of gravitational torsion in our universe, we will assume from
now on that it vanishes identically. Einstein made the same assumption when he originally created general relativity, although he
and Cartan later tinkered with non-torsion-free theories in a failed
attempt to unify gravity with electromagnetism. Some models that
include torsion remain viable. For example, it has been argued that
the torsion tensor should fall off quickly with distance from the
Chapter 5
Carroll and Field,
5.9 From metric to curvature
5.9.1 Finding the Christoffel symbol from the metric
We’ve already found the Christoffel symbol in terms of the metric
in one dimension. Expressing it in tensor notation, we have
Γdba = g cd (∂? g?? )
where inversion of the one-component matrix G has been replaced
by matrix inversion, and, more importantly, the question marks indicate that there would be more than one way to place the subscripts
so that the result would be a grammatical tensor equation. The
most general form for the Christoffel symbol would be
Γbac = g db (L∂c gab + M ∂a gcb + N ∂b gca )
where L, M , and N are constants. Consistency with the onedimensional expression requires L + M + N = 1, and vanishing
torsion gives L = M . The L and M terms have a different physical
significance than the N term.
Suppose an observer uses coordinates such that all objects are
described as lengthening over time, and the change of scale accumulated over one day is a factor of k > 1. This is described by the
derivative ∂t gxx < 1, which affects the M term. Since the metric is
used to calculate
squared distances, the gxx matrix element scales
down by 1/ k. To compensate for ∂t v x < 0, so we need to add a
positive correction term, M > 0, to the covariant derivative. When
the same observer measures the rate of change of a vector v t with
respect to space, the rate of change comes out to be too small, because the variable she differentiates with respect to is too big. This
requires N < 0, and the correction is of the same size as the M
correction, so |M | = |N |. We find L = M = −N = 1.
Self-check: Does the above argument depend on the use of space
for one coordinate and time for the other?
The resulting general expression for the Christoffel symbol in
terms of the metric is
Γcab = g cd (∂a gbd + ∂b gad − ∂d gab )
One can readily go back and check that this gives ∇c gab = 0. In fact,
the calculation is a bit tedious. For that matter, tensor calculations
in general can be infamously time-consuming and error-prone. Any
reasonable person living in the 21st century will therefore resort to
a computer algebra system. The most widely used computer algebra system is Mathematica, but it’s expensive and proprietary, and
it doesn’t have extensive built-in facilities for handling tensors. It
Section 5.9
From metric to curvature
turns out that there is quite a bit of free and open-source tensor software, and it falls into two classes: coordinate-based and coordinateindependent. The best open-source coordinate-independent facility available appears to be Cadabra, and in fact the verification of
∇c gab = 0 is the first example given in the Leo Brewin’s handy guide
to applications of Cadabra to general relativity.13
Self-check: In the case of 1 dimension, show that this reduces to
the earlier result of −(1/2)dG/dX.
Since Γ is not a tensor, it is not obvious that the covariant derivative, which is constructed from it, is a tensor. But if it isn’t obvious,
neither is it surprising – the goal of the above derivation was to get
results that would be coordinate-independent.
Christoffel symbols on the globe, quantitatively
Example: 10
In example 9 on page 177, we inferred the following properties
for the Christoffel symbol Γ θφφ on a sphere of radius R: Γ θφφ is
independent of φ and R, Γ θφφ < 0 in the northern hemisphere
(colatitude θ less than π/2), Γ θφφ = 0 on the equator, and Γ θφφ >
0 in the southern hemisphere.
The metric on a sphere is ds2 = R 2 dθ2 + R 2 sin2 θdφ2 . The only
nonvanishing term in the expression for Γ θφφ is the one involving
∂θ gφφ = 2R 2 sin θ cos θ. The result is Γ θφφ = − sin θ cos θ, which
can be verified to have the properties claimed above.
5.9.2 Numerical solution of the geodesic equation
On page 180 I gave an algorithm that demonstrated the uniqueness of the solutions to the geodesic equation. This algorithm can
also be used to find geodesics in cases where the metric is known.
The following program, written in the computer language Python,
carries out a very simple calculation of this kind, in a case where
we know what the answer should be; even without any previous
familiarity with Python, it shouldn’t be difficult to see the correspondence between the abstract algorithm presented on page 180
and its concrete realization below. For polar coordinates in a Euclidean plane, one can compute Γrφφ = −r and Γφrφ = 1/r (problem
2, page 199). Here we compute the geodesic that starts out tangent
to the unit circle at φ = 0.
import math
l = 0
# affine parameter lambda
dl = .001 # change in l with each iteration
l_max = 100.
# initial position:
Chapter 5
# initial derivatives of coordinates w.r.t. lambda
vr = 0
vphi = 1
k = 0 # keep track of how often to print out updates
while l<l_max:
l = l+dl
# Christoffel symbols:
Grphiphi = -r
Gphirphi = 1/r
# second derivatives:
= -Grphiphi*vphi*vphi
aphi = -2.*Gphirphi*vr*vphi
# ... factor of 2 because G^a_{bc}=G^a_{cb} and b
is not the same as c
# update velocity:
vr = vr + dl*ar
vphi = vphi + dl*aphi
# update position:
r = r + vr*dl
phi = phi + vphi*dl
if k%10000==0: # k is divisible by 10000
phi_deg = phi*180./math.pi
print "lambda=%6.2f
phi=%6.2f deg." % (l,r,phi_deg)
k = k+1
It is not necessary to worry about all the technical details of the
language (e.g., line 1, which makes available such conveniences as
math.pi for π). Comments are set off by pound signs. Lines 16-34
are indented because they are all to be executed repeatedly, until it
is no longer true that λ < λmax (line 15).
Self-check: By inspecting lines 18-22, find the signs of r̈ and φ̈
at λ = 0. Convince yourself that these signs are what we expect
The output is as follows:
Section 5.9
From metric to curvature
lambda= 90.00
r= 90.06
phi= 89.29 deg.
We can see that φ → 90 deg. as λ → ∞, which makes sense,
because the geodesic is a straight line parallel to the y axis.
A less trivial use of the technique is demonstrated on page 220,
where we calculate the deflection of light rays in a gravitational field,
one of the classic observational tests of general relativity.
5.9.3 The Riemann tensor in terms of the Christoffel symbols
The covariant derivative of a vector can be interpreted as the rate
of change of a vector in a certain direction, relative to the result of
parallel-transporting the original vector in the same direction. We
can therefore see that the definition of the Riemann curvature tensor
on page 168 is a measure of the failure of covariant derivatives to
(∇a ∇b − ∇b ∇a )Ac = Ad Rcdab
A tedious calculation now gives R in terms of the Γs:
Rabcd = ∂c Γadb − ∂d Γacb + Γace Γedb − Γade Γecb
This is given as another example later in Brewin’s manual for applying Cadabra to general relativity.14 (Brewin writes the upper index
in the second slot of R.)
5.9.4 Some general ideas about gauge
Let’s step back now for a moment and try to gain some physical insight by looking at the features that the electromagnetic and
relativistic gauge transformations have in common. We have the
following analogies:
Chapter 5
global symmetry
A constant phase
shift α has no observable effects.
Adding a constant
coordinate has no
observable effects.
local symmetry
A phase shift α
that varies from
point to point has
no observable effects.
An arbitrary coordinate transformation has no observable effects.
The gauge is described by . . .
. . . and differentiation of this gives the
gauge field. . .
A second differentiation gives the
directly observable
field(s) . . .
E and B
The interesting thing here is that the directly observable fields
do not carry all of the necessary information, but the gauge fields are
not directly observable. In electromagnetism, we can see this from
the Aharonov-Bohm effect, shown in figure a.15 The solenoid has
B = 0 externally, and the electron beams only ever move through
the external region, so they never experience any magnetic field. Experiments show, however, that turning the solenoid on and off does
change the interference between the two beams. This is because the
vector potential does not vanish outside the solenoid, and as we’ve
seen on page 138, the phase of the beams varies according to the
path integral of the Ab . We are therefore left with an uncomfortable, but unavoidable, situation. The concept of a field is supposed
to eliminate the need for instantaneous action at a distance, which
is forbidden by relativity; that is, (1) we want our fields to have only
local effects. On the other hand, (2) we would like our fields to be
directly observable quantities. We cannot have both 1 and 2. The
gauge field satisfies 1 but not 2, and the electromagnetic fields give
2 but not 1.
We describe the effect here in terms of an idealized, impractical experiment.
For the actual empirical status of the Aharonov-Bohm effect, see Batelaan and
Tonomura, Physics Today 62 (2009) 38.
Section 5.9
a / The
effect. An electron enters a beam
splitter at P, and is sent out in
two different directions. The two
parts of the wave are reflected so
that they reunite at Q. The arrows
represent the vector potential A.
The observable magnetic field
B is zero everywhere outside
the solenoid, and yet the interference observed at Q depends
on whether the field is turned
on. See page 137 for further
discussion of the A and B fields
of a solenoid.
From metric to curvature
Figure b shows an analog of the Aharonov-Bohm experiment in
differential geometry. Everywhere but at the tip, the cone has zero
curvature, as we can see by cutting it and laying it out flat. But even
an observer who never visits the tightly curved region at the tip can
detect its existence, because parallel-transporting a vector around
a closed loop can change the vector’s direction, provided that the
loop surrounds the tip.
b / The cone has zero intrinsic curvature everywhere except
at its tip. An observer who never
visits the tip can nevertheless
detect its existence, because
parallel transport around a path
that encloses the tip causes a
vector to change its direction.
In the electromagnetic example, integrating A around a closed
loop reveals, via Stokes’ theorem, the existence of a magnetic flux
through the loop, even though the magnetic field is zero at every
location where A has to be sampled. In the relativistic example,
integrating Γ around a closed loop shows that there is curvature
inside the loop, even though the curvature is zero at all the places
where Γ has to be sampled.
The fact that Γ is a gauge field, and therefore not locally observable, is simply a fancy way of expressing the ideas introduced
on pp. 176 and 177, that due to the equivalence principle, the gravitational field in general relativity is not locally observable. This nonobservability is local because the equivalence principle is a statement
about local Lorentz frames. The example in figure b is non-local.
Geodetic effect and structure of the source
Example: 11
. In section 5.5.1 on page 170, we estimated the geodetic effect
on Gravity Probe B and found a result that was only off by a factor
of 3π. The mathematically pure form of the 3π suggests that the
geodetic effect is insensitive to the distribution of mass inside the
earth. Why should this be so?
. The change in a vector upon parallel transporting it around a
closed loop can be expressed in terms of either (1) the area integral of the curvature within the loop or (2) the line integral of the
Christoffel symbol (essentially the gravitational field) on the loop
itself. Although I expressed the estimate as 1, it would have been
equally valid to use 2. By Newton’s shell theorem, the gravitational field is not sensitive to anything about its mass distribution
other than its near spherical symmetry. The earth spins, and this
does affect the stress-energy tensor, but since the velocity with
which it spins is everywhere much smaller than c, the resulting
effect, called frame dragging, is much smaller.
Chapter 5
5.10 Manifolds
This section can be omitted on a first reading.
5.10.1 Why we need manifolds
General relativity doesn’t assume a predefined background metric, and this creates a chicken-and-egg problem. We want to define
a metric on some space, but how do we even specify the set of points
that make up that space? The usual way to define a set of points
would be by their coordinates. For example, in two dimensions we
could define the space as the set of all ordered pairs of real numbers
(x, y). But this doesn’t work in general relativity, because space is
not guaranteed to have this structure. For example, in the classic
1979 computer game Asteroids, space “wraps around,” so that if
your spaceship flies off the right edge of the screen, it reappears
on the left, and similarly at the top and bottom. Even before we
impose a metric on this space, it has topological properties that differ from those of the Euclidean plane. By “topological” we mean
properties that are preserved if the space is thought of as a sheet
of rubber that can be stretched in any way, but not cut or glued
back together. Topologically, the space in Asteroids is equivalent to
a torus (surface of a doughnut), but not to the Euclidean plane.
a / In Asteroids, space “wraps
b / A coffee cup is topologically
equivalent to a torus.
Another useful example is the surface of a sphere. In example
10 on page 188, we calculated Γθφφ . A similar calculation gives
Γφθφ = cot θ/R. Now consider what happens as we drive our dogsled
north along the line of longitude φ = 0, cross the north pole at
θ = 0, and continue along the same geodesic. As we cross the pole,
our longitude changes discontinuously from 0 to π. Consulting the
geodesic equation, we see that this happens because Γφθφ blows up
at θ = 0. Of course nothing really special happens at the pole.
The bad behavior isn’t the fault of the sphere, it’s the fault of the
(θ, φ) coordinates we’ve chosen, that happen to misbehave at the
pole. Unfortunately, it is impossible to define a pair of coordinates
on a two-sphere without having them misbehave somewhere. (This
follows from Brouwer’s famous 1912 “Hairy ball theorem,” which
states that it is impossible to comb the hair on a sphere without
creating a cowlick somewhere.)
Section 5.10
5.10.2 Topological definition of a manifold
This motivates us to try to define a “bare-bones” geometrical
space in which there is no predefined metric or even any predefined
set of coordinates.
There is a general notion of a topological space, which is too
general for our purposes. In such a space, the only structure we are
guaranteed is that certain sets are defined as “open,” in the same
sense that an interval like 0 < x < 1 is called “open.” Any point
in an open set can be moved around without leaving the set. An
open set is essentially a set without a boundary, for in a set like
0 ≤ x ≤ 1, the boundary points 0 and 1 can only be moved in one
direction without taking them outside.
c / General
assume a predefined background
Therefore all we can
really know before we calculate
anything is that we’re working
on a manifold, without a metric
imposed on it.
A toplogical space is too general for us because it can include
spaces like fractals, infinite-dimensional spaces, and spaces that have
different numbers of dimensions in different regions. It is nevertheless useful to recognize certain concepts that can be defined using
only the generic apparatus of a topological space, so that we know
they do not depend in any way on the presence of a metric. An
open set surrounding a point is called a neighborhood of that point.
In a topological space we have a notion of getting arbitrarily close
to a certain point, which means to take smaller and smaller neighborhoods, each of which is a subset of the last. But since there is
no metric, we do not have any concept of comparing distances of
distant points, e.g., that P is closer to Q than R is to S. A continuous function is a purely topological idea; a continuous function
is one such that for any open subset U of its range, the set V of
points in its domain that are mapped to points in U is also open.
Although some definitions of continuous functions talk about real
numbers like and δ, the notion of continuity doesn’t depend on
the existence of any structure such as the real number system. A
homeomorphism is a function that is invertible and continuous in
both directions. Homeomorphisms formalize the informal notion of
“rubber-sheet geometry without cutting or gluing.” If a homeomorphism exists between two topological spaces, we say that they are
homeomorphic; they have the same structure and are in some sense
the same space.
The more specific type of topological space we want is called a
manifold. Without attempting any high level of mathematical rigor,
we define an n-dimensional manifold M according to the following
informal principles:16
For those with knowledge of topology, these can be formalized a little more:
we want a completely normal, second-countable, locally connected topological
space that has Lebesgue covering dimension n, is a homogeneous space under
its own homeomorphism group, and is a complete uniform space. I don’t know
whether this is sufficient to characterize a manifold completely, but it suffices to
rule out all the counterexamples of which I know.
Chapter 5
M1 Dimension: M’s dimension is n.
M2 Homogeneity: No point has any property that distinguishes it
from any other point.
M3 Completeness: M is complete, in the sense that specifying an
arbitrarily small neighborhood gives a unique definition of a
Example: 12
The set of all real numbers is a 1-manifold. Similarly, any line with
the properties specified in Euclid’s Elements is a 1-manifold. All
such lines are homeomorphic to one another, and we can therefore speak of “the line.”
A circle
Example: 13
A circle (not including its interior) is a 1-manifold, and it is not
homeomorphic to the line. To see this, note that deleting a point
from a circle leaves it in one connected piece, but deleting a point
from a line makes two. Here we use the fact that a homeomorphism is guaranteed to preserve “rubber-sheet” properties like the
number of pieces.
No changes of dimension
Example: 14
A “lollipop” formed by gluing an open 2-circle (i.e., a circle not
including its boundary) to an open line segment is not a manifold,
because there is no n for which it satisfies M1.
It also violates M2, because points in this set fall into three distinct
classes: classes that live in 2-dimensional neighborhoods, those
that live in 1-dimensional neighborhoods, and the point where the
line segment intersects the boundary of the circle.
No manifolds made from the rational numbers
Example: 15
The rational numbers are not a manifold,
arbitrarily small neighborhood around 2 excludes every rational
number, violating M3.
Similarly, the rational plane defined by rational-number coordinate
pairs (x, y) is not a 2-manifold. It’s good that we’ve excluded
this space, because it has the unphysical property that curves
can cross without having a point in common. For example, the
curve y = x 2 crosses from one side of the line y = 2 to the other,
but never intersects it. This is physically undesirable because it
doesn’t match up with what we have in mind when we talk about
collisions between particles as intersections of their world-lines,
or when we say that electric field lines aren’t supposed to intersect.
No boundary
Example: 16
The open half-plane y > 0 in the Cartesian plane is a 2-manifold.
The closed half-plane y ≥ 0 is not, because it violates M2; the
Section 5.10
boundary points have different properties than the ones on the
Disconnected manifolds
Example: 17
Two nonintersecting lines are a 1-manifold. Physically, disconnected manifolds of this type would represent a universe in which
an observer in one region would never be able to find out about
the existence of the other region.
No bad glue jobs
Example: 18
Hold your hands like you’re pretending you know karate, and then
use one hand to karate-chop the other. Suppose we want to join
two open half-planes in this way. As long as they’re separate,
then we have a perfectly legitimate disconnected manifold. But if
we want to join them by adding the point P where their boundaries
coincide, then we violate M2, because this point has special properties not possessed by any others. An example of such a property is that there exist points Q and R such that every continuous
curve joining them passes through P. (Cf. problem 5, p. 329.)
5.10.3 Local-coordinate definition of a manifold
An alternative way of characterizing an n-manifold is as an object that can locally be described by n real coordinates. That is,
any sufficiently small neighborhood is homeomorphic to an open set
in the space of real-valued n-tuples of the form (x1 , x2 , . . . , xn ). For
example, a closed half-plane is not a 2-manifold because no neighborhood of a point on its edge is homeomorphic to any open set in
the Cartesian plane.
Self-check: Verify that this alternative definition of a manifold
gives the same answers as M1-M3 in all the examples above.
Roughly speaking, the equivalence of the two definitions occurs
because we’re using n real numbers as coordinates for the dimensions
specified by M1, and the real numbers are the unique number system
that has the usual arithmetic operations, is ordered, and is complete
in the sense of M3.
As usual when we say that something is “local,” a question arises
as to how local is local enough. The language in the definition above
about “any sufficiently small neighborhood” is logically akin to the
Weierstrass -δ approach: if Alice gives Bob a manifold and a point
on a manifold, Bob can always find some neighborhood around that
point that is compatible with coordinates, but it may be an extremely small neighborhood. As discussed in section 3.3, a method
that is equally rigorous — and usually much more convenient in differential geometry — is to use infinitesimals. For example, suppose
that we want to write down a metric in the form ds2 = gab dxa dxb .
Infinitesimal distances like dxa are always small enough to fit in
any open set of real numbers. In fact, an alternative definition of
an open set, in a space with a Euclidean metric, is one in which
Chapter 5
every point can be surrounded by an infinitesimal ball. Similarly, if
we want to calculate Christoffel symbols, the Riemann tensor, etc.,
then all we need is the ability to take derivatives, and this only
requires infinitesimal coordinate changes. Likewise the equivalence
principle says that a spacetime is compatible with a Lorentzian metric at every point, and this only requires an infinitesimal amount of
In practice, we never have to break up a manifold into infinitely
many pieces, each of them infinitesimally small, in order to have
well-behaved coordinates.
Coordinates on a circle
Example: 19
If we are to define coordinates on a circle, they should be continuous functions. The angle φ about the center therefore doesn’t
quite work as a global coordinate, because it has a discontinuity
where φ = 0 is identified with φ = 2π. We can get around this by
using different coordinates in different regions, as is guaranteed
to be possible by the local-coordinate definition of a manifold. For
example, we can cover the circle with two open sets, one on the
left and one on the right. The left one, L, is defined by deleting
only the φ = 0 point from the circle. The right one, R, is defined
by deleting only the one at φ = π. On L, we use coordinates
0 < φL < 2π, which are always a continuous function from L to
the real numbers. On R, we use −π < φR < π.
In examples like this one, the sets like L and R are referred to
as patches. We require that the coordinate maps on the different
patches match up smoothly. In this example, we would like all
four of the following functions, known as transition maps, to be
• φL as a function of φR on the domain 0 < φR < π
• φL as a function of φR on the domain −π < φR < 0
• φR as a function of φL on the domain 0 < φL < π
• φR as a function of φL on the domain π < φL < 2π
The local-coordinate definition only states that a manifold can
be coordinatized. That is, the functions that define the coordinate
maps are not part of the definition of the manifold, so, for example,
if two people define coordinates patches on the unit circle in different
ways, they are still talking about exactly the same manifold.
We conclude with a few examples relating to homeomorphism.
Open line segment homeomorphic to a line
Example: 20
Let L be an open line segment, such as the open interval (0, 1). L
is homeomorphic to a line, because we can map (0, 1) to the real
line through the function f (x) = tan(πx − π/2).
Section 5.10
Closed line segment not homeomorphic to a line Example: 21
A closed line segment (which is not a manifold) is not homeomorphic to a line. If we map it to a line, then the endpoints have to
go to two special points A and B. There is then no way for the
mapping to visit the points exterior to the interval [A, B] without
visiting A and B more than once.
Open line segment not homeomorphic to the interior of a circle
Example: 22
If the interior of a circle could be mapped by a homeomorphism f
to an open line segment, then consider what would happen if we
took a closed curve lying inside the circle and found its image. By
the intermediate value theorem, f would not be one-to-one, but
this is a contradiction since f was assumed to be a homeomorphism. This is an example of a more general fact that homeomorphism preserves the dimensionality of a manifold.
d / Example 22.
Chapter 5
Example 6 on p. 167 discussed some examples in electrostatics
where the charge density on the surface of a conductor depends on
the Gaussian curvature, when the curvature is positive. In the case
of a knife-edge formed by two half-planes at an exterior angle β > π,
there is a standard result17 that the charge density at the edge blows
up to infinity as Rπ/β−1 . Does this match up with the hypothesis
that Gaussian curvature determines the charge density?
. Solution, p. 371
Show, as claimed on page 188, that for polar coordinates in
a Euclidean plane, Γrφφ = −r and Γφrφ = 1/r.
Partial derivatives commute with partial derivatives. Covariant derivatives don’t commute with covariant derivatives. Do
covariant derivatives commute with partial derivatives?
Show that if the differential equation for geodesics on page
178 is satisfied for one affine parameter λ, then it is also satisfied for
any other affine parameter λ0 = aλ + b, where a and b are constants.
Equation [2] on page 110 gives a flat-spacetime metric in
rotating polar coordinates. (a) Verify by explicit computation that
this metric represents a flat spacetime. (b) Reexpress the metric in
rotating Cartesian coordinates, and check your answer by verifying
that the Riemann tensor vanishes.
The purpose of this problem is to explore the difficulties
inherent in finding anything in general relativity that represents a
uniform gravitational field g. In example 12 on page 59, we found,
based on elementary arguments about the equivalence principle and
photons in elevators, that gravitational time dilation must be given
by eΦ , where Φ = gz is the gravitational potential. This results in
a metric
ds2 = e2gz dt2 − dz 2
On the other hand, example 19 on page 140 derived the metric
ds2 = (1 + gz)2 dt2 − dz 2
by transforming from a Lorentz frame to a frame whose origin moves
with constant proper acceleration g. (These are known as Rindler
coordinates.) Prove the following facts. None of the calculations
are so complex as to require symbolic math software, so you might
want to perform them by hand first, and then check yourself on a
(a) The metrics [1] and [2] are approximately consistent with one
another for z near 0.
Jackson, Classical Electrodynamics
(b) When a test particle is released from rest in either of these metrics, its initial proper acceleration is g.
(c) The two metrics are not exactly equivalent to one another under
any change of coordinates.
(d) Both spacetimes are uniform in the sense that the curvature is
constant. (In both cases, this can be proved without an explicit
computation of the Riemann tensor.)
Remark: The incompatibility between [1] and [2] can be interpreted as showing
that general relativity does not admit any spacetime that has all the global
properties we would like for a uniform gravitational field. This is related to Bell’s
spaceship paradox (example 16, p. 66). Some further properties of the metric [1]
are analyzed in subsection 7.4 on page 255.
. Solution, p. 371
In a topological space T, the complement of a subset U is
defined as the set of all points in T that are not members of U. A
set whose complement is open is referred to as closed. On the real
line, give (a) one example of a closed set and (b) one example of
a set that is neither open nor closed. (c) Give an example of an
inequality that defines an open set on the rational number line, but
a closed set on the real line.
Prove that a double cone (e.g., the surface r = z in cylindrical
coordinates) is not a manifold.
. Solution, p. 372
Prove that a torus is a manifold.
. Solution, p. 372
Prove that a sphere is not homeomorphic to a torus.
. Solution, p. 372
Curvature on a Riemannian space in 2 dimensions is a
topic that goes back to Gauss and has a simple interpretation: the
only intrinsic measure of curvature is a single number, the Gaussian
curvature. What about 1+1 dimensions? The simplest metrics I
can think of are of the form ds2 = dt2 − f (t)dx2 . (Something like
ds2 = f (t)dt2 −dx2 is obviously equivalent to Minkowski space under
a change of coordinates, while ds2 = f (x)dt2 − dx2 is the same as
the original example except that we’ve swapped x and t.) Playing
around with simple examples, one stumbles across the seemingly
mysterious fact that the metric ds2 = dt2 − t2 dx2 is flat, while ds2 =
dt2 − tdx2 is not. This seems to require some simple explanation.
Consider the metric ds2 = dt2 − tp dx2 .
(a) Calculate the Christoffel symbols by hand.
(b) Use a computer algebra system such as Maxima to show that
the Ricci tensor vanishes only when p = 2.
Remark: The explanation is that in the case p = 2, the x coordinate is expanding
in proportion to the t coordinate. This can be interpreted as a situation in which
our length scale is defined by a lattice of test particles that expands inertially.
Since their motion is inertial, no gravitational fields are required in order to
explain the observed change in the length scale; cf. the Milne universe, p. 299.
. Solution, p. 372
Chapter 5
Chapter 6
Vacuum solutions
In this chapter we investigate general relativity in regions of space
that have no matter to act as sources of the gravitational field.
We will not, however, limit ourselves to calculating spacetimes in
cases in which the entire universe has no matter. For example,
we will be able to calculate general-relativistic effects in the region
surrounding the earth, including a full calculation of the geodetic
effect, which was estimated in section 5.5.1 only to within an order
of magnitude. We can have sources, but we just won’t describe the
metric in the regions where the sources exist, e.g., inside the earth.
The advantage of accepting this limitation is that in regions of empty
space, we don’t have to worry about the details of the stress-energy
tensor or how it relates to curvature. As should be plausible based
on the physical motivation given in section 5.1, page 160, the field
equations in a vacuum are simply Rab = 0.
coin shows the vacuum field
6.1 Event horizons
One seemingly trivial way to generate solutions to the field equations
in vacuum is simply to start with a flat Lorentzian spacetime and do
a change of coordinates. This might seem pointless, since it would
simply give a new description (and probably a less convenient and
descriptive one) of the same old, boring, flat spacetime. It turns
out, however, that some very interesting things can happen when
we do this.
6.1.1 The event horizon of an accelerated observer
Consider the uniformly accelerated observer described in examples 4 on page 126 and 19 on page 140. Recalling these earlier results,
we have for the ship’s equation of motion in an inertial frame
1 p
1 + a2 t2 − 1
and for the metric in the ship’s frame
gt0 0 t0 = (1 + ax0 )2
gx0 0 x0 = −1
Since this metric was derived by a change of coordinates from a flatspace metric, and the Ricci curvature is an intrinsic property, we
expect that this one also has zero Ricci curvature. This is straightforward to verify. The nonvanishing Christoffel symbols are
and Γx t0 t0 = a(1 + ax0 )
Γt x0 t0 =
1 + ax
The only elements of the Riemann tensor that look like they might
be nonzero are Rt t0 x0 x0 and Rx t0 x0 t0 , but both of these in fact vanish.
Self-check: Verify these facts.
This seemingly routine exercise now leads us into some very interesting territory. Way back on page 12, we conjectured that not all
events could be time-ordered: that is, that there might exists events
in spacetime 1 and 2 such that 1 cannot cause 2, but neither can 2
cause 1. We now have enough mathematical tools at our disposal
to see that this is indeed the case.
a / A spaceship (curved worldline) moves with an acceleration
perceived as constant by its
passengers. The photon (straight
world-line) come closer and
closer to the ship, but will never
quite catch up.
We observe that x(t) approaches the asymptote x = t−1/a. This
asymptote has a slope of 1, so it can be interpreted as the world-line
of a photon that chases the ship but never quite catches up to it.
Any event to the left of this line can never have a causal relationship
with any event on the ship’s world-line. Spacetime, as seen by an
observer on the ship, has been divided by a curtain into two causally
disconnected parts. This boundary is called an event horizon. Its
existence is relative to the world-line of a particular observer. An
observer who is not accelerating along with the ship does consider
an event horizon to exist. Although this particular example of the
indefinitely accelerating spaceship has some physically implausible
features (e.g., the ship would have to run out of fuel someday), event
horizons are real things. In particular, we will see in section 6.3.2
that black holes have event horizons.
Interpreting everything in the (t0 , x0 ) coordinates tied to the ship,
the metric’s component gt0 0 t0 vanishes at x0 = −1/a. An observer
aboard the ship reasons as follows. If I start out with a head-start
of 1/a relative to some event, then the timelike part of the metric at
that event vanishes. If the event marks the emission of a material
particle, then there is no possible way for that particle’s world-line
to have ds2 > 0. If I were to detect a particle emitted at that event,
it would violate the laws of physics, since material particles must
have ds2 > 0, so I conclude that I will never observe such a particle.
Since all of this applies to any material particle, regardless of its
mass m, it must also apply in the limit m → 0, i.e., to photons and
other massless particles. Therefore I can never receive a particle
emitted from this event, and in fact it appears that there is no way
for that event, or any other event behind the event horizon, to have
any effect on me. In my frame of reference, it appears that light
cones near the horizon are tipped over so far that their future lightcones lie entirely in the direction away from me.
We’ve already seen in example 15 on page 65 that a naive Newtonian argument suggests the existence of black holes; if a body is
Chapter 6
Vacuum solutions
sufficiently compact, light cannot escape from it. In a relativistic
treatment, this should be described as an event horizon.
6.1.2 Information paradox
The existence of event horizons in general relativity has deep
implications, and in particular it helps to explain why it is so difficult to reconcile general relativity with quantum mechanics, despite
nearly a century of valiant attempts. Quantum mechanics has a
property called unitarity. Mathematically, this says that if the state
of a quantum mechanical system is given, at a certain time, in the
form of a vector, then its state at some point in the future can be
predicted by applying a unitary matrix to that vector. A unitary
matrix is the generalization to complex numbers of the ordinary
concept of an orthogonal matrix, and essentially it just represents a
change of basis, in which the basis vectors have unit length and are
perpendicular to one another.
To see what this means physically, consider the following nonexamples. The matrix
1 0
0 0
is not unitary, because its rows and columns are not orthogonal vectors with unit lengths. If this matrix represented the time-evolution
of a quantum mechanical system, then its meaning would be that
any particle in state number 1 would be left alone, but any particle
in state 2 would disappear. Any information carried by particles in
state 2 is lost forever and can never be retrieved. This also violates
the time-reversal symmetry of quantum mechanics.
Another nonunitary matrix is:
1 √0
any particle in state 2 is increased in amplitude by a factor of
2, meaning that it is doubled in probability. That is, the particle
is cloned. This is the opposite problem compared to the one posed
by the first matrix, and it is equally problematic in terms of timereversal symmetry and conservation of information. Actually, if we
could clone a particle in this way, it would violate the Heisenberg
uncertainty principle. We could make two copies of the particle,
and then measure the position of one copy and the momentum of
the other, each with unlimited precision. This would violate the
uncertainty principle, so we believe that it cannot be done. This is
known as the no-cloning theorem.1
The existence of event horizons in general relativity violates unitarity, because it allows information to be destroyed. If a particle is
thrown behind an event horizon, it can never be retrieved.
Ahn et al. have shown that the no-cloning theorem is violated in the presence
of closed timelike curves:
Section 6.1
Event horizons
6.1.3 Radiation from event horizons
b / Bill Unruh (1945-).
In interesting twist on the situation was introduced by Bill Unruh in 1976. Observer B aboard the accelerating spaceship believes
in the equivalence principle, so she knows that the local properties of space at the event horizon would seem entirely normal and
Lorentzian to a local observer A. (The same applies to a black hole’s
horizon.) In particular, B knows that A would see pairs of virtual
particles being spontaneously created and destroyed in the local vacuum. This is simply a manifestation of the time-energy form of the
uncertainty principle, ∆E∆t . h. Now suppose that a pair of particles is created, but one is created in front of the horizon and one
behind it. To A these are virtual particles that will have to be annihilated within the time ∆t, but according to B the one created in
front of the horizon will eventually catch up with the spaceship, and
can be observed there,p
althoughpit will be red-shifted. The amount
of redshift is given by gt0 0 t0 = (1 + ax0 )2 . Say the pair is created
right near the horizon, at x0 = −1/a. By the uncertainty principle, each of the two particles is spread out over a region of space
of size ∆x0 . Since these are photons, which travel at the speed of
light, the uncertainty in position is essentially the same as the uncertainty in time. The forward-going photon’s redshift comes out
to be a∆x0 = a∆t0 , which by the uncertainty principle should be at
least ha/E, so that when the photon is observed by B, its energy is
E(ha/E) = ha.
Now B sees a uniform background of photons, with energies of
around ha, being emitted randomly from the horizon. They are
being emitted from empty space, so it seems plausible to believe
that they don’t encode any information at all; they are completely
random. A surface emitting a completely random (i.e., maximumentropy) hail of photons is a black-body radiator, so we expect that
the photons will have a black-body spectrum, with its peak at an
energy of about ha. This peak is related to the temperature of
the black body by E ∼ kT , where k is Boltzmann’s constant. We
conclude that the horizon acts like a black-body radiator with a
temperature T ∼ ha/k. The more careful treatment by Unruh shows
that the exact relation is T = ha/4π 2 k, or ha/4π 2 kc in SI units.
An important observation here is that not only do different observers disagree about the number of quanta that are present (which
is true in the case of ordinary Doppler shifts), but about the number
of quanta in the vacuum as well. B sees photons that according to
A do not exist.
Let’s consider some real-world examples of large accelerations:
Chapter 6
Vacuum solutions
(m/s2 )
temperature of
horizon (K)
bullet fired from a gun
electron in a CRT
plasmas produced by intense
laser pulses
proton in a helium nucleus
To detect Unruh radiation experimentally, we would ideally like to
be able to accelerate a detector and let it detect the radiation. This
is clearly impractical. The third line shows that it is possible to
impart very large linear accelerations to subatomic particles, but
then one can only hope to infer the effect of the Unruh radiation
indirectly by its effect on the particles. As shown on the final line,
examples of extremely large nonlinear accelerations are not hard to
find, but the interpretation of Unruh radiation for nonlinear motion
is unclear. A summary of the prospects for direct experimental detection of this effect is given by Rosu.2 This type of experiment is
clearly extremely difficult, but it is one of the few ways in which one
could hope to get direct empirical insight, under controlled conditions, into the interface between gravity and quantum mechanics.
6.2 The Schwarzschild metric
We now set ourselves the goal of finding the metric describing the
static spacetime outside a spherically symmetric, nonrotating, body
of mass m. This problem was first solved by Karl Schwarzschild
in 1915.3 One byproduct of finding this metric will be the ability
to calculate the geodetic effect exactly, but it will have more farreaching consequences, including the existence of black holes.
The problem we are solving is similar to calculating the spherically symmetric solution to Gauss’s law in a vacuum. The solution to the electrical problem is of the form r̂/r2 , with an arbitrary
constant of proportionality that turns out to be proportional to
the charge creating the field. One big difference, however, is that
whereas Gauss’s law is linear, the equation Rab = 0 is highly nonlinear, so that the solution cannot simply be scaled up and down in
proportion to m.
The reason for this nonlinearity is fundamental to general relativity. For example, when the earth condensed out of the primordial
solar nebula, large amounts of heat were produced, and this energy
was then gradually radiated into outer space, decreasing the total
mass of the earth. If we pretend, as in figure a, that this process
“On the gravitational field of a point mass according to Einstein’s theory,” Sitzungsberichte der K oniglich Preussischen Akademie der Wissenschaften
1 (1916) 189. An English translation is available at
Section 6.2
a / The field equations of general
relativity are nonlinear.
The Schwarzschild metric
involved the merging of only two bodies, each with mass m, then
the net result was essentially to take separated masses m and m at
rest, and bring them close together to form close-neighbor masses
m and m, again at rest. The amount of energy radiated away was
proportional to m2 , so the gravitational mass of the combined system has been reduced from 2m to 2m − (. . .)m2 , where . . . is roughly
G/c2 r. There is a nonlinear dependence of the gravitational field on
the masses.
Self-check: The signature of a metric is defined as the list of
positive and negative signs that occur when it is diagonalized.4 The
equivalence principle requires that the signature be + − −− (or
− + ++, depending on the choice of sign conventions). Verify that
any constant metric (including a metric with the “wrong” signature,
e.g., 2+2 dimensions rather than 3+1) is a solution to the Einstein
field equation in vacuum.
The correspondence principle tells us that our result must have
a Newtonian limit, but the only variables involved are m and r, so
this limit must be the one in which r/m is large. Large compared
to what? There is nothing else available with which to compare,
so it can only be large compared to some expression composed of
the unitless constants G and c. We have already chosen units such
that c = 1, and we will now set G = 1 as well. Mass and distance
are now comparable, with the conversion factor being G/c2 = 7 ×
10−28 m/kg, or about a mile per solar mass. Since the earth’s radius
is thousands of times more than a mile, and its mass hundreds of
thousands of times less than the sun’s, its r/m is very large, and
the Newtonian approximation is good enough for all but the most
precise applications, such as the GPS network or the Gravity Probe
B experiment.
6.2.1 The zero-mass case
First let’s demonstrate the trivial solution with flat spacetime.
In spherical coordinates, we have
ds2 = dt2 − dr2 − r2 dθ2 − r2 sin2 θdφ2
The nonvanishing Christoffel symbols (ignoring swaps of the lower
indices) are:
Γθrθ =
Γ rφ =
Γ θθ = −r
Γrφφ = −r sin2 θ
Γθφφ = − sin θ cos θ
Γφθφ = cot θ
Chapter 6
See p. 234 for a different but closely related use of the same term.
Vacuum solutions
Self-check: If we’d been using the (− + ++) metric instead of
(+ − −−), what would have been the effect on the Christoffel symbols? What if we’d expressed the metric in different units, rescaling
all the coordinates by a factor k?
Use of ctensor
In fact, when I calculated the Christoffel symbols above by hand,
I got one of them wrong, and missed calculating one other because I
thought it was zero. I only found my mistake by comparing against
a result in a textbook. The computation of the Riemann tensor is
an even bigger mess. It’s clearly a good idea to resort to a computer algebra system here. Cadabra, which was discussed earlier, is
specifically designed for coordinate-independent calculations, so it
won’t help us here. A good free and open-source choice is ctensor,
which is one of the standard packages distributed along with the
computer algebra system Maxima, introduced on page 77.
The following Maxima program calculates the Christoffel symbols found in section 6.2.1.
Line 1 loads the ctensor package. Line 2 sets up the names of the
coordinates. Line 3 defines the gab , with lg meaning “the version of
g with lower indices.” Line 7 tells Maxima to do some setup work
with gab , including the calculation of the inverse matrix g ab , which
is stored in ug. Line 8 says to calculate the Christoffel symbols.
The notation mcs refers to the tensor Γ0 bca with the indices swapped
around a little compared to the convention Γabc followed in this
book. On a Linux system, we put the program in a file flat.mac
and run it using the command maxima -b flat.mac. The relevant
part of the output is:
= 2, 3, 3
= 2, 4, 4
Section 6.2
The Schwarzschild metric
= - r
3, 3, 2
= ---------3, 4, 4
4, 4, 2
= - r sin (theta)
= - cos(theta) sin(theta)
4, 4, 3
Adding the command ricci(true); at the end of the program results in the output THIS SPACETIME IS EMPTY AND/OR FLAT, which
saves us hours of tedious computation. The tensor ric (which here
happens to be zero) is computed, and all its nonzero elements are
printed out. There is a similar command riemann(true); to compute the Riemann rensor riem. This is stored so that riem[i,j,k,l]
is what we would call Rlikj . Note that l is moved to the end, and j
and k are also swapped.
6.2.2 Geometrized units
If the mass creating the gravitational field isn’t zero, then we
need to decide what units to measure it in. It has already proved
very convenient to adopt units with c = 1, and we will now also set
the gravitational constant G = 1. Previously, with only c set to 1,
the units of time and length were the same, [T ] = [L], and so were
the units of mass and energy, [M ] = [E]. With G = 1, all of these
become the same units, [T ] = [L] = [M ] = [E].
Self-check: Verify this statement by combining Newton’s law of
gravity with Newton’s second law of motion.
The resulting system is referred to as geometrized, because units
like mass that had formerly belonged to the province of mechanics
are now measured using the same units we would use to do geometry.
6.2.3 A large-r limit
Now let’s think about how to tackle the real problem of finding
the non-flat metric. Although general relativity lets us pick any coordinates we like, the spherical symmetry of the problem suggests
using coordinates that exploit that symmetry. The flat-space coordinates θ and φ can stil be defined in the same way, and they have
the same interpretation. For example, if we drop a test particle
toward the mass from some point in space, its world-line will have
constant θ and φ. The r coordinate is a little different. In curved
Chapter 6
Vacuum solutions
spacetime, the circumference of a circle is not equal to 2π times the
distance from the center to the circle; in fact, the discrepancy between these two is essentially the definition of the Ricci curvature.
This gives us a choice of two logical ways to define r. We’ll define it as the circumference divided by 2π, which has the advantage
that the last two terms of the metric are the same as in flat space:
−r2 dθ2 − r2 sin2 θdφ2 . Since we’re looking for static solutions, none
of the elements of the metric can depend on t. Also, the solution is
going to be symmetric under t → −t, θ → −θ, and φ → −φ, so we
can’t have any off-diagonal elements.5 The result is that we have
narrowed the metric down to something of the form
ds2 = h(r)dt2 − k(r)dr2 − r2 dθ2 − r2 sin2 θdφ2
where both h and k approach 1 for r → ∞, where spacetime is flat.
For guidance in how to construct h and k, let’s consider the
acceleration of a test particle at r m, which we know to be
−m/r2 , since nonrelativistic physics applies there. We have
∇t v r = ∂t v r + Γrtc v c
An observer free-falling along with the particle observes its acceleration to be zero, and a tensor that is zero in one coordinate system
is zero in all others. Since the covariant derivative is a tensor, we
conclude that ∇t v r = 0 in all coordinate systems, including the
(t, r, . . .) system we’re using. If the particle is released from rest,
then initially its velocity four-vector is (1, 0, 0, 0), so we find that its
acceleration in (t, r) coordinates is −Γrtt = − 21 g rr ∂r gtt = − 12 h0 /k.
Setting this equal to −m/r2 , we find h0 /k = 2m/r2 for r m.
Since k ≈ 1 for large r, we have
h0 ≈
for r m
The interpretation of this calculation is as follows. We assert the
equivalence principle, by which the acceleration of a free-falling particle can be said to be zero. After some calculations, we find that
the rate at which time flows (encoded in h) is not constant. It is
different for observers at different heights in a gravitational potential well. But this is something we had already deduced, without
the index gymnastics, in example 7 on page 129.
Integrating, we find that for large r, h = 1 − 2m/r.
6.2.4 The complete solution
A series solution
We’ve learned some interesting things, but we still have an extremely nasty nonlinear differential equation to solve. One way to
For more about time-reversal symmetry, see p. 211.
Section 6.2
The Schwarzschild metric
attack a differential equation, when you have no idea how to proceed, is to try a series solution. We have a small parameter m/r to
expand around, so let’s try to write h and k as series of the form
m n
h = Σ∞
n=0 k
k = Σn=0 bk
We already know a0 , a1 , and b0 . Let’s try to find b1 . In the
following Maxima code I omit the factor of m in h1 for convenience.
In other words, we’re looking for the solution for m = 1.
I won’t reproduce the entire output of the Ricci tensor, which
is voluminous. We want all four of its nonvanishing components to
vanish as quickly as possible for large values of r, so I decided to
fiddle with Rtt , which looked as simple as any of them. It
to vary as r−4 for large r, so let’s evaluate limr→∞ r4 Rtt :
The result is (b1 −2)/2, so let’s set b1 = 2. The approximate solution
we’ve found so far (reinserting the m’s),
ds ≈ 1 −
dt − 1 +
dr2 − r2 dθ2 − r2 sin2 θdφ2
was first derived by Einstein in 1915, and he used it to solve the
problem of the non-Keplerian relativistic correction to the orbit of
Mercury, which was one of the first empirical tests of general relativity.
Continuing in this fashion, the results are as follows:
= −2
The closed-form solution
The solution is unexpectedly simple, and can be put into closed
form. The approximate result we found for h was in fact exact.
Chapter 6
Vacuum solutions
For k we have a geometric series 1/(1 − 2/r), and when we reinsert
the factor of m in the only way that makes the units work, we get
1/(1 − 2m/r). The result for the metric is
ds = 1 −
dt −
dr2 − r2 dθ2 − r2 sin2 θdφ2
1 − 2m/r
This is called the Schwarzschild metric. A quick calculation in Maxima demonstrates that it is an exact solution for all r, i.e., the Ricci
tensor vanishes everywhere, even at r < 2m, which is outside the
radius of convergence of the geometric series.
Time-reversal symmetry
The Schwarzschild metric is invariant under time reversal, since
time occurs only in the form of dt2 , which stays the same under
dt → −dt. This is the same time-reversal symmetry that occurs in
Newtonian gravity, where the field is described by the gravitational
acceleration g, and accelerations are time-reversal invariant.
Fundamentally, this is an example of general relativity’s coordinate independence. The laws of physics provided by general relativity, such as the vacuum field equation, are invariant under any
smooth coordinate transformation, and t → −t is such a coordinate
transformation, so general relativity has time-reversal symmetry.
Since the Schwarzschild metric was found by imposing time-reversalsymmetric boundary conditions on a time-reversal-symmetric differential equation, it is an equally valid solution when we time-reverse
it. Furthermore, we expect the metric to be invariant under time
reversal, unless spontaneous symmetry breaking occurs (see p. 314).
This suggests that we ask the more fundamental question of what
global symmetries general relativity has. Does it have symmetry
under parity inversion, for example? Or can we take any solution
such as the Schwarzschild spacetime and transform it into a frame
of reference in which the source of the field is moving uniformly in a
certain direction? Because general relativity is locally equivalent to
special relativity, we know that these symmetries are locally valid.
But it may not even be possible to define the corresponding global
symmetries. For example, there are some spacetimes on which it
is not even possible to define a global time coordinate. On such a
spacetime, which is described as not time-orientable, there does not
exist any smooth vector field that is everywhere timelike, so it is
not possible to define past versus future light-cones at all points in
space without having a discontinuous change in the definition occur
somewhere. This is similar to the way in which a Möbius strip does
not allow an orientation of its surface (an “up” direction as seen by
an ant) to be defined globally.
Suppose that our spacetime is time-orientable, and we are able
to define coordinates (p, q, r, s) such that p is always the timelike
coordinate. Because q → −q is a smooth coordinate transforma-
Section 6.2
The Schwarzschild metric
tion, we are guaranteed that our spacetime remains a valid solution
of the field equations under this change. But that doesn’t mean
that what we’ve found is a symmetry under parity inversion in a
plane. Our coordinate q is not necessarily interpretable as distance
along a particular “q axis.” Such axes don’t even exist globally in
general relativity. A coordinate does not even have to have units
of time or distance; it could be an angle, for example, or it might
not have any geometrical significance at all. Similarly, we could do
a transformation q → q 0 = q + kp. If we think of q as measuring
spatial position and p time, then this looks like a Galilean transformation, with k being the velocity. The solution to the field equations
obtained after performing this transformation is still a valid solution, but that doesn’t mean that relativity has Galilean symmetry
rather than Lorentz symmetry. There is no sensible way to define
a Galilean transformation acting on an entire spacetime, because
when we talk about a Galilean transformation we assume the existence of things like global coordinate axes, which do not even exist
in general relativity.
6.2.5 Geodetic effect
As promised in section 5.5.1, we now calculate the geodetic effect
on Gravity Probe B, including all the niggling factors of 3 and π. To
make the physics clear, we approach the actual calculation through
a series of warmups.
Flat space
As a first warmup, consider two spatial dimensions, represented
by Euclidean polar coordinates (r, φ). Parallel-transport of a gyroscope’s angular momentum around a circle of constant r gives
∇φ Lφ = 0
∇φ Lr = 0
Computing the covariant derivatives, we have
0 = ∂φ Lφ + Γφφr Lr
0 = ∂φ Lr + Γrφφ Lφ
The Christoffel symbols are Γφφr = 1/r and Γrφφ = −r. This is
all made to look needlessly complicated because Lφ and Lr are expressed in different units. Essentially the vector is staying the same,
but we’re expressing it in terms of basis vectors in the r and φ directions that are rotating. To see this more transparently, let r = 1,
and write P for Lφ and Q for Lr , so that
P 0 = −Q
Q0 = P
which have solutions such as P = sin φ, Q = cos φ. For each orbit
(2π change in φ), the basis vectors rotate by 2π, so the angular
Chapter 6
Vacuum solutions
momentum vector once again has the same components. In other
words, it hasn’t really changed at all.
Spatial curvature only
The flat-space calculation above differs in two ways from the
actual result for an orbiting gyroscope: (1) it uses a flat spatial
geometry, and (2) it is purely spatial. The purely spatial nature of
the calculation is manifested in the fact that there is nothing in the
result relating to how quickly we’ve moved the vector around the
circle. We know that if we whip a gyroscope around in a circle on
the end of a rope, there will be a Thomas precession (section 2.5.4),
which depends on the speed.
As our next warmup, let’s curve the spatial geometry, but continue to omit the time dimension. Using the Schwarzschild metric,
we replace the flat-space Christoffel symbol Γrφφ = −r with −r+2m.
The differential equations for the components of the L vector, again
evaluated at r = 1 for convenience, are now
P 0 = −Q
Q0 = (1 − )P
where = 2m. The solutions rotate with frequency ω 0 = 1 − .
The result is that when the basis vectors rotate by 2π, the components
√ no longer return to their original values; they lag by a factor
of 1 − ≈ 1 − m. Putting the factors of r back in, this is 1 − m/r.
The deviation from unity shows that after one full revolution, the L
vector no longer has quite the same components expressed in terms
of the (r, φ) basis vectors.
To understand the sign of the effect, let’s imagine a counterclockwise rotation. The (r, φ) rotate counterclockwise, so relative
to them, the L vector rotates clockwise. After one revolution, it has
not rotated clockwise by a full 2π, so its orientation is now slightly
counterclockwise compared to what it was. Thus the contribution
to the geodetic effect arising from spatial curvature is in the same
direction as the orbit.
Comparing with the actual results from Gravity Probe B, we see
that the direction of the effect is correct. The magnitude, however,
is off. The precession accumulated over n periods is 2πnm/r, or,
in SI units, 2πnGm/c2 r. Using the data from section 2.5.4, we find
∆θ = 2 × 10−5 radians, which is too small compared to the data
shown in figure b on page 171.
2+1 Dimensions
To reproduce the experimental results correctly, we need to include the time dimension. The angular momentum vector now has
components (Lφ , Lr , Lt ). The physical interpretation of the Lt component is obscure at this point; we’ll return to this question later.
Section 6.2
The Schwarzschild metric
Writing down the total derivatives of the three components, and
notating dt/dφ as ω −1 , we have
= ∂φ Lφ + ω −1 ∂t Lφ
= ∂φ Lr + ω −1 ∂t Lr
= ∂φ Lt + ω −1 ∂t Lt
Setting the covariant derivatives equal to zero gives
0 = ∂φ Lφ + Γφφr Lr
0 = ∂φ Lr + Γrφφ Lφ
0 = ∂t Lr + Γrtt Lt
0 = ∂t Lt + Γttr Lr
Self-check: There are not just four but six covariant derivatives
that could in principle have occurred, and in these six covariant
derivatives we could have had a total of 18 Christoffel symbols. Of
these 18, only four are nonvanishing. Explain based on symmetry
arguments why the following Christoffel symbols must vanish: Γφφt ,
Γttt .
Putting all this together in matrix form, we have L0 = M L,
−(1 − )/2ω 
M = 1−
−/2ω(1 − )
The solutions of this differential equation oscillate like eiΩt , where
iΩ is an eigenvalue of the matrix.
Self-check: The frequency in the purely spatial calculation was
found by inspection. Verify the result by applying the eigenvalue
technique to the relevant 2 × 2 submatrix.
To lowest order, we can use the Newtonian relation ω 2 r = Gm/r
and neglect terms of order 2 , so that theptwo new off-diagonal matrix elements are both approximated as /2. The three resulting
eigenfrequencies are zero and Ω = ±[1 − (3/2)m/r].
The presence of the mysterious zero-frequency solution can now
be understood by recalling the earlier mystery of the physical interpretation of the angular momentum’s Lt component. Our results
come from calculating parallel transport, and parallel transport is
a purely geometric process, so it gives the same result regardless
of the physical nature of the four-vector. Suppose that we had
instead chosen the velocity four-vector as our guinea pig. The definition of a geodesic is that it parallel-transports its own tangent
Chapter 6
Vacuum solutions
vector, so the velocity vector has to stay constant. If we inspect
the eigenvector corresponding to the zero-frequency eigenfrequency,
we find a timelike vector that is parallel to the velocity four-vector.
In our 2+1-dimensional space, the other two eigenvectors, which
are spacelike, span the subspace of spacelike vectors, which are the
ones that can physically be realized as the angular momentum of
a gyroscope. These two eigenvectors, which vary as e±iΩ , can be
superposed to make real-valued spacelike solutions that match the
initial conditions, and these lag the rotation of the basis vectors
by ∆Ω = (3/2)mr. This is greater than the purely spatial result
by a factor of 3/2. The resulting precession angle, over n orbits of
Gravity Probe B, is 3πnGm/c2 r = 3 × 10−5 radians, in excellent
agreement with experiment.
One will see apparently contradictory statements in the literature about whether Thomas precession occurs for a satellite: “The
Thomas precession comes into play for a gyroscope on the surface
of the Earth . . . , but not for a gyroscope in a freely moving satellite.”6 But: “The total effect, geometrical and Thomas, gives the
well-known Fokker-de Sitter precession of 3πm/r, in the same sense
as the orbit.”7 The second statement arises from subtracting the
purely spatial result from the 2+1-dimensional result, and noting
that the absolute value of this difference is the same as the Thomas
precession that would have been obtained if the gyroscope had been
whirled at the end of a rope. In my opinion this is an unnatural
way of looking at the physics, for two reasons. (1) The signs don’t
match, so one is forced to say that the Thomas precession has a
different sign depending on whether the rotation is the result of
gravitational or nongravitational forces. (2) Referring to observation, it is clearly artificial to treat the spatial curvature and Thomas
effects separately, since neither one can be disentangled from the
other by varying the quantities n, m, and r. For more discussion,
6.2.6 Orbits
The main event of Newton’s Principia Mathematica is his proof
of Kepler’s laws. Similarly, Einstein’s first important application in
general relativity, which he began before he even had the exact form
of the Schwarzschild metric in hand, was to find the non-Newtonian
behavior of the planet Mercury. The planets deviate from Keplerian
behavior for a variety of Newtonian reasons, and in particular there
is a long list of reasons why the major axis of a planet’s elliptical
orbit is expected to gradually rotate. When all of these were taken
into account, however, there was a remaining discrepancy of about
40 seconds of arc per century, or 6.6 × 10−7 radians per orbit. The
direction of the effect was in the forward direction, in the sense that
Misner, Thorne, and Wheeler, Gravitation, p. 1118
Rindler, Essential Relativity, 1969, p. 141
Section 6.2
The Schwarzschild metric
if we view Mercury’s orbit from above the ecliptic, so that it orbits
in the counterclockwise direction, then the gradual rotation of the
major axis is also counterclockwise. In other words, Mercury spends
more time near perihelion than it should nonrelativistically. During this time, it sweeps out a greater angle than nonrelativistically
expected, so that when it flies back out and away from the sun, its
orbit has rotated counterclockwise.
We can at least qualitatively understand the reason for such an
effect based on the spatial part of the curvature of the spacetime
surrounding the sun. This spatial curvature is positive, so a circle’s
circumference is less than 2π times its radius. This causes Mercury
to get back to a previously visited angular position before it has had
time to complete its Newtonian cycle of radial motion.
Based on the examples in section 5.5, we also expect that the
effect will be of order m/r, where m is the mass of the sun and r
is the radius of Mercury’s orbit. This works out to be 2.5 × 10−8 ,
which is smaller than the observed precession by a factor of about
Conserved quantities
If Einstein had had a computer on his desk, he probably would
simply have integrated the motion numerically using the geodesic
equation. But it is possible to simplify the problem enough to attack it with pencil and paper, if we can find the relevant conserved
quantities of the motion. Nonrelativistically, these are energy and
angular momentum.
Consider a rock falling directly toward the sun. The Schwarzschild
metric is of the special form
ds2 = h(r)dt2 − k(r)dr2 − . . .
The rock’s trajectory is a geodesic, so it extremizes the proper time
s between any two events fixed in spacetime, just as a piece of string
stretched across a curved surface extremizes its length. Let the rock
pass through distance r1 in coordinate time t1 , and then through r2
in t2 . (These should really be notated as ∆r1 , . . . or dr1 , . . . , but we
avoid the ∆’s or d’s for convenience.) Approximating the geodesic
using two line segments, the proper time is
s = s1 + s2
= h1 t21 − k1 r12 + h2 t22 − k2 r22
= h1 t21 − k1 r12 + h2 (T − t1 )2 − k2 r22
where T = t1 + t2 is fixed. If this is to be extremized with respect
to t1 , then ds/dt1 = 0, which leads to
Chapter 6
Vacuum solutions
h1 t1 h2 t2
which means that
= gtt
is a constant of the motion. Except for an irrelevant factor of m,
this is the same as pt , the timelike component of the covariant momentum vector. We’ve already seen that in special relativity, the
timelike component of the momentum four-vector is interpreted as
the mass-energy E, and the quantity pt has a similar interpretation
here. Note that no special assumption was made about the form of
the functions h and k. In addition, it turns out that the assumption
of purely radial motion was unnecessary. All that really mattered
was that h and k were independent of t. Therefore we will have
a similar conserved quantity pµ any time the metric’s components,
expressed in a particular coordinate system, are independent of xµ .
(This is generalized on p. 246.) In particular, the Schwarzschild
metric’s components are independent of φ as well as t, so we have a
second conserved quantity pφ , which is interpreted as angular momentum.
b / Proof that if the metric’s
components are independent of
t , the geodesic of a test particle
conserves pt .
Writing these two quantities out explicitly in terms of the contravariant coordinates, in the case of the Schwarzschild spacetime,
we have
2m dt
E = 1−
L = r2
for the conserved energy per unit mass and angular momentum per
unit mass.
In interpreting the energy per unit mass E, it is important to
understand that in the general-relativistic context, there is no useful way of separating the rest mass, kinetic energy, and potential
energy into separate terms, as we could in Newtonian mechanics.
E includes contributions from all of these, and turns out to be less
than the contribution due to the rest mass (i.e., less than 1) for a
planet orbiting the sun. It turns out that E can be interpreted as a
measure of the additional gravitational mass that the solar system
possesses as measured by a distant observer, due to the presence of
the planet. It then makes sense that E is conserved; by analogy
with Newtonian mechanics, we would expect that any gravitational
effects that depended on the detailed arrangement of the masses
within the solar system would decrease as 1/r4 , becoming negligible
at large distances and leaving a constant field varying as 1/r2 .
One way of seeing that it doesn’t make sense to split E into parts
is that although the equation given above for E involves a specific set
of coordinates, E can actually be expressed as a Lorentz-invariant
Section 6.2
The Schwarzschild metric
scalar (see p. 246). This property makes E especially interesting and
useful (and different from the energy in Newtonian mechanics, which
is conserved but not frame-independent). On the other hand, the
kinetic and potential energies depend on the velocity and position.
These are completely dependent on the coordinate system, and there
is nothing physically special about the coordinate system we’ve used
here. Suppose a particle is falling directly toward the earth, and an
astronaut in a space-suit is free-falling along with it and monitoring
its progress. The astronaut judges the particle’s kinetic energy to
be zero, but other observers say it’s nonzero, so it’s clearly not a
Lorentz scalar. And suppose the astronaut insists on defining a
potential energy to go along with this kinetic energy. The potential
energy must be decreasing, since the particle is getting closer to the
earth, but then there is no way that the sum of the kinetic and
potential energies could be constant.
Perihelion advance
For convenience, let the mass of the orbiting rock be 1, while m
stands for the mass of the gravitating body.
The unit mass of the rock is a third conserved quantity, and
since the magnitude of the momentum vector equals the square of
the mass, we have for an orbit in the plane θ = π/2,
1 = g tt p2t − g rr p2r − g φφ p2φ
= g tt p2t − grr (pr )2 − g φφ p2φ
E2 −
1 − 2m/r
1 − 2m/r
1 2
Rearranging terms and writing ṙ for dr/ds, this becomes
ṙ2 = E 2 − (1 − 2m/r)(1 + L2 /r2 )
ṙ2 = E 2 − U 2
U 2 = (1 − 2m/r)(1 + L2 /r2 )
There is a varied and strange family of orbits in the Schwarzschild
field, including bizarre knife-edge trajectories that take several nearly
circular turns before suddenly flying off. We turn our attention instead to the case of an orbit such as Mercury’s which is nearly
Newtonian and nearly circular.
Nonrelativistically, a circular orbit has radius r = L2 /m and
period T = 2πL3 /m2 .
Chapter 6
Vacuum solutions
Relativistically, a circular orbit occurs when there is only one
turning point at which ṙ = 0. This requires that E 2 equal the
minimum value of U 2 , which occurs at
L2 1 + 1 − 12m2 /L2
(1 − )
where = 3(m/L)2 . A planet in a nearly circular orbit oscillates
between perihelion and aphelion with a period that depends on the
curvature of U 2 at its minimum. We have
d2 (U 2 )
2m L2 2mL2
= 2 1−
+ 2 −
4m 6L2 24mL2
=− 3 + 4 −
−6 4
= 2L m (1 + 2)
The period of the oscillations is
∆sosc = 2π 2/k
= 2πL3 m−2 (1 − )
The period of the azimuthal motion is
∆saz = 2πr2 /L
= 2πL3 m−2 (1 − 2)
The periods are slightly mismatched because of the relativistic correction terms. The period of the radial oscillations is longer, so
that, as expected, the perihelion shift is in the forward direction.
The mismatch is ∆s, and because of it each orbit rotates the major axis by an angle 2π = 6π(m/L)2 = 6πm/r. Plugging in the
data for Mercury, we obtain 5.8 × 10−7 radians per orbit, which
agrees with the observed value to within about 10%. Eliminating
some of the approximations we’ve made brings the results in agreement to within the experimental error bars, and Einstein recalled
that when the calculation came out right, “for a few days, I was
beside myself with joyous excitement.”
Further attempts were made to improve on the precision of this
historically crucial test of general relativity. Radar now gives the
most precise orbital data for Mercury. At the level of about one part
per thousand, however, an effect creeps in due to the oblateness of
the sun, which is difficult to measure precisely.
In 1974, astronomers J.H. Taylor and R.A. Hulse of Princeton,
working at the Arecibo radio telescope, discovered a binary star
Section 6.2
The Schwarzschild metric
system whose members are both neutron stars. The detection of
the system was made possible because one of the neutron stars is
a pulsar: a neutron star that emits a strong radio pulse in the
direction of the earth once per rotational period. The orbit is highly
elliptical, and the minimum separation between the two stars is very
small, about the same as the radius of our sun. Both because the
r is small and because the period is short (about 8 hours), the rate
of perihelion advance per unit time is very large, about 4.2 degrees
per year. The system has been compared in great detail with the
predictions of general relativity,8 giving extremely good agreement,
and as a result astronomers have been confident enough to reason in
the opposite direction and infer properties of the system, such as its
total mass, from the general-relativistic analysis. The system’s orbit
is decaying due to the radiation of energy in the form of gravitational
waves, which are predicted to exist by relativity.
6.2.7 Deflection of light
As discussed on page 171, one of the first tests of general relativity was Eddington’s measurement of the deflection of rays of
light by the sun’s gravitational field. The deflection measured by
Eddington was 1.6 seconds of arc. For a light ray that grazes the
sun’s surface, the only physically relevant parameters are the sun’s
mass m and radius r. Since the deflection is unitless, it can only
depend on m/r, the unitless ratio of the sun’s mass to its radius.
Expressed in SI units, this is Gm/c2 r, which comes out to be about
10−6 . Roughly speaking, then, we expect the order of magnitude of
the effect to be about this big, and indeed 10−6 radians comes out
to be in the same ball-park as a second of arc. We get a similar
estimate in Newtonian physics by treating a photon as a (massive)
particle moving at speed c.
It is possible to calculate a precise value for the deflection using methods very much like those used to determine the perihelion
advance in section 6.2.6. However, some of the details would have
to be changed. For example, it is no longer possible to parametrize
the trajectory using the proper time s, since a light ray has ds = 0;
we must use an affine parameter. Let us instead use this an an
example of the numerical technique for solving the geodesic equation, first demonstrated in section 5.9.2 on page 188. Modifying our
earlier program, we have the following:
import math
# constants, in SI units:
G = 6.67e-11
# gravitational constant
c = 3.00e8
# speed of light
m_kg = 1.99e30
# mass of sun
Chapter 6
Vacuum solutions
r_m = 6.96e8
# radius of sun
# From now on, all calculations are in units of the
# radius of the sun.
# mass of sun, in units of the radius of the sun:
m_sun = (G/c**2)*(m_kg/r_m)
m = 1000.*m_sun
print "m/r=",m
# Start at point of closest approach.
# initial position:
r=1 # closest approach, grazing the sun’s surface
# initial derivatives of coordinates w.r.t. lambda
vr = 0
vt = 1
vphi = math.sqrt((1.-2.*m/r)/r**2)*vt # gives ds=0, lightlike
l = 0
# affine parameter lambda
l_max = 20000.
epsilon = 1e-6 # controls how fast lambda varies
while l<l_max:
dl = epsilon*(1.+r**2) # giant steps when farther out
l = l+dl
# Christoffel symbols:
Gttr = m/(r**2-2*m*r)
Grtt = m/r**2-2*m**2/r**3
Grrr = -m/(r**2-2*m*r)
Grphiphi = -r+2*m
Gphirphi = 1/r
# second derivatives:
# The factors of 2 are because we have, e.g., G^a_{bc}=G^a_{cb}
= -2.*Gttr*vt*vr
= -(Grtt*vt*vt + Grrr*vr*vr + Grphiphi*vphi*vphi)
aphi = -2.*Gphirphi*vr*vphi
# update velocity:
vt = vt + dl*at
vr = vr + dl*ar
vphi = vphi + dl*aphi
# update position:
r = r + vr*dl
t = t + vt*dl
phi = phi + vphi*dl
# Direction of propagation, approximated in asymptotically flat coords.
# First, differentiate (x,y)=(r cos phi,r sin phi) to get vx and vy:
Section 6.2
The Schwarzschild metric
vx = vr*math.cos(phi)-r*math.sin(phi)*vphi
vy = vr*math.sin(phi)+r*math.cos(phi)*vphi
prop = math.atan2(vy,vx) # inverse tan of vy/vx, in the proper quadra
prop_sec = prop*180.*3600/math.pi
print "final direction of propagation = %6.2f arc-seconds" % prop_sec
At line 14, we take the mass to be 1000 times greater than the
mass of the sun. This helps to make the deflection easier to calculate accurately without running into problems with rounding errors.
Lines 17-25 set up the initial conditions to be at the point of closest
approach, as the photon is grazing the sun. This is easier to set
up than initial conditions in which the photon approaches from far
away. Because of this, the deflection angle calculated by the program is cut in half. Combining the factors of 1000 and one half, the
final result from the program is to be interpreted as 500 times the
actual deflection angle.
The result is that the deflection angle is predicted to be 870
seconds of arc. As a check, we can run the program again with
m = 0; the result is a deflection of −8 seconds, which is a measure
of the accumulated error due to rounding and the finite increment
used for λ.
Dividing by 500, we find that the predicted deflection angle is
1.74 seconds, which, expressed in radians, is exactly 4Gm/c2 r. The
unitless factor of 4 is in fact the correct result in the case of small
deflections, i.e., for m/r 1.
Although the numerical technique has the disadvantage that it
doesn’t let us directly prove a nice formula, it has some advantages
as well. For one thing, we can use it to investigate cases for which
the approximation m/r 1 fails. For m/r = 0.3, the numerical
techique gives a deflection of 222 degrees, whereas the weak-field
approximation 4Gm/c2 r gives only 69 degrees. What is happening
here is that we’re getting closer and closer to the event horizon of a
black hole. Black holes are the topic of section 6.3, but it should be
intuitively reasonable that something wildly nonlinear has to happen
as we get close to the point where the light wouldn’t even be able
to escape.
The precision of Eddington’s original test was only about ± 30%,
and has never been improved on significantly with visible-light astronomy. A better technique is radio astronomy, which allows measurements to be carried out without waiting for an eclipse. One
merely has to wait for the sun to pass in front of a strong, compact
radio source such as a quasar. These techniques have now verified
the deflection of light predicted by general relativity to a relative
precision of about 10−5 .9
For a review article on this topic, see Clifford Will, “The Confrontation between General Relativity and Experiment,” http://relativity.
Chapter 6
Vacuum solutions
6.3 Black holes
6.3.1 Singularities
A provocative feature of the Schwarzschild metric is that it has
elements that blow up at r = 0 and at r = 2m. If this is a description
of the sun, for example, then these singularities are of no physical
significance, since we only solved the Einstein field equation for the
vacuum region outside the sun, whereas r = 2m would lie about 3
km from the sun’s center. Furthermore, it is possible that one or
both of these singularities is nothing more than a spot where our
coordinate system misbehaves. This would be known as a coordinate
singularity. For example, the metric of ordinary polar coordinates
in a Euclidean plane has g θθ → ∞ as r → 0.
One way to test whether a singularity is a coordinate singularity
is to calculate a scalar measure of curvature, whose value is independent of the coordinate system. We can take the trace of the
Ricci tensor, Raa , known as the scalar curvature or Ricci scalar,
but since the Ricci tensor is zero, it’s not surprising that that is
zero. A different scalar we can construct is the product Rabcd Rabcd
of the Riemann tensor with itself. This is known as the Kretchmann invariant. The Maxima command lriemann(true) displays
the nonvanishing components of Rabcd . The component that misbehaves the most severely at r = 0 is Rtrrt = 2m/r3 . Because of this,
the Kretchmann invariant blows up like r−6 as r → 0. This shows
that the singularity at r = 0 is a real, physical singularity.
The singularity at r = 2m, on the other hand, turns out to
be only a coordinate singularity. To prove this, we have to use
some technique other than constructing scalar measures of curvature. Even if every such scalar we construct is finite at r = 2m, that
doesn’t prove that every such scalar we could construct is also well
behaved. We can instead search for some other coordinate system
in which to express the solution to the field equations, one in which
no such singularity appears. A partially successful change of coordinates for the Schwarzschild metric, found by Eddington in 1924, is
t → t0 = t − 2m ln(r − 2m) (see problem 7 on page 223). This makes
the covariant metric finite at r = 2m, although the contravariant
metric still blows up there. A more complicated change of coordinates that completely eliminates the singularity at r = 2m was
found by Eddington and Finkelstein in 1958, establishing that the
singularity was only a coordinate singularity. Thus, if an observer
is so unlucky as to fall into a black hole, he will not be subjected
to infinite tidal stresses — or infinite anything — at r = 2m. He
may not notice anything special at all about his local environment.
(Or he may already be dead because the tidal stresses at r > 2m,
although finite, were nevertheless great enough to kill him.)
Section 6.3
Black holes
6.3.2 Event horizon
Even though r = 2m isn’t a real singularity, interesting things
do happen there. For r < 2m, the sign of gtt becomes negative,
while grr is positive. In our + − −− signature, this has the following interpretation. For the world-line of a material particle, ds2
is supposed to be the square of the particle’s proper time, and it
must always be positive. If a particle had a constant value of r, for
r < 2m, it would have ds2 < 0, which is impossible.
The timelike and spacelike characters of the r and t coordinates
have been swapped, so r acts like a time coordinate.
Thus for an object compact enough that r = 2m is exterior,
r = 2m is an event horizon: future light cones tip over so far that
they do not allow causal relationships to connect with the spacetime
outside. In relativity, event horizons do not occur only in the context
of black holes; their properties, and some of the implications for
black holes, have already been discussed in section 6.1.
The gravitational time dilation in the Schwarzschild field, relative to a clock at infinity, is given by the square root of the gtt
component of the metric. This goes to zero at the event horizon,
meaning that, for example, a photon emitted from the event horizon
will be infinitely redshifted when it reaches an observer at infinity.
This makes sense, because the photon is then undetectable, just as
it would be if it had been emitted from inside the event horizon.
If matter is falling into a black hole, then due to time dilation an
observer at infinity “sees” that matter as slowing down more and
more as it approaches the horizon. This has some counterintuitive
effects. A radially infalling particle has d2 r/dt2 > 0 once it falls
past a certain point, which could be interpreted as a gravitational
repulsion. The observer at infinity may also be led to describe the
black hole as consisting of an empty, spherical shell of matter that
never quite made it through the horizon. If asked what holds the
shell up, the observer could say that it is held up by gravitational
There is actually nothing wrong with any of this, but one should
realize that it is only one possible description in one possible coordinate system. An observer hovering just outside the event horizon
sees a completely different picture, with matter falling past at velocities that approach the speed of light as it comes to the event
horizon. If an atom emits a photon from the event horizon, the
hovering observer sees it as being infinitely red-shifted, but explains
the red-shift as a kinematic one rather than a gravitational one.
We can imagine yet a third observer, one who free-falls along
with the infalling matter. According to this observer, the gravitational field is always zero, and it takes only a finite time to pass
through the event horizon.
Chapter 6
Vacuum solutions
If a black hole has formed from the gravitational collapse of a
cloud of matter, then some of our observers can say that “right now”
the matter is located in a spherical shell at the event horizon, while
others can say that it is concentrated at an infinitely dense singularity at the center. Since simultaneity isn’t well defined in relativity,
it’s not surprising that they disagree about what’s happening “right
now.” Regardless of where they say the matter is, they all agree on
the spacetime curvature. In fact, Birkhoff’s theorem (p. 251) tells us
that any spherically symmetric vacuum spacetime is Schwarzschild
in form, so it doesn’t matter where we say the matter is, as long as
it’s distributed in a spherically symmetric way and surrounded by
6.3.3 Expected formation
Einstein and Schwarzschild did not believe, however, that any of
these features of the Schwarzschild metric were more than a mathematical curiosity, and the term “black hole” was not invented until
the 1967, by John Wheeler. Although there is quite a bit of evidence these days that black holes do exist, there is also the related
question of what sizes they come in.
We might expect naively that since gravity is an attractive force,
there would be a tendency for any primordial cloud of gas or dust
to spontaneously collapse into a black hole. But clouds of less than
about 0.1M (0.1 solar masses) form planets, which achieve a permanent equilibrium between gravity and internal pressure. Heavier
objects initiate nuclear fusion, but those with masses above about
100M are immediately torn apart by their own solar winds. In the
range from 0.1 to 100M , stars form. As discussed in section 4.4.3,
those with masses greater than about a few M are expected to
form black holes when they die. We therefore expect, on theoretical
grounds, that the universe should contain black holes with masses
ranging from a few solar masses to a few tens of solar masses.
6.3.4 Observational evidence
A black hole is expected to be a very compact object, with a
strong gravitational field, that does not emit any of its own light. A
bare, isolated black hole would be difficult to detect, except perhaps
via its lensing of light rays that happen to pass by it. But if a black
hole occurs in a binary star system, it is possible for mass to be
transferred onto the black hole from its companion, if the companion’s evolution causes it to expand into a giant and intrude upon
the black hole’s gravity well. The infalling gas would then get hot
and emit radiation before disappearing behind the event horizon.
The object known as Cygnus X-1 is the best-studied example. This
X-ray-emitting object was discovered by a rocket-based experiment
in 1964. It is part of a double-star system, the other member being
a blue supergiant. They orbit their common center of mass with
a period of 5.6 days. The orbit is nearly circular, and has a semi-
Section 6.3
Black holes
a / A black hole accretes matter
from a companion star.
major axis of about 0.2 times the distance from the earth to the sun.
Applying Kepler’s law of periods to these data constrains the sum
of the masses, and knowledge of stellar structure fixes the mass of
the supergiant. The result is that the mass of Cygnus X-1 is greater
than about 10 solar masses, and this is confirmed by multiple methods. Since this is far above the Tolman-Oppenheimer-Volkoff limit,
Cygnus X-1 is believed to be a black hole, and its X-ray emissions
are interpreted as the radiation from the disk of superheated material accreting onto it from its companion. It is believed to have
more than 90% of the maximum possible spin for a black hole of its
Around the turn of the 21st century, new evidence was found
for the prevalence of supermassive black holes near the centers of
nearly all galaxies, including our own. Near our galaxy’s center is
an object called Sagittarius A*, detected because nearby stars orbit
around it. The orbital data show that Sagittarius A* has a mass
of about four million solar masses, confined within a sphere with
a radius less than 2.2 × 107 km. There is no known astrophysical
model that could prevent the collapse of such a compact object into
a black hole, nor is there any plausible model that would allow this
much mass to exist in equilibrium in such a small space, without
emitting enough light to be observable.
The existence of supermassive black holes is surprising. Gas
clouds with masses greater than about 100 solar masses cannot form
stable stars, so supermassive black holes cannot be the end-point of
the evolution of heavy stars. Mergers of multiple stars to form more
massive objects are generally statistically unlikely, since a star is
such a small target in relation to the distance between the stars.
Once astronomers were confronted with the empirical fact of their
existence, a variety of mechanisms was proposed for their formation.
Little is known about which of these mechanisms is correct, although
the existence of quasars in the early universe is interpreted as evidence that mass accreted rapidly onto supermassive black holes in
the early stages of the evolution of the galaxies.
A skeptic could object that although Cygnus X-1 and Sagittarius A* are more compact than is believed possible for a neutron
star, this does not necessarily prove that they are black holes. Indeed, speculative theories have been proposed in which exotic objects could exist that are intermediate in compactness between black
holes and neutron stars. These hypothetical creatures have names
like black stars, gravastars, quark stars, boson stars, Q-balls, and
electroweak stars. Although there is no evidence that these theories are right or that these objects exist, we are faced with the
question of how to determine whether a given object is really a
Gou et al., “The Extreme Spin of the Black Hole in Cygnus X-1,” http:
Chapter 6
Vacuum solutions
black hole or one of these other species. The defining characteristic of a black hole is that it has an event horizon rather than a
physical surface. If an object is not a black hole, than by conservation of energy any matter that falls onto it must release its
gravitational potential energy when it hits that surface. Cygnus X1 has a copious supply of matter falling onto it from its supergiant
companion, and Sagittarius A* likewise accretes a huge amount of
gas from the stellar wind of nearby stars. By analyzing millimeter
and infrared very-long-baseline-interferometry observations, Broderick, Loeb, and Narayan11 have shown that if Sagittarius A* had a
surface, then the luminosity of this surface must be less than 0.3% of
the luminosity of the accretion disk. But this is not physically possible, because there are fundamental limits on the efficiency with
which the gas can radiate away its energy before hitting the surface. We can therefore conclude that Sagittarius A* must have an
event horizon. Its event horizon may be imaged directly in the near
6.3.5 Singularities and cosmic censorship
Informal ideas
Since we observe that black holes really do exist, maybe we
should take the singularity at r = 0 seriously. Physically, it says
that the mass density and tidal forces blow up to infinity there.
Generally when a physical theory says that observable quantities
blow up to infinity at a particular point, it means that the theory has
reached the point at which it can no longer make physical predictions. For instance, Maxwell’s theory of electromagnetism predicts
that the electric field blows up like r−2 near a point charge, and
this implies that infinite energy is stored in the field within a finite
radius around the charge. Physically, this can’t be right, because
we know it only takes 511 keV of energy to create an electron out
of nothing, e.g., in nuclear beta decay. The paradox is resolved by
quantum electrodynamics, which modifies the description of the vacuum around the electron to include a sea of virtual particles popping
into and out of existence.
In the case of the singularity at the center of a black hole, it is
possible that quantum mechanical effects at the Planck scale prevent
the formation of a singularity. Unfortunately, we are unlikely to find
any empirical evidence about this, since black holes always seem
to come clothed in event horizons, so we outside observers cannot
extract any data about the singularity inside. Even if we take a
suicidal trip into a black hole, we get no data about the singularity,
because the singularity in the Schwarzschild metric is spacelike, not
timelike, and therefore it always lies in our future light cone, never
in our past.
Section 6.3
Black holes
In a way, the inaccessibility of singularities is a good thing. If a
singularity exists, it is a point at which all the known laws of physics
break down, and physicists therefore have no way of predicting anything about its behavior. There is likewise no great crisis for physics
due to the Big Bang singularity or the Big Crunch singularity that
occurs in some cosmologies in which the universe recollapses; we
have no reasonable expectation of being able to make and test predictions or retrodictions that extend beyond the beginning or end
of the universe.
What would be a crushing blow to the enterprise of physics would
be a singularity that could sit on someone’s desk. As John Earman
of the University of Pittsburgh puts it, anything could pop out of
such a singularity, including green slime or your lost socks. In more
technical language, a singularity would constitute an extreme violation of unitarity and an acute instance of the information paradox
(see page 203).
There is no obvious reason that general relativity should not allow naked singularities, but neither do we know of any real-world
process by which one could be formed by gravitational collapse. Penrose’s cosmic censorship hypothesis states that the laws of physics
prevent the formation of naked singularities from nonsingular and
generic initial conditions. “Generic” is a necessary addition to Penrose’s original 1969 formulation, since Choptuik showed in 1993 that
certain perfectly fine-tuned initial conditions allowed collapse to a
naked singularity.13
Formal definitions
The remainder of this subsection provides a more formal exposition of the definitions relating to singularities. It can be skipped
without loss of continuity.
The reason we care about singularities is that they indicate an
incompleteness of the theory, and the theory’s inability to make
predictions. One of the simplest things we could ask any theory
to do would be to predict the trajectories of test particles. For
example, Maxwell’s equations correctly predict the motion of an
electron in a uniform magnetic field, but they fail to predict the
motion of an electron that collides head-on with a positron. It might
have been natural for someone in Maxwell’s era (assuming they were
informed about the existence of positrons and told to assume that
both particles were pointlike) to guess that the two particles would
scatter through one another at θ = 0, their velocities momentarily
becoming infinite. But it would have been equally natural for this
person to refuse to make a prediction.
Similarly, if a particle hits a black hole singularity, we should not
expect general relativity to make a definite prediction. It doesn’t.
Chapter 6
Phys. Rev. Lett. 70, p. 9
Vacuum solutions
Not only does the geodesic equation break down, but if we were to
naively continue the particle’s geodesic by assuming that it scatters
in the forward direction, the continuation would be a world-line
whose future-time direction pointed into the singularity rather than
back out of it.
We would therefore like to define a singularity as a situation in
which the geodesics of test particles can’t be extended indefinitely.
But what does “indefinitely” mean? If the test particle is a photon,
then the metric length of its world-line is zero. We get around this
by defining length in terms of an affine parameter.
Definition: A spacetime is said to be geodesically incomplete if
there exist timelike or lightlike geodesics that cannot be extended
past some finite affine parameter into the past or future.
Geodesic incompleteness defines what we mean by a singularity.
A geodesically incomplete spacetime has one or more singularities
in it. The Schwarzschild spacetime has a singularity at r = 0, but
not at the event horizon, since geodesics continue smoothly past
the event horizon. Cosmological spacetimes contain a Big Bang
singularity which prevents geodesics from being extended beyond a
certain point in the past.
There are two types of singularities, curvature singularities and
conical singularities. The examples above are curvature singularities. Figure b shows an example of a conical singularity. (Cf. figure
b, 192.) As one approaches a curvature singularity, the curvature of
spacetime diverges to infinity, as measured by a curvature invariant
such as the Ricci scalar. In 2+1-dimensional relativity, curvature
vanishes identically, and the only kind of gravity that exists is due
to conical singularities. Conical singularities are not expected to
be present in our universe, since there is no known mechanism by
which they could be formed by gravitational collapse.
b / A conical singularity.
cone has zero intrinsic curvature
everywhere except at its tip.
Geodesic 1 can be extended infinitely far, but geodesic 2 cannot;
since the metric is undefined at
the tip, there is no sensible way
to define how geodesic 2 should
be extended.
Actual singularities involving geodesic incompleteness are to be
distinguished from coordinate singularities, which are not really singularities at all. In the Schwarzschild spacetime, as described in
Schwarzschild’s original coordinates, some components of the metric blow up at the event horizon, but this is not an actual singularity.
This coordinate system can be replaced with a different one in which
the metric is well behaved.
The reason curvature scalars are useful as tests for an actual curvature singularity is that since they’re scalars, they can’t diverge in
one coordinate system but stay finite in another. We define a singularity to be a curvature singularity if timelike or lightlike geodesics
can only be extended to some finite affine parameter, and some curvature scalar (not necessarily every such scalar) approaches infinity
as we approach this value of the affine parameter. Anything that is
not a curvature singularity is considered a conical singularity.
Section 6.3
Black holes
A singularity is not considered to be a point in a spacetime; it’s
more like a hole in the topology of the manifold. For example, the
Big Bang didn’t occur at a point.
Because a singularity isn’t a point or a point-set, we can’t define
its timelike or spacelike character in quite the way we would with,
say, a curve. A timelike singularity is one such that an observer
with a timelike world-line can have the singularity sometimes in his
future light-cone and sometimes in his past light-cone.14
Schwarzschild and Big Bang singularities are spacelike. (Note
that in the Schwarzschild metric, the Schwarzschild r and t coordinates swap their timelike and spacelike characters inside the event
The definition of a timelike singularity is local. A timelike singularity would be one that you could have sitting on your desk, where
you could look at it and poke it with a stick.
A naked singularity is one from which timelike or lightlike worldlines can originate and then escape to infinity. The Schwarzschild
metric’s singularity is not naked. This notion is global.
If either a timelike or a naked singularity can be formed by gravitational collapse from realistic initial conditions, then it would create severe difficulties for physicists wishing to make predictions using
the laws of physics.
6.3.6 Hawking radiation
Radiation from black holes
Since event horizons are expected to emit blackbody radiation,
a black hole should not be entirely black; it should radiate. This is
called Hawking radiation. Suppose observer B just outside the event
horizon blasts the engines of her rocket ship, producing enough acceleration to keep from being sucked in. By the equivalence principle, what she observes cannot depend on whether the acceleration
she experiences is actually due to a gravitational field. She therefore detects radiation, which she interprets as coming from the event
horizon below her. As she gets closer and closer to the horizon, the
acceleration approaches infinity, so the intensity and frequency of
the radiation grows without limit.
A distant observer A, however, sees a different picture. According to A, B’s time is extremely dilated. A sees B’s acceleration
as being only ∼ 1/m, where m is the mass of the black hole; A
does not perceive this acceleration as blowing up to infinity as B
approaches the horizon. When A detects the radiation, it is extremely red-shifted, and it has the spectrum that one would expect
Penrose, Gravitational radiation and gravitational collapse; Proceedings of
the Symposium, Warsaw, 1973. Dordrecht, D. Reidel Publishing Co. pp. 82-91,
free online at
Chapter 6
Vacuum solutions
for a horizon characterized by an acceleration a ∼ 1/m. The result
for a 10-solar-mass black hole is T ∼ 10−8 K, which is so low that
the black hole is actually absorbing more energy from the cosmic
microwave background radiation than it emits.
Direct observation of black-hole radiation is therefore probably
only possible for black holes of very small masses. These may have
been produced soon after the big bang, or it is conceivable that
they could be created artificially, by advanced technology. If blackhole radiation does exist, it may help to resolve the information
paradox, since it is possible that information that goes into a black
hole is eventually released via subtle correlations in the black-body
radiation it emits.
Particle physics
Hawking radiation has some intriguing properties from the point
of view of particle physics. In a particle accelerator, the list of
particles one can create in appreciable quantities is determined by
coupling constants. In Hawking radiation, however, we expect to
see a representative sampling of all types of particles, biased only
by the fact that massless or low-mass particles are more likely to be
produced than massive ones. For example, it has been speculated
that some of the universe’s dark matter exists in the form of “sterile”
particles that do not couple to any force except for gravity. Such
particles would never be produced in particle accelerators, but would
be seen in radiation.
Hawking radiation would violate many cherished conservation
laws of particle physics. Let a hydrogen atom fall into a black hole.
We’ve lost a lepton and a baryon, but if we want to preserve conservation of lepton number and baryon number, we cover this up
with a fig leaf by saying that the black hole has simply increased
its lepton number and baryon number by +1 each. But eventually
the black hole evaporates, and the evaporation is probably mostly
into zero-mass particles such as photons. Once the hole has evaporated completely, our fig leaf has evaporated as well. There is now
no physical object to which we can attribute the +1 units of lepton
and baryon number.
Black-hole complementarity
A very difficult question about the relationship between quantum mechanics and general relativity occurs as follows. In our example above, observer A detects an extremely red-shifted spectrum
of light from the black hole. A interprets this as evidence that
the space near the event horizon is actually an intense maelstrom
of radiation, with the temperature approaching infinity as one gets
closer and closer to the horizon. If B returns from the region near
the horizon, B will agree with this description. But suppose that
observer C simply drops straight through the horizon. C does not
Section 6.3
Black holes
feel any acceleration, so by the equivalence principle C does not
detect any radiation at all. Passing down through the event horizon, C says, “A and B are liars! There’s no radiation at all.” A
and B, however, C see as having entered a region of infinitely intense radiation. “Ah,” says A, “too bad. C should have turned
back before it got too hot, just as I did.” This is an example of a
principle we’ve encountered before, that when gravity and quantum
mechanics are combined, different observers disagree on the number
of quanta present in the vacuum. We are presented with a paradox,
because A and B believe in an entirely different version of reality
that C. A and B say C was fricasseed, but C knows that that didn’t
happen. One suggestion is that this contradiction shows that the
proper logic for describing quantum gravity is nonaristotelian, as
described on page 68. This idea, suggested by Susskind et al., goes
by the name of black-hole complementarity, by analogy with Niels
Bohr’s philosophical description of wave-particle duality as being
“complementary” rather than contradictory. In this interpretation,
we have to accept the fact that C experiences a qualitatively different reality than A and B, and we comfort ourselves by recognizing
that the contradiction can never become too acute, since C is lost
behind the event horizon and can never send information back out.
6.3.7 Black holes in d dimensions
It has been proposed that our universe might actually have not
d = 4 dimensions but some higher number, with the d − 4 “extra”
ones being spacelike, and curled up on some small scale ρ so that
we don’t see them in ordinary life. One candidate for such a scale
ρ is the Planck length, and we then have to talk about theories of
quantum gravity such as string theory. On the other hand, it could
be the 1 TeV electroweak scale; the motivation for such an idea is
that it would allow the unification of electroweak interactions with
gravity. This idea goes by the name of “large extra dimensions” —
“large” because ρ is bigger than the Planck length. In fact, in such
theories the Planck length is the electroweak unification scale, and
the number normally referred to as the Planck length is not really
the Planck length.15
In d dimensions, there are d−1 spatial dimensions, and a surface
of spherical symmetry has d − 2. In the Newtonian weak-field limit,
the density of gravitational field lines falls off like m/rd−2 with distance from a source m, and we therefore find that Newton’s law of
gravity has an exponent of −(d − 2). If d 6= 3, we can integrate to
find that the gravitational potential varies as Φ ∼ −mr−(d−3) . Passing back to the weak-field limit of general relativity, the equivalence
principle dictates that the gtt term of the metric be approximately
1 + 2Φ, so we find that the metric has the form
ds2 ≈ (1 − 2mr−(d−3) )dt2 − (. . .)dr2 − r2 dθ2 − r2 sin2 θdφ2
Chapter 6
Vacuum solutions
This looks like the Schwarzschild form with no other change than
a generalization of the exponent, and in fact Tangherlini showed
in 1963 that for d > 4, one obtains the exact solution simply by
applying the same change of exponent to grr as well.16
If large extra dimensions do exist, then this is the actual form
of any black-hole spacetime for r ρ, where the background curvature of the extra dimensions is negligible. Since the exponents are
all changed, gravitational forces become stronger than otherwise expected at small distances, and it becomes easier to make black holes.
It has been proposed that if large extra dimensions exist, microscopic
black holes would be observed at the Large Hadron Collider. They
would immediately evaporate into Hawking radiation (p. 230), with
an experimental signature of violating the standard conservation
laws of particle physics. As of 2010, the empirical results seem to
be negative.17
The reasoning given above fails in the case of d = 3, i.e., 2+1dimensional spacetime, both because the integral of r−1 is not r0
and because the Tangherlini-Schwarzschild metric is not a vacuum
solution. As shown in problem 11 on p. 239, there is no counterpart of the Schwarzschild metric in 2+1 dimensions. This is essentially because for d = 3 mass is unitless, so given a source having
a certain mass, there is no way to set the distance scale at which
Newtonian weak-field behavior gives way to the relativistic strong
field. Whereas for d ≥ 4, Newtonian gravity is the limiting case
of relativity, for d = 3 they are unrelated theories. In fact, the
relativistic theory of gravity for d = 3 is somewhat trivial. Spacetime does not admit curvature in vacuum solutions,18 so that the
only nontrivial way to make non-Minkowski 2+1-dimensional spacetimes is by gluing together Minkowski pieces in various topologies,
like gluing pieces of paper to make things like cones and Möbius
strips. 2+1-dimensional gravity has conical singularities, but not
Schwarzschild-style ones that are surrounded by curved spacetime.
If black-hole solutions exist in d dimensions, then one can extend
such a solution to d+1 dimensions with cylindrical symmetry, forming a “black string.” The nonexistence of d = 3 black holes implies
that black string solutions do not exist in our own d = 4 universe.
However, different considerations arise in a universe with a negative
cosmological constant (p. 285). There are then 2+1-dimensional solutions known as BTZ black holes.19 Since our own universe has a
positive cosmological constant, not a negative one, we still find that
black strings cannot exist.
Emparan and Reall, “Black Holes in Higher Dimensions,” relativity.
Section 6.3
Black holes
6.4 Degenerate solutions
This section can be omitted on a first reading.
At the event horizon of the Schwarzschild spacetime, the timelike
and spacelike roles of the Schwarzschild r and t coordinates get
swapped around, so that the signs in the metric change from +−−−
to − + −−. In discussing cases like this, it becomes convenient to
define a new usage of the term “signature,” as s = p − q, where p is
the number of positive signs and q the number of negative ones. This
can also be represented by the pair of numbers (p, q). The example
of the Schwarzschild horizon is not too disturbing, both because
the funny behavior arises at a singularity that can be removed by a
change of coordinates and because the signature stays the same. An
observer who free-falls through the horizon observes that the local
properties of spacetime stay the same, with |s| = 2, as required by
the equivalence principle.
But this only makes us wonder whether there are other examples
in which an observer would actually detect a change in the metric’s
signature. We are encouraged to think of the signature as something
empirically observable because, for example, it has been proposed
that our universe may have previously unsuspected additional spacelike dimensions, and these theories make testable predictions. Since
we don’t notice the extra dimensions in ordinary life, they would
have to be wrapped up into a cylindrical topology. Some such theories, like string theory, are attempts to create a theory of quantum
gravity, so the cylindrical radius is assumed to be on the order of
the Planck length, which corresponds quantum-mechanically to an
energy scale that we will not be able to probe using any foreseeable technology. But it is also possible that the radius is large —
a possibility that goes by the name of “large extra dimensions” —
so that we could see an effect at the Large Hadron Collider. Nothing in the formulation of the Einstein field equations requires a 3+1
(i.e., (1, 3)) signature, and they work equally well if the signature
is instead 4+1, 5+1, . . . . Newton’s inverse-square law of gravity is
described by general relativity as arising from the three-dimensional
nature of space, so on small scales in a theory with n large extra dimensions, the 1/r2 behavior changes over to 1/r2+n , and it becomes
possible that the LHC could produce microscopic black holes, which
would immediately evaporate into Hawking radiation in a characteristic way.
So it appears that the signature of spacetime is something that
is not knowable a priori, and must be determined by experiment.
When a thing is supposed to be experimentally observable, general
relativity tells us that it had better be coordinate-independent. Is
this so? A proposition from linear algebra called Sylvester’s law of
inertia encourages us to believe that it is. The theorem states that
when a real matrix A is diagonalized by a real, nonsingular change
Chapter 6
Vacuum solutions
of basis (a similarity transformation S −1 AS), the number of positive, negative, and zero diagonal elements is uniquely determined.
Since a change of coordinates has the effect of applying a similarity transformation on the metric, it appears that the signature is
This is not quite right, however, as shown by the following paradox. The coordinate invariance of general relativity tells us that if
all clocks, everywhere in the universe, were to slow down simultaneously (with simultaneity defined in any way we like), there would
be no observable consequences. This implies that the spacetime
ds2 = −tdt2 − d`2 , where d`2 = dx2 + dy 2 + dz 2 , is empirically
indistinguishable from a flat spacetime. Starting from t = −∞,
the positive gtt component of the metric shrinks uniformly, which
should be harmless. We can indeed verify by direct evaluation of
the Riemann tensor that this is a flat spacetime (problem 9, p. 239).
But for t > 0 the signature of the metric switches from + − −− to
− − −−, i.e., from Lorentzian (|s| = 2) to Euclidean (|s| = 4). This
is disquieting. For t < 0, the metric is a perfectly valid description
of our own universe (which is approximately flat). Time passes, and
there is no sign of any impending disaster. Then, suddenly, at some
point in time, the entire structure of spacetime undergoes a horrible
spasm. This is a paradox, because we could just as well have posed
our initial conditions using some other coordinate system, in which
the metric had the familiar form ds2 = dt2 − d`2 . General relativity
is supposed to be agnostic about coordinates, but a choice of coordinate leads to a differing prediction about the signature, which is
a coordinate-independent quantity.
We are led to the resolution of the paradox if we explicitly
construct the coordinate transformation involved. In coordinates
(t, x, y, z), we have ds2 = −tdt2 − d`2 . We would like to find the
relationship between t and some other coordinate u such that we
recover the familiar form ds2 = du2 − d`2 for the metric. The tensor
transformation law gives
gtt =
−t =
with solution
u = ± t3/2
There is no solution for t > 0.
If physicists living in this universe, at t < 0, for some reason
choose t as their time coordinate, there is in fact a way for them to
tell that the cataclysmic event at t = 0 is not a reliable prediction.
Section 6.4
a / The change of coordinates is
degenerate at t = 0.
Degenerate solutions
At t = 0, their metric’s time component vanishes, so its signature
changes from + − −− to 0 − −−. At that moment, the machinery of the standard tensor formulation of general relativity breaks
down. For example, one can no longer raise indices, because g ab is
the matrix inverse of gab , but gab is not invertible. Since the field
equations are ultimately expressed in terms of the metric using machinery that includes raising and lowering of indices, there is no way
to apply them at t = 0. They don’t make a false prediction of the
end of the world; they fail to make any prediction at all. Physicists
accustomed to working in terms of the t coordinate can simply throw
up their hands and say that they have no way to predict anything
at t > 0. But they already know that their spacetime is one whose
observables, such as curvature, are all constant with respect to time,
so they should ask why this perfect symmetry is broken by singling
out t = 0. There is physically nothing that should make one moment in time different than any other, so choosing a particular time
to call t = 0 should be interpreted merely as an arbitrary choice of
the placement of the origin of the coordinate system. This suggests
to the physicists that all of the problems they’ve been having are not
problems with any physical meaning, but merely problems arising
from a poor choice of coordinates. They carry out the calculation
above, and discover the u time coordinate. Expressed in terms of u,
the metric is well behaved, and the machinery of prediction never
breaks down.
The paradox posed earlier is resolved because Sylvester’s law
of inertia only applies to a nonsingular transformation S. If S had
been singular, then the S −1 referred to in the theorem wouldn’t even
have existed. But the transformation from u to t has ∂t/∂u = 0 at
u = t = 0, so it is singular. This is all in keeping with the general
philosophy of coordinate-invariance in relativity, which is that only
smooth, one-to-one coordinate transformations are allowed. Someone who has found a lucky coordinate like u, and who then contemplates transforming to t, should realize that it isn’t a good idea,
because the transformation is not smooth and one-to-one. Someone
who has started by working with an unlucky coordinate like t finds
that the machinery breaks down at t = 0, and concludes that it
would be a good idea to search for a more useful set of coordinates.
This situation can actually arise in practical calculations.
What about our original question: could the signature of spacetime actually change at some boundary? The answer is now clear.
Such a change of signature is something that could conceivably
have intrinsic physical meaning, but if so, then the standard formulation of general relativity is not capable of making predictions
about it. There are other formulations of general relativity, such as
Ashtekar’s, that are ordinarily equivalent to Einstein’s, but that are
capable of making predictions about changes of signature. However,
there is more than one such formulation, and they do not agree on
Chapter 6
Vacuum solutions
their predictions about signature changes.
Section 6.4
Degenerate solutions
Show that in geometrized units, power is unitless. Find the
equivalent in watts of a power that equals 1 in geometrized units.
The metric of coordinates (θ, φ) on the unit sphere is ds2 =
dθ2 + sin2 θdφ2 . (a) Show that there is a singular point at which
g ab → ∞. (b) Verify directly that the scalar curvature R = Raa
constructed from the trace of the Ricci tensor is never infinite. (c)
Prove that the singularity is a coordinate singularity.
(a) Space probes in our solar system often use a slingshot
maneuver. In the simplest case, the probe is scattered gravitationally through an angle of 180 degrees by a planet. Show that in some
other frame such as the rest frame of the sun, in which the planet
has speed u toward the incoming probe, the maneuver adds 2u to
the speed of the probe. (b) Suppose that we replace the planet with
a black hole, and the space probe with a light ray. Why doesn’t this
accelerate the ray to a speed greater than c? . Solution, p. 373
The curve given parametrically by (cos3 t, sin3 t) is called an
astroid. The arc length along this curve is given by s = (3/2) sin2 t,
and its curvature by k = −(2/3) csc 2t. By rotating this astroid
about the x axis, we form a surface of revolution that can be described by coordinates (t, φ), where φ is the angle of rotation. (a)
Find the metric on this surface. (b) Identify any singularities, and
classify them as coordinate or intrinsic singularities.
. Solution, p. 373
(a) Section 3.5.4 (p. 108) gave a flat-spacetime metric in
rotating polar coordinates,
ds2 = (1 − ω 2 r2 )dt2 − dr2 − r2 dθ02 − 2ωr2 dθ0 dt
Identify the two values of r at which singularities occur, and classify
them as coordinate or non-coordinate singularities.
(b) The corresponding spatial metric was found to be
ds2 = −dr2 −
1 − ω2 r2
Identify the two values of r at which singularities occur, and classify
them as coordinate or non-coordinate singularities.
(c) Consider the following argument, which is intended to provide
an answer to part b without any computation. In two dimensions,
there is only one measure of curvature, which is equivalent (up to a
constant of proportionality) to the Gaussian curvature. The Gaussian curvature is proportional to the angular deficit of a triangle.
Since the angular deficit of a triangle in a space with negative curvature satisfies the inequality −π < < 0, we conclude that the
Gaussian curvature can never be infinite. Since there is only one
Chapter 6
Vacuum solutions
measure of curvature in a two-dimensional space, this means that
there is no non-coordinate singularity. Is this argument correct,
and is the claimed result consistent with your answers to part b?
. Solution, p. 373
The first experimental verification of gravitational redshifts
was a measurement in 1925 by W.S. Adams of the spectrum of light
emitted from the surface of the white dwarf star Sirius B. Sirius B
has a mass of 0.98M and a radius of 5.9×106 m. Find the redshift.
Show that, as claimed on page 223, applying the change of
coordinates t0 = t−2m ln(r−2m) to the Schwarzschild metric results
0 0
in a metric for which grr and gt0 t0 never blow up, but that g t t does
blow up.
Use the geodesic equation to show that, in the case of a
circular orbit in a Schwarzschild metric, d2 t/ds2 = 0. Explain why
this makes sense.
Verify by direct calculation, as asserted on p. 235, that the
Riemann tensor vanishes for the metric ds2 = −tdt2 − d`2 , where
d`2 = dx2 + dy 2 + dz 2 .
. Solution, p. 374
Suppose someone proposes that the vacuum field equation
of general relativity isn’t Rab = 0 but rather Rab = k, where k is
some constant that describes an innate tendency of spacetime to
have tidal distortions. Explain why this is not a good proposal.
. Solution, p. 374
Prove, as claimed on p. 232, that in 2+1 dimensions, with a
vanishing cosmological constant, there is no nontrivial Schwarzschild
. Solution, p. 374
On p. 211 I argued that there is no way to define a timereversal operation in general relativity so that it applies to all spacetimes. Why can’t we define it by picking some arbitrary spacelike surface that covers the whole universe, flipping the velocity of
every particle on that surface, and evolving a new version of the
spacetime backward and forward from that surface using the field
. Solution, p. 375
Chapter 6
Vacuum solutions
Chapter 7
This chapter is not required in order to understand the later material.
7.1 Killing vectors
The Schwarzschild metric is an example of a highly symmetric spacetime. It has continuous symmetries in space (under rotation) and in
time (under translation in time). In addition, it has discrete symmetries under spatial reflection and time reversal. In section 6.2.6,
we saw that the two continuous symmetries led to the existence of
conserved quantities for the trajectories of test particles, and that
these could be interpreted as mass-energy and angular momentum.
Generalizing, we want to consider the idea that a metric may
be invariant when every point in spacetime is systematically shifted
by some infinitesimal amount. For example, the Schwarzschild metric is invariant under t → t + dt. In coordinates (x0 , x1 , x2 , x3 ) =
(t, r, θ, φ), we have a vector field (dt,0,0,0) that defines the timetranslation symmetry, and it is conventional to split this into two
factors, a finite vector field ξ and an infinitesimal scalar, so that the
displacement vector is
ξdt = (1, 0, 0, 0)dt
a / The two-dimensional space
has a symmetry which can be
visualized by imagining it as a
surface of revolution embedded
in three-space. Without reference
to any extrinsic features such
as coordinates or embedding,
an observer on this surface can
detect the symmetry, because
there exists a vector field ξdu
such that translation by ξdu
doesn’t change the distance
between nearby points.
Such a field is called a Killing vector field, or simply a Killing vector,
after Wilhelm Killing. When all the points in a space are displaced
as specified by the Killing vector, they flow without expansion or
compression. The path of a particular point, such as the dashed line
in figure a, under this flow is called its orbit. Although the term
“Killing vector” is singular, it refers to the entire field of vectors,
each of which differs in general from the others. For example, the
ξ shown in figure a has a greater magnitude than a ξ near the neck
of the surface.
The infinitesimal notation is designed to describe a continuous
symmetry, not a discrete one. For example, the Schwarzschild spacetime also has a discrete time-reversal symmetry t → −t. This can’t
be described by a Killing vector, because the displacement in time
is not infinitesimal.
b / Wilhelm Killing (1847-1923).
The Euclidean plane
Example: 1
The Euclidean plane has two Killing vectors corresponding to
translation in two linearly independent directions, plus a third Killing
vector for rotation about some arbitrarily chosen origin O. In Cartesian coordinates, one way of writing a complete set of these is is
ξ1 = (1, 0)
ξ2 = (0, 1)
ξ3 = (−y , x)
A theorem from classical geometry1 states that any transformation in the Euclidean plane that preserves distances and handedness can be expressed either as a translation or as a rotation about some point. The transformations that do not preserve
handedness, such as reflections, are discrete, not continuous.
This theorem tells us that there are no more Killing vectors to be
found beyond these three, since any translation can be accomplished using ξ1 and ξ2 , while a rotation about a point P can be
done by translating P to O, rotating, and then translating O back
to P.
In the example of the Schwarzschild spacetime, the components
of the metric happened to be independent of t when expressed in
our coordinates. This is a sufficient condition for the existence of a
Killing vector, but not a necessary one. For example, it is possible
to write the metric of the Euclidean plane in various forms such as
ds2 = dx2 +dy 2 and ds2 = dr2 +r2 dφ2 . The first form is independent
of x and y, which demonstrates that x → x + dx and y → y + dy
are Killing vectors, while the second form gives us φ → φ + dφ.
Although we may be able to find a particular coordinate system in
which the existence of a Killing vector is manifest, its existence is an
intrinsic property that holds regardless of whether we even employ
coordinates. In general, we define a Killing vector not in terms of a
particular system of coordinates but in purely geometrical terms: a
space has a Killing vector ξ if translation by an infinitesimal amount
ξdu doesn’t change the distance between nearby points. Statements
such as “the spacetime has a timelike Killing vector” are therefore
intrinsic, since both the timelike property and the property of being
a Killing vector are coordinate-independent.
c / Vectors at a point P on a
sphere can be visualized as
occupying a Euclidean plane that
is particular to P.
Killing vectors, like all vectors, have to live in some kind of
vector space. On a manifold, this vector space is particular to a
given point, figure c. A different vector space exists at every point,
so that vectors at different points, occupying different spaces, can be
compared only by parallel transport. Furthermore, we really have
two such spaces at a given point, a space of contravariant vectors
and a space of covariant ones. These are referred to as the tangent
and cotangent spaces. The infinitesimal displacements we’ve been
Chapter 7
Coxeter, Introduction to Geometry, ch. 3
discussing belong to the contravariant (upper-index) space, but by
lowering and index we can just as well discuss them as covariant
vectors. The customary way of notating Killing vectors makes use
of the fact that the partial derivative operators ∂0 , ∂1 , ∂2 , ∂3 form
the basis for a vector space. In this notation, the Killing vector
of the Schwarzschild metric we’ve been discussing can be notated
simply as
ξ = ∂t
The partial derivative notation, like the infinitesimal notation,
implicitly refers to continuous symmetries rather than discrete ones.
If a discrete symmetry carries a point P1 to some distant point P2 ,
then P1 and P2 have two different tangent planes, so there is not a
uniquely defined notion of whether vectors ξ 1 and ξ 2 at these two
points are equal — or even approximately equal. There can therefore be no well-defined way to construe a statement such as, “P1 and
P2 are separated by a displacement ξ.” In the case of a continuous
symmetry, on the other hand, the two tangent planes come closer
and closer to coinciding as the distance s between two points on an
orbit approaches zero, and in this limit we recover an approximate
notion of being able to compare vectors in the two tangent planes.
They can be compared by parallel transport, and although parallel
transport is path-dependent, the difference bewteen paths is proportional to the area they enclose, which varies as s2 , and therefore
becomes negligible in the limit s → 0.
Self-check: Find another Killing vector of the Schwarzschild metric, and express it in the tangent-vector notation.
It can be shown that an equivalent condition for a field to be a
Killing vector is ∇a ξ b + ∇b ξ a = 0. This relation, called the Killing
equation, is written without reference to any coordinate system, in
keeping with the coordinate-independence of the notion.
When a spacetime has more than one Killing vector, any linear combination of them is also a Killing vector. This means that
although the existence of certain types of Killing vectors may be
intrinsic, the exact choice of those vectors is not.
Euclidean translations
Example: 2
The Euclidean plane has two translational Killing vectors (1, 0)
and (0, 1), i.e., ∂x and ∂y . These same vectors could be expressed as (1, 1) and (1, −1) in coordinate system that was rescaled
and rotated by 45 degrees.
A cylinder
Example: 3
The local properties of a cylinder, such as intrinsic flatness, are
the same as the local properties of a Euclidean plane. Since the
definition of a Killing vector is local and intrinsic, a cylinder has
the same three Killing vectors as a plane, if we consider only a
patch on the cylinder that is small enough so that it doesn’t wrap
Section 7.1
Killing vectors
all the way around. However, only two of these — the translations
— can be extended to form a smooth vector field on the entire
surface of the cylinder. These might be more naturally notated in
(φ, z) coordinates rather than (x, y), giving ∂z and ∂φ .
d / Example 3:
A cylinder
has three local symmetries, but
only two that can be extended
globally to make Killing vectors.
A sphere
Example: 4
A sphere is like a plane or a cylinder in that it is a two-dimensional
space in which no point has any properties that are intrinsically
different than any other. We might expect, then, that it would
have two Killing vectors. Actually it has three, ξx , ξy , and ξz , corresponding to infinitesimal rotations about the x, y , and z axes.
To show that these are all independent Killing vectors, we need
to demonstrate that we can’t, for example, have ξx = c1 ξy + c2 ξz
for some constants c1 and c2 . To see this, consider the actions of
ξy and ξz on the point P where the x axis intersects the sphere.
(References to the axes and their intersection with the sphere are
extrinsic, but this is only for convenience of description and visualization.) Both ξy and ξz move P around a little, and these
motions are in orthogonal directions, wherease ξx leaves P fixed.
This proves that we can’t have ξx = c1 ξy + c2 ξz . All three Killing
vectors are linearly independent.
This example shows that linear independence of Killing vectors
can’t be visualized simply by thinking about the vectors in the
tangent plane at one point. If that were the case, then we could
have at most two linearly independent Killing vectors in this twodimensional space. When we say “Killing vector” we’re really referring to the Killing vector field, which is defined everywhere on
the space.
Proving nonexistence of Killing vectors
. Find all Killing vectors of these two metrics:
Example: 5
ds2 = e−x dx 2 + ex dy 2
ds2 = dx 2 + x 2 dy 2
. Since both metrics are manifestly independent of y , it follows
that ∂y is a Killing vector for both of them. Neither one has any
other manifest symmetry, so we can reasonably conjecture that
this is the only Killing vector either one of them has. However, one
can have symmetries that are not manifest, so it is also possible
that there are more.
One way to attack this would be to use the Killing equation to find
a system of differential equations, and then determine how many
linearly independent solutions there were.
But there is a simpler approach. The dependence of these metrics on x suggests that the spaces may have intrinsic properties
that depend on x; if so, then this demonstrates a lower symmetry
than that of the Euclidean plane, which has three Killing vectors.
Chapter 7
One intrinsic property we can check is the scalar curvature R.
The following Maxima code calculates R for the first metric.
R:scurvature(); /* scalar curvature */
The result is R = −ex , which demonstrates that points that differ
in x have different intrinsic properties. Since the flow of a Killing
field ξ can never connect points that have different properties,
we conclude that ξx = 0. If only ξy can be nonzero, the Killing
equation ∇a ξb + ∇b ξa = 0 simplifies to ∇x ξy = ∇y ξy = 0. These
equations constrain both ∂x ξy and ∂y ξy , which means that given
a value of ξy at some point in the plane, its value everywhere
else is determined. Therefore the only possible Killing vectors
are scalar multiples of the Killing vector already found. Since we
don’t consider Killing vectors to be distinct unless they are linearly
independent, the first metric only has one Killing vector.
A similar calculation for the second metric shows that R = 0, and
an explicit calculation of its Riemann tensor shows that in fact
the space is flat. It is simply the Euclidean plane written in funny
coordinates. This metric has the same three Killing vectors as the
Euclidean plane.
It would have been tempting to leap to the wrong conclusion about
the second metric by the following reasoning. The signature of a
metric is an intrinsic property. The metric has signature ++ everywhere in the plane except on the y axis, where it has signature
+0. This shows that the y axis has different intrinsic properties
than the rest of the plane, and therefore the metric must have a
lower symmetry than the Euclidean plane. It can have at most two
Killing vectors, not three. This contradicts our earlier conclusion.
The resolution of this paradox is that this metric has a removable
degeneracy of the same type as the one described in section 6.4.
As discussed in that section, the signature is invariant only under
nonsingular transformations, but the transformation that converts
these coordinates to Cartesian ones is singular.
7.1.1 Inappropriate mixing of notational systems
Confusingly, it is customary to express vectors and dual vectors
by summing over basis vectors like this:
v = v µ ∂µ
ω = ωµ dxµ
Section 7.1
Killing vectors
This is an abuse of notation, driven by the desire to have up-down
pairs of indices to sum according to the usual rules of the Einstein
notation convention. But by that convention, a quantity like v or
ω with no indices is a scalar, and that’s not the case here. The
products on the right are not tensor products, i.e., the indices aren’t
being contracted.
This muddle is the result of trying to make the Einstein notation
do too many things at once and of trying to preserve a clumsy and
outdated system of notation and terminology originated by Sylvester
in 1853. In pure abstract index notation, there are not six flavors of
objects as in the two equations above but only two: vectors like v a
and dual vectors like ωa . The Sylvester notation is the prevalent one
among mathematicians today, because their predecessors committed
themselves to it a century before the development of alternatives
like abstract index notation and birdtracks. The Sylvester system is
inconsistent with the way physicists today think of vectors and dual
vectors as being defined by their transformation properties, because
Sylvester considers v and ω to be invariant.
Mixing the two systems leads to the kinds of notational clashes
described above. As a particularly absurd example, a physicist who
is asked to suggest a notation for a vector will typically pick up a
pen and write v µ . We are then led to say that a vector is written in
a concrete basis as a linear combination of dual vectors ∂µ !
7.1.2 Conservation laws
Whenever a spacetime has a Killing vector, geodesics have a
constant value of v b ξb , where v b is the velocity four-vector. For
example, because the Schwarzschild metric has a Killing vector ξ =
∂t , test particles have a conserved value of v t , and therefore we also
have conservation of pt , interpreted as the mass-energy.
Energy-momentum in flat 1+1 spacetime
Example: 6
A flat 1+1-dimensional spacetime has Killing vectors ∂x and ∂t .
Corresponding to these are the conserved momentum and massenergy, p and E. If we do a Lorentz boost, these two Killing vectors get mixed together by a linear transformation, corresponding
to a transformation of p and E into a new frame.
In addition, one can define a globally conserved quantity found
by integrating the flux density P a = T ab ξb over the boundary of any
compact orientable region.2 In case of a flat spacetime, there are
enough Killing vectors to give conservation of energy-momentum
and angular momentum.
Hawking and Ellis, The Large Scale Structure of Space-Time, p. 62, give
a succinct treatment that describes the flux densities and proves that Gauss’s
theorem, which ordinarily fails in curved spacetime for a non-scalar flux, holds in
the case where the appropriate Killing vectors exist. For an explicit description
of how one can integrate to find a scalar mass-energy, see Winitzki, Topics in
General Relativity, section 3.1.5, available for free online.
Chapter 7
7.2 Spherical symmetry
A little more work is required if we want to link the existence of
Killing vectors to the existence of a specific symmetry such as spherical symmetry. When we talk about spherical symmetry in the context of Newtonian gravity or Maxwell’s equations, we may say, “The
fields only depend on r,” implicitly assuming that there is an r coordinate that has a definite meaning for a given choice of origin. But
coordinates in relativity are not guaranteed to have any particular physical interpretation such as distance from a particular origin.
The origin may not even exist as part of the spacetime, as in the
Schwarzschild metric, which has a singularity at the center. Another
possibility is that the origin may not be unique, as on a Euclidean
two-sphere like the earth’s surface, where a circle centered on the
north pole is also a circle centered on the south pole; this can also
occur in certain cosmological spacetimes that describe a universe
that wraps around on itself spatially.
We therefore define spherical symmetry as follows. A spacetime
S is spherically symmetric if we can write it as a union S = ∪sr,t of
nonintersecting subsets sr,t , where each s has the structure of a twosphere, and the real numbers r and t have no preassigned physical
interpretation, but sr,t is required to vary smoothly as a function
of them. By “has the structure of a two-sphere,” we mean that no
intrinsic measurement on s will produce any result different from the
result we would have obtained on some two-sphere. A two-sphere
has only two intrinsic properties: (1) it is spacelike, i.e., locally its
geometry is approximately that of the Euclidean plane; (2) it has
a constant positive curvature. If we like, we can require that the
parameter r be the corresponding radius of curvature, in which case
t is some timelike coordinate.
To link this definition to Killing vectors, we note that condition
2 is equivalent to the following alternative condition: (20 ) The set
s should have three Killing vectors (which by condition 1 are both
spacelike), and it should be possible to choose these Killing vectors
such that algebraically they act the same as the ones constructed
explicitly in example 4 on p. 244. As an example of such an algebraic
property, figure a shows that rotations are noncommutative.
Section 7.2
Spherical symmetry
a / Performing the rotations in one
order gives one result, 3, while reversing the order gives a different
result, 5.
A cylinder is not a sphere
Example: 7
. Show that a cylinder does not have the structure of a twosphere.
. The cylinder passes condition 1. It fails condition 2 because its
Gaussian curvature is zero. Alternatively, it fails condition 20 because it has only two independent Killing vectors (example 3).
A plane is not a sphere
Example: 8
. Show that the Euclidean plane does not have the structure of a
. Condition 2 is violated because the Gaussian curvature is zero.
Or if we wish, the plane violates 20 because ∂x and ∂y commute,
but none of the Killing vectors of a 2-sphere commute.
7.3 Static and stationary spacetimes
7.3.1 Stationary spacetimes
When we set out to describe a generic spacetime, the Alice
in Wonderland quality of the experience is partly because coordinate invariance allows our time and distance scales to be arbitrarily
rescaled, but also partly because the landscape can change from one
moment to the next. The situation is drastically simplified when the
spacetime has a timelike Killing vector. Such a spacetime is said to
be stationary. Two examples are flat spacetime and the spacetime
surrounding the rotating earth (in which there is a frame-dragging
effect). Non-examples include the solar system, cosmological models, gravitational waves, and a cloud of matter undergoing gravitational collapse.
Can Alice determine, by traveling around her spacetime and
Chapter 7
carrying out observations, whether it is stationary? If it’s not, then
she might be able to prove it. For example, suppose she visits a
certain region and finds that the Kretchmann invariant Rabcd Rabcd
varies with time in her frame of reference. Maybe this is because
an asteroid is coming her way, in which case she could readjust her
velocity vector to match that of the asteroid. Even if she can’t see
the asteroid, she can still try to find a velocity that makes her local
geometry stop changing in this particular way. If the spacetime is
truly stationary, then she can always “tune in” to the right velocity
vector in this way by searching systematically. If this procedure ever
fails, then she has proved that her spacetime is not stationary.
Self-check: Why is the timelike nature of the Killing vector important in this story?
Proving that a spacetime is stationary is harder. This is partly
just because spacetime is infinite, so it will take an infinite amount
of time to check everywhere. We aren’t inclined to worry too much
about this limitation on our geometrical knowledge, which is of a
type that has been familiar since thousands of years ago, when it
upset the ancient Greeks that the parallel postulate could only be
checked by following lines out to an infinite distance. But there is a
new type of limitation as well. The Schwarzschild spacetime is not
stationary according to our definition. In the coordinates used in
section 6.2, ∂t is a Killing vector, but is only timelike for r > 2m; for
r < 2m it is spacelike. Although the solution describes a black hole
that is going to sit around forever without changing, no observer
can ever verify that fact, because once she strays inside the horizon
she must follow a timelike world-line, which will end her program of
observation within some finite time.
7.3.2 Isolated systems
Asymptotic flatness
This unfortunate feature of our definition of stationarity — its
empirical unverifiability — is something that in general we just have
to live with. But there is an alternative in the special case of an
isolated system, such as our galaxy or a black hole. It may be a
good approximation to ignore distant matter, modeling such a system with a spacetime that is almost flat everywhere except in the
region nearby. Such a spacetime is called asymptotically flat. Formulating the definition of this term rigorously and in a coordinateinvariant way involves a large amount of technical machinery, since
we are not guaranteed to be presented in advance with a special,
physically significant set of coordinates that would lead directly to
a quantitative way of defining words like “nearby.” The reader who
wants a rigorous definition is referred to Hawking and Ellis.
Section 7.3
Static and stationary spacetimes
Asymptotically stationary spacetimes
In the case of an asymptotically flat spacetime, we say that it is
also asymoptotically stationary if it has a Killing vector that becomes timelike far away. Some authors (e.g., Ludvigsen) define
“stationary” to mean what I’m calling “asymoptotically stationary,”
others (Hawking and Ellis) define it the same way I do, and still others (Carroll) are not self-consistent. The Schwarzschild spacetime is
asymptotically stationary, but not stationary.
7.3.3 A stationary field with no other symmetries
Consider the most general stationary case, in which the only
Killing vector is the timelike one. The only ambiguity in the choice
of this vector is a rescaling; its direction is fixed. At any given point
in space, we therefore have a notion of being at rest, which is to
have a velocity vector parallel to the Killing vector. An observer at
rest detects no time-dependence in quantities such as tidal forces.
Points in space thus have a permanent identity. The gravitational field, which the equivalence principle tells us is normally an
elusive, frame-dependent concept, now becomes more concrete: it is
the proper acceleration required in order to stay in one place. We
can therefore use phrases like “a stationary field,” without the usual
caveats about the coordinate-dependent meaning of “field.”
Space can be sprinkled with identical clocks, all at rest. Furthermore, we can compare the rates of these clocks, and even compensate for mismatched rates, by the following procedure. Since the
spacetime is stationary, experiments are reproducible. If we send a
photon or a material particle from a point A in space to a point B,
then identical particles emitted at later times will follow identical
trajectories. The time lag between the arrival of two such particls
tells an observer at B the amount of time at B that corresponds to
a certain interval at A. If we wish, we can adjust all the clocks so
that their rates are matched. An example of such rate-matching is
the GPS satellite system, in which the satellites’ clocks are tuned
to 10.22999999543 MHz, matching the ground-based clocks at 10.23
MHz. (Strictly speaking, this example is out of place in this subsection, since the earth’s field has an additional azimuthal symmetry.)
It is tempting to conclude that this type of spacetime comes
equipped with a naturally preferred time coordinate that is unique
up to a global affine transformation t → at + b. But to construct
such a time coordinate, we would have to match not just the rates of
the clocks, but also their phases. The best method relativity allows
for doing this is Einstein synchronization (p. 347), which involves
trading a photon back and forth between clocks A and B and adjusting the clocks so that they agree that each clock gets the photon
at the mid-point in time between its arrivals at the other clock. The
trouble is that for a general stationary spacetime, this procedure is
Chapter 7
not transitive: synchronization of A with B, and of B with C, does
not guarantee agreement between A with C. This is because the time
it takes a photon to travel clockwise around triangle ABCA may be
different from the time it takes for the counterclockwise itinerary
ACBA. In other words, we may have a Sagnac effect, which is generally interpreted as a sign of rotation. Such an effect will occur,
for example, in the field of the rotating earth, and it cannot be
eliminated by choosing a frame that rotates along with the earth,
because the surrounding space experiences a frame-dragging effect,
which falls off gradually with distance.
Although a stationary spacetime does not have a uniquely preferred time, it does prefer some time coordinates over others. In a
stationary spacetime, it is always possible to find a “nice” t such
that the metric can be expressed without any t-dependence in its
7.3.4 A stationary field with additional symmetries
Most of the results given above for a stationary field with no
other symmetries also hold in the special case where additional symmetries are present. The main difference is that we can make linear
combinations of a particular timelike Killing vector with the other
Killing vectors, so the timelike Killing vector is not unique. This
means that there is no preferred notion of being at rest. For example, in a flat spacetime we cannot define an observer to be at rest if
she observes no change in the local observables over time, because
that is true for any inertial observer. Since there is no preferred rest
frame, we can’t define the gravitational field in terms of that frame,
and there is no longer any preferred definition of the gravitational
7.3.5 Static spacetimes
In addition to synchronizing all clocks to the same frequency, we
might also like to be able to match all their phases using Einstein
synchronization, which requires transitivity. Transitivity is framedependent. For example, flat spacetime allows transitivity if we use
the usual coordinates. However, if we change into a rotating frame
of reference, transitivity fails (see p. 108). If coordinates exist in
which a particular spacetime has transitivity, then that spacetime
is said to be static. In these coordinates, the metric is diagonalized,
and since there are no space-time cross-terms like dxdt in the metric,
such a spacetime is invariant under time reversal. Roughly speaking,
a static spacetime is one in which there is no rotation.
7.3.6 Birkhoff’s theorem
Birkhoff’s theorem, proved below, states that in the case of
spherical symmetry, the vacuum field equations have a solution,
the Schwarzschild spacetime, which is unique up to a choice of coordinates and the value of m. Let’s enumerate the assumptions
Section 7.3
Static and stationary spacetimes
that went into our derivation of the Schwarzschild metric on p. 209.
These were: (1) the vacuum field equations, (2) spherical symmetry,
(3) asymptotic staticity, (4) a certain choice of coordinates, and (5)
Λ = 0. Birkhoff’s theorem says that the assumption of staticity was
not necessary. That is, even if the mass distribution contracts and
expands over time, the exterior solution is still the Schwarzschild
solution. Birkhoff’s theorem holds because gravitational waves are
transverse, not longitudinal (see p. 337), so the mass distribution’s
radial throbbing cannot generate a gravitational wave. Birkhoff’s
theorem can be viewed as the simplest of the no-hair theorems describing black holes. The most general no-hair theorem states that
a black hole is completely characterized by its mass, charge, and
angular momentum. Other than these three numbers, nobody on
the outside can recover any information that was possessed by the
matter and energy that were sucked into the black hole.
It has been proposed3 that the no-hair theorem for nonzero angular momentum and zero charge could be tested empirically by
observations of Sagittarius A*. If the observations are consistent
with the no-hair theorem, it would be taken as supporting the validity of general relativity and the interpretation of this object as a
supermassive black hole. If not, then there are various possibilities,
including a failure of general relativity to be the correct theory of
strong gravitational fields, or a failure of one of the theorem’s other
assumptions, such as the nonexistence of closed timelike curves in
the surrounding universe.
Proof of Birkhoff’s theorem: Spherical symmetry guarantees
that we can introduce coordinates r and t such that the surfaces
of constant r and t have the structure of a sphere with radius r. On
one such surface we can introduce colatitude and longitude coordinates θ and φ. The (θ, φ) coordinates can be extended in a natural
way to other values of r by choosing the radial lines to lie in the
direction of the covariant derivative vector4 ∇a r, and this ensures
that the metric will not have any nonvanishing terms in drdθ or
drdφ, which could only arise if our choice had broken the symmetry
between positive and negative values of dθ and dφ. Just as we were
free to choose any way of threading lines of constant (θ, φ, t) between
spheres of different radii, we can also choose how to thread lines of
constant (θ, φ, r) between different times, and this can be done so as
Johannsen and Psaltis,
It may seem backwards to start talking about the covariant derivative of
a particular coordinate before a complete coordinate system has even been introduced. But (excluding the trivial case of a flat spacetime), r is not just an
arbitrary coordinate, it is something that an observer at a certain point in spacetime can determine by mapping out a surface of geometrically identical points,
and then determining that surface’s radius of curvature. Another worry is that it
is possible for ∇a r to misbehave on certain surfaces, such as the event horizon of
the Schwarzschild spacetime, but we can simply require that radial lines remain
continuous as they pass through these surfaces.
Chapter 7
to keep the metric free of any time-space cross-terms such as dθdt.
The metric can therefore be written in the form5
ds2 = h(t, r)dt2 − k(t, r)dr2 − r2 (dθ2 + sin2 θdφ2 )
This has to be a solution of the vacuum field equations, Rab = 0,
and in particular a quick calculation with Maxima shows that Rrt =
−∂t k/k 2 r, so k must be independent of time. With this restriction,
we find Rrr = −∂r h/hkr − 1/r2 − 1/kr2 = 0, and since k is timeindependent, ∂r h/h is also time-independent. This means that for
a particular time to , the function f (r) = h(to , r) has some universal
shape set by a differential equation, with the only possible ambiguity
being an over-all scaling that depends on to . But since h is the timetime component of the metric, this scaling corresponds physically to
a situation in which every clock, all over the universe, speeds up and
slows down in unison. General relativity is coordinate-independent,
so this has no observable effects, and we can absorb it into a redefinition of t that will cause h to be time-independent. Thus the
metric can be expressed in the time-independent diagonal form
ds2 = h(r)dt2 − k(r)dr2 − r2 (dθ2 + sin2 θdφ2 )
We have already solved the field equations for a metric of this form
and found as a solution the Schwarzschild spacetime.6 Since the
metric’s components are all independent of t, ∂t is a Killing vector, and it is timelike for large r, so the Schwarzschild spacetime is
asymptotically static.
The no-hair theorems say that relativity only has a small repertoire of types of black-hole singularities, defined as singularities inside regions of space that are causally disconnected from the universe, in the sense that future light-cones of points in the region
do not extend to infinity.7 That is, a black hole is defined as a
singularity hidden behind an event horizon, and since the definition of an event horizon is dependent on the observer, we specify
an observer infinitely far away. The theorems cannot classify naked
singularities, i.e., those not hidden behind horizons, because the role
On the same surfaces referred to in the preceding footnote, the functions h
and k may to go to 0 or ∞. These turn out to be nothing more serious than
coordinate singularities.
The Schwarzschild spacetime is the uniquely defined geometry found by removing the coordinate singularities from this form of the Schwarzschild metric.
For a more formal statement of this, see Hawking and Ellis, “The Large Scale
Structure of Space-Time,” p. 315. Essentially, the region must be a connected
region on a spacelike three-surface, and there must be no lightlike world-lines
that connect points in that region to null infinity. Null infinity is defined formally
using conformal techniques, but basically refers to points that are infinitely far
away in both space and time, and have the two infinities equal in a certain
sense, so that a free light ray could end up there. The definition is based on the
assumption that the surrounding spacetime is asymptotically flat, since otherwise
null infinity can’t be defined. It is not actually necessary to assume a singularity
as part of the definition; the no-hair theorems guarantee that one exists.
Section 7.3
Static and stationary spacetimes
of naked singularities in relativity is the subject of the cosmic censorship hypothesis, which is an open problem. The theorems do
not rule out the Big Bang singularity, because we cannot define the
notion of an observer infinitely far from the Big Bang. We can also
see that Birkhoff’s theorem does not prohibit the Big Bang, because
cosmological models are not vacuum solutions with Λ = 0. Black
string solutions are not ruled out by Birkhoff’s theorem because they
would lack spherical symmetry, so we need the arguments given on
p. 232 to show that they don’t exist.
7.3.7 The gravitational potential
When Pound and Rebka made the first observation of gravitational redshifts, these shifts were interpreted as evidence of gravitational time dilation, i.e., a mismatch in the rates of clocks. We are
accustomed to connecting these two ideas by using the expression
e−∆Φ for the ratio of the rates of two clocks (example 12, p. 59),
where Φ is a function of the spatial coordinates, and this is in fact
the most general possible definition of a gravitational potential Φ
in relativity. Since a stationary field allows us to compare rates
of clocks, it seems that we should be able to define a gravitational
potential for any stationary field. There is a problem, however, because when we talk about a potential, we normally have in mind
something that has encoded within it all there is to know about the
field. We would therefore expect to be able to find the metric from
the potential. But the example of the rotating earth shows that this
need not be the case for a general stationary field. In that example,
there are effects like frame-dragging that clearly cannot be deduced
from Φ; for by symmetry, Φ is independent of azimuthal angle, and
therefore it cannot distinguish between the direction of rotation and
the contrary direction. The best we can do in a general stationary
spacetime is to specify a pair of scalar potentials, sometimes known
as Hansen’s potentials, one analogous to Φ and the other giving information about angular momentum. Only a static spacetime can
be described by a single potential.
It is also important to step back and think about why relativity
does not offer a gravitational potential with the same general utility
as its Newtonian counterpart. There are two main issues.
The Einstein field equations are nonlinear. Therefore one cannot, in general, find the field created by a given set of sources by
adding up the potentials. At best this is a possible weak-field approximation. In particular, although Birkhoff’s theorem is in some
ways analogous to the Newtonian shell theorem, it cannot be used
to find the metric of an arbitrary spherically symmetric mass distribution by breaking it up into spherical shells.
It is also not meaningful to talk about any kind of gravitational
potential for spacetimes that aren’t static or stationary. For example, consider a cosmological model describing our expanding uni-
Chapter 7
verse. Such models are usually constructed according to the Copernican principle that no position in the universe occupies a privileged
place. In other words, they are homogeneous in the sense that they
have Killing vectors describing arbitrary translations and rotations.
Because of this high degree of symmetry, a gravitational potential
for such a model would have to be independent of position, and then
it clearly could not encode any information about the spatial part of
the metric. Even if we were willing to make the potential a function
of time, Φ(t), the results would still be nonsense. The gravitational
potential is defined in terms of rate-matching of clocks, so a potential that was purely a function of time would describe a situation
in which all clocks, everywhere in the universe, were changing their
rates in a uniform way. But this is clearly just equivalent to a redefinition of the time coordinate, which has no observable consequences
because general relativity is coordinate-invariant. A corollary is that
in a cosmological spacetime, it is not possible to give a natural prescription for deciding whether a particular redshift is gravitational
(measured by Φ) or kinematic, or some combination of the two (see
also p. 305).
7.4 The uniform gravitational field revisited
This section gives a somewhat exotic example. It is not necessary
to read it in order to understand the later material.
In problem 6 on page 199, we made a wish list of desired properties for a uniform gravitational field, and found that they could not
all be satisfied at once. That is, there is no global solution to the
Einstein field equations that uniquely and satisfactorily embodies
all of our Newtonian ideas about a uniform field. We now revisit
this question in the light of our new knowledge.
The 1+1-dimensional metric
ds2 = e2gz dt2 − dz 2
is the one that uniquely satisfies our expectations based on the
equivalence principle (example 12, p. 59), and it is a vacuum solution. We might logically try to generalize this to 3+1 dimensions
as follows:
ds2 = e2gz dt2 − dx2 − dy 2 − dz 2
But a funny thing happens now — simply by slapping on the two
new Cartesian axes x and y, it turns out that we have made our
vacuum solution into a non-vacuum solution, and not only that,
but the resulting stress-energy tensor is unphysical (ch. 8, problem
8, p. 330).
One way to proceed would be to relax our insistence on making
the spacetime one that exactly embodies the equivalence principle’s
Section 7.4
The uniform gravitational field revisited
requirements for a uniform field.8 This can be done by taking gtt =
e2Φ , where Φ is not necessarily equal to 2gz. By requiring that the
metric be a 3+1 vacuum solution, we arrive at a differential equation
whose solution is Φ = ln(z + k1 ) + k2 , which recovers the flat-space
metric that we found in example 19 on page 140 by applying a
change of coordinates to the Lorentz metric.
What if we want to carry out the generalization from 1+1 to 3+1
without violating the equivalence principle? For physical motivation
in how to get past this obstacle, consider the following argument
made by Born in 1920.9 Take a frame of reference tied to a rotating
disk, as in the example from which Einstein originally took much
of the motivation for creating a geometrical theory of gravity (subsection 3.5.4, p. 108). Clocks near the edge of the disk run slowly,
and by the equivalence principle, an observer on the disk interprets
this as a gravitational time dilation. But this is not the only relativistic effect seen by such an observer. Her rulers are also Lorentz
contracted as seen by a non-rotating observer, and she interprets
this as evidence of a non-Euclidean spatial geometry. There are
some physical differences between the rotating disk and our default
conception of a uniform field, specifically in the question of whether
the metric should be static (i.e., lacking in cross-terms between the
space and time variables). But even so, these considerations make
it natural to hypothesize that the correct 3+1-dimensional metric
should have transverse spatial coefficients that decrease with height.
With this motivation, let’s consider a metric of the form
ds2 = e2z dt2 − e−2jz dx2 − e−2kz dy 2 − dz 2
where j and k are constants, and I’ve taken g = 1 for convenience.10
The following Maxima code calculates the scalar curvature and the
Einstein tensor:
Thanks to user Mentz114 for suggesting this approach
and demonstrating the following calculation.
Max Born, Einstein’s Theory of Relativity, 1920. In the 1962 Dover edition,
the relevant passage is on p. 320
A metric of this general form is referred to as a Kasner metric. One usually
sees it written with a logarithmic change of variables, so that z appears in the
base rather than in the exponent.
Chapter 7
The output from line 9 shows that the scalar curvature is constant,
which is a necessary condition for any spacetime that we want to
think of as representing a uniform field. Inspecting the Einstein
tensor output by line 10, we find that√in order to get Gxx and Gyy
to vanish, we need j and k to be (1 ± 3i)/2. By trial and error, we
find that assigning the complex-conjugate values to j and k makes
Gtt and Gzz vanish as well, so that we have a vacuum solution.
This solution is, unfortunately, complex, so it is not of any obvious
value as a physically meaningful result. Since the field equations
are nonlinear, we can’t use the usual trick of forming real-valued
superpositions of the complex solutions. We could try simply
√ tak−z
ing the real part
√ of the metric. This gives gxx = e cos 3z and
gyy = e−z sin 3z, and is unsatisfactory because the
√ metric becomes
degenerate (has a zero determinant) at z = nπ/2 3, where n is an
It turns out, however, that there is a very similar solution, found
by Petrov in 1962,11 that is real-valued. The Petrov metric, which
describes a spacetime with cylindrical symmetry, is:
ds2 = −dr2 − e−2r dz 2 + er [2 sin 3rdφdt − cos 3r(dφ2 − dt2 )]
Note that it has many features in common with the complex oscillatory solution we found above. There are transverse length contractions that decay and oscillate in exactly the same way. The presence
of the dφdt term tells us that this is a non-static, rotating solution
— exactly like the one that Einstein and Born had in mind in their
prototypical example! We typically obtain this type of effect due
to frame dragging by some rotating massive body (see p. 149), and
the Petrov solution can indeed be interpreted as the spacetime that
exists in the vacuum on the exterior of an infinite, rigidly rotating
cylinder of “dust” (see p. 132).
The complicated Petrov metric might seem like the furthest possible thing from a uniform gravitational field, but in fact it is about
the closest thing general relativity provides to such a field. We
first note that the metric has Killing vectors ∂z , ∂φ , and ∂r , so it
has at least three out of the four translation symmetries we expect from a uniform field. By analogy with electromagnetism, we
would expect this symmetry to be absent in the radial direction,
since by Gauss’s law the electric field of a line of charge falls off
like 1/r. But surprisingly, the Petrov metric is also uniform radially. It is possible√to give the fourth killing
vector explicitly (it
is ∂r + z∂z + (1/2)( 3t − φ)∂φ − (1/2)( 3φ + t)∂t ), but it is perhaps more transparent to check that it represents a field of constant
strength (problem 4, p. 260).
Petrov, in Recent Developments in General Relativity, 1962, Pergamon, p.
383. For a presentation that is freely accessible online, see Gibbons and Gielen,
“The Petrov and Kaigorodov-Ozsváth Solutions: Spacetime as a Group Manifold,”
Section 7.4
The uniform gravitational field revisited
For insight into this surprising result, recall that in our attempt
at constructing the Cartesian version of this metric, we ran√into the
problem that the metric became degenerate at z = nπ/2 3. The
presence of the dφdt term prevents this from happening in Petrov’s
cylindrical version; two of the metric’s diagonal components can
vanish at certain values of r, but the presence of the off-diagonal
component prevents the determinant from going to zero. (The determinant is in fact equal to −1 everywhere.) What is happening
physically is that although the labeling of the φ and t coordinates
suggests a time and an azimuthal angle, these two coordinates are
in fact treated completely symmetrically. At values of r where the
cosine factor equals 1, the metric is diagonal, and has signature
(t, φ, r, z) = (+, −, −, −), but when the cosine equals −1, this becomes (−, +, −, −), so that φ is now the timelike coordinate. This
perfect symmetry between φ and t is an extreme example of framedragging, and is produced because of the specially chosen rate of
rotation of the dust cylinder, such that the velocity of the dust at
the outer surface is exactly c (or approaches it).
Classically, we would expect that a test particle released close
enough to the cylinder would be pulled in by the gravitational attraction and destroyed on impact, while a particle released farther
away would fly off due to the centrifugal force, escaping and eventually approaching a constant velocity. Neither of these would be
anything like the experience of a test particle released in a uniform
field. But consider a particle√released at rest in the rotating frame
at a radius r1 for which cos 3r1 = 1, so that t is the timelike coordinate. The particle accelerates (let’s say outward), but at some
point it arrives at an r2 where the cosine equals zero, and the φ − t
part of the metric is purely of the form dφdt. At this location, we
can define local coordinates u = φ − t and v = φ + t, so that the
metric depends only on du2 − dv 2 . One of the coordinates, say u, is
now the timelike one. Since our particle is material, its world-line
must be timelike, so it is swept along in the −φ direction. Gibbons
and Gielen show that the particle will now come back inward, and
continue forever by oscillating back and forth between two radii at
which the cosine vanishes.
7.4.1 Closed timelike curves
This oscillation still doesn’t sound like the motion of a particle
in a uniform field, but another strange thing happens, as we can
see by taking another look at the values of r at which the cosine
vanishes. At such a value of r, construct a curve of the form (t =
constant, r = constant, φ, z = constant). This is a closed curve, and
its proper length is zero, i.e., it is lightlike. This violates causality.
A photon could travel around this path and arrive at its starting
point at the same time when it was emitted. Something similarly
weird hapens to the test particle described above: whereas it seems
Chapter 7
to fall sometimes up and sometimes down, in fact it is always falling
down — but sometimes it achieves this by falling up while moving
backward in time!
Although the Petov metric violates causality, Gibbons and Gielen have shown that it satisfies the chronology protection conjecture:
“In the context of causality violation we have shown that one cannot
create CTCs [closed timelike curves] by spinning up a cylinder beyond its critical angular velocity by shooting in particles on timelike
or null curves.”
We have an exact vacuum solution to the Einstein field equations
that violates causality. This raises troublesome questions about the
logical self-consistency of general relativity. A very readable and
entertaining overview of these issues is given in the final chapter of
Kip Thorne’s Black Holes and Time Warps: Einstein’s Outrageous
Legacy. In a toy model constructed by Thorne’s students, involving a billiard ball and a wormhole, it turned out that there always
seemed to be self-consistent solutions to the ball’s equations of motion, but they were not unique, and they often involved disquieting
possibilities in which the ball went back in time and collided with its
earlier self. Among other things, this seems to lead to a violation of
conservation of mass-energy, since no mass was put into the system
to create extra copies of the ball. This would then be an example
of the fact that, as discussed in section 4.5.1, general relativity does
not admit global conservation laws. However, there is also an argument that the mouths of the wormhole change in mass in such a
way as to preserve conservation of energy.12
Section 7.4
The uniform gravitational field revisited
Example 3 on page 243 gave the Killing vectors ∂z and ∂φ of
a cylinder. If we express these instead as two linearly independent
Killing vectors that are linear combinations of these two, what is
the geometrical interpretation?
Section 7.3 told the story of Alice trying to find evidence that
her spacetime is not stationary, and also listed the following examples of spacetimes that were not stationary: (a) the solar system,
(b) cosmological models, (c) gravitational waves propagating at the
speed of light, and (d) a cloud of matter undergoing gravitational
collapse. For each of these, show that it is possible for Alice to
accomplish her mission.
If a spacetime has a certain symmetry, then we expect that
symmetry to be detectable in the behavior of curvature scalars such
as the scalar curvature R = Raa and the Kretchmann invariant
k = Rabcd Rabcd .
(a) Show that the metric
ds2 = e2gz dt2 − dx2 − dy 2 − dz 2
from page 255 has constant values of R = 1/2 and k = 1/4. Note
that Maxima’s ctensor package has built-in functions for these; you
have to call the lriemann and uriemann before calling them.
(b) Similarly, show that the Petrov metric
ds2 = −dr2 − e−2r dz 2 + er [2 sin 3rdφdt − cos 3r(dφ2 − dt2 )]
(p. 257) has R = 0 and k = 0.
Remark: Surprisingly, one can have a spacetime on which every possible curvature invariant vanishes identically, and yet which is not flat. See Coley, Hervik,
and Pelavas, “Spacetimes characterized by their scalar curvature invariants,”
Section 7.4 on page 255 presented the Petrov metric. The
purpose of this problem is to verify that the gravitational field it
represents does not fall off with distance. For simplicity, let’s
√ restrict
our attention to a particle released at an r such that cos 3r = 1,
so that t is the timelike coordinate. Let the particle be released at
rest in the sense that initially it has ż = ṙ = φ̇ = 0, where dots
represent differentiation with respect to the particle’s proper time.
Show that the magnitude of the proper acceleration is independent
of r.
. Solution, p. 375
The idea that a frame is “rotating” in general relativity can
be formalized by saying that the frame is stationary but not static.
Suppose someone says that any rotation must have a center. Give
a counterexample.
. Solution, p. 375
Chapter 7
Chapter 8
8.1 Sources in general relativity
8.1.1 Point sources in a background-independent theory
The Schrödinger equation and Maxwell’s equations treat spacetime as a stage on which particles and fields act out their roles.
General relativity, however, is essentially a theory of spacetime itself. The role played by atoms or rays of light is so peripheral
that by the time Einstein had derived an approximate version of
the Schwarzschild metric, and used it to find the precession of Mercury’s perihelion, he still had only vague ideas of how light and matter would fit into the picture. In his calculation, Mercury played the
role of a test particle: a lump of mass so tiny that it can be tossed
into spacetime in order to measure spacetime’s curvature, without
worrying about its effect on the spacetime, which is assumed to be
negligible. Likewise the sun was treated as in one of those orchestral pieces in which some of the brass play from off-stage, so as to
produce the effect of a second band heard from a distance. Its mass
appears simply as an adjustable parameter m in the metric, and if
we had never heard of the Newtonian theory we would have had no
way of knowing how to interpret m.
When Schwarzschild published his exact solution to the vacuum
field equations, Einstein suffered from philosophical indigestion. His
strong belief in Mach’s principle led him to believe that there was a
paradox implicit in an exact spacetime with only one mass in it. If
Einstein’s field equations were to mean anything, he believed that
they had to be interpreted in terms of the motion of one body relative to another. In a universe with only one massive particle, there
would be no relative motion, and so, it seemed to him, no motion
of any kind, and no meaningful interpretation for the surrounding
Not only that, but Schwarzschild’s solution had a singularity
at its center. When a classical field theory contains singularities,
Einstein believed, it contains the seeds of its own destruction. As
we’ve seen on page 228, this issue is still far from being resolved, a
century later.
However much he might have liked to disown it, Einstein was
now in possession of a solution to his field equations for a point
source. In a linear, background-dependent theory like electromag-
netism, knowledge of such a solution leads directly to the ability to
write down the field equations with sources included. If Coulomb’s
law tells us the 1/r2 variation of the electric field of a point charge,
then we can infer Gauss’s law. The situation in general relativity
is not this simple. The field equations of general relativity, unlike
the Gauss’s law, are nonlinear, so we can’t simply say that a planet
or a star is a solution to be found by adding up a large number of
point-source solutions. It’s also not clear how one could represent a
moving source, since the singularity is a point that isn’t even part
of the continuous structure of spacetime (and its location is also
hidden behind an event horizon, so it can’t be observed from the
8.1.2 The Einstein field equation
The Einstein tensor
Given these difficulties, it’s not surprising that Einstein’s first
attempt at incorporating sources into his field equation was a dead
end. He postulated that the field equation would have the Ricci
tensor on one side, and the stress-energy tensor T ab (page 161) on
the other,
Rab = 8πTab
where a factor of G/c4 on the right is suppressed by our choice
of units, and the 8π is determined on the basis of consistency with
Newtonian gravity in the limit of weak fields and low velocities. The
problem with this version of the field equations can be demonstrated
by counting variables. R and T are symmetric tensors, so the field
equation contains 10 constraints on the metric: 4 from the diagonal
elements and 6 from the off-diagonal ones.
In addition, local conservation of mass-energy requires the divergence-free property ∇b T ab = 0. In order to construct an example,
we recall that the only component of T for which we have so far
introduced any physical interpretation is T tt , which gives the density of mass-energy. Suppose we had a stress-energy tensor whose
components were all zero, except for a time-time component varying
as T tt = kt. This would describe a region of space in which massenergy was uniformly appearing or disappearing everywhere at a
constant rate. To forbid such examples, we need the divergencefree property to hold. This is exactly analogous to the continuity
equation in fluid mechanics or electromagnetism, ∂ρ/∂t + ∇ · J = 0
(or ∇a J a = 0), which states that the quantity of fluid or charge is
But imposing the divergence-free condition adds 4 more constraints on the metric, for a total of 14. The metric, however, is a
symmetric rank-2 tensor itself, so it only has 10 independent components. This overdetermination of the metric suggests that the
proposed field equation will not in general allow a solution to be
evolved forward in time from a set of initial conditions given on a
Chapter 8
spacelike surface, and this turns out to be true. It can in fact be
shown that the only possible solutions are those in which the traces
R = Raa and T = T aa are constant throughout spacetime.
The solution is to replace Rab in the field equations with the a
different tensor Gab , called the Einstein tensor, defined by Gab =
Rab − (1/2)Rgab ,
Gab = 8πTab
The Einstein tensor is constructed exactly so that it is divergencefree, ∇b Gab = 0. (This is not obvious, but can be proved by direct
computation.) Therefore any stress-energy tensor that satisfies the
field equation is automatically divergenceless, and thus no additional
constraints need to be applied in order to guarantee conservation of
Self-check: Does replacing Rab with Gab invalidate the Schwarzschild metric?
This procedure of making local conservation of mass-energy “baked
in” to the field equations is analogous to the way conservation of
charge is treated in electricity and magnetism, where it follows from
Maxwell’s equations rather than having to be added as a separate
Interpretation of the stress-energy tensor
The stress-energy tensor was briefly introduced in section 5.2 on
page 161. By applying the Newtonian limit of the field equation
to the Schwarzschild metric, we find that T tt is to be identified as
the mass density ρ. The Schwarzschild metric describes a spacetime
using coordinates in which the mass is at rest. In the cosmological
applications we’ll be considering shortly, it also makes sense to adopt
a frame of reference in which the local mass-energy is, on average,
at rest, so we can continue to think of T tt as the (average) mass
density. By symmetry, T must be diagonal in such a frame. For
example, if we had T tx 6= 0, then the positive x direction would
be distinguished from the negative x direction, but there is nothing
that would allow such a distinction.
Dust in a different frame
Example: 1
As discussed in example 14 on page 132, it is convenient in
cosmology to distinguish between radiation and “dust,” meaning
noninteracting, nonrelativistic materials such as hydrogen gas or
galaxies. Here “nonrelativistic” means that in the comoving frame,
in which the average flow of dust vanishes, the dust particles all
have |v | 1. What is the stress-energy tensor associated with
Since the dust is nonrelativistic, we can obtain the Newtonian limit
by using units in which c 6= 1, and letting c approach infinity. In
Cartesian coordinates, the components of the stress-energy have
Section 8.1
Sources in general relativity
units that cause them to scale like
1/c 1/c 1/c
1/c 1/c 2 1/c 2 1/c 2 
T µν ∝ 
1/c 1/c 2 1/c 2 1/c 2 
1/c 1/c 2 1/c 2 1/c 2
In the limit of c → ∞, we can therefore take the only source of
gravitational fields to be T tt , which in Newtonian gravity must be
the mass density ρ, so
T µν
Under a Lorentz boost by v in the x direction, the tensor transformation law gives
γ2 ρ γ2 v ρ
γ2 v ρ γ2 v 2 ρ
 0
0 0
Tµ ν
The over-all factor of γ2 arises because of the combination of
two effects: each dust particle’s mass-energy is increased by a
factor of γ, and length contraction also multiplies the density of
dust particles by a factor of γ. In the limit of small boosts, the
stress-energy tensor becomes
0 0
Tµ ν
ρ vρ
v ρ 0
This motivates the interpretation of the time-space components
of T as the flux of mass-energy along each axis. In the primed
frame, mass-energy with density ρ flows in the x direction at velocity v , so that the rate at which mass-energy passes through a
window of area A in the y − z plane is given by ρv A.
This is also consistent with our imposition of the divergence-free
property, by which we were essentially stating T tx to be the rate
of flow of T tt .
The center of mass-energy
Example: 2
In Newtonian mechanics, for motion in one dimension, the total momentum of a system of particles is given by ptot = Mvcm ,
where M is the total mass and vcm the velocity of the center of
mass. Is there such a relation in relativity?
Chapter 8
Since mass and energy are equivalent, we expect that the relativistic equivalent of the center of mass would have to be a center
of mass-energy.
It should also be clear that a center of mass-energy can only be
well defined for a region of spacetime that is small enough so that
effects due to curvature are negligible. For example, we can have
cosmological models in which space is finite, and expands like the
surface of a balloon being blown up. If the model is homogeneous
(there are no “special points” on the surface of the balloon), then
there is no point in space that could be a center. (A real balloon
has a center, but in our metaphor only the balloon’s spherical surface correponds to physical space.) The fundamental issue here
is the same geometrical one that caused us to conclude that there
is no global conservation of mass-energy in general relativity (see
section 4.5.1). In a curved spacetime, parallel transport is pathdependent, so we can’t unambiguously define a way of adding
vectors that occur in different places. The center of mass is defined by a sum of position vectors. From these considerations we
conclude that the center of mass-energy is only well defined in
special relativity, not general relativity.
For simplicity, let’s restrict ourselves to 1+1 dimensions, and adopt
a frame of reference in which the center of mass is at rest at x = 0.
Since T tt is interpreted as the density of mass-energy, the position of the center of mass must be given by
0 = xT tt dx
By analogy with the Newtonian relation ptot = Mvcm , let’s see
what happens when we differentiate with respect toRtime. The
velocity of the center of mass is then 0 = dxcm /dt = ∂t T tt xdx.
Applying the divergence-free
property ∂t T tt + ∂x T tx = 0, this beR
comes 0 = − ∂x T xdx. Integration by parts gives us finally
0 = T tx dx
We’ve already interpreted T tx as the rate of flow of mass-energy,
which is another way of describing momentum. We can therefore
interpret T tx as the density of momentum, and the right-hand side
of this equation as the total momentum. The interpretation is that
a system’s center of mass-energy is at rest if and only if it has
zero total momentum.
Suppose, for example, that we prepare a uniform metal rod so
that one end is hot and the other cold. We then deposit it in outer
space, initially motionless relative to some observer. Although
the rod itself is uniform, its mass-energy is very slightly nonuniform, so its center of mass-energy must be displaced a tiny bit
Section 8.1
Sources in general relativity
away from the center, toward the hot end. As the rod approaches
thermal equilibrium, the observer sees it accelerate very slightly
and then come to rest again, so that its center of mass-energy
remains fixed! An even stranger case is described in example 9
on p. 280.
Since the Einstein tensor is symmetric, the Einstein field equation requires that the stress-energy tensor be symmetric as well. It
is reassuring that according to example 1 the tensor is symmetric
for dust, and that symmetry is preserved by changes of coordinates
and by superpositions of sources. Besides dust, the other cosmologically significant sources of gravity are electromagnetic radiation
and the cosmological constant, and one can also check that these
give symmetry. Belinfante noted in 1939 that symmetry seemed to
fail in the case of fields with intrinsic spin, but he found that this
problem could be avoided by modifying the previously assumed way
of connecting T to the properties of the field. This shows that it can
be rather subtle to interpret the stress-energy tensor and connect it
to experimental observables. For more on this connection, and the
case of electromagnetic fields, see examples 7 and 8 on p. 277.
In example 1, we found that T xt had to be interpreted as the
flux of T tt (i.e., the flux of mass-energy) across the x axis. Lorentz
invariance requires that we treat t, x, y, and z symmetrically, and
this forces us to adopt the following interpretation: T µν , where µ is
spacelike, is the flux of the density of the mass-energy four-vector
in the µ direction. In the comoving frame, in Cartesian coordinates, this means that T xx , T yy , and T zz should be interpreted as
pressures. For example, T xx is the flux in the x direction of xmomentum. This is simply the pressure, P , that would be exerted
on a surface with its normal in the x direction, so in the comoving frame we have T µν = diag(ρ, P , P , P ). For a fluid that is not
in equilibrium, the pressure need not be isotropic, and the stress
exerted by the fluid need not be perpendicular to the surface on
which it acts. The space-space components of T would then be the
classical stress tensor, whose diagonal elements are the anisotropic
pressure, and whose off-diagonal elements are the shear stress. This
is the reason for calling T the stress-energy tensor.
The prediction of general relativity is then that pressure acts as a
gravitational source with exactly the same strength as mass-energy
density. This has important implications for cosmology, since the
early universe was dominated by radiation, and a photon gas has
P = ρ/3 (example 14, p. 132).
Experimental tests
But how do we know that this prediction is even correct? Can
it be verified in the laboratory? The classic laboratory test of the
strength of a gravitational source is the 1797 Cavendish experiment,
in which a torsion balance was used to measure the very weak grav-
Chapter 8
itational attractions between metal spheres. We could test this aspect of general relativity by doing a Cavendish experiment with
boxes full of photons, so that the pressure is of the same order
of magnitude as the mass-energy. This is unfortunately utterly impractical, since both P and ρ for a well-lit box are ridiculously small
compared to ρ for a metal ball.
However, the repulsive electromagnetic pressure inside an atomic
nucleus is quite large by ordinary standards — about 1033 Pa! To
see how big this is compared to the nuclear mass density of ρ ∼
1018 kg/m3 , we need to take into account the factor of c2 6= 1 in SI
units, the result being that P/ρ is about 10−2 , which is not too small.
Thus if we measure gravitational interactions of nuclei with different
values of P/ρ, we should be able to test this prediction of general
relativity. This was done in a Princeton PhD-thesis experiment by
Kreuzer1 in 1966.
Before we can properly describe and interpret the Kreuzer experiment, we need to distinguish the several different types of mass
that could in principle be different from one another in a theory of
gravity. We’ve already encountered the distinction between inertial
and gravitational mass, which Eötvös experiments (p. 22) show to
be equivalent to about one part in 1012 . But there is also a distinction between an object’s active gravitational mass ma , which
measures its ability to create gravitational fields, and its passive
gravitational mass mp , which measures the force it feels when placed
in an externally generated field. For experiments using laboratoryscale material objects at nonrelativistic velocities, the Newtonian
limit applies, and we can think of active gravitational mass as a
scalar, with a density T tt = ρ.
a / A Cavendish balance, used
to determine the gravitational
To understand how this relates to pressure as a source of gravitational fields, it is helpful to consider a case where P is about the
same as ρ, which occurs for light. Light is inherently relativistic, so
the Newtonian concept of a scalar gravitational mass breaks down,
but we can still use “mass” in quotes to talk qualitatively about
an electromagnetic wave’s active and passive participation in gravitational effects. Experiments show that general relativity correctly
predicts the deflection of light by the sun to about one part in 105
(p. 220). This is the electromagnetic equivalent of an Eötvös experiment; it shows that general relativity predicts the right thing about
the proportion between a light wave’s inertial and passive gravitational “masses.” Now suppose that general relativity was wrong,
and pressure was not a source of gravitational fields. This would
cause a drastic decrease in the active gravitational “mass” of an
electromagnetic wave.
The Kreuzer experiment actually dealt with static electric fields
inside nuclei, not electromagnetic waves, but it is still clear what we
Kreuzer, Phys. Rev. 169 (1968) 1007
Section 8.1
Sources in general relativity
should expect in general: if pressure does not act as a gravitational
source, then the ratio ma /mp should be different for different nuclei.
Specifically, it should be lower for a nucleus with a higher atomic
number Z, in which the electrostatic pressures are higher.
b / A simplified diagram of
Kreuzer’s modification.
moving teflon mass is submerged
in a liquid with nearly the same
Kreuzer did a Cavendish experiment, figure b, using masses
made of two different substances. The first substance was teflon.
The second substance was a mixture of the liquids trichloroethylene
and dibromoethane, with the proportions chosen so as to give a density as close as possible to that of teflon. Teflon is 76% fluorine by
weight, and the liquid is 74% bromine. Fluorine has atomic number
Z = 9, bromine Z = 35, and since the electromagnetic force has a
long range, the pressure within a nucleus scales upward roughly like
Z 1/3 (because any given proton is acted on by Z − 1 other protons,
and the size of a nucleus scales like Z 1/3 , so P ∝ Z/(Z 1/3 )2 ). The
solid mass was immersed in the liquid, and the combined gravitational field of the solid and the liquid was detected by a Cavendish
Ideally, one would formulate the liquid mixture so that its passivemass density was exactly equal to that of teflon, as determined by
buoyancy. Any oscillation in the torque measured by the Cavendish
balance would then indicate an inequivalence between active and
passive gravitational mass.
c / The Kreuzer experiment. 1. There are two passive masses, P, and an active mass A consisting of
a single 23-cm diameter teflon cylinder immersed in a fluid. The teflon cylinder is driven back and forth
with a period of 400 s. The resulting deflection of the torsion beam is monitored by an optical lever and
canceled actively by electrostatic forces from capacitor plates (not shown). The voltage required for this active
cancellation is a measure of the torque exerted by A on the torsion beam. 2. Active mass as a function of
temperature. 3. Passive mass as a function of temperature. In both 2 and 3, temperature is measured in units
of ohms, i.e., the uncalibrated units of a thermistor that was immersed in the liquid.
In reality, the two substances involved had different coefficients
of thermal expansion, so slight variations in temperature made their
passive-mass densities unequal. Kreuzer therefore measured both
the buoyant force and the gravitational torque as functions of temperature. He determined that these became zero at the same tem-
Chapter 8
perature, to within experimental errors, which verified the equivalence of active and passive gravitational mass to within a certain
mp ∝ ma
to within 5 × 10−5
Kreuzer intended this exeriment only as a test of mp ∝ ma ,
but it was reinterpreted in 1976 by Will2 as a test of the coupling
of sources to gravitational fields as predicted by general relativity
and other theories of gravity. Crudely, we’ve already argued that
mp ∝ ma would be substance-dependent if pressure did not couple to gravitational fields. Will actually carried out a more careful
calculation, of which I present a simplified summary. Suppose that
pressure does not contribute as much to gravitational fields as is
claimed by general relativity; its coupling is reduced by a factor
1 − x, where x = 0 in general relativity.3 Will considers a model
consisting of pointlike particles interacting through static electrical
forces, and shows that for such a system,
ma = mp + xUe
where Ue is the electrical energy. The Kreuzer experiment then
requires |x| < 6 × 10−2 , meaning that pressure does contribute to
gravitational fields as predicted by general relativity, to within a
precision of 6%.
One of the important ways in which Will’s calculation goes beyond my previous crude argument is that it shows that when x = 0,
as it does for general relativity, the correction term xUe /2 vanishes,
and ma = mp exactly. This is interpreted as follows. Let a bromine
nucleus be referred to with a capital M , fluorine with the lowercase
m. Then when a bromine nucleus and a fluorine nucleus interact
gravitationally at a distance r, the Newtonian approximation applies, and the total internal force acting on the pair of nuclei taken
as a whole equals (mp Ma − Mp ma )/r2 (in units where the Newtonian gravitational constant G equals 1). This vanishes only if
mp Ma − Mp ma = 0, which is equivalent to mp /Mp = ma /Ma . If
this proportionality fails, then the system violates Newton’s third
law and conservation of momentum; its center of mass will accelerate along the line connecting the two nuclei, either in the direction
of M or in the direction of m, depending on the sign of x.
Will, “Active mass in relativistic gravity: Theoretical interpretation of
the Kreuzer experiment,” Ap. J. 204 (1976) 234, available online at adsabs. A broader review of experimental tests of general relativity is
given in Will, “The Confrontation between General Relativity and Experiment,” The Kreuzer experiment is discussed in section 3.7.3.
In Will’s notation, ζ4 measures nonstandard coupling to pressure, ζ3 to
internal energy, and ζ1 to kinetic energy. By requiring that point-particle models
agree with perfect-fluid models, one obtains (−2/3)ζ1 = ζ3 = −ζ4 = x.
Section 8.1
Sources in general relativity
Thus the vanishing of the correction term xUe /2 tells us that
general relativity predicts exact conservation of momentum in this
interaction. This is comforting, but a little susprising on the face
of it. Newtonian gravity treats active and passive massive perfectly
symmetrically, so that there is a perfect guarantee of conservation of
momentum. But relativity incorporates them in a completely asymmetric manner, so there is no obvious reason that we should have
perfect conservation of momentum. In fact we don’t have any general guarantee of conservation of momentum, since, as discussed in
section 4.5.1 on page 148, the language of general relativity doesn’t
even give us the symbols we would need in order to state a global
conservation law for a vector. General relativity does, however,
allow local conservation laws. We will have local conservation of
mass-energy and momentum provided that the stress-energy tensor’s divergence ∇b T ab vanishes.
Bartlett and van Buren4 used this connection to conservation of
momentum in 1986 to derive a tighter limit on x. Since the moon
has an asymmetrical distribution of iron and aluminum, a nonzero
x would cause it to have an anomalous acceleration along a certain
line. Because lunar laser ranging gives extremely accurate data on
the moon’s orbit, the constraint is tightened to |x| < 1 × 10−8 .
These are tests of general relativity’s predictions about the gravitational fields generated by the pressure of a static electric field. In
addition, there is indirect confirmation (p. 298) that general relativity is correct when it comes to electromagnetic waves.
d / The Apollo 11 mission left
behind this mirror, which in this
photo shows the reflection of the
black sky. The mirror is used
for lunar laser ranging measurements, which have an accuracy
of about a centimeter.
Energy of gravitational fields not included in the stress-energy
Summarizing the story of the Kreuzer and Bartlett-van Buren
results, we find that observations verify to high precision one of the
defining properties of general relativity, which is that all forms of
energy are equivalent to mass. That is, Einstein’s famous E = mc2
can be extended to gravitational effects, with the proviso that the
source of gravitational fields is not really a scalar m but the stressenergy tensor T .
But there is an exception to this even-handed treatment of all
types of energy, which is that the energy of the gravitational field
itself is not included in T , and is not even generally a well-defined
concept locally. In Newtonian gravity, we can have conservation of
energy if we attribute to the gravitational field a negative potential
energy density −g2 /8π. But the equivalence principle tells us that
g is not a tensor, for we can always make g vanish locally by going
into the frame of a free-falling observer, and yet the tensor transformation laws will never change a nonzero tensor to a zero tensor
Phys. Rev. Lett. 57 (1986) 21. The result is summarized in section 3.7.3 of
the review by Will.
Chapter 8
under a change of coordinates. Since the gravitational field is not
a tensor, there is no way to add a term for it into the definition of
the stress-energy, which is a tensor. The grammar and vocabulary
of the tensor notation are specifically designed to prevent writing
down such a thing, so that the language of general relativity is not
even capable of expressing the idea that gravitational fields would
themselves contribute to T .
Self-check: (1) Convince yourself that the negative sign in the
expression −g2 /8π makes sense, by considering the case where two
equal masses start out far apart and then fall together and combine
to make a single body with twice the mass. (2) The Newtonian
gravitational field is the gradient of the gravitational potential φ,
which corresponds in the Newtonian limit to the time-time component of the metric. With this motivation, suppose someone proposes
generalizing the Newtonian energy density −(∇φ)2 /8π to a similar
expression such as −(∇a g ab )(∇c gc b ), where ∇ is now the covariant
derivative, and g is the metric, not the Newtonian field strength.
What goes wrong?
As a concrete example, we observe that the Hulse-Taylor binary
pulsar system (p. 220) is gradually losing orbital energy, and that
the rate of energy loss exactly matches general relativity’s prediction
of the rate of gravitational radiation. There is a net decrease in
the forms of energy, such as rest mass and kinetic energy, that are
accounted for in the stress energy tensor T . We can account for
the missing energy by attributing it to the outgoing gravitational
waves, but that energy is not included in T , and we have to develop
special techniques for evaluating that energy. Those techniques only
turn out to apply to certain special types of spacetimes, such as
asymptotically flat ones, and they do not allow a uniquely defined
energy density to be attributed to a particular small region of space
(for if they did, that would violate the equivalence principle).
Gravitational energy is locally unmeasurable.
Example: 3
When a new form of energy is discovered, the way we establish that it is a form of energy is that it can be transformed to or
from other forms of energy. For example, Becquerel discovered
radioactivity by noticing that photographic plates left in a desk
drawer along with radium salts became clouded: some new form
of energy had been converted into previously known forms such
as chemical energy. It is only in this limited sense that energy is
ever locally observable, and this limitation prevents us from meaningfully defining certain measures of energy. For example we can
never measure the local electrical potential in the same sense
that we can measure the local barometric pressure; a potential
of 137 volts only has meaning relative to some other region of
space taken to be at ground. Let’s use the acronym MELT to refer to measurement of energy by the local transformation of that
energy from one form into another.
Section 8.1
Sources in general relativity
The reason MELT works is that energy (or actually the momentum four-vector) is locally conserved, as expressed by the zerodivergence property of the stress-energy tensor. Without conservation, there is no notion of transformation. The Einstein field
equations imply this zero-divergence property, and the field equations have been well verified by a variety of observations, including many observations (such as solar system tests and observation of the Hulse-Taylor system) that in Newtonian terms would
be described as involving (non-local) transformations between kinetic energy and the energy of the gravitational field. This agreement with observation is achieved by taking T = 0 in vacuum,
regardless of the gravitational field. Therefore any local transformation of gravitational field energy into another form of energy
would be inconsistent with previous observation. This implies that
MELT is impossible for gravitational field energy.
In particular, suppose that observer A carries out a local MELT
of gravitational field energy, and that A sees this as a process in
which the gravitational field is reduced in intensity, causing the
release of some other form of energy such as heat. Now consider the situation as seen by observer B, who is free-falling in
the same local region. B says that there was never any gravitational field in the first place, and therefore sees heat as violating
local conservation of energy. In B’s frame, this is a nonzero divergence of the stress-energy tensor, which falsifies the Einstein
field equations.
Some examples
We conclude this introduction to the stress-energy tensor with
some illustrative examples.
A perfect fluid
For a perfect fluid, we have
Example: 4
Tab = (ρ + P)va vb − sPgab
where s = 1 for our + − −− signature or −1 for the signature
− + ++, and v represents the coordinate velocity of the fluid’s rest
Suppose that the metric is diagonal, but its components are varying, gαβ = diag(A2 , −B 2 , . . .). The properly normalized velocity
vector of an observer at (coordinate-)rest is v α = (A−1 , 0, 0, 0).
Lowering the index gives vα = (sA, 0, 0, 0). The various forms of
the stress-energy tensor then look like the following:
T00 = A2 ρ
T11 = B 2 P
T 00 = sρ
T 11 = −sP
T 00 = A−2 ρ
Chapter 8
T 11 = B −2 P
A rope dangling in a Schwarzschild spacetime
Example: 5
Suppose we want to lower a bucket on a rope toward the event
horizon of a black hole. We have already made some qualitative
remarks about this idea in example 15 on p. 65. This seemingly
whimsical example turns out to be a good demonstration of some
techniques, and can also be used in thought experiments that
illustrate the definition of mass in general relativity and that probe
some ideas about quantum gravity.5
The Schwarzschild metric (p. 210) is
ds2 = f 2 dt 2 − f −2 dr 2 + . . .
where f = (1 − 2m/r )1/2 , and . . . represents angular terms. We
will end up needing the following Christoffel symbols:
Γ ttr = f 0 /f
Γ θθr = Γ φφr = r −1
Since the spacetime has spherical symmetry, it ends up being
more convenient to consider a rope whose shape, rather than
being cylindrical, is a cone defined by some set of (θ, φ). For
convenience we take this set to cover unit solid angle. The final
results obtained in this way can be readily converted into statements about a cylindrical rope. We let µ be the mass per unit
length of the rope, and T the tension. Both of these may depend
on r . The corresponding energy density and tensile stress are
ρ = µ/A = µ/r 2 and S = T /A. To connect this to the stress-energy
tensor, we start by comparing to the case of a perfect fluid from
example 4. Because the rope is made of fibers that have stength
only in the radial direction, we will have T θθ = T φφ = 0. Furthermore, the stress is tensile rather than compressional, corresponding to a negative pressure. The Schwarzschild coordinates are
orthogonal but not orthonormal, so the properly normalized velocity of a static observer has a factor of f in it: v α = (f −1 , 0, 0, 0),
or, lowering an index, vα = (f , 0, 0, 0). The results of example 4
show that the mixed-index form of T will be the most convenient,
since it can be expressed without messy factors of f . We have
T κν = diag(ρ, S, 0, 0) = r −2 diag(µ, T , 0, 0)
By writing the stress-energy tensor in this form, which is independent of t, we have assumed static equilibrium outside the event
horizon. Inside the horizon, the r coordinate is the timelike one,
the spacetime itself is not static, and we do not expect to find
static solutions, for the reasons given on p. 65.
Brown, “Tensile Strength and the Mining of Black Holes,”
Section 8.1
Sources in general relativity
Conservation of energy is automatically satisfied, since there is
no time dependence. Conservation of radial momentum is expressed by
∇κ T κr = 0
0 = ∇r T rr + ∇t T tr + ∇θ T θr + ∇φ T φr
It would be tempting to throw away all but the first term, since
T is diagonal, and therefore T tr = T θr = T φr = 0. However, a
covariant derivative can be nonzero even when the symbol being
differentiated vanishes identically. Writing out these four terms,
we have
0 =∂r T rr + Γ rr r T rr − Γ rr r T rr
+Γ ttr T rr − Γ ttr T tt
+Γ θθr T rr
+Γ φφr T rr
where each line corresponds to one covariant derivative. Evaluating this, we have
0 = T0 +
T− µ
where primes denote differentiation with respect to r . Note that
since no terms of the form ∂r T tt occur, this expression is valid
regardless of whether we take µ to be constant or varying. Thus
we are free to take ρ ∝ r −2 , so that µ is constant, and this means
that our result is equally applicable to a uniform cylindrical rope.
This result is checked using computer software in example 6.
This is a differential equation that tells us how the tensile stress in
the rope varies along its length. The coefficient f 0 /f = m/r (r −2m)
blows up at the event horizon, which is as expected, since we do
not expect to be able to lower the rope to or below the horizon.
Let’s check the Newtonian limit, where the gravitational field is g
and the potential is Φ. In this limit, we have f ≈ 1 − Φ, f 0 /f ≈ g
(with g > 0), and µ T , resulting in
0 = T 0 − gµ
which is the expected Newtonian relation.
Returning to the full general-relativistic result, it can be shown that
for a loaded rope with no mass of its own, we have a finite result
for limr →∞ S, even when the bucket is brought arbitrarily close to
the horizon. However, this is misleading without the caveat that
for µ < T , the speed of transverse waves in the rope is greater
than c, which is not possible for any known form of matter — it
would violate the null energy condition, discussed in the following
Chapter 8
The rope, using computer algebra
Example: 6
The result of example 5 can be checked with the following Maxima
depends(ten,r); /* tension depends on r */
depends(mu,r); /* mass/length depends on r */
/* stress-energy tensor, T^mu_nu */
Compute covariant derivative of the stress-energy
tensor with respect to its first index. The
function checkdiv is defined so that the first
index has to be covariant (lower); the T I’m
putting in is T^mu_nu, and since it’s symmetric,
that’s the same as T_mu^nu.
8.1.3 Energy conditions
Physical theories are supposed to answer questions. For example:
1. Does a small enough physical object always have a world-line
that is approximately a geodesic?
2. Do massive stars collapse to form black-hole singularities?
3. Did our universe originate in a Big Bang singularity?
4. If our universe doesn’t currently have violations of causality
such as the closed timelike curves exhibited by the Petrov metric (p. 257), can we be assured that it will never develop causality violation in the future?
Section 8.1
Sources in general relativity
We would like to “prove” whether the answers to questions like these
are yes or no, but physical theories are not formal mathematical
systems in which results can be “proved” absolutely. For example,
the basic structure of general relativity isn’t a set of axioms but a
list of ingredients like the equivalence principle, which has evaded
formal definition.6
Even the Einstein field equations, which appear to be completely
well defined, are not mathematically formal predictions of the behavior of a physical system. The field equations are agnostic on
the question of what kinds of matter fields contribute to the stressenergy tensor. In fact, any spacetime at all is a solution to the
Einstein field equations, provided we’re willing to admit the corresponding stress-energy tensor. We can never answer questions like
the ones above without assuming something about the stress-energy
In example 14 on page 132, we saw that radiation has P = ρ/3
and dust has P = 0. Both have ρ ≥ 0. If the universe is made out
of nothing but dust and radiation, then we can obtain the following
four constraints on the energy-momentum tensor:
trace energy condition
strong energy condition
dominant energy condition
weak energy condition
null energy condition
ρ − 3P ≥ 0
ρ + 3P ≥ 0 and ρ + P ≥ 0
ρ ≥ 0 and |P | ≤ ρ
ρ ≥ 0 and ρ + P ≥ 0
ρ+P ≥0
These are arranged roughly in order from strongest to weakest. They
all have to do with the idea that negative mass-energy doesn’t seem
to exist in our universe, i.e., that gravity is always attractive rather
than repulsive. With this motivation, it would seem that there
should only be one way to state an energy condition: ρ > 0. But
the symbols ρ and P refer to the form of the stress-energy tensor
in a special frame of reference, interpreted as the one that is at rest
relative to the average motion of the ambient matter. (Such a frame
is not even guaranteed to exist unless the matter acts as a perfect
fluid.) In this frame, the tensor is diagonal. Switching to some other
frame of reference, the ρ and P parts of the tensor would mix, and
it might be possible to end up with a negative energy density. The
weak energy condition is the constraint we need in order to make
sure that the energy density is never negative in any frame.
The dominant energy condition is like the weak energy condition,
but it also guarantees that no observer will see a flux of energy
flowing at speeds greater than c.
The strong energy condition essentially states that gravity is
never repulsive; it is violated by the cosmological constant (see
“Theory of gravitation theories: a no-progress report,” Sotiriou, Faraoni,
and Liberati,
Chapter 8
p. 288).
An electromagnetic wave
Example: 7
In example 1 on p. 263, we saw that dust boosted along the x
axis gave a stress-energy tensor
1 v
Tµν = γ ρ
v v2
where we now suppress the y and z parts, which vanish. For
v → 1, this becomes
0 1 1
Tµν = ρ
1 1
where ρ0 is the energy density as measured in the new frame. As
a source of gravitational fields, this ultrarelativistic dust is indistinguishable from any other form of matter with v = 1 along the
x axis, so this is also the stress-energy tensor of an electromagnetic wave with local energy-density ρ0 , propagating along the x
axis. (For the full expression for the stress-energy tensor of an
arbitrary electromagnetic field, see the Wikipedia article “Electromagnetic stress-energy tensor.”)
This is a stress-energy tensor that represents a flux of energy at a
speed equal to c, so we expect it to lie at exactly the limit imposed
by the dominant energy condition (DEC). Our statement of the
DEC, however, was made for a diagonal stress-energy tensor,
which is what is seen by an observer at rest relative to the matter.
But we know that it’s impossible to have an observer who, as the
teenage Einstein imagined, rides alongside an electromagnetic
wave on a motorcycle. One way to handle this is to generalize our
definition of the energy condition. For the DEC, it turns out that
this can be done by requiring that the matrix T , when multiplied
by a vector on or inside the future light-cone, gives another vector
on or inside the cone.
A less elegant but more concrete workaround is as follows. Returning to the original expression for the T of boosted dust at
velocity v , we let v = 1 + , where || 1. This gives a stressenergy tensor that (ignoring multiplicative constants) looks like:
1 + 1 + 2
If is negative, we have ultrarelativistic dust, and we can verify
that it satisfies the DEC by un-boosting back to the rest frame.
To do this explicitly, we can find the matrix’s eigenvectors, which
(ignoring terms of order 2 ) are (1, 1+) and (1, 1−), with eigenvalues 2 + 2 and 0, respectively. For < 0, the first of these is
timelike, the second spacelike. We interpret them simply as the
Section 8.1
Sources in general relativity
t and x basis vectors of the rest frame in which we originally described the dust. Using them as a basis, the stress-energy tensor
takes on the form diag(2 + 2, 0). Except for a constant factor that
we didn’t bother to keep track of, this is the original form of the
T in the dust’s rest frame, and it clearly satisfies the DEC, since
P = 0.
For > 0, v = 1 + is a velocity greater than the speed of light,
and there is no way to construct a boost corresponding to −v . We
can nevertheless find a frame of reference in which the stressenergy tensor is diagonal, allowing us to check the DEC. The
expressions found above for the eigenvectors and eigenvalues
are still valid, but now the timelike and spacelike characters of the
two basis vectors have been interchanged. The stress-energy
tensor has the form diag(0, 2 + 2), with ρ = 0 and P > 0, which
violates the DEC. As in this example, any flux of mass-energy at
speeds greater than c will violate the DEC.
The DEC is obeyed for < 0 and violated for > 0, and since =
0 gives a stress-energy tensor equal to that of an electromagnetic
wave, we can tell that light is exactly on the border between forms
of matter that fulfill the DEC and those that don’t. Since the DEC
is formulated as a non-strict inequality, it follows that light obeys
the DEC.
No “speed of flux”
Example: 8
The foregoing discussion may have encouraged the reader to believe that it is possible in general to read off a “speed of energy
flux” from the value of T at a point. This is not true.
The difficulty lies in the distinction between flow with and without
accumulation, which is sometimes valid and sometimes not. In
springtime in the Sierra Nevada, snowmelt adds water to alpine
lakes more rapidly than it can flow out, and the water level rises.
This is flow with accumulation. In the fall, the reverse happens,
and we have flow with depletion (negative accumulation).
Figure e/1 shows a second example in which the distinction seems
valid. Charge is flowing through the lightbulb, but because there
is no accumulation of charge in the DC circuit, we can’t detect the
flow by an electrostatic measurement; the wire does not attract
the tiny bits of paper below it on the table.
But we know that with different measurements, we could detect
the flow of charge in e/1. For example, the magnetic field from
the wire would deflect a nearby magnetic compass. This shows
that the distinction between flow with and without accumulation
may be sometimes valid and sometimes invalid. Flow without
accumulation may or may not be detectable; it depends on the
physical context.
e / Example 8.
Chapter 8
In figure e/2, an electric charge and a magnetic dipole are superimposed at a point. The Poynting vector P defined as E × B is
used in electromagnetism as a measure of the flux of energy, and
it tells the truth, for example, when the sun warms your sun on
a hot day. In e/2, however, all the fields are static. It seems as
though there can be no flux of energy. But that doesn’t mean that
the Poynting vector is lying to us. It tells us that there is a pattern of flow, but it’s flow without accumulation; the Poynting vector
forms circular loops that close upon themselves, and electromagnetic energy is transported in and out of any volume at the same
rate. We would perhaps prefer to have a mathematical rule that
gave zero for the flux in this situation, but it’s acceptable that our
rule P = E × B gives a nonzero result, since it doesn’t incorrectly
predict an accumulation, which is what would be detectable.
Now suppose we’re presented with this stress-energy tensor, measured at a single point and expressed in some units:
4.037 ± 0.002 4.038 ± 0.002
T µν =
4.036 ± 0.002 4.036 ± 0.002
To within the experimental error bars, it has the right form to be
many different things: (1) We could have a universe filled with
perfectly uniform dust, moving along the x axis at some ultrarelativistic speed v so great that the in v = 1 − , as in example
7, is not detectably different from zero. (2) This could be a point
sampled from an electromagnetic wave traveling along the x axis.
(3) It could be a point taken from figure e/2. (In cases 2 and 3,
the off-diagonal elements are simply the Poynting vector.)
In cases 1 and 2, we would be inclined to interpret this stressenergy tensor by saying that its off-diagonal part measures the
flux of mass-energy along the x axis, while in case 3 we would reject such an interpretation. The trouble here is not so much in our
interpretation of T as in our Newtonian expectations about what is
or isn’t observable about fluxes that flow without accumulation. In
Newtonian mechanics, a flow of mass is observable, regardless
of whether there is accumulation, because it carries momentum
with it; a flow of energy, however, is undetectable if there is no
accumulation. The trouble here is that relativistically, we can’t
maintain this distinction between mass and energy. The Einstein
field equations tell us that a flow of either will contribute equally to
the stress-energy, and therefore to the surrounding gravitational
The flow of energy in e/2 contributes to the gravitational field, and
its contribution is changed, for example, if the magnetic field is reversed. The figure is in fact not a bad qualitative representation of
the spacetime around a rotating, charged black hole. At large distances, however, the gravitational effect of the off-diagonal terms
in T becomes small, because they average to nearly zero over a
Section 8.1
Sources in general relativity
sufficiently large spherical region. The distant gravitational field
approaches that of a point mass with the same mass-energy.
Momentum in static fields
Example: 9
Continuing the train of thought described in example 8, we can
come up with situations that seem even more paradoxical. In figure e/2, the total momentum of the fields vanishes by symmetry.
This symmetry can, however, be broken by displacing the electric
charge by ∆R perpendicular to the magnetic dipole vector D. The
total momentum no longer vanishes, and now lies in the direction
of D × ∆R. But we have proved in example 2 on p. 264 that a
system’s center of mass-energy is at rest if and only if its total
momentum is zero. Since this system’s center of mass-energy is
certainly at rest, where is the other momentum that cancels that
of the electric and magnetic fields?
Suppose, for example, that the magnetic dipole consists of a loop
of copper wire with a current running around it. If we open a
switch and extinguish the dipole, it appears that the system must
recoil! This seems impossible, since the fields are static, and an
electric charge does not interact with a magnetic dipole.
Babson et al.7 have analyzed a number of examples of this type.
In the present one, the mysterious “other momentum” can be attributed to a relativistic imbalance between the momenta of the
electrons in the different parts of the wire. A subtle point about
these examples is that even in the case of an idealized dipole of
vanishingly small size, it makes a difference what structure we assume for the dipole. In particular, the field’s momentum is nonzero
for a dipole made from a current loop of infinitesimal size, but zero
for a dipole made out of two magnetic monopoles.8
Geodesic motion of test particles
Question 1 on p. 275 was: “Does a small enough physical object
always have a world-line that is approximately a geodesic?” In other
words, do Eötvös experiments give null results when carried out in
laboratories using real-world apparatus of small enough size? We
would like something of this type to be true, since general relativity
is based on the equivalence principle, and the equivalence principle is
motivated by the null results of Eötvös experiments. Nevertheless, it
is fairly easy to show that the answer to the question is no, unless we
make some more specific assumption, such as an energy condition,
about the system being modeled.
Before we worry about energy conditions, let’s consider why the
small size of the apparatus is relevant. Essentially this is because of
gravitational radiation. In a gravitationally radiating system such
as the Hulse-Taylor binary pulsar (p. 220), the material bodies lose
Chapter 8
Am. J. Phys. 77 (2009) 826
Milton and Meille,
energy, and as with any radiation process, the radiated power depends on the square of the strength of the source. The world-line of
a such a body therefore depends on its mass, and this shows that its
world-line cannot be an exact geodesic, since the initially tangent
world-lines of two different masses diverge from one another, and
these two world-lines can’t both be geodesics.
Let’s proceed to give a rough argument for geodesic motion and
then try to poke holes in it. When we test geodesic motion, we do
an Eötvös experiment that is restricted to a certain small region
of spacetime S. Our test-body’s world-line enters S with a certain
energy-momentum vector p and exits with p0 . If spacetime was
flat, then Gauss’s theorem would hold exactly, and the vanishing
divergence ∇b T ab of the stress-energy tensor would require that the
incoming flux represented by p be exactly canceled by the outgoing
flux due to p0 . In reality spacetime isn’t flat, and it isn’t even possible
to compare p and p0 except by parallel-transporting one into the
same location as the other. Parallel transport is path-dependent,
but if we make the reasonable restriction to paths that stay within S,
we expect the ambiguity due to path-dependence to be proportional
to the area enclosed by any two paths, so that if S is small enough,
the ambiguity can be made small. Ignoring this small ambiguity,
we can see that one way for the fluxes to cancel would be for the
particle to travel along a geodesic, since both p and p0 are tangent
to the test-body’s world-line, and a geodesic is a curve that paralleltransports its own tangent vector. Geodesic motion is therefore one
solution, and we expect the solution to be nearly unique when S is
Although this argument is almost right, it has some problems.
First we have to ask whether “geodesic” means a geodesic of the full
spacetime including the object’s own fields, or of the background
spacetime B that would have existed without the object. The latter
is the more sensible interpretation, since the question is basically
asking whether a spacetime can really be defined geometrically, as
the equivalence principle claims, based on the motion of test particles inserted into it. We also have to define words like “small
enough” and “approximately;” to do this, we imagine a sequence of
objects On that get smaller and smaller as n increases. We then
form the following conjecture, which is meant to formulate question
1 more exactly: Given a vacuum background spacetime B, and a
timelike world-line ` in B, consider a sequence of spacetimes Sn ,
formed by inserting the On into B, such that: (i) the metric of Sn is
defined on the same points as the metric of B; (ii) On moves along
`, and for any r > 0, there exists some n such that for m ≥ n, Om
is smaller than r;9 (iii) the metric of Sn approaches the metric of B
i.e., at any point P on `, an observer moving along ` at P defines a surface
of simultaneity K passing through P, and sees the stress-energy tensor of On as
vanishing outside of a three-sphere of radius r within K and centered on P
Section 8.1
Sources in general relativity
as n → ∞. Then ` is a geodesic of B.
This is almost right but not quite, as shown by the following
counterexample. Papapetrou10 has shown that a spinning body in
a curved background spacetime deviates from a geodesic with an
acceleration that is proportional to LR, where L is its angular momentum and R is the Riemann curvature. Let all the On have a fixed
value of L, but let the spinning mass be concentrated into a smaller
and smaller region as n increases, so as to satisfy (ii). As the radius
r decreases, the motion of the particles composing an On eventually
has to become ultrarelativistic, so that the main contribution to the
gravitational field is from the particles’ kinetic energy rather than
their rest mass. We then have L ∼ pr ∼ Er, so that in order to
keep L constant, we must have E ∝ 1/r. This causes two problems.
First, it makes the gravitational field blow up at small distances,
violating (iii). Also, we expect that for any known form of matter,
there will come a point (probably the Tolman-Oppenheimer-Volkoff
limit) at which we get a black hole; the singularity is then not part
of the spacetime Sn , violating (i). But our failed counterexample
can be patched up. We obtain a supply of exotic matter, whose
gravitational mass is negative, and we mix enough of this mysterious stuff into each On so that the gravitational field shrinks rather
than growing as n increases, and no black hole is ever formed.
Ehlers and Geroch11 have proved that it suffices to require an
additional condition: (iv) The On satisfy the dominant energy condition. This rules out our counterexample.
The Newtonian limit
f / Negative mass.
g / The black sphere is made
of ordinary matter.
crosshatched sphere has positive
gravitational mass and negative
inertial mass. If the two of them
are placed side by side in empty
space, they will both accelerate
steadily to the right, gradually
approaching the speed of light.
Conservation of momentum is
preserved, because the exotic
sphere has leftward momentum
when it moves to the right, so the
total momentum is always zero.
Chapter 8
In units with c 6= 1, a quantity like ρ+P is expressed as ρ+P/c2 .
The Newtonian limit is recovered as c → ∞, which makes the pressure term negligible, so that all the energy conditions reduce to
ρ ≥ 0. What would it mean if this was violated? Would ρ < 0
describe an object with negative inertial mass, which would accelerate east when you pushed it to the west? Or would it describe
something with negative gravitational mass, which would repel ordinary matter? We can imagine various possiblities, as shown in
figure f. Anything that didn’t lie on the main diagonal would violate the equivalence principle, and would therefore be impossible
to accomodate within general relativity’s geometrical description of
gravity. If we had “upsidasium” matter such as that described by
the second quadrant of the figure (example 2, p. 26), gravity would
be like electricity, except that like masses would attract and opposites repel; we could have gravitational dielectrics and gravitational
Faraday cages. The fourth quadrant leads to amusing possibilities
like figure g.
Proc. Royal Soc. London A 209 (1951) 248. The relevant result is summarized in Misner, Thorne, and Wheeler, Gravitation, p. 1121.
No gravitational shielding
Example: 10
Electric fields can be completely excluded from a Faraday cage,
and magnetic fields can be very strongly blocked with high-permeability materials such as mu-metal. It would be fun if we could do
the same with gravitational fields, so that we could have zerogravity or near-zero-gravity parties in a specially shielded room.
It would be a form of antigravity, but a different one than the “upsidasium” type. Unfortunately this is difficult to do, and the reason
it’s difficult turns out to be related to the unavailability of materials
that violate energy conditions.
First we need to define what we mean by shielding. We restrict
ourselves to the Newtonian limit, and to one dimension, so that a
gravitational field is specified by a function of one variable g(x).
The best kind of shielding would be some substance that we
could cut with shears and form into a box, and that would exclude gravitational fields from the interior of the box. This would
be analogous to a Faraday cage; no matter what external field it
was embedded in, it would spontaneously adjust itself so that the
internal field was canceled out. A less desirable kind of shielding
would be one that we could set up on an ad hoc basis to null out a
specific, given, externally imposed field. Once we know what the
external field is, we try to choose some arrangement of masses
such that the field is nulled out. We will show that even this kind
of shielding is unachievable, if nulling out the field is interpreted
to mean this: at some point, which for convenience we take to be
the origin, we wish to have a gravitational field such that g(0) = 0,
dg/dx(0) = 0, . . . dn g/dx n (0) = 0, where n is arbitrarily specified.
For comparison, magnetic fields can be nulled out according to
this definition by building an appropriately chosen configuration of
coils such as a Helmholtz coil.
Since we’re only doing the Newtonian limit, the gravitational field
is the sum of the fields made by all the sources, and we can take
this as a sum over point sources. For a point source m placed at
xo , the field g(x) is odd under reflection about xo . The derivative
of the field g 0 (x) is even. Since g 0 is even, we can’t control its sign
at x = 0 by choosing xo > 0 or xo < 0. The only way to control
the sign of g 0 is by choosing the sign of m. Therefore if the sign
of the externally imposed field’s derivative is wrong, we can never
never null it out. Figure h shows a special case of this theorem.
The theorem does not apply to three dimensions, and it does not
prove that all fields are impossible to null out, only that some are.
For example, the field inside a hemispherical shell can be nulled
by adding another hemispherical shell to complete the sphere. I
thank P. Allen for helpful discussion of this topic.
Section 8.1
h / Nulling out a gravitational
field is impossible in one dimension without exotic matter. 1. The
planet imposes a nonvanishing
gravitational field with a nonvanishing gradient. 2. We can null
the field at one point in space, by
placing a sphere of very dense,
but otherwise normal, matter
overhead. The stick figure still
experiences a tidal force, g 0 6= 0.
3. To change the field’s derivative
without changing the field, we
can place two additional masses
above and below the given point.
But to change its derivative in the
desired direction — toward zero
— we would have to make these
masses negative.
Sources in general relativity
Singularity theorems
An important example of the use of the energy conditions is that
Hawking and Ellis have proved that under the assumption of the
strong energy condition, any body that becomes sufficiently compact will end up forming a singularity. We might imagine that
the formation of a black hole would be a delicate thing, requiring
perfectly symmetric initial conditions in order to end up with the
perfectly symmetric Schwarzschild metric. Many early relativists
thought so, for good reasons. If we look around the universe at
various scales, we find that collisions between astronomical bodies
are extremely rare. This is partly because the distances are vast
compared to the sizes of the objects, but also because conservation
of angular momentum has a tendency to make objects swing past
one another rather than colliding head-on. Starting with a cloud of
objects, e.g., a globular cluster, Newton’s laws make it extremely
difficult, regardless of the attractive nature of gravity, to pick initial
conditions that will make them all collide the future. For one thing,
they would have to have exactly zero total angular momentum.
Most relativists now believe that this is not the case. General
relativity describes gravity in terms of the tipping of light cones.
When the field is strong enough, there is a tendency for the light
cones to tip over so far that the entire future light-cone points at the
source of the field. If this occurs on an entire surface surrounding
the source, it is referred to as a trapped surface.
To make this notion of light cones “pointing at the source” more
rigorous, we need to define the volume expansion Θ. Let the set of
all points in a spacetime (or some open subset of it) be expressed as
the union of geodesics. This is referred to as a foliation in geodesics,
or a congruence. Let the velocity vector tangent to such a curve
be ua . Then we define Θ = ∇a ua . This is exactly analogous to
the classical notion of the divergence of the velocity field of a fluid,
which is a measure of compression or expansion. Since Θ is a scalar,
it is coordinate-independent. Negative values of Θ indicate that the
geodesics are converging, so that volumes of space shrink. A trapped
surface is one on which Θ is negative when we foliate with lightlike
geodesics oriented outward along normals to the surface.
When a trapped surface forms, any lumpiness or rotation in
the initial conditions becomes irrelevant, because every particle’s
entire future world-line lies inward rather than outward. A possible loophole in this argument is the question of whether the light
cones will really tip over far enough. We could imagine that under extreme conditions of high density and temperature, matter
might demonstrate unusual behavior, perhaps including a negative
energy density, which would then give rise to a gravitational repulsion. Gravitational repulsion would tend to make the light cones tip
outward rather than inward, possibly preventing the collapse to a
Chapter 8
singularity. We can close this loophole by assuming an appropriate
energy condition. Penrose and Hawking have formalized the above
argument in the form of a pair of theorems, known as the singularity
theorems. One of these applies to the formation of black holes, and
another one to cosmological singularities such as the Big Bang.
In a cosmological model, it is natural to foliate using worldlines that are at rest relative to the Hubble flow (or, equivalently,
the world-lines of observers who see a vanishing dipole moment in
the cosmic microwave background). The Θ we then obtain is positive, because the universe is expanding. The volume expansion
is Θ = 3Ho , where Ho ≈ 2.3 × 10−18 s−1 is the Hubble constant
(the fractional rate of change of the scale factor of cosmological distances). The factor of three occurs because volume is proportional
to the cube of the linear dimensions.
Current status
The current status of the energy conditions is shaky. Although
it is clear that all of them hold in a variety of situations, there are
strong reasons to believe that they are violated at both microscopic
and cosmological scales, for reasons both classical and quantummechanical.12 We will see such a violation in the following section.
8.1.4 The cosmological constant
Having included the source term in the Einstein field equations,
our most important application will be to cosmology. Some of the
relevant ideas originate long before Einstein. Once Newton had
formulated a theory of gravity as a universal attractive force, he
realized that there would be a tendency for the universe to collapse.
He resolved this difficulty by assuming that the universe was infinite
in spatial extent, so that it would have no center of symmetry, and
therefore no preferred point to collapse toward. The trouble with
this argument is that the equilibrium it describes is unstable. Any
perturbation of the uniform density of matter breaks the symmetry,
leading to the collapse of some pocket of the universe. If the radius
of such a collapsing region is r, then its gravitational is proportional
to r3 , and its gravitational field is proportional to r3 /r2 = r. Since
its acceleration is proportional to its own size, the time it takes to
collapse is independent of its size. The prediction is that the universe will have a self-similar structure, in which the clumping on
small scales behaves in the same way as clumping on large scales;
zooming in or out in such a picture gives a landscape that appears
the same. With modern hindsight, this is actually not in bad agreement with reality. We observe that the universe has a hierarchical
structure consisting of solar systems, galaxies, clusters of galaxies,
superclusters, and so on. Once such a structure starts to condense,
Barcelo and Visser, “Twilight for the energy conditions?,” http://arxiv.
Section 8.1
Sources in general relativity
the collapse tends to stop at some point because of conservation of
angular momentum. This is what happened, for example, when our
own solar system formed out of a cloud of gas and dust.
Einstein confronted similar issues, but in a more acute form.
Newton’s symmetry argument, which failed only because of its instability, fails even more badly in relativity: the entire spacetime
can simply contract uniformly over time, without singling out any
particular point as a center. Furthermore, it is not obvious that
angular momentum prevents total collapse in relativity in the same
way that it does classically, and even if it did, how would that apply
to the universe as a whole? Einstein’s Machian orientation would
have led him to reject the idea that the universe as a whole could
be in a state of rotation, and in any case it was sensible to start the
study of relativistic cosmology with the simplest and most symmetric possible models, which would have no preferred axis of rotation.
Because of these issues, Einstein decided to try to patch up his
field equation so that it would allow a static universe. Looking back
over the considerations that led us to this form of the equation,
we see that it is very nearly uniquely determined by the following
• It should be consistent with experimental evidence for local
conservation of mass-energy and momentum.
• It should satisfy the equivalence principle.
• It should be coordinate-independent.
• It should be equivalent to Newtonian gravity in the appropriate limit.
• It should not be overdetermined.
This is not meant to be a rigorous proof, just a general observation
that it’s not easy to tinker with the theory without breaking it.
A failed attempt at tinkering
Example: 11
As an example of the lack of “wiggle room” in the structure of the
field equations, suppose we construct the scalar T aa , the trace of
the stress-energy tensor, and try to insert it into the field equations as a further source term. The first problem is that the field
equation involves rank-2 tensors, so we can’t just add a scalar.
To get around this, suppose we multiply by the metric. We then
have something like Gab = c1 Tab + c2 gab T cc , where the two constants c1 and c2 would be constrained by the requirement that the
theory agree with Newtonian gravity in the classical limit.
To see why this attempt fails, consider a beam of light directed
along the x axis. Its momentum is equal to its energy (see page
126), so its contributions to the local energy density and pressure
Chapter 8
are equal. Thus its contribution to the stress-energy tensor is of
the form T µν = (constant) × diag(−1, 1, 0, 0). The trace vanishes,
so the beam of light’s coupling to gravity in the c2 term is zero.
As discussed on pp. 267-270, empirical tests of conservation of
momentum would therefore constrain c2 to be . 10−8 .
One way in which we can change the field equation without
violating any of these is to add a term Λgab , giving
Gab = 8πTab + Λgab
which is what we will refer to as the Einstein field equation.13 The
universal constant Λ is called the cosmological constant. Einstein
originally introduced a positive cosmological constant because he
wanted relativity to be able to describe a static universe. To see
why it would have this effect, compare its behavior with that of
an ordinary fluid. When an ordinary fluid, such as the exploding
air-gas mixture in a car’s cylinder, expands, it does work on its environment, and therefore by conservation of energy its own internal
energy is reduced. A positive cosmological constant, however, acts
like a certain amount of mass-energy built into every cubic meter of
vacuum. Thus when it expands, it releases energy. Its pressure is
negative. Another way of verifying these statements is by observing
that for a given cosmological constant, we can always observe the
Λgab term in the field equations into the 8πTab , as if the cosmological
constant were some form of matter.
Now consider the following pseudo-classical argument. Although
we’ve already seen (page 217) that there is no useful way to separate the roles of kinetic and potential energy in general relativity,
suppose that there are some quantities analogous to them in the
description of the universe as a whole. (We’ll see below that the
universe’s contraction and expansion is indeed described by a set of
differential equations that can be interpreted in essentially this way.)
If the universe contracts, a cubic meter of space becomes less than
a cubic meter. The cosmological-constant energy associated with
that volume is reduced, so some energy has been consumed. The
kinetic energy of the collapsing matter goes down, and the collapse
is decelerated.
The addition of the Λ term constitutes a change to the vacuum
field equations, and the good agreement between theory and experiment in the case of, e.g., Mercury’s orbit puts an upper limit on
Λ then implies that Λ must be small. For an order-of-magnitude
estimate, consider that Λ has units of mass density, and the only
parameters with units that appear in the description of Mercury’s
orbit are the mass of the sun, m, and the radius of Mercury’s orbit,
r. The relativistic corrections to Mercury’s orbit are on the order
In books that use a − + ++ metric rather then our + − −−, the sign of the
cosmological constant term is reversed relative to ours.
Section 8.1
Sources in general relativity
of v 2 , or about 10−8 , and they come out right. Therefore we can estimate that the cosmological constant could not have been greater
than about (10−8 )m/r3 ∼ 10−10 kg/m3 , or it would have caused
noticeable discrepancies. This is a very poor bound; if Λ was this
big, we might even be able to detect its effects in laboratory experiments. Looking at the role played by r in the estimate, we see
that the upper bound could have been made tighter by increasing
r. Observations on galactic scales, for example, constrain it much
more tightly. This justifies the description of Λ as cosmological: the
larger the scale, the more significant the effect of a nonzero Λ would
Since the right-hand side of the field equation is 8πTab + Λgab , it
is possible to consider the cosmological constant as a type of matter
contributing to the stress-energy tensor. We then have ρ = −P =
Λ/8π. As described in more detail in section 8.2.11 on p. 319, we
now know that Λ is positive. With Λ > 0, the weak and dominant energy conditions are both satisfied, so that in every frame
of reference, ρ is positive and there is no flux of energy flowing at
speeds greater than c. The negative pressure does violate the strong
energy condition, meaning that the constant acts as a form of gravitational repulsion. If the cosmological constant is a product of the
quantum-mechanical structure of the vacuum, then this violation
is not too surprising, because quantum fields are known to violate
various energy conditions. For example, the energy density between
two parallel conducting plates is negative due to the Casimir effect.
If Λ is thought of as a form of matter, then it becomes natural to ask whether it’s spread more thickly in some places than
others: is the cosmological “constant” really constant? The following argument shows that it cannot vary. The field equations are
Gab = 8πTab + Λgab . Taking the divergence of both sides, we have
∇a Gab = 8π∇a Tab + ∇a (Λgab ). The left-hand side vanishes (see
p. 263). Since laboratory experiments have verified conservation of
mass-energy to high precision for all the forms of matter represented
by T , we have ∇a Tab = 0 as well. Applying the product rule to the
term ∇a (Λgab ), we get gab ∇a Λ + Λ∇a gab . But the covariant derivative of the metric vanishes, so the result is simply ∇b Λ. Thus any
variation in the cosmological constant over space or time violates the
field equations, and the violation is equivalent to the violation we
would get from a form of matter than didn’t conserve mass-energy
8.2 Cosmological solutions
We are thus led to pose two interrelated questions. First, what
can empirical observations about the universe tell us about the laws
of physics, such as the zero or nonzero value of the cosmological
constant? Second, what can the laws of physics, combined with
Chapter 8
observation, tell us about the large-scale structure of the universe,
its origin, and its fate?
8.2.1 Evidence for the finite age of the universe
We have a variety of evidence that the universe’s existence does
not stretch for an unlimited time into the past.
When astronomers view light from the deep sky that has been
traveling through space for billions of years, they observe a universe that looks different from today’s. For example, quasars were
common in the early universe but are uncommon today.
In the present-day universe, stars use up deuterium nuclei, but
there are no known processes that could replenish their supply. We
therefore expect that the abundance of deuterium in the universe
should decrease over time. If the universe had existed for an infinite
time, we would expect that all its deuterium would have been lost,
and yet we observe that deuterium does exist in stars and in the
interstellar medium.
The second law of thermodynamics predicts that any system
should approach a state of thermodynamic equilibrium, and yet our
universe is very far from thermal equilibrium, as evidenced by the
fact that our sun is hotter than interstellar space, or by the existence
of functioning heat engines such as your body or an automobile
With hindsight, these observations suggest that we should not
look for cosmological models that persist for an infinite time into
the past.
8.2.2 Evidence for expansion of the universe
We don’t only see time-variation in locally observable quantities
such as quasar abundance, deuterium abundance, and entropy. In
addition, we find empirical evidence for global changes in the universe. By 1929, Edwin Hubble at Mount Wilson had determined
that the universe was expanding, and historically this was the first
convincing evidence that Einstein’s original goal of modeling a static
cosmology had been a mistake. Einstein later referred to the cosmological constant as the “greatest blunder of my life,” and for the
next 70 years it was commonly assumed that Λ was exactly zero.
Since we observe that the universe is expanding, the laws of thermodynamics require that it also be cooling, just as the exploding
air-gas mixture in a car engine’s cylinder cools as it expands. If the
universe is currently expanding and cooling, it is natural to imagine
that in the past it might have been very dense and very hot. This is
confirmed directly by looking up in the sky and seeing radiation from
the hot early universe. In 1964, Penzias and Wilson at Bell Laboratories in New Jersey detected a mysterious background of microwave
radiation using a directional horn antenna. As with many acciden-
Section 8.2
Cosmological solutions
a / The horn antenna
by Penzias and Wilson.
tal discoveries in science, the important thing was to pay attention
to the surprising observation rather than giving up and moving on
when it confounded attempts to understand it. They pointed the
antenna at New York City, but the signal didn’t increase. The radiation didn’t show a 24-hour periodicity, so it couldn’t be from a
source in a certain direction in the sky. They even went so far as to
sweep out the pigeon droppings inside. It was eventually established
that the radiation was coming uniformly from all directions in the
sky and had a black-body spectrum with a temperature of about 3
This is now interpreted as follows. At one time, the universe
was hot enough to ionize matter. An ionized gas is opaque to light,
since the oscillating fields of an electromagnetic wave accelerate the
charged particles, depositing kinetic energy into them. Once the
universe became cool enough, however, matter became electrically
neutral, and the universe became transparent. Light from this time
is the most long-traveling light that we can detect now. The latest
data show that transparency set in when the temperature was about
3000 K. The surface we see, dating back to this time, is known as
the surface of last scattering. Since then, the universe has expanded
by about a factor of 1000, causing the wavelengths of photons to
be stretched by the same amount due to the expansion of the underlying space. This is equivalent to a Doppler shift due to the
source’s motion away from us; the two explanations are equivalent.
We therefore see the 3000 K optical black-body radiation red-shifted
to 3 K, in the microwave region.
It is logically possible to have a universe that is expanding but
whose local properties are nevertheless static, as in the steady-state
model of Fred Hoyle, in which some novel physical process spontaneously creates new hydrogen atoms, preventing the infinite dilution
of matter over the universe’s history, which in this model extends
infinitely far into the past. But we have already seen strong empirical evidence that the universe’s local properties (quasar abundance,
etc.) are changing over time. The CMB is an even more extreme
and direct example of this; the universe full of hot, dense gas that
emitted the CMB is clearly nothing like today’s universe.14
8.2.3 Evidence for homogeneity and isotropy
These observations demonstrate that the universe is not homogeneous in time, i.e., that one can observe the present conditions of
the universe (such as its temperature and density), and infer what
epoch of the universe’s evolution we inhabit. A different question
is the Copernican one of whether the universe is homogeneous in
space. Surveys of distant quasars show that the universe has very
little structure at scales greater than a few times 1025 m. (This can
For a detailed review of the evidence that rules out various variations on the
Hoyle theme, see
Chapter 8
be seen on a remarkable logarithmic map constructed by Gott et
al., This suggests that we can,
to a good approximation, model the universe as being isotropic (the
same in all spatial directions) and homogeneous (the same at all
locations in space).
Further evidence comes from the extreme uniformity of the cosmic microwave background radiation, once one subtracts out the
dipole anisotropy due to the Doppler shift arising from our galaxy’s
motion relative to the CMB. When the CMB was first discovered,
there was doubt about whether it was cosmological in origin (rather
than, say, being associated with our galaxy), and it was expected
that its isotropy would be as large as 10%. As physicists began to
be convinced that it really was a relic of the early universe, interest
focused on measuring this anisotropy, and a series of measurements
put tighter and tighter upper bounds on it.
Other than the dipole term, there are two ways in which one
might naturally expect anisotropy to occur. There might have
been some lumpiness in the early universe, which might have served
as seeds for the condensation of galaxy clusters out of the cosmic
medium. Furthermore, we might wonder whether the universe as a
whole is rotating. The general-relativistic notion of rotation is very
different from the Newtonian one, and in particular, it is possible
to have a cosmology that is rotating without having any center of
rotation (see problem 5, p. 260). In fact one of the first exact solutions discovered for the Einstein field equations was the Gödel metric, which described a bizarre rotating universe with closed timelike
curves, i.e., one in which causality was violated. In a rotating universe, one expects that radiation received from great cosmological
distances will have a transverse Doppler shift, i.e., a shift originating from the time dilation due to the motion of the distant matter
across the sky. This shift would be greatest for sources lying in the
plane of rotation relative to us, and would vanish for sources lying
along the axis of rotation. The CMB would therefore show variation with the form of a quadrupole term, 3 cos2 θ − 1. In 1977 a U-2
spyplane (the same type involved in the 1960 U.S.-Soviet incident)
was used by Smoot et al.15 to search for anisotropies in the CMB.
This experiment was the first to definitively succeed in detecting
the dipole anisotropy. After subtraction of the dipole component,
the CMB was found to be uniform at the level of ∼ 3 × 10−4 . This
provided strong support for homogeneous cosmological models, and
ruled out rotation of the universe with ω & 10−22 Hz.
G. F. Smoot, M. V. Gorenstein, and R. A. Muller, “Detection of Anisotropy
in the Cosmic Blackbody Radiation,” Phys. Rev. Lett. 39 (1977) 898. The
interpretation of the CMB measurements is somewhat model-dependent; in the
early years of observational cosmology, it was not even universally accepted that
the CMB had a cosmological origin. The best model-independent limit on the
rotation of the universe comes from observations of the solar system, Clemence,
“Astronomical Time,” Rev. Mod. Phys. 29 (1957) 2.
Section 8.2
Cosmological solutions
8.2.4 The FRW cosmologies
The FRW metric and the standard coordinates
Motivated by Hubble’s observation that the universe is expanding, we hypothesize the existence of solutions of the field equation
in which the properties of space are homogeneous and isotropic, but
the over-all scale of space is increasing as described by some scale
function a(t). Because of coordinate invariance, the metric can still
be written in a variety of forms. One such form is
ds2 = dt2 − a(t)2 d`2
where the spatial part is
d`2 = f (r)dr2 + r2 dθ2 + r2 sin2 θdφ2
To interpret the coordinates, we note that if an observer is able
to determine the functions a and f for her universe, then she can
always measure some scalar curvature such as the Ricci scalar or
the Kretchmann invariant, and since these are proportional to a
raised to some power, she can determine a and t. This shows that
t is a “look-out-the-window” time, i.e., a time coordinate that we
can determine by looking out the window and observing the present
conditions in the universe. Because the quantity being measured directly is a scalar, the result is independent of the observer’s state of
motion. (In practice, these scalar curvatures are difficult to measure
directly, so we measure something else, like the sky-wide average
temperature of the cosmic microwave background.) Simultaneity is
supposed to be ill-defined in relativity, but the look-out-the-window
time defines a notion of simultaneity that is the most naturally interesting one in this spacetime. With this particular definition of
simultaneity, we can also define a preferred state of rest at any location in spacetime, which is the one in which t changes as slowly as
possible relative to one’s own clock. This local rest frame, which is
more easily determined in practice as the one in which the microwave
background is most uniform across the sky, can also be interpreted
as the one that is moving along with the Hubble flow, i.e., the average motion of the galaxies, photons, or whatever else inhabits the
spacetime. The time t is interpreted as the proper time of a particle
that hasRalways been locally at rest. The spatial distance measured
by L = ad` is called the proper distance. It is the distance that
would be measured by a chain of rulers, each of them “at rest” in
the above sense.
These coordinates are referred as the “standard” cosmological
coordinates; one will also encounter other choices, such as the comoving and conformal coordinates, which are more convenient for
certain purposes. Historically, the solution for the functions a and
f was found by de Sitter in 1917.
Chapter 8
The spatial metric
The unknown function f (r) has to give a 3-space metric d`2
with a constant Einstein curvature tensor. The following Maxima
program computes the curvature.
Line 2 tells Maxima that we’re working in a space with three dimensions rather than its default of four. Line 4 tells it that f is a
function of time. Line 9 uses its built-in function for computing the
Einstein tensor Gab . The result has only one nonvanishing component, Gtt = (1 − 1/f )/r2 . This has to be constant, and since scaling
can be absorbed in the factor a(t) in the 3+1-dimensional metric,
we can just set the value of Gtt more or less arbitrarily, except for
its sign. The result is f = 1/(1 − kr2 ), where k = −1, 0, or 1.
The resulting metric, called the Robertson-Walker metric, is
ds = dt − a
+ r dθ + r sin θdφ
1 − kr2
The form of d`2 shows us that k can be interpreted in terms of
the sign of the spatial curvature. We recognize the k = 0 metric
as a flat spacetime described in spherical coordinates. To interpret
the k 6= 0 cases, we note that a circle at coordinate Rr has
circumference C = 2πar and proper radius R = a 0 f (r0 )dr0 .
For k < 0, we have f < 1 and C > 2πR, indicating negative spatial
curvature. For k > 0 there is positive curvature.
Let’s examine the positive-curvature case more closely. Suppose
we select a particular plane of simultaneity defined by t = constant
and φ = π/2, and we start doing geometry in this plane. In two spatial dimensions, the Riemann tensor only has a single independent
component, which can be identified with the Gaussian curvature
(sec. 5.4, p. 168), and when this Gaussian curvature is positive and
constant, it can be interpreted as the angular defect of a triangle
per unit area (sec. 5.3, p. 162). Since the sum of the interior angles of a triangle can never be greater than 3π, we have an upper
limit on the area of any triangle. This happens because the positivecurvature Robertson-Walker metric represents a cosmology that is
spatially finite. At a given t, it is the three-dimensional analogue
Section 8.2
Cosmological solutions
of a two-sphere. On a two-sphere, if we set up polar coordinates
with a given point arbitrarily chosen as the origin, then we know
that the r coordinate must “wrap around” when we get to the antipodes. That is, there is a coordinate singularity there. (We know
it can only be a coordinate singularity, because if it wasn’t, then the
antipodes would have special physical characteristics, but the FRW
model was constructed to be spatially homogeneous.) This “wraparound” behavior is described by saying that the model is closed.
b / 1. In the Euclidean plane, this
triangle can be scaled by any
factor while remaining similar to
itself. 2. In a plane with positive
curvature, geometrical figures
have a maximum area and maximum linear dimensions. This
triangle has almost the maximum
area, because the sum of its
angles is nearly 3π. 3. In a plane
with negative curvature, figures
have a maximum area but no
maximum linear dimensions. This
triangle has almost the maximum
area, because the sum of its
angles is nearly zero. Its vertices,
however, can still be separated
from one another without limit.
In the negative-curvature case, there is no limit on distances,
b/3. Such a universe is called open. In the case of an open universe,
it is particularly easy to demonstrate a fact that bothers many students, which is that proper distances can grow at rates exceeding c.
Let particles A and B both be at rest relative to the Hubble flow.
The Rproper distance between them is then given by L = a`, where
` = A d` is constant. Then differentiating L with respect to the
look-out-the-window time t gives dL/dt = ȧ`. In an open universe,
there is no limit on the size of `, so at any given time, we can make
dL/dt as large as we like. This does not violate special relativity,
since it is only locally that special relativity is a valid approximation
to general relativity. Because GR only supplies us with frames of
reference that are local, the velocity of two objects relative to one
another is not even uniquely defined; our choice of dL/dt was just
one of infinitely many possible definitions.
The distinction between closed and open universes is not just
a matter of geometry, it’s a matter of topology as well. Just as a
two-sphere cannot be made into a Euclidean plane without cutting
or tearing, a closed universe is not topologically equivalent to an
open one. The correlation between local properties (curvature) and
global ones (topology) is a general theme in differential geometry.
A universe that is open is open forever, and similarly for a closed
The Friedmann equations
Having fixed f (r), we can now see what the field equation tells
us about a(t). The next program computes the Einstein tensor for
the full four-dimensional spacetime:
Chapter 8
The result is
Grr = Gθθ = Gφφ
+ 3ka−2
+ ka−2
=2 +
where dots indicate differentiation with respect to time.
Since we have Gab with mixed upper and lower indices, we either
have to convert it into Gab , or write out the field equations in this
mixed form. The latter turns out to be simpler. In terms of mixed
indices, g ab is always simply diag(1, 1, 1, 1). Arbitrarily singling out
r = 0 for simplicity, we have g = diag(1, −a2 , 0, 0). The stressenergy tensor is T µν = diag(−ρ, P , P , P ). Substituting into Gab =
8πT ab + Λg ab , we find
+ 3ka−2 − Λ = 8πρ
2 +
+ ka−2 − Λ = −8πP
Rearranging a little, we have a set of differential equations known
as the Friedmann equations,
= Λ−
= Λ+
(ρ + 3P )
ρ − ka−2
The cosmology that results from a solution of these differential
equations is known as the Friedmann-Robertson-Walker (FRW) or
Friedmann-Lemaı̂tre-Robertson-Walker (FLRW) cosmology.
The first Friedmann equation describes the rate at which cosmological expansion accelerates or decelerates. Let’s refer to it as the
acceleration equation. It expresses the basic idea of the field equations, which is that non-tidal curvature (left-hand side) is caused
by the matter that is present locally (right-hand side). Example 12
illustrates this in a simple case.
The second Friedmann equation tells us the magnitude of the
rate of expansion or contraction. Call it the velocity equation. The
quantity ȧ/a, evaluated at the present cosmological time, is the
Hubble constant Ho (which is constant only in the sense that at a
fixed time, it is a constant of proportionality between distance and
recession velocity).
To the practiced eye, it seems odd to have two dynamical laws,
one predicting velocity and one acceleration. The analogous laws in
freshman mechanics would be Newton’s second law, which predicts
Section 8.2
c / Alexander Friedmann (18881925).
Cosmological solutions
acceleration, and conservation of energy, which predicts velocity.
Newton’s laws and conservation of energy are not independent, and
for mechanical systems either can be derived from the other. The
Friedmann equations, however, are not overdetermined or redundant. They are underdetermined, because we want to predict three
unknown functions of time: a, ρ, and P . Since there are only two
equations, they are not sufficient to uniquely determine a solution
for all three functions. The third constraint comes in the form of
some type of equation of state for the matter described by ρ and P ,
which in simple models can often be written in the form P = wρ.
For example, dust has w = 0.
Unlike a, ρ, and P , the cosmological constant Λ is not free to
vary with time; if it did, then the stress-energy tensor would have a
nonvanishing divergence, which is not consistent with the Einstein
field equations (see p. 288).
Although general relativity does not provide any scalar, globally
conserved measure of mass-energy that is conserved in all spacetimes, the Friedmann velocity equation can be loosely interpreted
as a statement of conservation of mass-energy in an FRW spacetime.
The left-hand side acts like kinetic energy. In a cosmology that expands and then recontracts in a Big Crunch, the turn-around point
is defined by the time at which the right-hand side equals zero. The
origin of the velocity equation is in fact the time-time part of the
field equations, whose source term is the mass-energy component of
the stress-energy tensor.
Scooping out a hole
Example: 12
This example illustrates the connection between cosmological acceleration and local density of matter given by the Friedmann acceleration equation. Consider two cosmologies, each with Λ = 0.
Cosmology 1 is an FRW spacetime in which all matter is in the
form of nonrelativistic particles such as atoms or galaxies. 2 is
identical to 1, except that all the matter has been scooped out of a
small spherical region S, leaving a vacuum. (“Small” means small
compared to the Hubble scale 1/Ho .) Within S, we introduce test
particles A and B. Because an FRW spacetime is homogeneous
and isotropic, cosmology 2 retains spherical symmetry about the
center of S. Since Λ = 0, Birkhoff’s theorem applies to 2, so 2 is
flat inside S. Therefore in 2, the relative acceleration a of the test
particles equals zero.
d / Example 12.
Because S is small compared to cosmological distances, and because the dust is nonrelativistic, local observers can accurately
attibute the difference in behavior between 1 and 2 to the Newtonian gravitational force from the dust that was present in 1 but not
in 2. For convenience, let A and B both be initially at rest relative
to the local dust (i.e., having θ̇ = φ̇ = 0). By the definition of the
scale factor (i.e., by inspection of the FRW metric), the distance
Chapter 8
between them varies as const × a(t). If one of these particles is
an observer, she sees a “force” acting on the other particle that
causes an acceleration (ä/a)r, where r is the displacement between the particles.
Since a = 0 in 2, it follows that the acceleration in 1 can be calculated accurately by finding the Newtonian gravitational force due
to the added dust. This results in a connection between ä/a, on
the left-hand side of the Friedmann acceleration equation, and ρ,
on the right side.
For consistency, we can verify that the Newtonian gravitational
force exerted by a uniform sphere, at a point on its interior, is
proportional to r. This is a classic result that is easily derived
from Newton’s shell theorem.
8.2.5 A singularity at the Big Bang
The Friedmann equations only allow a constant a in the case
where Λ is perfectly tuned relative to the other parameters, and
even this artificially fine-tuned equilibrium turns out to be unstable. These considerations make a static cosmology implausible on
theoretical grounds, and they are also consistent with the observed
Hubble expansion (p. 289).
Since the universe is not static, what happens if we use general
relativity to extrapolate farther and farther back in time?
If we extrapolate the Friedmann equations backward in time, we
find that they always have a = 0 at some point in the past, and this
occurs regardless of the details of what we assume about the matter and radiation that fills the universe. To see this, note that, as
discussed in example 14 on page 132, radiation is expected to dominate the early universe, for generic reasons that are not sensitive
to the (substantial) observational uncertainties about the universe’s
present-day mixture of ingredients. Under radiation-dominated conditions, we can approximate Λ = 0 and P = ρ/3 (example 14, p. 132)
in the first Friedmann equation, finding
=− ρ
where ρ is the density of mass-energy due to radiation. Since ä/a
is always negative, the graph of a(t) is always concave down, and
since a is currently increasing, there must be some time in the past
when a = 0. One can readily verify that this is not just a coordinate singularity; the Ricci scalar curvature Raa diverges, and the
singularity occurs at a finite proper time in the past.
In section 6.3.1, we saw that a black hole contains a singularity,
but it appears that such singularities are always hidden behind event
horizons, so that we can never observe them from the outside. The
FRW singularity, however, is not hidden behind an event horizon.
Section 8.2
e / Georges Lemaı̂tre (18941966) proposed in 1927 that our
universe be modeled in general
relativity as a spacetime in which
space expanded over time.
Lemaı̂tre’s ideas were initially
treated skeptically by Eddington
and Einstein, who told him,
“Your calculations are correct,
but your physics is abominable.”
Later, as Hubble’s observational
evidence for cosmological expansion became widely accepted,
both Einstein and Eddington
became converts, helping to
bring Lemaı̂tre’s ideas to the
attention of the community. In
1931, an emboldened Lemaı̂tre
described the idea that the
universe began from a “Primeval
Atom” or “Cosmic Egg.”
name that eventually stuck was
“Big Bang,” coined by Fred Hoyle
as a derisive term.
Cosmological solutions
It lies in our past light-cone, and our own world-lines emerged from
it. The universe, it seems, originated in a Big Bang, a concept
that originated with the Belgian Roman Catholic priest Georges
Self-check: Why is it not correct to think of the Big Bang as an
explosion that occurred at a specific point in space?
Does the FRW singularity represent something real about our
One thing to worry about is the accuracy of our physical modeling of the radiation-dominated universe. The presence of an initial
singularity in the FRW solutions does not depend sensitively on on
assumptions like P = ρ/3, but it is still disquieting that no laboratory experiment has ever come close to attaining the conditions
under which we could test whether a gas of photons produces gravitational fields as predicted by general relativity. We saw on p. 267
that static electric fields do produce gravitational fields as predicted,
but this is not the same as an empirical confirmation that electromagnetic waves also act as gravitational sources in exactly the
manner that general relativity claims. We do, however, have a consistency check in the form of the abundances of nuclei. Calculations
of nuclear reactions in the early, radiation-dominated universe predict certain abundances of hydrogen, helium, and deuterium. In
particular, the relative abundance of helium and deuterium is a sensitive test of the relationships among a, ȧ, and ä predicted by the
FRW equations, and they confirm these relationships to a precision
of about 5 ± 4%.16
An additional concern is whether the Big Bang singularity is
just a product of the unrealistic assumption of perfect symmetry
that went into the FRW cosmology. One of the Penrose-Hawking
singularity theorems proves that it is not.17 This particular singularity theorm requires three conditions: (1) the strong energy condition
holds; (2) there are no closed timelike curves; and (3) a trapped surface exists in the past timelike geodesics originating at some point.
The requirement of a trapped surface can fail if the universe is inhomogeneous to & 10−4 , but observations of the cosmic microwave
background rule out any inhomogeneity this large (see p. 290). The
other possible failure of the assumptions is that if the cosmological
constant is large enough, it violates the strong energy equation, and
we can have a Big Bounce rather than a Big Bang (see p. 311).
Steigman, Ann. Rev. Nucl. Part. Sci. 57 (2007) 463. These tests are
stated in terms of the Hubble “constant” H = ȧ/a, which is actually varying
over cosmological time-scales. The nuclear helium-deuterium ratio is sensitive
to Ḣ/H.
Hawking and Ellis, “The Cosmic Black-Body Radiation and the Existence of
Singularities in Our Universe,” Astrophysical Journal, 152 (1968) 25. Available
online at
Chapter 8
An exceptional case: the Milne universe
There is still a third loophole in our conclusion that the Big Bang
singularity must have existed. Consider the special case of the FRW
analysis, found by Milne in 1932 (long before FRW), in which the
universe is completely empty, with ρ = 0 and Λ = 0. This is of
course not consistent with the fact that the universe contains stars
and galaxies, but we might wonder whether it could tell us anything
interesting as a simplified approximation to a very dilute universe.
The result is that the scale factor a varies linearly with time (problem 3, p. 329). If a is not constant, then there exists a time at which
a = 0, but this doesn’t turn out to be a real singularity (which isn’t
surprising, since there is no matter to create gravitational fields).
Let this universe have a scattering of test particles whose masses
are too small to invalidate the approximation of ρ = 0, and let
the test particles be at rest in the (r, θ, φ) coordinates. The linear
dependence of a on t means that these particles simply move inertially and without any gravitational interactions, spreading apart
from one another at a constant rate like the raisins in a rising loaf
of raisin bread. The Friedmann equations require k = −1, so the
spatial geometry is one of constant negative curvature.
The Milne universe is in fact flat spacetime described in tricky
coordinates. The connection can be made as follows. Let a spherically symmetric cloud of test particles be emitted by an explosion
that occurs at some arbitrarily chosen event in flat spacetime. Make
the cloud’s density be nonuniform in a certain specific way, so that
every observer moving along with a test particle (called a comoving
observer) sees the same local conditions in his own frame; due to
Lorentz contraction by a factor γ, this requires that the density be
proportional to γ as described by the observer O who remained at
the origin. This scenario turns out to be identical to the Milne universe under the change of coordinates from spatially flat coordinates
(T , R) to FRW coordinates (t, r), where t = T /γ is the proper time
and r = vγ. (Cf. problem 11, p. 200.)
The Milne universe may be useful as an innoculation against
the common misconception that the Big Bang was an explosion of
matter spreading out into a preexisting vacuum. Such a description
seems obviously incompatible with homogeneity, since, for example,
an observer at the edge of the cloud sees the cloud filling only half
of the sky. But isn’t this a logical contradiction, since the Milne
universe does have an explosion into vacuum, and yet it was derived
as a special case of the FRW analysis, which explicitly assumed homogeneity? It is not a contradiction, because a comoving observer
never actually sees an edge. In the limit as we approach the edge,
the density of the cloud (as seen by the observer who stayed at
the origin) approaches infinity, and the Lorentz contraction also approaches infinity, so that O considers them to be like Hamlet saying,
“I could be bounded in a nutshell, and count myself a king of infinite
Section 8.2
Cosmological solutions
space.” This logic only works in the case of the Milne universe. The
explosion-into-preexisting-vacuum interpretation fails in Big Bang
cosmologies with ρ 6= 0.
8.2.6 Observability of expansion
Brooklyn is not expanding!
The proper interpretation of the expansion of the universe, as
described by the Friedmann equations, can be tricky. The example
of the Milne universe encourages us to imagine that the expansion
would be undetectable, since the Milne universe can be described as
either expanding or not expanding, depending on the choice of coordinates. A more general consequence of coordinate-independence is
that relativity does not pick out any preferred distance scale. That
is, if all our meter-sticks expand, and the rest of the universe expands
as well, we would have no way to detect the expansion. The flaw
in this reasoning is that the Friedmann equations only describe the
average behavior of spacetime. As dramatized in the classic Woody
Allen movie “Annie Hall:” “Well, the universe is everything, and
if it’s expanding, someday it will break apart and that would be
the end of everything!” “What has the universe got to do with it?
You’re here in Brooklyn! Brooklyn is not expanding!”
To organize our thoughts, let’s consider the following hypotheses:
1. The distance between one galaxy and another increases at the
rate given by a(t) (assuming the galaxies are sufficiently distant from one another that they are not gravitationally bound
within the same galactic cluster, supercluster, etc.).
2. The wavelength of a photon increases according to a(t) as it
travels cosmological distances.
3. The size of the solar system increases at this rate as well (i.e.,
gravitationally bound systems get bigger, including the earth
and the Milky Way).
4. The size of Brooklyn increases at this rate (i.e., electromagnetically bound systems get bigger).
5. The size of a helium nucleus increases at this rate (i.e., systems
bound by the strong nuclear force get bigger).
We can imagine that:
• All the above hypotheses are true.
• All the above hypotheses are false, and in fact none of these
sizes increases at all.
• Some are true and some false.
Chapter 8
If all five hypotheses were true, the expansion would be undetectable, because all available meter-sticks would be expanding together. Likewise if no sizes were increasing, there would be nothing
to detect. These two possibilities are really the same cosmology,
described in two different coordinate systems. But the Ricci and
Einstein tensors were carefully constructed so as to be intrinsic.
The fact that the expansion affects the Einstein tensor shows that
it cannot interpreted as a mere coordinate expansion. Specifically,
suppose someone tells you that the FRW metric can be made into
a flat metric by a change of coordinates. (I have come across this
claim on internet forums.) The linear structure of the tensor transformation equations guarantees that a nonzero tensor can never be
made into a zero tensor by a change of coordinates. Since the Einstein tensor is nonzero for an FRW metric, and zero for a flat metric,
the claim is false.
Self-check: The reasoning above implicitly assumed a non-empty
universe. Convince yourself that it fails in the special case of the
Milne universe.
We can now see some of the limitations of a common metaphor
used to explain cosmic expansion, in which the universe is visualized as the surface of an expanding balloon. The metaphor correctly
gets across several ideas: that the Big Bang is not an explosion that
occurred at a preexisting point in empty space; that hypothesis 1
above holds; and that the rate of recession of one galaxy relative
to another is proportional to the distance between them. Nevertheless the metaphor may be misleading, because if we take a laundry
marker and draw any structure on the balloon, that structure will
expand at the same rate. But this implies that hypotheses 1-5 all
hold, which cannot be true.
Since some of the five hypotheses must be true and some false,
and we would like to sort out which are which. It should also be
clear by now that these are not five independent hypotheses. For
example, we can test empirically whether the ratio of Brooklyn’s
size to the distances between galaxies changes like a(t), remains
constant, or changes with some other time dependence, but it is
only the ratio that is actually observable.
Empirically, we find that hypotheses 1 and 2 are true (i.e., the
photon’s wavelength maintains a constant ratio with the intergalactic distance scale), while 3, 4, and 5 are false. For example, the
orbits of the planets in our solar system have been measured extremely accurately by radar reflection and by signal propagation
times to space probes, and no expanding trend is detected.
General-relativistic predictions
Does general relativity correctly reproduce these observations?
General relativity is mainly a theory of gravity, so it should be well
Section 8.2
Cosmological solutions
within its domain to explain why the solar system does not expand detectably while intergalactic distances do. It is impractical
to solve the Einstein field equations exactly so as to describe the
internal structure of all the bodies that occupy the universe: galaxies, superclusters, etc. We can, however, handle simple cases, as
in example 17 on page 311, where we display an exact solution for
the case of a universe containing only two things: an isolated black
hole, and an energy density described by a cosmological constant.
We find that the characteristic scale of the black hole, i.e., the radius
of its event horizon, does not increase with time. A fuller treatment
of these issues is given on p. 315, after some facts about realistic cosmologies have been established. The result is that although
bound systems like the solar system are in some cases predicted to
expand, the expansion is absurdly small, too small to measure, and
much smaller than the rate of expansion of the universe in general as
represented by the scale factor a(t). This agrees with observation.
It is easy to show that atoms and nuclei do not steadily expand over time. because such an expansion would violate either
the equivalence principle or the basic properties of quantum mechanics. One way of stating the equivalence principle is that the
local geometry of spacetime is always approximately Lorentzian, so
that the laws of physics do not depend on one’s position or state of
motion. Among these laws of physics are the principles of quantum
mechanics, which imply that an atom or a nucleus has a well-defined
ground state, with a certain size that depends only on fundamental
constants such as Planck’s constant and the masses of the particles
involved. Atoms and nuclei do experience deformation due to gravitational strains (examples 20-21, p. 317), but these deformations do
not increase with time, and would only be detectable if cosmological
expansion were to accelerate radically (example 22, p. 317).
This is different from the case of a photon traveling across the
universe. The argument given above fails, because the photon does
not have a ground state. The photon does expand, and this is
required by the correspondence principle. If the photon did not expand, then its wavelength would remain constant, and this would
be inconsistent with the classical theory of electromagnetism, which
predicts a Doppler shift due to the relative motion of the source
and the observer. One can choose to describe cosmological redshifts
either as Doppler shifts or as expansions of wavelength due to cosmological expansion.
A nice way of discussing atoms, nuclei, photons, and solar systems all on the same footing is to note that in geometrized units,
the units of mass and length are the same. Therefore the existence
of any fundamental massive particle sets a universal length scale,
one that will be known to any intelligent species anywhere in the
universe. Since photons are massless, they can’t be used to set a
universal scale in this way; a photon has a certain mass-energy, but
Chapter 8
that mass-energy can take on any value. Similarly, a solar system
sets a length scale, but not a universal one; the radius of a planet’s
orbit can take on any value. A universe without massive fundamental particles would be a universe without length measurement. It
would obey the laws of conformal geometry, in which angles and
light-cones were the only measures. This is the reason that atoms
and nuclei, which are made of massive fundamental particles, do not
More than one dimension required
Another good way of understanding why a photon expands,
while an atom does not, is to recall that a one-dimensional space
can never have any intrinsic curvature. If the expansion of atoms
were to be detectable, we would need to detect it by comparing
against some other meter-stick. Let’s suppose that a hydrogen atom
expands more, while a more tightly bound uranium atom expands
less, so that over time, we can detect a change in the ratio of the two
atoms’ sizes. The world-lines of the two atoms are one-dimensional
curves in spacetime. They are housed in a laboratory, and although
the laboratory does have some spatial extent, the equivalence principle guarantees that to a good approximation, this small spatial
extent doesn’t matter. This implies an intrinsic curvature in a onedimensional space, which is mathematically impossible, so we have
a proof by contradiction that atoms do not expand streadily.
Now why does this one-dimensionality argument fail for photons
and galaxies? For a pair of galaxies, it fails because the galaxies are
not sufficiently close together to allow them both to be covered by
a single Lorentz frame, and therefore the set of world-lines comprising the observation cannot be approximated well as lying within
a one-dimensional space. Similar reasoning applies for cosmological redshifts of photons received from distant galaxies. One could
instead propose flying along in a spaceship next to an electromagnetic wave, and monitoring the change in its wavelength while it is
in flight. All the world-lines involved in such an experiment would
indeed be confined to a one-dimensional space. The experiment is
impossible, however, because the measuring apparatus cannot be
accelerated to the speed of light. In reality, the speed of the light
wave relative to the measuring apparatus will always equal c, so the
two world-lines involved in the experiment will diverge, and will not
be confined to a one-dimensional region of spacetime.
A cosmic girdle
Example: 13
Since cosmic expansion has no significant effect on Brooklyn, nuclei, and solar systems, we might be tempted to infer that its effect on any solid body would also be negligible. To see that this
is not true, imagine that we live in a closed universe, and the universe has a leather belt wrapping around it on a closed spacelike
geodesic. All parts of the belt are initially at rest relative to the
Section 8.2
Cosmological solutions
local galaxies, and the tension is initially zero everywhere. The
belt must stretch and eventually break: for if not, then it could
not remain everywhere at rest with respect to the local galaxies,
and this would violate the symmetry of the initial conditions, since
there would be no way to pick the direction in which a certain part
of the belt should begin accelerating.
Østvang’s quasi-metric relativity
Example: 14
Over the years, a variety of theories of gravity have been proposed as alternatives to general relativity. Some of these, such
as the Brans-Dicke theory, remain viable, i.e., they are consistent with all the available experimental data that have been used
to test general relativity. One of the most important reasons for
trying to construct such theories is that it can be impossible to
interpret tests of general relativity’s predictions unless one also
possesses a theory that predicts something different. This issue,
for example, has made it impossible to test Einstein’s century-old
prediction that gravitational effects propagate at c, since there is
no viable theory available that predicts any other speed for them
(see section 9.1).
Østvang ( has proposed an alternative theory of gravity, called quasi-metric relativity, which,
unlike general relativity, predicts a significant cosmological expansion of the solar system, and which is claimed to be able
to explain the observation of small, unexplained accelerations of
the Pioneer space probes that remain after all accelerations due
to known effects have been subtracted (the “Pioneer anomaly”).
We’ve seen above that there are a variety of arguments against
such an expansion of the solar system, and that many of these
arguments do not require detailed technical calculations but only
knowledge of certain fundamental principles, such as the structure of differential geometry (no intrinsic curvature in one dimension), the equivalence principle, and the existence of ground states
in quantum mechanics. We therefore expect that Østvang’s theory, if it is logically self-consistent, will probably violate these assumptions, but that the violations must be relatively small if the
theory is claimed to be consistent with existing observations. This
is in fact the case. The theory violates the strictest form of the
equivalence principle.
Over the years, a variety of explanations have been proposed
for the Pioneer anomaly, including both glamorous ones (a modification of the 1/r 2 law of gravitational forces) and others more
pedestrian (effects due to outgassing of fuel, radiation pressure
from sunlight, or infrared radiation originating from the spacecrafts radioisotope thermoelectric generator). Calculations by Iorio18 in 2006-2009 show that if the force law for gravity is modified
Chapter 8
in order to explain the Pioneer anomalies, and if gravity obeys the
equivalence principle, then the results are inconsistent with the
observed orbital motion of the satellites of Neptune. This makes
gravitational explanations unlikely, but does not obviously rule out
Østvang’s theory, since the theory is not supposed to obey the
equivalence principle. Østvang says19 that his theory predicts an
expansion of ∼ 1m/yr in the orbit of Triton’s moon Nereid, which
is consistent with observation.
In December 2010, the original discoverers of the effect made
a statement in the popular press that they had a new analysis,
which they were preparing to publish in a scientific paper, in which
the size of the anomaly would be drastically revised downward,
with a far greater proportion of the acceleration being accounted
for by thermal effects. In my opinion this revision, combined with
the putative effect’s violation of the equivalence principle, make it
clear that the anomaly is not gravitational.
Does space expand?
Finally, the balloon metaphor encourages us to interpret cosmological expansion as a phenomenon in which space itself expands,
or perhaps one in which new space is produced. Does space really
expand? Without posing the question in terms of more rigorously
defined, empirically observable quantities, we can’t say yes or no. It
is merely a matter of which definitions one chooses and which conceptual framework one finds easier and more natural to work within.
Bunn and Hogg have stated the minority view against expansion of
space20 , while the opposite opinion is given by Francis et al.21
As an example of a self-consistent set of definitions that lead
to the conclusion that space does expand, Francis et al. give the
following. Define eight observers positioned at the corners of a cube,
at cosmological distances from one another. Let each observer be
at rest relative to the local matter and radiation that were used as
ingredients in the FRW cosmology. (For example, we know that our
own solar system is not at rest in this sense, because we observe
that the cosmic microwave background radiation is slightly Doppler
shifted in our frame of reference.) Then these eight observers will
observe that, over time, the volume of the cube grows as expected
according to the cube of the function a(t) in the FRW model.
This establishes that expansion of space is a plausible interpretation. To see that it is not the only possible interpretation, consider
the following example. A photon is observed after having traveled
to earth from a distant galaxy G, and is found to be red-shifted. Alice, who likes expansion, will explain this by saying that while the
photon was in flight, the space it occupied expanded, lengthening
private communication, Jan. 4, 2010
Section 8.2
Cosmological solutions
its wavelength. Betty, who dislikes expansion, wants to interpret it
as a kinematic red shift, arising from the motion of galaxy G relative to the Milky Way Malaxy, M. If Alice and Betty’s disagreement
is to be decided as a matter of absolute truth, then we need some
objective method for resolving an observed redshift into two terms,
one kinematic and one gravitational. But we’ve seen in section 7.3
on page 248 that this is only possible for a stationary spacetime,
and cosmological spacetimes are not stationary: regardless of an
observer’s state of motion, he sees a change over time in observables
such as density of matter and curvature of spacetime. As an extreme example, suppose that Betty, in galaxy M, receives a photon
without realizing that she lives in a closed universe, and the photon has made a circuit of the cosmos, having been emitted from her
own galaxy in the distant past. If she insists on interpreting this as
a kinematic red shift, the she must conclude that her galaxy M is
moving at some extremely high velocity relative to itself. This is in
fact not an impossible interpretation, if we say that M’s high velocity is relative to itself in the past. An observer who sets up a frame
of reference with its origin fixed at galaxy G will happily confirm
that M has been accelerating over the eons. What this demonstrates
is that we can split up a cosmological red shift into kinematic and
gravitational parts in any way we like, depending on our choice of
coordinate system (see also p. 255).
A cosmic whip
Example: 15
The cosmic girdle of example 13 on p. 303 does not transmit any
information from one part of the universe to another, for its state
is the same everywhere by symmetry, and therefore an observer
near one part of the belt gets no information that is any different
from what would be available to an observer anywhere else.
Now suppose that the universe is open rather than closed, but
we have a rope that, just like the belt, stretches out over cosmic
distances along a spacelike geodesic. If the rope is initially at
rest with respect to a particular galaxy G (or, more strictly speaking, with respect to the locally averaged cosmic medium), then by
symmetry the rope will always remain at rest with respect to G,
since there is no way for the laws of physics to pick a direction in
which it should accelerate. Now the residents of G cut the rope,
release half of it, and tie the other half securely to one of G’s spiral arms using a square knot. If they do this smoothly, without
varying the rope’s tension, then no vibrations will propagate, and
everything will be as it was before on that half of the rope. (We
assume that G is so massive relative to the rope that the rope
does not cause it to accelerate significantly.)
Can observers at distant points observe the tail of the rope whipping by at a certain speed, and thereby infer the velocity of G
relative to them? This would produce all kinds of strange conclusions. For one thing, the Hubble law says that this velocity is
Chapter 8
directly proportional to the length of the rope, so by making the
rope long enough we could make this velocity exceed the speed
of light. We’ve also convinced ourselves that the relative velocity of cosmologically distant objects is not even well defined in
general relativity, so it clearly can’t make sense to interpret the
rope-end’s velocity in that way.
The way out of the paradox is to recognize that disturbances can
only propagate along the rope at a certain speed v . Let’s say that
the information is transmitted in the form of longitudinal vibrations,
in which case it propagates at the speed of sound. For a rope
made out of any known material, this is far less than the speed
of light, and we’ve also seen in example 15 on page 65 and in
problem 4 on page 85 that relativity places fundamental limits
on the properties of all possible materials, guaranteeing v < c.
We can now see that all we’ve accomplished with the rope is to
recapitulate using slower sound waves the discussion that was
carried out on page 305 using light waves. The sound waves
may perhaps preserve some information about the state of motion
of galaxy G long ago, but all the same ambiguities apply to its
interpretation as in the case of light waves — and in addition, we
suspect that the rope has long since parted somewhere along its
8.2.7 The vacuum-dominated solution
For 70 years after Hubble’s discovery of cosmological expansion,
the standard picture was one in which the universe expanded, but
the expansion must be decelerating. The deceleration is predicted
by the special cases of the FRW cosmology that were believed to
be applicable, and even if we didn’t know anything about general
relativity, it would be reasonable to expect a deceleration due to the
mutual Newtonian gravitational attraction of all the mass in the
But observations of distant supernovae starting around 1998 introduced a further twist in the plot. In a binary star system consisting of a white dwarf and a non-degenerate star, as the nondegenerate star evolves into a red giant, its size increases, and it
can begin dumping mass onto the white dwarf. This can cause the
white dwarf to exceed the Chandrasekhar limit (page 144), resulting
in an explosion known as a type Ia supernova. Because the Chandrasekhar limit provides a uniform set of initial conditions, the behavior of type Ia supernovae is fairly predictable, and in particular
their luminosities are approximately equal. They therefore provide
a kind of standard candle: since the intrinsic brightness is known,
the distance can be inferred from the apparent brightness. Given
the distance, we can infer the time that was spent in transit by the
light on its way to us, i.e. the look-back time. From measurements
of Doppler shifts of spectral lines, we can also find the velocity at
Section 8.2
Cosmological solutions
which the supernova was receding from us. The result is that we
can measure the universe’s rate of expansion as a function of time.
Observations show that this rate of expansion has been accelerating.
The Friedmann equations show that this can only occur for Λ & 4ρ.
This picture has been independently verified by measurements of the
cosmic microwave background (CMB) radiation. A more detailed
discussion of the supernova and CMB data is given in section 8.2.11
on page 319.
With hindsight, we can see that in a quantum-mechanical context, it is natural to expect that fluctuations of the vacuum, required
by the Heisenberg uncertainty principle, would contribute to the cosmological constant, and in fact models tend to overpredict Λ by a
factor of about 10120 ! From this point of view, the mystery is why
these effects cancel out so precisely. A correct understanding of the
cosmological constant presumably requires a full theory of quantum
gravity, which is presently far out of our reach.
The latest data show that our universe, in the present epoch, is
dominated by the cosmological constant, so as an approximation we
can write the Friedmann equations as
= Λ
= Λ
This is referred to as a vacuum-dominated universe. The solution is
"r #
a = exp
where observations show that Λ ∼ 10−26 kg/m3 , giving
1011 years.
3/Λ ∼
The implications for the fate of the universe are depressing. All
parts of the universe will accelerate away from one another faster and
faster as time goes on. The relative separation between two objects,
say galaxy A and galaxy B, will eventually be increasing faster than
the speed of light. (The Lorentzian character of spacetime is local,
so relative motion faster than c is only forbidden between objects
that are passing right by one another.) At this point, an observer
in either galaxy will say that the other one has passed behind an
event horizon. If intelligent observers do actually exist in the far
future, they may have no way to tell that the cosmos even exists.
They will perceive themselves as living in island universes, such as
we believed our own galaxy to be a hundred years ago.
When I introduced the standard cosmological coordinates on
page 292, I described them as coordinates in which events that
are simultaneous according to this t are events at which the local
Chapter 8
properties of the universe are the same. In the case of a perfectly
vacuum-dominated universe, however, this notion loses its meaning.
The only observable local property of such a universe is the vacuum
energy described by the cosmological constant, and its density is always the same, because it is built into the structure of the vacuum.
Thus the vacuum-dominated cosmology is a special one that maximally symmetric, in the sense that it has not only the symmetries of
homogeneity and isotropy that we’ve been assuming all along, but
also a symmetry with respect to time: it is a cosmology without
history, in which all times appear identical to a local observer. In
the special case of this cosmology, the time variation of the scaling
factor a(t) is unobservable, and may be thought of as the unfortunate result of choosing an inappropriate set of coordinates, which
obscure the underlying symmetry. When I argued in section 8.2.6
for the observability of the universe’s expansion, note that all my
arguments assumed the presence of matter or radiation. These are
completely absent in a perfectly vacuum-dominated cosmology.
For these reasons de Sitter originally proposed this solution as a
static universe in 1927. But by 1920 it was realized that this was an
oversimplification. The argument above only shows that the time
variation of a(t) does not allow us to distinguish one epoch of the
universe from another. That is, we can’t look out the window and
infer the date (e.g., from the temperature of the cosmic microwave
background radiation). It does not, however, imply that the universe is static in the sense that had been assumed until Hubble’s
observations. The r-t part of the metric is
ds2 = dt2 − a2 dr2
where a blows up exponentially with time, and the k-dependence
has been neglected, as it was in the approximation to the Friedmann
equations used to derive a(t).22 Let a test particle travel in the radial
direction, starting at event A = (0, 0) and ending at B = (t0 , r0 ). In
flat space, a world-line of the linear form r = vt would be a geodesic
connecting A and B; it would maximize the particle’s proper time.
But in the this metric, it cannot be a geodesic. The curvature of
geodesics relative to a line on an r-t plot is most easily understood
to the time-scale T =
p the limit where t is fairly long compared
3/Λ of the exponential, so that a(t0 ) is huge. The particle’s best
strategy for maximizing its proper time is to make sure that its dr
is extremely small when a is extremely large. The geodesic must
therefore have nearly constant r at the end. This makes it sound as
A computation of the Einstein tensor with ds2 = dt2 − a2 (1 − kr2 )−1 dr2
shows that k enters only via a factor the form (. . .)e(...)t + (. . .)k. For large t, the
k term becomes negligible, and the Einstein tensor becomes Gab = g ab Λ, This is
consistent with the approximation we used in deriving the solution, which was
to ignore both the source terms and the k term in the Friedmann equations.
The exact solutions with Λ > 0 and k = −1, 0, and 1 turn out in fact to be
equivalent except for a change of coordinates.
Section 8.2
Cosmological solutions
though the particle was decelerating, but in fact the opposite is true.
If r is constant, then the particle’s spacelike distance from the origin
is just ra(t), which blows up exponentially. The near-constancy of
the coordinate r at large t actually means that the particle’s motion
at large t isn’t really due to the particle’s inertial memory of its
original motion, as in Newton’s first law. What happens instead
is that the particle’s initial motion allows it to move some distance
away from the origin during a time on the order of T , but after that,
the expansion of the universe has become so rapid that the particle’s
motion simply streams outward because of the expansion of space
itself. Its initial motion only mattered because it determined how
far out the particle got before being swept away by the exponential
Geodesics in a vacuum-dominated universe
Example: 16
In this example we confirm the above interpretation in the special
case where the particle, rather than being released in motion at
the origin, is released at some nonzero radius r , with dr /dt = 0
initially. First we recall the geodesic equation
d2 x i
dx j dx k
= Γ ijk
dλ dλ
from page 178. The nonvanishing Christoffel symbols for the 1+1dimensional metric ds2 = dt 2 − a2 dr 2 are Γ rtr = ȧ/a and Γ tr r =
−ȧa. Setting T = 1 for convenience, we have Γ rtr = 1 and Γ tr r =
−e−2t .
We conjecture that the particle remains at the Rsame value of r .
Given this conjecture, the particle’s proper time ds is simply the
same as its time coordinate t, and we can therefore use t as an
affine coordinate. Letting λ = t, we have
d2 t
dt 2
0 − Γ tr r ṙ 2 = 0
ṙ = 0
r = constant
This confirms the self-consistency of the conjecture that r = constant
is a geodesic.
Note that we never actually had to use the actual expressions for
the Christoffel symbols; we only needed to know which of them
vanished and which didn’t. The conclusion depended only on
the fact that the metric had the form ds2 = dt 2 − a2 dr 2 for some
function a(t). This provides a rigorous justification for the interpretation of the cosmological scale factor a as giving a universal
time-variation on all distance scales.
The calculation also confirms that there is nothing special about
r = 0. A particle released with r = 0 and ṙ = 0 initially stays at
Chapter 8
r = 0, but a particle released at any other value of r also stays
at that r . This cosmology is homogeneous, so any point could
have been chosen as r = 0. If we sprinkle test particles, all at
rest, across the surface of a sphere centered on this arbitrarily
chosen point, then they will all accelerate outward relative to one
another, and the volume of the sphere will increase. This is exactly what we expect. The Ricci curvature is interpreted as the
second derivative of the volume of a region of space defined by
test particles in this way. The fact that the second derivative is
positive rather than negative tells us that we are observing the
kind of repulsion provided by the cosmological constant, not the
attraction that results from the existence of material sources.
Schwarzschild-de Sitter space
Example: 17
The metric
2m 1 2
dr 2
ds = 1 −
− Λr dt 2 −
−r 2 dθ2 −r 2 sin2 θdφ2
1 − 2m
is an exact solution to the Einstein field equations with cosmological constant Λ, and can be interpreted as a universe in which
the only mass is a black hole of mass m located at r = 0. Near
the black hole, the Λ terms become negligible, and this is simply
the Schwarzschild metric. As argued in section 8.2.6, page 300,
this is a simple example of how cosmological expansion does not
cause all structures in the universe to grow at the same rate.
The Big Bang singularity in a universe with a cosmological
On page 298 we discussed the possibility that the Big Bang
singularity was an artifact of the unrealistically perfect symmetry
assumed by our cosmological models, and we found that this was
not the case: the Penrose-Hawking singularity theorems demonstrate that the singularity is real, provided that the cosmological
constant is zero. The cosmological constant is not zero, however.
Models with a very large positive cosmological constant can also
display a Big Bounce rather than a Big Bang. If we imagine using the Friedmann equations to evolve the universe backward in
time from its present state, the scaling arguments of example 14 on
page 132 suggest that at early enough times, radiation and matter should dominate over the cosmological constant. For a large
enough value of the cosmological constant, however, it can happen
that this switch-over never happens. In such a model, the universe
is and always has been dominated by the cosmological constant,
and we get a Big Bounce in the past because of the cosmological
constant’s repulsion. In this book I will only develop simple cosmological models in which the universe is dominated by a single
component; for a discussion of bouncing models with both matter
Section 8.2
Cosmological solutions
and a cosmological constant, see Carroll, “The Cosmological Constant,” By 2008, a
variety of observational data had pinned down the cosmological constant well enough to rule out the possibility of a bounce caused by
a very strong cosmological constant.
8.2.8 The matter-dominated solution
Our universe is not perfectly vacuum-dominated, and in the past
it was even less so. Let us consider the matter-dominated epoch,
in which the cosmological constant was negligible compared to the
material sources. The equation of state for nonrelativistic matter
(p. 132) is
P =0
The dilution of the dust with cosmological expansion gives
ρ ∝ a−3
(see example 19). The Friedmann equations become
=− ρ
ρ − ka−2
where for compactness ρ’s dependence on a, with some constant
of proportionality, is not shown explicitly. A static solution, with
constant a, is impossible, and ä is negative, which we can interpret
in Newtonian terms as the deceleration of the matter in the universe
due to gravitational attraction. There are three cases to consider,
according to the value of k.
The closed universe
We’ve seen that k = +1 describes a universe in which the spatial
curvature is positive, i.e., the circumference of a circle is less than
its Euclidean value. By analogy with a sphere, which is the twodimensional surface of constant positive curvature, we expect that
the total volume of this universe is finite.
The second Friedmann equation also shows us that at some value
of a, we will have ȧ = 0. The universe will expand, stop, and then
recollapse, eventually coming back together in a “Big Crunch” which
is the time-reversed version of the Big Bang.
Suppose we were to describe an initial-value problem in this
cosmology, in which the initial conditions are given for all points in
the universe on some spacelike surface, say t = constant. Since the
universe is assumed to be homogeneous at all times, there are really
only three numbers to specify, a, ȧ, and ρ: how big is the universe,
how fast is it expanding, and how much matter is in it? But these
three pieces of data may or may not be consistent with the second
Chapter 8
Friedmann equation. That is, the problem is overdetermined. In
particular, we can see that for small enough values of ρ, we do
not have a valid solution, since the square of ȧ/a would have to be
negative. Thus a closed universe requires a certain amount of matter
in it. The present observational evidence (from supernovae and the
cosmic microwave background, as described above) is sufficient to
show that our universe does not contain this much matter.
The flat universe
The case of k = 0 describes a universe that is spatially flat.
It represents a knife-edge case lying between the closed and open
universes. In a Newtonian analogy, it represents the case in which
the universe is moving exactly at escape velocity; as t approaches
infinity, we have a → ∞, ρ → 0, and ȧ → 0. This case, unlike the
others, allows an easy closed-form solution to the motion. Let the
constant of proportionality in the equation of state ρ ∝ a−3 be fixed
by setting −4πρ/3 = −ca−3 . The Friedmann equations are
ä = −ca−2
ȧ = 2ca−1/2
Looking for a solution of the form a ∝ tp , we find that by choosing
p = 2/3 we can simultaneously satisfy both equations. The constant
c is also fixed, and we can investigate this most transparently by
recognizing that ȧ/a is interpreted as the Hubble constant, H, which
is the constant of proportionality relating a far-off galaxy’s velocity
to its distance. Note that H is a “constant” in the sense that it is
the same for all galaxies, in this particular model with a vanishing
cosmological constant; it does not stay constant with the passage
of cosmological time. Plugging back into the original form of the
Friedmann equations, we find that the flat universe can only exist if
the density of matter satisfies ρ = ρcrit = 3H 2 /8π = 3H 2 /8πG. The
observed value of the Hubble constant is about 1/(14 × 109 years),
which is roughly interpreted as the age of the universe, i.e., the
proper time experienced by a test particle since the Big Bang. This
gives ρcrit ∼ 10−26 kg/m3 .
As discussed in subsection 8.2.11, our universe turns out to
be almost exactly spatially flat. Although it is presently vacuumdominated, the flat and matter-dominated FRW cosmology is a useful description of its matter-dominated era.
The open universe
The k = −1 case represents a universe that has negative spatial
curvature, is spatially infinite, and is also infinite in time, i.e., even
if the cosmological constant had been zero, the expansion of the universe would have had too little matter in it to cause it to recontract
and end in a Big Crunch.
The time-reversal symmetry of general relativity was discussed
Section 8.2
Cosmological solutions
on p. 211 in connection with the Schwarzschild metric.23 Because
of this symmetry, we expect that solutions to the field equations
will be symmetric under time reversal (unless asymmetric boundary
conditions were imposed). The closed universe has exactly this type
of time-reversal symmetry. But the open universe clearly breaks this
symmetry, and this is why we speak of the Big Bang as lying in the
past, not in the future. This is an example of spontaneous symmetry
breaking. Spontaneous symmetry breaking happens when we try to
balance a pencil on its tip, and it is also an important phenomenon
in particle physics. The time-reversed version of the open universe
is an equally valid solution of the field equations. Another example
of spontaneous symmetry breaking in cosmological solutions is that
the solutions have a preferred frame of reference, which is the one at
rest relative to the cosmic microwave background and the average
motion of the galaxies. This is referred to as the Hubble flow.
Size and age of the observable universe
Example: 18
The observable universe is defined by the region from which
light has had time to reach us since the Big Bang. Many people
are inclined to assume that its radius in units of light-years must
therefore be equal to the age of the universe expressed in years.
This is not true. Cosmological distances like these are not even
uniquely defined, because general relativity only has local frames
of reference, not global ones.
Suppose we adopt the proper distance L defined on p. 292 as
our measure of radius. By this measure, realistic cosmological
models say that our 14-billion-year-old universe has a radius of
46 billion light years.
For a flat universe, f = 1, so by inspecting the FRW metric we find
that a photon moving radially with ds = 0 has |dr /dt| = a−1 , giving
r = ± t12 dt/a. Suppressing signs, the proper distance the phoR
ton traverses starting soon after the Big Bang is L = a(t2 ) d` =
a(t2 ) dr = a(t2 )r = a(t2 ) t12 dt/a.
In the matter-dominated case, a ∝ t 2/3 , so this results in L = 3t2
in the limit where t1 is small. Our universe has spent most of
its history being matter-dominated, so it’s encouraging that the
matter-dominated calculation seems to do a pretty good job of
reproducing the actual ratio of 46/14=3.3 between L and t2 .
While we’re at it, we can see what happens in the purely
p vacuumdominated case, which has a ∝ et/T , where T = 3/Λ. This
cosmology doesn’t have a Big Bang, but we can think of it as an
approximation to the more recent history of the universe, glued
an earlier
on(t to
matter-dominated solution. Here we find L =
− 1 T , where t1 is the time when the switch to vacuum23
Problem 5 on p. 329 shows that this symmetry is also exhibited by the
Friedmann equations.
Chapter 8
domination happened. This function grows more quickly with t2
than the one obtained in the matter-dominated case, so it makes
sense that the real-world ratio of L/t2 is somewhat greater than
the matter-dominated value of 3.
The radiation-dominated version is handled in problem 12 on p. 330.
Local conservation of mass-energy
Example: 19
Any solution to the Friedmann equations is a solution of the field
equations, and therefore locally conserves mass-energy. We saved
work above by applying this condition in advance in the form
ρ ∝ a−3 to make the dust dilute itself properly with cosmological expansion. In this example we prove the same proportionality
by explicit calculation.
Local conservation of mass-energy is expressed by the zero divergence of the stress-energy tensor, ∇j T jb = 0. The definition of
the covariant derivative gives
b dc
c bd
∇j T bc = ∂j T bc + Γjd
T + Γjd
For convenience, we carry out the calculation at r = 0; if conservation holds here, then it holds everywhere by homogeneity.
In a local Cartesian frame (t 0 , x 0 , y 0 , z 0 ) at rest relative to the dust,
0 0
the stress-energy tensor is diagonal with T t t = ρ. At r = 0,
the transformation from FLRW coordinates into these coordinates
doesn’t mix t or t 0 with the other coordinates, so by the tensor
transformation law we still have T tt = ρ.
There are a number of Christoffel symbols involved, but the only
three of relevance that don’t vanish at r = 0 turn out to be Γrrt =
= ȧ/a. The result is
Γθtθ = Γφt
∇µ T tµ = ∂t T tt + 3 T tt
or ρ̇/ρ = −3ȧ/a, which can be rewritten as
ln ρ = −3 ln a
producing the proportionality originally claimed.
8.2.9 The radiation-dominated solution
For the reasons discussed in example 14 on page 132, the early
universe was dominated by radiation. The solution of the Friedmann
equations for this case is taken up in problem 11 on page 330.
8.2.10 Local effects of expansion
In this section we discuss the predictions of general relativity
concerning the effect of cosmological expansion on small, gravitationally bound systems such as the solar system or clusters of galaxies. The short answer is that in most realistic cosmologies (but not
Section 8.2
Cosmological solutions
necessarily in “Big Rip” scenarios, p. 317) the effect of expansion
is not zero, but is many orders of magnitude too small to measure.
Many readers will probably be willing to accept these assertions
while skipping the following demonstrations.
To begin with, we observe that there are two qualitatively distinct types of effects that could exist. Suppose that a loaf of raisin
bread is rising. Let’s say that the loaf’s scale factor a doubles by
the time the yeast’s efforts are spent. By definition, this means that
the raisins (galaxies, test particles) get farther apart by a factor of
2. We could imagine that in addition: (1) the strain of expansion
could cause each raisin to puff up by, say, 1%, and to maintain this
increased size over the entire course of expansion; or that (2) expansion could could cause each raisin to expand gradually, to 0.2% more
than its original size, then 0.4% more than its original size, and so
on, until, at the end of the process, each had grown beyond its original size by some amount such as 3.8%, which, while less than the
100% growth of the inter-raisin distances, was nevertheless nonzero.
Astronomers refer to the second possibility as a “secular” trend. For
example, simulations of solar systems often show that over billions
of years, planets gradually migrate either inward or outward, under
the influence of their gravitational interactions with other planets.
As an example of an expansion without a secular trend, asteroids
may experience a nonnegligible 1/r2 force due to radiation pressure
from the sun. The effect is exactly as if the sun’s mass or the gravitational constant had been slightly reduced. Kepler’s elliptical orbit
law holds, the law of periods is slightly off, and the orbital radius
shows zero trend over time.
If either type of effect exists, an observer in some local inertial
frame will interpret it as a “force.” (The scare quotes are a reminder
that general relativity doesn’t describe gravity as Newton-style linearly additive, instantaneous action at a distance.) Such a force,
if it exists, cannot simply be proportional to the rate of expansion
ȧ/a. As a counterexample, the Milne universe is just flat spacetime
described in silly coordinates, and it has ȧ 6= 0.
It would make more sense for the force to depend on the second
derivative of the scale factor. To justify this more precisely, imagine
releasing two test particles, initially separated by some distance that
is much less than the Hubble scale. They are initially at rest relative
to the Hubble flow, and no locally gravitating bodies are present.
As discussed in example 12 on p. 296, the acceleration of one test
particle relative to the other is given by (ä/a)r, where r is their
relative displacement.
Thus if we are to observe any nonzero effects of expansion on a
local system, they are not really effects of expansion at all, but effects
of the acceleration of expansion. The factor ä/a is on the order of
the inverse square of the age of the universe, i.e., Ho2 ∼ 10−35 s−2 .
Chapter 8
The smallness of this factor is what makes the effect on a system
such as the solar system so absurdly tiny.
A human body
Example: 20
Let’s estimate the effect of cosmological expansion on the length
L of your thigh bone. The body is made of atoms, and for the reasons given on p. 302, there can be no steady trend in the sizes of
these atoms or the lengths of the chemical bonds between them.
The bone experiences a stress due to cosmological expansion,
but it is in equilibrium, and the strain will disappear if the gravitational stress is removed (e.g., if other gravitational stresses are
superimposed on top of the cosmological one in order to cancel
it). The anomalous acceleration between the ends of the bone
is (ä/a)L, which is observed as an anomalous stress. Taking
ä/a ∼ Ho2 , the anomalous acceleration of one end of the bone
relative to the other is ∼ LH 2 . The corresponding compression
or tension is ∼ mLH 2 , where m is your body mass. The resulting
strain is ∼ mLH 2 /AE, where E is the Young’s modulus of bone
(about 1010 Pa) and A is the bone’s cross-sectional area.
Putting in numbers, the result for the strain is about 10−40 , which
is much too small to be measurable by any imaginable technique,
and would in reality be swamped by other effects. Since the sign
of ä is currently positive, this strain is tensile, not compressive. In
the earlier, matter-dominated era of the universe, it would have
been compressive.
There is no “secular trend,” i.e., your leg bone is not expanding
over time. It’s in equilibrium, and is simply elongated imperceptibly compared to the length if would have had without the effect of
cosmological expanson.
Strain on an atomic nucleus
Example: 21
The estimate in example 20 can also be applied to an atomic
nucleus, which has a nuclear “Young’s modulus” on the order of
1 MeV/fm3 ∼ 1032 Pa. The result is a strain ∼ 10−52 .
A Big Rip
Example: 22
Known forms of matter are believed to have equations of state
P = wρ with w ≥ −1. The value for a vacuum-dominated universe would be w = −1. Cosmological observations24 show that
empirically the present-day universe behaves as if it is made out
of stuff with w = −1.03 ± .16, and this leaves open the possibility
of w < −1. In this case, the solution to the Friedmann equations
gives a scale factor a(t) that blows up to infinity at some finite t. In
such a scenario, known as a “Big Rip,” (d/dt)(ä/a) diverges, and
any system, no matter how tightly bound, is ripped apart.25
Carnero et al.,
Caldwell et al.,
Section 8.2
Cosmological solutions
Examples 20-22 show that except under hypothetical extreme
cosmological conditions, there is no hope of detecting any effect of
cosmological expansion on systems made of condensed matter. We
need to look at much larger systems to see any effect, and such
systems are held together by gravity. For concreteness, let’s keep
talking about the earth-sun system. Not only is the anomalous
force on the earth small, it is not guaranteed to produce any secular
trend, which is what would be most likely to be detectable. The
direction of the anomalous force on the earth is outward for an
accelerating cosmological expansion, as we now know is the case for
the present epoch. As an example in which no secular trend occurs, a
vacuum-dominated cosmology gives a constant value for ä/a, so the
outward force is constant. As with the effect of radiation pressure,
the existence of this constant, outward force is very nearly equivalent
to rescaling the sun’s gravitational force by a tiny amount, so the
motion is still very nearly Keplerian, but with a slightly “wrong”
constant of proportionality in Kepler’s law of periods. The rate of
change ṙ in the radius of the circular orbit is therefore zero in this
But in most cosmologies ä/a is not exactly constant, and the
anomalous force on the earth varies. In a matter-dominated cosmology with Λ = 0, in its expanding phase, the force is inward
but decreasing over time, so the orbit expands over time. What
really matters then, is (d/dt)(ä/a). If we were free to pick any
function for a(t), we could make up examples in which ȧ > 0 but
(d/dt)(ä/a) < 0, so that the solar system would respond to cosmological expansion by shrinking!
The function a(t), however, has to satisfy the Friedmann equations, one of which is (in units with G 6= 1)
=G Λ−
(ρ + 3P )
The present epoch of the universe seems to be well modeled by dark
energy described by a constant Λ plus dust with P ρ. Differentiating both sides with respect to time gives
d ä
∝ ρ̇
dt a
with a negative constant of proportionality. This ensures that the
sign of the effect is always as expected from the naive Manichean image of binding forces struggling against cosmological expansion (or
perhaps cooperating during the contracting phase of a Big Crunch
One way of understanding why this reduces so nicely to a dependence on ρ̇ is the reasoning given in example 12 on p. 296,
in which we found that the relative acceleration of two test particles A and B in a matter-dominated FRW cosmology could be
Chapter 8
calculated accurately by pretending that it was due to the presence of the dust in any given sphere S surrounding the two particles. We now let A be the sun, B the earth, and S a sphere
centered on the sun whose radius equals the radius of the earth’s
circular orbit. Due to cosmological expansion, the dust inside S
thins out with time, reducing its density ρ. Applying Newton’s
laws to the orbit of the earth gives ω 2 r = GM/r2 , and conservation of angular momentum results in ωr2 = const. A calculation
gives r/ro = [M + (4π/3)ρo ro3 ]/[M + (4π/3)ρr3 ], which results in
ṙ/ro ≈ −(4π/3)Gωo−2 ρ̇. Application of the Friedmann equations
ṙ/ro = ωo−2 (d/dt)(ä/a)
which is valid generally, not just for P = 0. The ωo−2 factor shows
that the effect is smaller for more tightly bound systems.
We know that the universe in the present era has (d/dt)(ä/a) > 0
because ρ̇ < 0, and for purposes of an order-of-magnitude estimate
we can take (d/dt)(ä/a) ∼ Ho3 . Plugging in numbers for the earthsun system, we find that since the age of the dinosaurs, the radius of
the earth’s orbit has grown by less than the diameter of an atomic
8.2.11 Observation
Historically, it was believed that the cosmological constant was
zero, that nearly all matter in the universe was in the form of atoms,
and that there was therefore only one interesting cosmological parameter to measure, which was the average density of matter. This
density was very difficult to determine, even to within an order of
magnitude, because most of the matter in the universe probably
doesn’t emit light, making it difficult to detect. Astronomical distance scales were also very poorly calibrated against absolute units
such as the SI. Starting around 1995, however, a new set of techniques led to an era of high-precision cosmology.
Spatial curvature from CMB fluctuations
A strong constraint on the models comes from accurate measurements of the cosmic microwave background, especially by the
1989-1993 COBE probe, and its 2001-2009 successor, the Wilkinson
Microwave Anisotropy Probe, positioned at the L2 Lagrange point
of the earth-sun system, beyond the Earth on the line connecting
sun and earth.27 The temperature of the cosmic microwave background radiation is not the same in all directions, and its can be
measured at different angles. In a universe with negative spatial
curvature, the sum of the interior angles of a triangle is less than
The picturesque image comes from Cooperstock et al.,
abs/astro-ph/9803097v1, who give a different calculation leading to a result for
ṙ exactly equivalent to the one derived here.
Komatsu et al., 2010,
Section 8.2
f / The angular scale of fluctuations in the cosmic microwave
background can be used to infer
the curvature of the universe.
Cosmological solutions
the Euclidean value of 180 degrees. Therefore if we observe a variation in the CMB over some angle, the distance between two points
on the surface of last scattering is actually greater than would have
been inferred from Euclidean geometry. The distance scale of such
variations is limited by the speed of sound in the early universe, so
one can work backward and infer the universe’s spatial curvature
based on the angular scale of the anisotropies. The measurements
of spatial curvature are usually stated in terms of the parameter Ω,
defined as the total average density of all source terms in the Einstein field equations, divided by the critical density that results in
a flat universe. Ω includes contributions from matter, ΩM , the cosmological constant, ΩΛ , and radiation (negligible in the present-day
unverse). The results from WMAP, combined with other data from
other methods, gives Ω = 1.005 ± .006. In other words, the universe
is very nearly spatially flat.
Accelerating expansion from supernova data
g / A Hubble plot for distant
Each data point
represents an average over
several different supernovae with
nearly the same z .
The supernova data described on page 307 complement the CMB
data because they are mainly sensitive to the difference ΩΛ − ΩM ,
rather than their sum Ω = ΩΛ + ΩM . This is because these data
measure the acceleration or deceleration of the universe’s expansion.
Matter produces deceleration, while the cosmological constant gives
acceleration. Figure g shows some recent supernova data.28 The
horizontal axis gives the redshift factor z = (λ0 − λ)/λ, where λ0 is
the wavelength observed on earth and λ the wavelength originally
emitted. It measures how fast the supernova’s galaxy is receding
from us. The vertical axis is ∆(m − M ) = (m − M ) − (m − M )empty ,
where m is the apparent magnitude, M is the absolute magnitude,
and (m − M )empty is the value expected in a model of an empty
universe, with Ω = 0. The difference m−M is a measure of distance,
so essentially this is a graph of distance versus recessional velocity, of
the same general type used by Hubble in his original discovery of the
expansion of the universe. Subtracting (m − M )empty on the vertical
axis makes it easier to see small differences. Since the WMAP data
require Ω = 1, we need to fit the supernova data with values of ΩM
and ΩΛ that add up to one. Attempting to do so with ΩM = 1 and
ΩΛ = 0 is clearly inconsistent with the data, so we can conclude
that the cosmological constant is definitely positive.
Density of matter from baryonic acoustic oscillations
h / The acoustic peak in the
New Astronomy Reviews.
(2005) 360, as reproduced
in Bassett and Hlozek, 2009,
Chapter 8
Efforts such as the Sloan Digital Sky Survey have made threedimensional maps of the density of luminous matter in the universe.29 The distribution is clumpy. Measuring the average correlation ξ between the density at points separated by some distance
Riess et al., 2007, A larger data set is
analyzed in Kowalski et al., 2008,
Sanchez et al., 2012,
s (measured in the comoving frame), one would expect that the
function ξ(s) would be largest when s was small and would simply taper off with increasing s. By analogy, we don’t usually find
a Manhattan-style landscape of skyscrapers side by side with an
uninhabited mountainous wilderness. On the other hand, imagine
constructing such a correlation function for houses in a subdivision
in which the roads do not form any regular grid, but zoning regulations prohibit construction of houses on lots of less than a certain
size. In this situation, there would be zero probability of finding
houses separated by very small distances, and ξ(s) would exhibit
a peak at some larger scale set by the legal code. The actual results of the sky surveys do show such a peak, which is due to well
known physics referred to as baryon acoustic oscillations (BAO).30
In the early universe, any region of overdensity would tend to create
a radiating sound wave like the bang of a firecracker. Such waves
propagated at a known speed (about half the speed of light) for a
known time (about 400,000 years, until matter became deionized
and transparent to radiation, making it immune to the photon pressure that drove the oscillations). This leads to a known distance
s, which forms a standard ruler at which the peak in ξ(s) occurs.
In cosmological models, these results strongly constrain ΩM , while
being relatively insensitive to ΩΛ , and they are therefore complementary to both the supernova data and the CMB results.
Conclusions about cosmology
Figure i summarizes what we can conclude about our universe,
parametrized in terms of a model with both ΩM and ΩΛ nonzero.31
We can tell that it originated in a Big Bang singularity, that it will
go on expanding forever, and that it is very nearly flat. Note that
in a cosmology with nonzero values for both ΩM and ΩΛ , there is
no strict linkage between the spatial curvature and the question of
recollapse, as there is in a model with only matter and no cosmological constant; therefore even though we know that the universe
will not recollapse, we do not know whether its spatial curvature is
slightly positive (closed) or negative (open).
Consistency checks
Astrophysical considerations provide further constraints and consistency checks. In the era before the advent of high-precision cosmology, estimates of the age of the universe ranged from 10 billion
to 20 billion years, and the low end was inconsistent with the age
of the oldest globular clusters. This was believed to be a problem
either for observational cosmology or for the astrophysical models
used to estimate the age of the clusters: “You can’t be older than
your ma.” Current data have shown that the low estimates of the
i / The
9812133 and Kowalski, 2008, The
three shaded regions represent
the 95% confidence regions for
the three types of observations.
Bassett and Hlozek, 2009,
See Carroll, “The Cosmological Constant,” http://www.livingreviews.
org/lrr-2001-1 for a full mathematical treatment of such models.
Section 8.2
Cosmological solutions
age were incorrect, so consistency is restored.
That only a small fraction of the universe’s matter was luminous
had been suspected by astronomers such as Zwicky as early as 1933,
based on the inability to reconcile the observed kinematics with
Newton’s laws if all matter was assumed to be luminous.
Dark matter
Another constraint comes from models of nucleosynthesis during the era shortly after the Big Bang (before the formation of the
first stars). The observed relative abundances of hydrogen, helium,
and deuterium cannot be reconciled with the density of “dust” (i.e.,
nonrelativistic matter) inferred from the observational data. If the
inferred mass density were entirely due to normal “baryonic” matter
(i.e., matter whose mass consisted mostly of protons and neutrons),
then nuclear reactions in the dense early universe should have proceeded relatively efficiently, leading to a much higher ratio of helium
to hydrogen, and a much lower abundance of deuterium. The conclusion is that most of the matter in the universe must be made of
an unknown type of exotic non-baryonic matter, known generically
as “dark matter.”
The existence of nonbaryonic matter is also required in order to
reconcile the observed density of galaxies with the observed strength
of the CMB fluctuations, and in merging galaxy clusters it has been
observed that the gravitational potential is offset from the radiating
plasma. A 2012 review paper on dark matter is Roos,
A number of experiments are under way to detect dark matter
directly. As of 2013, the most sensitive experiment has given null
At one time it was widely expected that dark matter would consist of the lightest supersymmetric particle, which might for example
be the neutralino. However, results from the LHC seem to make it
unlikely that our universe exhibits supersymmetry, assuming that
the energy scale is the electroweak scale, which is the only scale that
has strong motivation. It now appears more likely that dark matter
consists of some other particle such as sterile neutrinos or axions.
Even with the inclusion of dark matter, there is a problem
with the abundance of lithium-7 relative to hydrogen, which models
greatly overpredict.32
8.3 Mach’s principle revisited
Chapter 8
8.3.1 The Brans-Dicke theory
Mach himself never succeeded in stating his ideas in the form of
a precisely testable physical theory, and we’ve seen that to the extent that Einstein’s hopes and intuition had been formed by Mach’s
ideas, he often felt that his own theory of gravity came up short.
The reader has so far encountered Mach’s principle in the context of
certain thought experiments that are obviously impossible to realize,
involving a hypothetical universe that is empty except for certain
apparatus (e.g., section 3.6.2, p. 115). It would be easy, then, to get
an impression of Mach’s principle as one of those theories that is
“not even wrong,” i.e., so ill-defined that it cannot even be falsified
by experiment, any more than Christianity can be.
But in 1961, Robert Dicke and his student Carl Brans came up
with a theory of gravity that made testable predictions, and that
was specifically designed to be more Machian than general relativity.
Their paper33 is extremely readable, even for the non-specialist.
In this theory, the seemingly foolproof operational definition of a
Lorentz frame given on p. 26 fails. On the first page, Brans and
Dicke propose one of those seemingly foolish thought experiments
about a nearly empty universe:
The imperfect expression of [Mach’s ideas] in general
relativity can be seen by considering the case of a space
empty except for a lone experimenter in his laboratory.
[...] The observer would, according to general relativity,
observe normal behavior of his apparatus in accordance
with the usual laws of physics. However, also according to general relativity, the experimenter could set his
laboratory rotating by leaning out a window and firing
his 22-caliber rifle tangentially. Thereafter the delicate
gyroscope in the laboratory would continue to point in a
direction nearly fixed relative to the direction of motion
of the rapidly receding bullet. The gyroscope would rotate relative to the walls of the laboratory. Thus, from
the point of view of Mach, the tiny, almost massless,
very distant bullet seems to be more important than the
massive, nearby walls of the laboratory in determining
inertial coordinate frames and the orientation of the gyroscope.
They then proceed to construct a mathematical and more Machian theory of gravity. From the Machian point of view, the correct
local definition of an inertial frame must be determined relative
to the bulk of the matter in the universe. We want to retain the
Lorentzian local character of spacetime, so this influence can’t be
C. Brans and R. H. Dicke, “Mach’s Principle and a Relativistic Theory of
Gravitation,” Physical Review 124 (1961) 925
Section 8.3
Mach’s principle revisited
transmitted via instantaneous action at a distance. It must propagate via some physical field, at a speed less than or equal to c.
It is implausible that this field would be the gravitational field as
described by general relativity. Suppose we divide the cosmos up
into a series of concentric spherical shells centered on our galaxy.
In Newtonian mechanics, the gravitational field obeys Gauss’s law,
so the field of such a shell vanishes identically on the interior. In
relativity, the corresponding statement is Birkhoff’s theorem, which
states that the Schwarzschild metric is the unique spherically symmetric solution to the vacuum field equations. Given this solution
in the exterior universe, we can set a boundary condition at the outside surface of the shell, use the Einstein field equations to extend
the solution through it, and find a unique solution on the interior,
which is simply a flat space.
Since the Machian effect can’t be carried by the gravitational
field, Brans and Dicke took up an idea earlier proposed by Pascual
Jordan34 of hypothesizing an auxiliary field φ. The fact that such a
field has never been detected directly suggests that it has no mass or
charge. If it is massless, it must propagate at exactly c, and this also
makes sense because if it were to propagate at speeds less than c,
there would be no obvious physical parameter that would determine
that speed. How many tensor indices should it have? Since Mach’s
principle tries to give an account of inertia, and inertial mass is a
scalar,35 φ should presumably be a scalar (quantized by a spin-zero
particle). Theories of this type are called tensor-scalar theories,
because they use a scalar field in addition to the metric tensor.
The wave equation for a massless scalar field, in the absence of
sources, is simply ∇i ∇i φ = 0. The solutions of this wave equation fall off as φ ∼ 1/r. This is gentler than the 1/r2 variation of
the gravitational field, so results like Newton’s shell theorem and
Birkhoff’s theorem no longer apply. If a spherical shell of mass acts
as a source of φ, then φ can be nonzero and varying inside the shell.
The φ that you experience right now as you read this book should be
a sum of wavelets originating from all the masses whose world-lines
intersected the surface of your past light-cone. In a static universe,
this sum would diverge linearly, so a self-consistency requirement for
Brans-Dicke gravity is that it should produce cosmological solutions
that avoid such a divergence, e.g., ones that begin with Big Bangs.
Masses are the sources of the field φ. How should they couple to
it? Since φ is a scalar, we need to construct a scalar as its source,
and the only reasonable scalar that can play this role is the trace of
Jordan was a member of the Nazi Sturmabteilung or “brown shirts” who
nevertheless ran afoul of the Nazis for his close professional relationships with
A limit of 5 × 10−23 has been placed on the anisotropy of the inertial mass
of the proton: R.W.P. Drever, “A search for anisotropy of inertial mass using a
free precession technique,” Philosophical Magazine, 6:687 (1961) 683.
Chapter 8
the stress-energy tensor, T ii . As discussed in example 11 on page
286, this vanishes for light, so the only sources of φ are material
particles.36 Even so, the Brans-Dicke theory retains a form of the
equivalence principle. As discussed on pp. 39 and 33, the equivalence principle is a statement about the results of local experiments,
and φ at any given location in the universe is dominated by contributions from matter lying at cosmological distances. Objects of
different composition will have differing fractions of their mass that
arise from internal electromagnetic fields. Two such objects will still
follow identical geodesics, since their own effect on the local value
of φ is negligible. This is unlike the behavior of electrically charged
objects, which experience significant back-reaction effects in curved
space (p. 39). However, the strongest form of the equivalence principle requires that all experiments in free-falling laboratories produce
identical results, no matter where and when they are carried out.
Brans-Dicke gravity violates this, because such experiments could
detect differences between the value of φ at different locations —
but of course this is part and parcel of the purpose of the theory.
We now need to see how to connect φ to the local notion of
inertia so as to produce an effect of the kind that would tend to
fulfill Mach’s principle. In Mach’s original formulation, this would
entail some kind of local rescaling of all inertial masses, but Brans
and Dicke point out that in a theory of gravity, this is equivalent to
scaling the Newtonian gravitational constant G down by the same
factor. The latter turns out to be a better approach. For one thing,
it has a natural interpretation in terms of units. Since φ’s amplitude
falls off as 1/r, we can write φ ∼ Σmi /r, where the sum is over the
past light cone. If we then make the identification of φ with 1/G
(or c2 /G in a system wher c 6= 1), the units work out properly, and
the coupling constant between matter and φ can be unitless. If this
coupling constant, notated 1/ω, were not unitless, then the theory’s
predictive value would be weakened, because there would be no way
to know what value to pick for it. For a unitless constant, however,
there is a reasonable way to guess what it should be: “in any sensible
theory,” Brans and Dicke write, “ω must be of the general order of
magnitude of unity.” This is, of course, assuming that the BransDicke theory was correct. In general, there are other reasonable
values to pick for a unitless number, including zero and infinity. The
limit of ω → ∞ recovers the special case of general relativity. Thus
Mach’s principle, which once seemed too vague to be empirically
falsifiable, comes down to measuring a specific number, ω, which
This leads to an exception to the statement above that all Brans-Dicke
spacetimes are expected to look like Big Bang cosmologies. Any solution of the
GR field equations containing nothing but vacuum and electromagnetic fields
(known as an “elevtrovac” solution) is also a valid Brans-Dicke spacetime. In
such a spacetime, a constant φ can be set arbitrarily. Such a spacetime is in
some sense not generic for Brans-Dicke gravity.
Section 8.3
Mach’s principle revisited
quantifies how non-Machian our universe is.37
8.3.2 Predictions of the Brans-Dicke theory
Returning to the example of the spherical shell of mass, we can
see based on considerations of units that the value of φ inside should
be ∼ m/r, where m is the total mass of the shell and r is its radius.
There may be a unitless factor out in front, which will depend on ω,
but for ω ∼ 1 we expect this constant to be of order 1. Solving the
nasty set of field equations that result from their Lagrangian, Brans
and Dicke indeed found φ ≈ [2/(3 + 2ω)](m/r), where the constant
in square brackets is of order unity if ω is of order unity. In the
limit of ω → ∞, φ = 0, and the shell has no physical effect on its
interior, as predicted by general relativity.
Brans and Dicke were also able to calculate cosmological models,
and in a typical model with a nearly spatially flat universe, they
found φ would vary according to
4 + 3ω 2 t 2/(4+3ω)
φ = 8π
ρo t
6 + 4ω o to
where ρo is the density of matter in the universe at time t = to .
When the density of matter is small, G is large, which has the same
observational consequences as the disappearance of inertia; this is
exactly what one expects according to Mach’s principle. For ω → ∞,
the gravitational “constant” G = 1/φ really is constant.
Returning to the thought experiment involving the 22-caliber rifle fired out the window, we find that in this imaginary universe, with
a very small density of matter, G should be very large. This causes
a frame-dragging effect from the laboratory on the gyroscope, one
much stronger than we would see in our universe. Brans and Dicke
calculated this effect for a laboratory consisting of a spherical shell,
and although technical difficulties prevented the reliable extrapolation of their result to ρo → 0, the trend was that as ρo became
Another good technical reasons for thinking of φ as relating to the gravitational constant is that general relativity has a standard prescription for describing fields on a background of curved spacetime. The vacuum field equations of
general relativity can be derived from the principle of least action, and although
the details are beyond the scope of this book (see, e.g., Wald, General Relativity, appendix E), the general idea is that we define a Lagrangian density LG
that depends on the Ricci scalar curvature, and then extremize its integral over
all possible histories of the evolution of the gravitational field. If we want to
describe some other field, such as matter, light, or φ, we simply take the specialrelativistic Lagrangian LM for that field, change all the derivatives to covariant
derivatives, and form the sum (1/G)LG + LM . In the Brans-Dicke theory, we
have three pieces, (1/G)LG + LM + Lφ , where LM is for matter and Lφ for φ.
If we were to interpret φ as a rescaling of inertia, then we would have to have φ
appearing as a fudge factor modifying all the inner workings of LM . If, on the
other hand, we think of φ as changing the value of the gravitational constant G,
then the necessary modification is extremely simple. Brans and Dicke introduce
one further modification to Lφ so that the coupling constant ω between matter
and φ can be unitless. This modification has no effect on the wave equation of
φ in flat spacetime.
Chapter 8
small, the frame-dragging effect would get stronger and stronger,
presumably eventually forcing the gyroscope to precess in lock-step
with the laboratory. There would thus be no way to determine, once
the bullet was far away, that the laboratory was rotating at all —
in perfect agreement with Mach’s principle.
8.3.3 Hints of empirical support
Only six years after the publication of the Brans-Dicke theory,
Dicke himself, along with H.M. Goldenberg38 carried out a measurement that seemed to support the theory empirically. Fifty years
before, one of the first empirical tests of general relativity, which it
had seemed to pass with flying colors, was the anomalous perihelion
precession of Mercury. The word “anomalous,” which is often left
out in descriptions of this test, is required because there are many
nonrelativistic reasons why Mercury’s orbit precesses, including interactions with the other planets and the sun’s oblate shape. It is
only when these other effects are subtracted out that one sees the
general-relativistic effect calculated on page 215. The sun’s oblateness is difficult to measure optically, so the original analysis of the
data had proceeded by determining the sun’s rotational period by
observing sunspots, and then assuming that the sun’s bulge was the
one found for a rotating fluid in static equilibrium. The result was an
assumed oblateness of about 1 × 10−5 . But we know that the sun’s
dynamics are more complicated than this, since it has convection
currents and magnetic fields. Dicke, who was already a renowned
experimentalist, set out to determine the oblateness by direct optical measurements, and the result was (5.0 ± 0.7) × 10−5 , which,
although still very small, was enough to put the observed perihelion
precession out of agreement with general relativity by about 8%.
The perihelion precession predicted by Brans-Dicke gravity differs
from the general relativistic result by a factor of (4 + 3ω)/(6 + 3ω).
The data therefore appeared to require ω ≈ 6 ± 1, which would be
inconsistent with general relativity.
a / The apparatus used by
Dicke and Goldenberg to measure the oblateness of the sun
was essentially a telescope with
a disk inserted in order to black
out most of the light from the sun.
8.3.4 Mach’s principle is false.
The trouble with the solar oblateness measurements was that
they were subject to a large number of possible systematic errors,
and for this reason it was desirable to find a more reliable test of
Brans-Dicke gravity. Not until about 1990 did a consensus arise,
based on measurements of oscillations of the solar surface, that the
pre-Dicke value was correct. In the interim, the confusion had the
salutary effect of stimulating a renaissance of theoretical and experimental work in general relativity. Often if one doesn’t have an
alternative theory, one has no reasonable basis on which to design
and interpret experiments to test the original theory.
Dicke and Goldenberg, “Solar Oblateness and General Relativity,” Physical
Review Letters 18 (1967) 313
Section 8.3
Mach’s principle revisited
Currently, the best bound on ω is based on measurements39
of the propagation of radio signals between earth and the CassiniHuygens space probe in 2003, which require ω > 4 × 104 . This is
so much greater than unity that it is reasonable to take Brans and
Dicke at their word that “in any sensible theory, ω must be of the
general order of magnitude of unity.” Brans-Dicke fails this test, and
is no longer a “sensible” candidate for a theory of gravity. We can
now see that Mach’s principle, far from being a fuzzy piece of philosophical navel-gazing, is a testable hypothesis. It has been tested
and found to be false, in the following sense. Brans-Dicke gravity
is about as natural a formal implementation of Mach’s principle as
could be hoped for, and it gives us a number ω that parametrizes
how Machian the universe is. The empirical value of ω is so large
that it shows our universe to be essentially as non-Machian as general relativity.
Bertotti, Iess, and Tortora, “A test of general relativity using radio links
with the Cassini spacecraft,” Nature 425 (2003) 374
Chapter 8
Verify, as claimed on p. 267, that the electromagnetic pressure
inside a medium-weight atomic nucleus is on the order of 1033 Pa.
Is the Big Bang singularity removable by the coordinate
transformation t → 1/t?
. Solution, p. 376
Verify the claim made on p. 299 that a is a linear function of
time in the case of the Milne universe, and that k = −1.
Examples 13 on page 303 and 15 on page 306 discussed ropes
with cosmological lengths. Reexamine these examples in the case of
the Milne universe.
. Solution, p. 376
(a) Show that the Friedmann equations are symmetric under
time reversal. (b) The spontaneous breaking of this symmetry in
perpetually expanding solutions was discussed on page 314. Use
the definition of a manifold to show that this symmetry cannot be
restored by gluing together an expanding solution and a contracting
one “back to back” to create a single solution on a single, connected
. Solution, p. 376
The Einstein field equations are
Gab = 8πTab + Λgab
and when it is possible to adopt a frame of reference in which the
local mass-energy is at rest on average, we can interpret the stressenergy tensor as
T µν = diag(−ρ, P , P , P )
where ρ is the mass-energy density and P is the pressure. Fix some
point as the origin of a local Lorentzian coordinate system. Analyze
the properties of these relations under a reflection such as x → −x
or t → −t.
. Solution, p. 377
(a) Show that a positive cosmological constant violates the
strong energy condition in a vacuum. In applying the definition of
the strong energy condition, treat the cosmological constant as a
form of matter, i.e., “roll in” the cosmological constant term to the
stress-energy term in the field equations. (b) Comment on how this
affects the results of the following paper: Hawking and Ellis, “The
Cosmic Black-Body Radiation and the Existence of Singularities in
Our Universe,” Astrophysical Journal, 152 (1968) 25,
In problem 6 on page 199, we analyzed the properties of the
ds2 = e2gz dt2 − dz 2
(a) In that problem we found that this metric had the same properties at all points in space. Verify in particular that it has the same
scalar curvature R at all points in space.
(b) Show that this is a vacuum solution in the two-dimensional (t, z)
(c) Suppose we try to generalize this metric to four dimensions as
ds2 = e2gz dt2 − dx2 − dy 2 − dz 2
Show that this requires an Einstein tensor with unphysical properties.
. Solution, p. 377
Consider the following proposal for defeating relativity’s prohibition on velocities greater than c. Suppose we make a chain billions
of light-years long and attach one end of the chain to a particular
galaxy. At its other end, the chain is free, and it sweeps past the
local galaxies at a very high speed. This speed is proportional to
the length of the chain, so by making the chain long enough, we can
make the speed exceed c.
Debunk this proposal in the special case of the Milne universe.
Make a rigorous definition of the volume V of the observable universe. Suppose someone asks whether V depends on the
observer’s state of motion. Does this question have a well-defined
answer? If so, what is it? Can we calculate V ’s observer-dependence
by applying a Lorentz contraction?
. Solution, p. 378
For a perfect fluid, we have P = wρ, where w is a constant.
The cases w = 0 and w = 1/3 correspond, respectively, to dust and
radiation. Show that for a flat universe with Λ = 0 dominated by a
single component that is a perfect fluid, the solution to the Friedmann equations is of the form a ∝ tδ , and determine the exponent
δ. Check your result in the dust case against the one on p. 313,
then find the exponent in the radiation case. Although the w = −1
case corresponds to a cosmological constant, show that the solution
is not of this form for w = −1.
. Solution, p. 378
Apply the result of problem 11 to generalize the result
of example 18 on p. 314 for the size of the observable universe.
What is the result in the case of the radiation-dominated universe?
. Solution, p. 379
Chapter 8
Chapter 9
Gravitational waves
9.1 The speed of gravity
In Newtonian gravity, gravitational effects are assumed to propagate
at infinite speed, so that for example the lunar tides correspond at
any time to the position of the moon at the same instant. This
clearly can’t be true in relativity, since simultaneity isn’t something
that different observers even agree on. Not only should the “speed of
gravity” be finite, but it seems implausible that it would be greater
than c; in section 2.2 (p. 52), we argued based on empirically well
established principles that there must be a maximum speed of cause
and effect. Although the argument was only applicable to special
relativity, i.e., to a flat spacetime, it seems likely to apply to general
relativity as well, at least for low-amplitude waves on a flat background. As early as 1913, before Einstein had even developed the
full theory of general relativity, he had carried out calculations in
the weak-field limit showing that gravitational effects should propagate at c. This seems eminently reasonable, since (a) it is likely to
be consistent with causality, and (b) G and c are the only constants
with units that appear in the field equations (obscured by our choice
of units, in which G = 1 and c = 1), and the only velocity-scale that
can be constructed from these two constants is c itself.1
Although extremely well founded theoretically, this turns out to
be extremely difficult to test empirically. In a 2003 experiment,2 Fomalont and Kopeikin used a world-wide array of radio telescopes to
observe a conjunction in which Jupiter passed within 3.70 of a quasar,
so that the quasar’s radio waves came within about 3 light-seconds
of the planet on their way to the earth. Since Jupiter moves with
High-amplitude waves need not propagate at c. For example, general relativity predicts that a gravitational-wave pulse propagating on a background of
curved spacetime develops a trailing edge that propagates at less than c (Misner,
Thorne, and Wheeler, p. 957). This effect is weak when the amplitude is small
or the wavelength is short compared to the scale of the background curvature.
It makes sense that the effect vanishes when background curvature is absent,
since there is then no fixed scale. Dispersion requires that different wavelengths
propagate at different speeds, but without a scale there is no reason for any
wavelength to behave any differently from any other. At very high amplitudes,
one can even have such exotic phenomena as the formation of black holes when
enough wave energy is focused into a small region. None of these phenomena is
ever likely to be observed empirically, since all gravitational waves in our universe
have extremely small amplitudes.
v = 4 × 10−5 , one expects naively that the radio waves passing by it
should be deflected by the field produced by Jupiter at the position it
had 3 seconds earlier. This position differs from its present position
by about 10−4 light-seconds, and the result should be a difference
in propagation time, which should be different when observed from
different locations on earth. Fomalont and Kopeikin measured these
phase differences with picosecond precision, and found them to be
in good agreement with the predictions of general relativity. The
real excitement started when they published their result with the
interpretation that they had measured, for the first time, the speed
of gravity, and found it to be within 20% error bars of c. Samuel3
and Will4 published refutations, arguing that Kopeikin’s calculations contained mistakes, and that what had really been measured
was the speed of light, not the speed of gravity.
The reason that the interpretation of this type of experiment
is likely to be controversial is that although we do have theories of
gravity that are viable alternatives to general relativity (e.g., the
Brans-Dicke theory, in which the gravitational constant is a dynamically changing variable), such theories have generally been carefully
designed to agree with general relativity in the weak-field limit, and
in particular every such theory (or at least every theory that remains
viable given current experimental data) predicts that gravitational
effects propagate at c in the weak-field limit. Without an alternative
theory to act as a framework — one that disagrees with relativity
about the speed of gravity — it is difficult to know whether an observation that agrees with relativity is a test of this specific aspect
of relativity.
9.2 Gravitational radiation
9.2.1 Empirical evidence
So we still don’t know, a century after Einstein found the field
equations, whether gravitational “ripples” travel at c. Nevertheless,
we do have strong empirical evidence that such ripples exist. The
Hulse-Taylor system (page 220) contains two neutron stars orbiting
around their common center of mass, and the period of the orbit
is observed to be decreasing gradually over time (figure a). This is
interpreted as evidence that the stars are losing energy to radiation
of gravitational waves.5 As we’ll see in section 9.2.5, the rate of
energy loss is in excellent agreement with the predictions of general
An even more dramatic, if less clearcut, piece of evidence is
Stairs, “Testing General Relativity with Pulsar Timing,”
Chapter 9
Gravitational waves
a / The Hulse-Taylor pulsar’s orbital motion is gradually losing
energy due to the emission of
gravitational waves. The linear
decrease of the period is integrated on this plot, resulting in
a parabola. From Weisberg and
Komossa, Zhou, and Lu’s observation6 of a supermassive black hole
that appears to be recoiling from its parent galaxy at a velocity of
2650 km/s (projected along the line of sight). They interpret this as
evidence for the following scenario. In the early universe, galaxies
form with supermassive black holes at their centers. When two such
galaxies collide, the black holes can merge. The merger is a violent
process in which intense gravitational waves are emitted, and these
waves carry a large amount of momentum, causing the black holes
to recoil at a velocity greater than the escape velocity of the merged
Although the energy loss from systems such as the Hulse-Taylor
binary provide strong evidence that gravitational waves exist and
carry enery, we would also like to detect them directly. This would
not only be a definitive test of a century-old prediction of general relativity, it would also open a window into a completely new method
Section 9.2
Gravitational radiation
of astronomical observation. A series of attempts is under way to
observe gravitational waves directly using interferometers, which detect oscillations in the lengths of their own arms. These can be either
ground-based or space-based. The early iterations of the groundbased LIGO interferometer did not detect gravitational waves, but
a new version, Advanced LIGO, is under construction. A spacebased system, LISA, has been proposed for launch in 2020, but its
funding is uncertain. The two devices would operate in complementary frequency ranges (figure b). A selling point of LISA is that if
it is launched, there are a number of known sources in the sky that
are known to be easily within its range of sensitivity.7 One excellent candidate is HM Cancri, a pair of white dwarfs with an orbital
period of 5.4 minutes, shorter than that of any other known binary
b / Predicted sensitivities of LISA
and LIGO to gravitational waves
of various frequences.
9.2.2 Energy content
Even without performing the calculations for a system like the
Hulse-Taylor binary, it is easy to show that if such waves exist, they
must be capable of carrying away energy. Consider two equal masses
in highly elliptical orbits about their common center of mass, figure
c. The motion is nearly one-dimensional. As the masses recede from
one another, they feel a delayed version of the gravitational force
originating from a time when they were closer together and the force
was stronger. The result is that in the near-Newtonian limit, they
lose more kinetic and gravitational energy than they would have
lost in the purely Newtonian theory. Now they come back inward
G. Nelemans, “The Galactic Gravitational wave foreground,”
Roelofs et al., “Spectroscopic Evidence for a 5.4-Minute Orbital Period in
HM Cancri,”
Chapter 9
Gravitational waves
in their orbits. As they approach one another, the time-delayed
force is anomalously weak, so they gain less mechanical energy than
expected. The result is that with each cycle, mechanical energy is
lost. We expect that this energy is carried by the waves, in the
same way that radio waves carry the energy lost by a transmitting
Not only can these waves remove mechanical energy from a system, they can also deposit energy in a detector, as shown by the
nonmathematical “sticky bead argument” (figure d), which was originated by Feynman in 1957 and later popularized by Bondi.
Now strictly speaking, we have only shown that gravitational
waves can extract or donate mechanical energy, but not that the
waves themselves transmit this energy. The distinction isn’t one
that normally occurs to us, since we are trained to believe that
energy is always conserved. But we know that, for fundamental reasons, general relativity doesn’t have global conservation laws that
apply to all spacetimes (p. 148). Perhaps the energy lost by the
Hulse-Taylor system is simply gone, never to reappear, and the energy imparted to the sticky bead is simply generated out of nowhere.
On the other hand, general relativity does have global conservation
laws for certain specific classes of spacetimes, including, for example,
a conserved scalar mass-energy in the case of a stationary spacetime
(p. 246). Spacetimes containing gravitational waves are not stationary, but perhaps there is something similar we can do in some
appropriate special case.
Suppose we want an expression for the energy of a gravitational wave in terms of its amplitude. This seems like it ought
to be straightforward. We have such expressions in other classical field theories. In electromagnetism, we have energy densities
+(1/8πk)|E|2 and +(1/2µo )|B|2 associated with the electric and
magnetic fields. In Newtonian gravity, we can assign an energy
density −(1/8πG)|g|2 to the gravitational field g; the minus sign
indicates that when masses glom onto each other, they produce a
greater field, and energy is released.
c / As the two planets recede
from one another, each feels the
gravitational attraction that the
other one exerted in its previous
position, delayed by the time
it takes gravitational effects to
propagate at c . At time t , the
right-hand planet experiences
the stronger deceleration corresponding to the left-hand planet’s
closer position at the earlier
time t 0 , not its current position
at t . Mechanical energy is not
conserved, and the orbits will
d / The sticky bead argument
for the reality of gravitational
waves. As a gravitational wave
with the appropriate polarization
passes by, the bead vibrates back
and forth on the rod. Friction creates heat. This demonstrates that
gravitational waves carry energy,
and are thus real, observable
In general relativity, however, the equivalence principle tells us
that for any gravitational field measured by one observer, we can
One has to be careful with this type of argument. In particular, one can
obtain incorrect correct results by attempting to generalize this one-dimensional
argument to motion in more than one dimension, because the effective semiNewtonian interaction is not just a time-delayed version of Newton’s law; it also
includes velocity-dependent forces. It is easy to see why such velocity-dependence
must occur in the simpler case of electromagnetism. Suppose that charges A
and B are not at rest relative to one another. In B’s frame, the electric field
from A must come from the direction of the position that an observer comoving
with B would extrapolate linearly from A’s last known position and velocity,
as determined by light-speed calculation. This follows from Lorentz invariance,
since this is the direction that will be seen by an observer comoving with A. A
full discussion is given by Carlip,
Section 9.2
Gravitational radiation
find another observer, one who is free-falling, who says that the local
field is zero. It follows that we cannot associate an energy with the
curvature of a particular region of spacetime in any exact way. The
best we can do is to find expressions that give the energy density (1)
in the limit of weak fields, and (2) when averaged over a region of
space that is large compared to the wavelength. These expressions
are not unique. There are a number of ways to write them in terms
of the metric and its derivatives, and they all give the same result
in the appropriate limit. The reader who is interested in seeing the
subject developed in detail is referred to Carroll’s Lecture Notes on
General Relativity, Although this sort of thing is technically messy, we can accomplish
quite a bit simply by knowing that such results do exist, and that
although they are non-unique in general, they are uniquely well defined in certain cases. Specifically, when one wants to discuss gravitational waves, it is usually possible to assume an asymptotically
flat spacetime. In an asymptotically flat spacetime, there is a scalar
mass-energy, called the ADM mass, that is conserved. In this restricted sense, we are assured that the books balance, and that the
emission and absorption of gravitational waves really does mean the
transmission of a fixed amount of energy.
9.2.3 Expected properties
To see what properties we should expect for gravitational radiation, first consider the reasoning that led to the construction of the
Ricci and Einstein tensors. If a certain volume of space is filled with
test particles, then the Ricci and Einstein tensors measure the tendency for this volume to “accelerate;” i.e., −d2 V /dt2 is a measure
of the attraction of any mass lying inside the volume. A distant
mass, however, will exert only tidal forces, which distort a region
without changing its volume. This suggests that as a gravitational
wave passes through a certain region of space, it should distort the
shape of a given region, without changing its volume.
When the idea of gravitational waves was first discussed, there
was some skepticism about whether they represented an effect that
was observable, even in principle. The most naive such doubt is of
the same flavor as the one discussed in section 8.2.6 about the observability of the universe’s expansion: if everything distorts, then
don’t our meter-sticks distort as well, making it impossible to measure the effect? The answer is the same as before in section 8.2.6;
systems that are gravitationally or electromagnetically bound do
not have their scales distorted by an amount equal to the change in
the elements of the metric.
A less naive reason to be skeptical about gravitational waves is
that just because a metric looks oscillatory, that doesn’t mean its
Chapter 9
Gravitational waves
oscillatory behavior is observable. Consider the following example.
ds = dt − 1 +
sin x dx2 − dy 2 − dz 2
The Christoffel symbols depend on derivatives of the form ∂a gbc , so
here the only nonvanishing Christoffel symbol is Γxxx . It is then
straightforward to check that the Riemann tensor Rabcd = ∂c Γadb −
∂d Γacb + Γace Γedb − Γade Γecb vanishes by symmetry. Therefore this
metric must really just be a flat-spacetime metric that has been
subjected to a silly change of coordinates.
Self-check: R vanishes, but Γ doesn’t. Is there a reason for
paying more attention to one or the other?
To keep the curvature from vanishing, it looks like we need a
metric in which the oscillation is not restricted to a single variable.
For example, the metric
ds2 = dt2 − 1 +
sin y dx2 − dy 2 − dz 2
does have nonvanishing curvature. In other words, it seems like
we should be looking for transverse waves rather than longitudinal ones.10 On the other hand, this metric cannot be a solution
to the vacuum field equations, since it doesn’t preserve volume. It
also stands still, whereas we expect that solutions to the field equations should propagate at the velocity of light, at least for small
amplitudes. These conclusions are self-consistent, because a wave’s
polarization can only be constrained if it propagates at c (see p. 130).
e / As the gravitational wave
propagates in the z direction, the
metric oscillates in the x and y
directions, preserving volume.
Based on what we’ve found out, the following seems like a metric
that might have a fighting chance of representing a real gravitational
ds2 = dt2 − (1 + A sin(z − t)) dx2 −
dy 2
− dz 2
1 + A sin(z − t)
It is transverse, it propagates at c(= 1), and the fact that gxx is the
reciprocal of gyy makes it volume-conserving. The following Maxima
program calculates its Einstein tensor:
A more careful treatment shows that longitudinal waves can always be interpreted as physically unobservable coordinate waves, in the limit of large distances from the source. On the other hand, it is clear that no such prohibition
against longitudinal waves could apply universally, because such a constraint
can only be Lorentz-invariant if the wave propagates at c (see p. 130), whereas
high-amplitude waves need not propagate at c. Longitudinal waves near the
source are referred to as Type III solutions in a classification scheme due to
Petrov. Transverse waves, which are what we could actually observe in practical
experiments, are type N.
Section 9.2
Gravitational radiation
For a representative component of the Einstein tensor, we find
Gtt = −
A2 cos2 (z − t)
2 + 4A sin(z − t) + 2A2 sin2 (z − t)
For small values of A, we have |Gtt | . A2 /2. The vacuum field
equations require Gtt = 0, so this isn’t an exact solution. But all
the components of G, not just Gtt , are of order A2 , so this is an
approximate solution to the equations.
It is also straightforward to check that propagation at approximately c was a necessary feature. For example, if we replace the
factors of sin(z − t) in the metric with sin(z − 2t), we get a Gxx that
is of order unity, not of order A2 .
To prove that gravitational waves are an observable effect, we
would like to be able to display a metric that (1) is an exact solution
of the vacuum field equations; (2) is not merely a coordinate wave;
and (3) carries momentum and energy. As late as 1936, Einstein and
Rosen published a paper claiming that gravitational waves were a
mathematical artifact, and did not actually exist.11
9.2.4 Some exact solutions
In this section we study several examples of exact solutions to
the field equations. Each of these can readily be shown not to be
a mere coordinate wave, since in each case the Riemann tensor has
nonzero elements.
An exact solution
Example: 1
We’ve already seen, e.g., in the derivation of the Schwarzschild
metric in section 6.2.4, that once we have an approximate solution to the equations of general relativity, we may be able to find a
series solution. Historically this approach was only used as a last
resort, because the lack of computers made the calculations too
complex to handle, and the tendency was to look for tricks that
would make a closed-form solution possible. But today the series
method has the advantage that any mere mortal can have some
reasonable hope of success with it — and there is nothing more
boring (or demoralizing) than laboriously learning someone else’s
special trick that only works for a specific problem. In this example, we’ll see that such an approach comes tantalizingly close
Some of the history is related at
Chapter 9
Gravitational waves
to providing an exact, oscillatory plane wave solution to the field
Our best solution so far was of the form
dy 2
ds2 = dt 2 − (1 + f ) dx 2 −
− dz 2
where f = A sin(z − t). This doesn’t seem likely to be an exact
solution for large amplitudes, since the x and y coordinates are
treated asymmetrically. In the extreme case of |A| ≥ 1, there
would be singularities in gy y , but not in gxx . Clearly the metric will
have to have some kind of nonlinear dependence on f , but we just
haven’t found quite the right nonlinear dependence. Suppose we
try something of this form:
ds2 = dt 2 − 1 + f + cf 2 dx 2 − 1 − f + df 2 dy 2 − dz 2
This approximately conserves volume, since (1+f +. . .)(1−f +. . .)
equals unity, up to terms of order f 2 . The following program tests
this form.
f : A*exp(%i*k*(z-t));
In line 3, the motivation for using the complex exponential rather
than a sine wave in f is the usual one of obtaining simpler expressions; as we’ll see, this ends up causing problems. In lines 5
and 6, the symbols c and d have not been defined, and have not
been declared as depending on other variables, so Maxima treats
them as unknown constants. The result is Gtt ∼ (4d + 4c − 3)A2
for small A, so we can make the A2 term disappear by an appropriate choice of d and c. For symmetry, we choose c = d = 3/8.
With these values of the constants, the result for Gtt is of order
A4 . This technique can be extended to higher and higher orders
of approximation, resulting in an exact series solution to the field
Unfortunately, the whole story ends up being too good to be true.
The resulting metric has complex-valued elements. If general relativity were a linear field theory, then we could apply the usual
technique of forming linear combinations of expressions of the
form e+i... and e−i... , so as to give a real result. Unfortunately the
field equations of general relativity are nonlinear, so the resulting
linear combination is no longer a solution. The best we can do is
to make a non-oscillatory real exponential solution (problem 2).
Section 9.2
Gravitational radiation
An exact, oscillatory, non-monochromatic solution
Assume a metric of the form
Example: 2
ds2 = dt 2 − p(z − t)2 dx 2 − q(z − t)2 dy 2 − dz 2
where p and q are arbitrary functions. Such a metric would clearly
represent some kind of transverse-polarized plane wave traveling
at velocity c(= 1) in the z direction. The following Maxima code
calculates its Einstein tensor.
The result is proportional to q̈/q + p̈/p, so any functions p and
q that satisfy the differential equation q̈/q + p̈/p = 0 will result
in a solution to the field equations. Setting p(u) = 1 + A cos u, for
example, we find that q is oscillatory, but with a period longer than
2π (problem 3).
An exact, plane, monochromatic wave
Any metric of the form
Example: 3
ds2 = (1 − h)dt 2 − dx 2 − dy 2 − (1 + h)dz 2 + 2hdzdt
where h = f (z − t)xy , and f is any function, is an exact solution of
the field equations (problem 4).
Because h is proportional to xy, this does not appear at first
glance to be a uniform plane wave. One can verify, however, that
all the components of the Riemann tensor depend only on z − t,
not on x or y . Therefore there is no measurable property of this
metric that varies with x and y.
9.2.5 Rate of radiation
How can we find the rate of gravitational radiation from a system
such as the Hulse-Taylor pulsar?
Let’s proceed by analogy. The simplest source of sound waves
is something like the cone of a stereo speaker. Since typical sound
waves have wavelengths measured in meters, the entire speaker is
generally small compared to the wavelength. The speaker cone is
a surface of oscillating displacement x = xo sin ωt. Idealizing such
Chapter 9
Gravitational waves
a source to a radially pulsating spherical surface, we have an oscillating monopole that radiates sound waves uniformly in all directions. To find the power radiated, we note that the velocity of the
source-surface is proportional to xo ω, so the kinetic energy of the air
immediately in contact with it is proportional to ω 2 x2o . The power
radiated is therefore proportional to ω 2 x2o .
In electromagnetism, conservation of charge forbids the existence
of an oscillating electric monopole. The simplest radiating source is
therefore an oscillating electric dipole D = Do sin ωt. If the dipole’s
physical size is small compared to a wavelength of the radiation,
then the radiation is an inefficient process; at any point in space,
there is only a small difference in path length between the positive
and negative portions of the dipole, so there tends to be strong
cancellation of their contributions, which were emitted with opposite
phases. The result is that the wave’s electromagnetic potential fourvector (section 4.2.5) is proportional to Do ω, the fields to Do ω 2 , and
the radiated power to Do2 ω 4 . The factor of ω 4 can be broken down
into (ω 2 )(ω 2 ), where the first factor of ω 2 occurs for reasons similar
to the ones that explain the ω 2 factor for the monopole radiation
of sound, while the second ω 2 arises because the smaller ω is, the
longer the wavelength, and the greater the inefficiency in radiation
caused by the small size of the source compared to the wavelength.
AM radio
Example: 4
Commercial AM radio uses wavelengths of several hundred meters, so AM dipole antennas are usually orders of magnitude
shorter than a wavelength. This causes severe attentuation in
both transmission and reception. (There are theorems called
reciprocity theorems that relate efficiency of transmission to efficiency of reception.) Receivers therefore need to use of a large
amount of amplification. This doesn’t cause problems, because
the ambient sources of RF noise are attenuated by the short antenna just as severely as the signal.
f / The power emitted by a
multipole source of order m is
proportional to ω2(m+1) , when
the size of the source is small
compared to the wavelength. The
main reason for the ω dependence is that at low frequencies,
the wavelength is long, so the
number of wavelengths traveled
to a particular point in space is
nearly the same from any point
in the source; we therefore get
strong cancellation.
Since our universe doesn’t seem to have particles with negative
mass, we can’t form a gravitational dipole by putting positive and
negative masses on opposite ends of a stick — and furthermore,
such a stick will not spin freely about its center, because its center
of mass does not lie at its center! In a more realistic system, such as
the Hulse-Taylor pulsar, we have two unequal masses orbiting about
their common center of mass. By conservation of momentum, the
mass dipole moment of such a system is constant, so we cannot have
an oscillating mass dipole. The simplest source of gravitational radiation is therefore an oscillating mass quadrupole, Q = Qo sin ωt.
As in the case of the oscillating electric dipole, the radiation is suppressed if, as is usually the case, the source is small compared to
the wavelength. The suppression is even stronger in the case of a
quadrupole, and the result is that the radiated power is proportional
Section 9.2
Gravitational radiation
to Q2o ω 6 .
This result has the interesting property of being invariant under a rescaling of coordinates. In geometrized units, mass, distance, and time all have the same units, so that Q2o has units
of (length3 )2 while ω 6 has units of (length)−6 . This is exactly
what is required, because in geometrized units, power is unitless,
energy/time = length/length = 1.
We can also tie the ω 6 dependence to our earlier argument, on
p. 334, for the dissipation of energy by gravitational waves. The
argument was that gravitating bodies are subject to time-delayed
gravitational forces, with the result that orbits tend to decay. This
argument only works if the forces are time-varying; if the forces
are constant over time, then the time delay has no effect. For example, in the semi-Newtonian limit the field of a sheet of mass is
independent of distance from the sheet. (The electrical analog of
this fact is easily proved using Gauss’s law.) If two parallel sheets
fall toward one another, then neither is subject to a time-varying
force, so there will be no radiation. In general, we expect that there
will be no gravitational radiation from a particle unless the third
derivative of its position d3 x/dt3 is nonzero. (The same is true for
electric quadrupole radiation.) In the special case where the position
oscillates sinusoidally, the chain rule tells us that taking the third
derivative is equivalent to bringing out a factor of ω 3 . Since the
amplitude of gravitational waves is proportional to d3 x/dt3 , their
energy varies as (d3 x/dt3 )2 , or ω 6 .
The general pattern we have observed is that for multipole radiation of order m (0=monopole, 1=dipole, 2=quadrupole), the radiated power depends on ω 2(m+1) . Since gravitational radiation must
always have m = 2 or higher, we have the very steep ω 6 dependence of power on frequency. This demonstrates that if we want
to see strong gravitational radiation, we need to look at systems
that are oscillating extremely rapidly. For a binary system with
unequal masses of order m, with orbits having radii of order r, we
have Qo ∼ mr2 . Newton’s laws give ω ∼ m1/2 r−3/2 , which is essentially Kepler’s law of periods. The result is that the radiated power
should depend on (m/r)5 . Reinserting the proper constants to give
an equation that allows practical calculation in SI units, we have
G4 m 5
P =k 5
where k is a unitless constant of order unity.
For the Hulse-Taylor pulsar,12 we have m ∼ 3 × 1030 kg (about
one and a half solar masses) and r ∼ 109 m. The binary pulsar is
made to order our purposes, since m/r is extremely large compared
to what one sees in almost any other astronomical system. The
resulting estimate for the power is about 1024 watts.
Chapter 9
Gravitational waves
The pulsar’s period is observed to be steadily lengthening at a
rate of α = 2.418 × 10−12 seconds per second. To compare this with
our crude theoretical estimate, we take the Newtonian energy of the
system Gm2 /r and multiply by ωα, giving 1025 W, which checks to
within an order of magnitude. A full general-relativistic calculation
reproduces the observed value of α to within the 0.1% error bars of
the data.
Section 9.2
Gravitational radiation
(a) Starting on page 21, we have associated geodesics with
the world-lines of low-mass objects (test particles). Use the HulseTaylor pulsar as an example to show that the assumption of low
mass was a necessary one. How is this similar to the issues encountered on pp. 39ff involving charged particles?
(b) Show that if low-mass, uncharged particles did not follow geodesics
(in a spacetime with no ambient electromagnetic fields), it would violate Lorentz invariance. Make sure that your argument explicitly
invokes the low mass and the lack of charge, because otherwise your
argument is wrong.
. Solution, p. 379
Show that the metric ds2 = dt2 − Adx2 − Bdy 2 − dz 2 with
A = 1 − f + f2 −
B = 1 + f + f2 +
f = Aek(t−z)
25 3
15211 5
f +
25 3
15211 5
f −
is an approximate solution to the vacuum field equations, provided
that k is real — which prevents this from being a physically realistic,
oscillating wave. Find the next nonvanishing term in each series.
Verify the claims made in example 2. Characterize the (somewhat complex) behavior of the function q obtained when p(u) =
1 + A cos u.
Verify the claims made in example 3 using Maxima. Although
the result holds for any function f , you may find it more convenient
to use some specific form of f , such as a sine wave, so that Maxima
will be able to simplify the result to zero at the end. Note that when
the metric is expressed in terms of the line element, there is a factor
of 2 in the 2hdzdt term, but when expressing it as a matrix, the 2 is
not present in the matrix elements, because there are two elements
in the matrix that each contribute an equal amount.
Chapter 9
Gravitational waves
Appendix 1: Excerpts from three papers by
The following English translations of excerpts from three papers by Einstein were originally
published in “The Principle of Relativity,” Methuen and Co., 1923. The translation was by
W. Perrett and G.B. Jeffery, and notes were provided by A. Sommerfeld. John Walker (www. has provided machine-readable versions of the first two and placed them in the
public domain. Some notation has been modernized, British spelling has been Americanized,
etc. Footnotes by Sommerfeld, Walker, and B. Crowell are marked with initials. B. Crowell’s
modifications to the present version are also in the public domain.
The paper “On the electrodynamics of moving bodies” contains two parts, the first dealing
with kinematics and the second with electrodynamics. I’ve given only the first part here, since
the second one is lengthy, and painful to read because of the cumbersome old-fashioned notation.
The second section can be obtained from John Walker’s web site.
The paper “Does the inertia of a body depend upon its energy content?,” which begins on
page 358, is very short and readable. A shorter and less general version of its main argument is
given on p. 135.
“The foundation of the general theory of relativity” is a long review article in which Einstein
systematically laid out the general theory, which he had previously published in a series of
shorter papers. The first three sections of the paper give the general physical reasoning behind
coordinate independence, referred to as general covariance. It begins on page 360.
The reader who is interested in seeing these papers in their entirety can obtain them inexpensively in a Dover reprint of the original Methuen anthology.
On the electrodynamics of moving bodies
A. Einstein, Annalen der Physik 17 (1905) 891.
It is known that Maxwell’s electrodynamics—as usually understood at the present time—
when applied to moving bodies, leads to asymmetries which do not appear to be inherent in
the phenomena.13 Take, for example, the reciprocal electrodynamic action of a magnet and
a conductor. The observable phenomenon here depends only on the relative motion of the
conductor and the magnet, whereas the customary view draws a sharp distinction between the
two cases in which either the one or the other of these bodies is in motion. For if the magnet
is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an
electric field with a certain definite energy, producing a current at the places where parts of
the conductor are situated. But if the magnet is stationary and the conductor in motion, no
electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an
electromotive force, to which in itself there is no corresponding energy, but which gives rise—
assuming equality of relative motion in the two cases discussed—to electric currents of the same
path and intensity as those produced by the electric forces in the former case.
Examples of this sort, together with the unsuccessful attempts to discover any motion of the
earth relative to the “light medium,” suggest that the phenomena of electrodynamics as well as
Einstein begins by giving an example involving electromagnetic induction, considered in two different frames
of reference. With modern hindsight, we would describe this by saying that a Lorentz boost mixes the electric
and magnetic fields, as described in section 4.2.4, p. 136. —BC
of mechanics possess no properties corresponding to the idea of absolute rest.14 They suggest
rather that, as has already been shown to the first order of small quantities,15 the same laws
of electrodynamics and optics will be valid for all frames of reference for which the equations
of mechanics hold good.16 We will raise this conjecture (the purport of which will hereafter be
called the “Principle of Relativity”) to the status of a postulate, and also introduce another
postulate, which is only apparently irreconcilable with the former, namely, that light is always
propagated in empty space with a definite velocity c which is independent of the state of motion
of the emitting body.17 These two postulates suffice for the attainment of a simple and consistent
theory of the electrodynamics of moving bodies based on Maxwell’s theory for stationary bodies.
The introduction of a “luminiferous ether” will prove to be superfluous inasmuch as the view
here to be developed will not require an “absolutely stationary space” provided with special
properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic
processes take place.
The theory to be developed is based—like all electrodynamics—on the kinematics of the
rigid body, since the assertions of any such theory have to do with the relationships between
rigid bodies (systems of coordinates), clocks, and electromagnetic processes.18 Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of
moving bodies at present encounters.
§1. Definition of Simultaneity
Let us take a system of coordinates in which the equations of Newtonian mechanics hold
good.19 In order to render our presentation more precise and to distinguish this system of
coordinates verbally from others which will be introduced hereafter, we call it the “stationary
Einstein knew about the Michelson-Morley experiment by 1905 (J. van Dongen,,
but it isn’t cited specifically here. The 1881 and 1887 Michelson-Morley papers are available online at en. —BC
I.e., to first order in v/c. Experimenters as early as Fresnel (1788-1827) had shown that there were no effects
of order v/c due to the earth’s motion through the aether, but they were able to interpret this without jettisoning
the aether, by contriving models in which solid substances dragged the aether along with them. The negative
result of the Michelson-Morley experiment showed a lack of an effect of order (v/c)2 . —BC
The preceding memoir by Lorentz was not at this time known to the author. —AS
The second postulate is redundant if we take the “laws of electrodynamics and optics” to refer to Maxwell’s
equations. Maxwell’s equations require that light move at c in any frame of reference in which they are valid, and
the first postulate has already claimed that they are valid in all inertial frames of reference. Einstein probably
states constancy of c as a separate postulate because his audience is accustomed to thinking of Maxwell’s equations
as a partial mathematical representation of certain aspects of an underlying aether theory. Throughout part I of
the paper, Einstein is able to derive all his results without assuming anything from Maxwell’s equations other than
the constancy of c. The use of the term “postulate” suggests the construction of a formal axiomatic system like
Euclidean geometry, but Einstein’s real intention here is to lay out a set of philosophical criteria for evaluating
candidate theories; he freely brings in other, less central, assumptions later in the paper, as when he invokes
homogeneity of spacetime on page 350. —BC
Essentially what Einstein means here is that you can’t have Maxwell’s equations without establishing position
and time coordinates, and you can’t have position and time coordinates without clocks and rulers. Therefore even
the description of a purely electromagnetic phenomenon such as a light wave depends on the existence of material
objects. He doesn’t spell out exactly what he means by “rigid,” and we now know that relativity doesn’t actually
allow the existence of perfectly rigid solids (see p. 109). Essentially he wants to be able to talk about rulers that
behave like solids rather than liquids, in the sense that if they are accelerated sufficiently gently from rest and
later brought gently back to rest, their properties will be unchanged. When he derives the length contraction
later, he wants it to be clear that this isn’t a dynamical phenomenon caused by an effect such as the drag of the
i.e., to the first approximation.—AS
Chapter 9
Gravitational waves
If a material point is at rest relative to this system of coordinates, its position can be defined
relative thereto by the employment of rigid standards of measurement and the methods of
Euclidean geometry, and can be expressed in Cartesian coordinates.
If we wish to describe the motion of a material point, we give the values of its coordinates
as functions of the time. Now we must bear carefully in mind that a mathematical description
of this kind has no physical meaning unless we are quite clear as to what we understand by
“time.” We have to take into account that all our judgments in which time plays a part are
always judgments of simultaneous events. If, for instance, I say, “That train arrives here at 7
o’clock,” I mean something like this: “The pointing of the small hand of my watch to 7 and the
arrival of the train are simultaneous events.”20
It might appear possible to overcome all the difficulties attending the definition of “time”
by substituting “the position of the small hand of my watch” for “time.” And in fact such a
definition is satisfactory when we are concerned with defining a time exclusively for the place
where the watch is located; but it is no longer satisfactory when we have to connect in time
series of events occurring at different places, or—what comes to the same thing—to evaluate
the times of events occurring at places remote from the watch.
We might, of course, content ourselves with time values determined by an observer stationed
together with the watch at the origin of the coordinates, and coordinating the corresponding
positions of the hands with light signals, given out by every event to be timed, and reaching him
through empty space. But this coordination has the disadvantage that it is not independent of
the standpoint of the observer with the watch or clock, as we know from experience. We arrive
at a much more practical determination along the following line of thought.
If at the point A of space there is a clock, an observer at A can determine the time values
of events in the immediate proximity of A by finding the positions of the hands which are
simultaneous with these events. If there is at the point B of space another clock in all respects
resembling the one at A, it is possible for an observer at B to determine the time values of
events in the immediate neighbourhood of B. But it is not possible without further assumption
to compare, in respect of time, an event at A with an event at B. We have so far defined only
an “A time” and a “B time.” We have not defined a common “time” for A and B, for the latter
cannot be defined at all unless we establish by definition that the “time” required by light to
travel from A to B equals the “time” it requires to travel from B to A. Let a ray of light start
at the “A time” tA from A towards B, let it at the “B time” tB be reflected at B in the direction
of A, and arrive again at A at the “A time” t0A .
In accordance with definition the two clocks synchronize21 if
tB − tA = t0A − tB .
We assume that this definition of synchronism is free from contradictions, and possible for
any number of points; and that the following relations are universally valid:—
1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the
clock at B.
We shall not here discuss the inexactitude which lurks in the concept of simultaneity of two events at approximately the same place, which can only be removed by an abstraction.—AS
The procedure described here is known as Einstein synchronization.—BC
2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks
at B and C also synchronize with each other.22
Thus with the help of certain imaginary physical experiments we have settled what is to
be understood by synchronous stationary clocks located at different places, and have evidently
obtained a definition of “simultaneous,” or “synchronous,” and of “time.” The “time” of an event
is that which is given simultaneously with the event by a stationary clock located at the place
of the event, this clock being synchronous, and indeed synchronous for all time determinations,
with a specified stationary clock.
In agreement with experience we further assume the quantity
= c,
− tA
to be a universal constant—the velocity of light in empty space.
It is essential to have time defined by means of stationary clocks in the stationary system,
and the time now defined being appropriate to the stationary system we call it “the time of the
stationary system.”
§ 2. On the Relativity of Lengths and Times
The following reflections are based on the principle of relativity and on the principle of the
constancy of the velocity of light. These two principles we define as follows:—
1. The laws by which the states of physical systems undergo change are not affected, whether
these changes of state be referred to the one or the other of two systems of coordinates in uniform
translatory motion.
2. Any ray of light moves in the “stationary” system of coordinates with the determined
velocity c, whether the ray be emitted by a stationary or by a moving body. Hence
velocity =
light path
time interval
where time interval is to be taken in the sense of the definition in § 1.
Let there be given a stationary rigid rod; and let its length be l as measured by a measuringrod which is also stationary. We now imagine the axis of the rod lying along the axis of x of
the stationary system of coordinates, and that a uniform motion of parallel translation with
velocity v along the axis of x in the direction of increasing x is then imparted to the rod. We
now inquire as to the length of the moving rod, and imagine its length to be ascertained by the
following two operations:—
(a) The observer moves together with the given measuring-rod and the rod to be measured,
and measures the length of the rod directly by superposing the measuring-rod, in just the same
way as if all three were at rest.
(b) By means of stationary clocks set up in the stationary system and synchronizing in
accordance with § 1, the observer ascertains at what points of the stationary system the two
This assumption fails in a rotating frame (see p. 111), but Einstein has restricted himself here to an approximately inertial frame of reference.—BC
Chapter 9
Gravitational waves
ends of the rod to be measured are located at a definite time. The distance between these two
points, measured by the measuring-rod already employed, which in this case is at rest, is also a
length which may be designated “the length of the rod.”
In accordance with the principle of relativity the length to be discovered by the operation
(a)—we will call it “the length of the rod in the moving system”—must be equal to the length
l of the stationary rod.
The length to be discovered by the operation (b) we will call “the length of the (moving)
rod in the stationary system.” This we shall determine on the basis of our two principles, and
we shall find that it differs from l.
Current kinematics tacitly assumes that the lengths determined by these two operations are
precisely equal, or in other words, that a moving rigid body at the epoch t may in geometrical
respects be perfectly represented by the same body at rest in a definite position.
We imagine further that at the two ends A and B of the rod, clocks are placed which synchronize with the clocks of the stationary system, that is to say that their indications correspond
at any instant to the “time of the stationary system” at the places where they happen to be.
These clocks are therefore “synchronous in the stationary system.”
We imagine further that with each clock there is a moving observer, and that these observers
apply to both clocks the criterion established in § 1 for the synchronization of two clocks. Let
a ray of light depart from A at the time23 tA , let it be reflected at B at the time tB , and reach
A again at the time t0A . Taking into consideration the principle of the constancy of the velocity
of light we find that
tB − tA =
and t0A − tB =
where rAB denotes the length of the moving rod—measured in the stationary system. Observers
moving with the moving rod would thus find that the two clocks were not synchronous, while
observers in the stationary system would declare the clocks to be synchronous.
So we see that we cannot attach any absolute signification to the concept of simultaneity, but
that two events which, viewed from a system of coordinates, are simultaneous, can no longer be
looked upon as simultaneous events when envisaged from a system which is in motion relative
to that system.
§ 3. Theory of the Transformation of coordinates and Times from a Stationary System to
another System in Uniform Motion of Translation Relative to the Former
Let us in “stationary” space take two systems of coordinates, i.e., two systems, each of three
rigid material lines, perpendicular to one another, and issuing from a point. Let the axes of X
of the two systems coincide, and their axes of Y and Z respectively be parallel. Let each system
be provided with a rigid measuring-rod and a number of clocks, and let the two measuring-rods,
and likewise all the clocks of the two systems, be in all respects alike.
Now to the origin of one of the two systems (k) let a constant velocity v be imparted in
the direction of the increasing x of the other stationary system (K), and let this velocity be
communicated to the axes of the coordinates, the relevant measuring-rod, and the clocks. To
“Time” here denotes “time of the stationary system” and also “position of hands of the moving clock situated
at the place under discussion.”—AS
any time of the stationary system K there then will correspond a definite position of the axes
of the moving system, and from reasons of symmetry we are entitled to assume that the motion
of k may be such that the axes of the moving system are at the time t (this “t” always denotes
a time of the stationary system) parallel to the axes of the stationary system.
We now imagine space to be measured from the stationary system K by means of the
stationary measuring-rod, and also from the moving system k by means of the measuring-rod
moving with it; and that we thus obtain the coordinates x, y, z, and ξ, η, ζ respectively. Further,
let the time t of the stationary system be determined for all points thereof at which there are
clocks by means of light signals in the manner indicated in § 1; similarly let the time τ of the
moving system be determined for all points of the moving system at which there are clocks at
rest relative to that system by applying the method, given in § 1, of light signals between the
points at which the latter clocks are located.
To any system of values x, y, z, t, which completely defines the place and time of an event
in the stationary system, there belongs a system of values ξ, η, ζ, τ , determining that event
relative to the system k, and our task is now to find the system of equations connecting these
In the first place it is clear that the equations must be linear on account of the properties of
homogeneity which we attribute to space and time.
If we place x0 = x − vt, it is clear that a point at rest in the system k must have a system of
values x0 , y, z, independent of time. We first define τ as a function of x0 , y, z, and t. To do this
we have to express in equations that τ is nothing else than the summary of the data of clocks
at rest in system k, which have been synchronized according to the rule given in § 1.
From the origin of system k let a ray be emitted at the time τ0 along the X-axis to x0 , and at
the time τ1 be reflected thence to the origin of the coordinates, arriving there at the time τ2 ; we
then must have 21 (τ0 + τ2 ) = τ1 , or, by inserting the arguments of the function τ and applying
the principle of the constancy of the velocity of light in the stationary system:—
τ (0, 0, 0, t) + τ 0, 0, 0, t +
= τ x , 0, 0, t +
c−v c+v
Hence, if x0 be chosen infinitesimally small,
c−v c+v
1 ∂τ
c − v ∂t
+ 2
= 0.
c − v ∂t
It is to be noted that instead of the origin of the coordinates we might have chosen any other
point for the point of origin of the ray, and the equation just obtained is therefore valid for all
values of x0 , y, z.
An analogous consideration—applied to the axes of Y and Z—it being borne in mind that
light is always
propagated along these axes, when viewed from the stationary system, with the
velocity c − v 2 gives us
Chapter 9
Gravitational waves
= 0,
= 0.
Since τ is a linear function, it follows from these equations that
τ =a t−
c2 − v 2
where a is a function φ(v) at present unknown, and where for brevity it is assumed that at the
origin of k, τ = 0, when t = 0.
With the help of this result we easily determine the quantities ξ, η, ζ by expressing in
equations that light (as required by the principle of the constancy of the velocity of light, in
combination with the principle of relativity) is also propagated with velocity c when measured
in the moving system. For a ray of light emitted at the time τ = 0 in the direction of the
increasing ξ
ξ = cτ or ξ = ac t −
x .
c2 − v 2
But the ray moves relative to the initial point of k, when measured in the stationary system,
with the velocity c − v, so that
= t.
If we insert this value of t in the equation for ξ, we obtain
x0 .
c2 − v 2
In an analogous manner we find, by considering rays moving along the two other axes, that
η = cτ = ac t −
c − v2
= t, x0 = 0.
c2 − v 2
η = a√
y and ζ = a √
c2 − v 2
c2 − v 2
Substituting for x0 its value, we obtain
= φ(v)β(t − vx/c2 ),
ξ = φ(v)β(x − vt),
η = φ(v)y,
ζ = φ(v)z,
1 − v 2 /c2
and φ is an as yet unknown function of v. If no assumption whatever be made as to the initial
position of the moving system and as to the zero point of τ , an additive constant is to be placed
on the right side of each of these equations.
We now have to prove that any ray of light, measured in the moving system, is propagated
with the velocity c, if, as we have assumed, this is the case in the stationary system; for we
have not as yet furnished the proof that the principle of the constancy of the velocity of light is
compatible with the principle of relativity.
At the time t = τ = 0, when the origin of the coordinates is common to the two systems,
let a spherical wave be emitted therefrom, and be propagated with the velocity c in system K.
If (x, y, z) be a point just attained by this wave, then
x2 + y 2 + z 2 = c2 t2 .
Transforming this equation with the aid of our equations of transformation we obtain after
a simple calculation
ξ 2 + η 2 + ζ 2 = c2 τ 2 .
The wave under consideration is therefore no less a spherical wave with velocity of propagation c when viewed in the moving system. This shows that our two fundamental principles are
In the equations of transformation which have been developed there enters an unknown
function φ of v, which we will now determine.
For this purpose we introduce a third system of coordinates K0 , which relative to the system
k is in a state of parallel translatory motion parallel to the axis of Ξ,25 such that the origin of
The equations of the Lorentz transformation may be more simply deduced directly from the condition that
in virtue of those equations the relation x2 + y 2 + z 2 = c2 t2 shall have as its consequence the second relation
ξ 2 + η 2 + ζ 2 = c2 τ 2 .—AS
In Einstein’s original paper, the symbols (Ξ, H, Z) for the coordinates of the moving system k were introduced
without explicitly defining them. In the 1923 English translation, (X, Y, Z) were used, creating an ambiguity
between X coordinates in the fixed system K and the parallel axis in moving system k. Here and in subsequent
references we use Ξ when referring to the axis of system k along which the system is translating with respect to
K. In addition, the reference to system K0 later in this sentence was incorrectly given as “k” in the 1923 English
Chapter 9
Gravitational waves
coordinates of system K0 moves with velocity −v on the axis of Ξ. At the time t = 0 let all
three origins coincide, and when t = x = y = z = 0 let the time t0 of the system K0 be zero. We
call the coordinates, measured in the system K0 , x0 , y 0 , z 0 , and by a twofold application of our
equations of transformation we obtain
φ(−v)β(−v)(τ + vξ/c2 )
φ(−v)β(−v)(ξ + vτ )
Since the relations between x0 , y 0 , z 0 and x, y, z do not contain the time t, the systems K
and K0 are at rest with respect to one another, and it is clear that the transformation from K
to K0 must be the identical transformation. Thus
φ(v)φ(−v) = 1.
We now inquire into the signification of φ(v). We give our attention to that part of the axis of
Y of system k which lies between ξ = 0, η = 0, ζ = 0 and ξ = 0, η = l, ζ = 0. This part of the
axis of Y is a rod moving perpendicularly to its axis with velocity v relative to system K. Its
ends possess in K the coordinates
x1 = vt, y1 =
, z1 = 0
x2 = vt, y2 = 0, z2 = 0.
The length of the rod measured in K is therefore l/φ(v); and this gives us the meaning of the
function φ(v). From reasons of symmetry it is now evident that the length of a given rod moving
perpendicularly to its axis, measured in the stationary system, must depend only on the velocity
and not on the direction and the sense of the motion. The length of the moving rod measured in
the stationary system does not change, therefore, if v and −v are interchanged. Hence follows
that l/φ(v) = l/φ(−v), or
φ(v) = φ(−v).
It follows from this relation and the one previously found that φ(v) = 1, so that the transformation equations which have been found become
= β(t − vx/c2 ),
ξ = β(x − vt),
η = y,
ζ = z,
β = 1/
1 − v 2 /c2 .
§ 4. Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and
Moving Clocks
We envisage a rigid sphere26 of radius R, at rest relative to the moving system k, and with
its centre at the origin of coordinates of k. The equation of the surface of this sphere moving
relative to the system K with velocity v is
ξ 2 + η 2 + ζ 2 = R2 .
The equation of this surface expressed in x, y, z at the time t = 0 is
+ y 2 + z 2 = R2 .
( 1 − v 2 /c2 )2
A rigid body which, measured in a state of rest, has the form of a sphere, therefore has in a
state of motion—viewed from the stationary system—the form of an ellipsoid of revolution with
the axes
R 1 − v 2 /c2 , R, R.
Thus, whereas the Y and Z dimensions of the sphere (and therefore of every rigid body of no
matter what form)
p do not appear modified by the motion, the X dimension appears shortened
in the ratio 1 : 1 − v 2 /c2 , i.e., the greater the value of v, the greater the shortening. For v = c
all moving objects—viewed from the “stationary” system—shrivel up into plane figures.27 For
velocities greater than that of light our deliberations become meaningless; we shall, however,
find in what follows, that the velocity of light in our theory plays the part, physically, of an
infinitely great velocity.
It is clear that the same results hold good of bodies at rest in the “stationary” system,
viewed from a system in uniform motion.
Further, we imagine one of the clocks which are qualified to mark the time t when at rest
relative to the stationary system, and the time τ when at rest relative to the moving system,
to be located at the origin of the coordinates of k, and so adjusted that it marks the time τ .
What is the rate of this clock, when viewed from the stationary system?
Between the quantities x, t, and τ , which refer to the position of the clock, we have, evidently,
x = vt and
(t − vx/c2 ).
1 − v 2 /c2
That is, a body possessing spherical form when examined at rest.—AS
In the 1923 English translation, this phrase was erroneously translated as “plain figures”. I have used the
correct “plane figures” in this edition.—JW
Chapter 9
Gravitational waves
τ = t 1 − v 2 /c2 = t − (1 − 1 − v 2 /c2 )t
p it follows that the time marked by the clock (viewed in the stationary system) is slow by
1 − 1 − v 2 /c2 seconds per second, or—neglecting magnitudes of fourth and higher order—by
1 2 2
2 v /c .
From this there ensues the following peculiar consequence. If at the points A and B of K
there are stationary clocks which, viewed in the stationary system, are synchronous; and if the
clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the
two clocks no longer synchronize, but the clock moved from A to B lags behind the other which
has remained at B by 12 tv 2 /c2 (up to magnitudes of fourth and higher order), t being the time
occupied in the journey from A to B.
It is at once apparent that this result still holds good if the clock moves from A to B in any
polygonal line, and also when the points A and B coincide.
If we assume that the result proved for a polygonal line is also valid for a continuously curved
line, we arrive at this result: If one of two synchronous clocks at A is moved in a closed curve
with constant velocity until it returns to A, the journey lasting t seconds, then by the clock
which has remained at rest the travelled clock on its arrival at A will be 12 tv 2 /c2 second slow.
Thence we conclude that a spring-clock at the equator must go more slowly, by a very small
amount, than a precisely similar clock situated at one of the poles under otherwise identical
§ 5. The Composition of Velocities
In the system k moving along the axis of X of the system K with velocity v, let a point move
in accordance with the equations
ξ = wξ τ , η = wη τ , ζ = 0,
where wξ and wη denote constants.
Required: the motion of the point relative to the system K. If with the help of the equations
of transformation developed in § 3 we introduce the quantities x, y, z, t into the equations of
motion of the point, we obtain
wξ + v
1 + vwξ /c2
1 − v 2 /c2
wη t,
1 + vwξ /c2
z = 0.
Einstein specifies a spring-clock (“unruhuhr”) because the effective gravitational field is weaker at the equator
than at the poles, so a pendulum clock at the equator would run more slowly by about two parts per thousand
than one at the north pole, for nonrelativistic reasons. This would completely mask any relativistic effect, which
he expected to be on the order of v 2 /c2 , or about 10−13 . In any case, it later turned out that Einstein was
mistaken about this example. There is also a gravitational time dilation that cancels the kinematic effect. See
example 11, p. 58. The two clocks would actually agree.—BC
Thus the law of the parallelogram of velocities is valid according to our theory only to a first
approximation. We set29
w2 = wξ2 + wη2 ,
a = tan−1 wη /wξ ,
a is then to be looked upon as the angle between the velocities v and w. After a simple calculation
we obtain
(v 2 + w2 + 2vw cos a) − (vw sin a/c)2
V =
1 + vw cos a/c2
It is worthy of remark that v and w enter into the expression for the resultant velocity in a
symmetrical manner. If w also has the direction of the axis of X, we get
V =
1 + vw/c2
It follows from this equation that from a composition of two velocities which are less than c,
there always results a velocity less than c. For if we set v = c − κ, w = c − λ, κ and λ being
positive and less than c, then
V =c
2c − κ − λ
< c.
2c − κ − λ + κλ/c
It follows, further, that the velocity of light c cannot be altered by composition with a
velocity less than that of light. For this case we obtain
V =
= c.
1 + w/c
We might also have obtained the formula for V, for the case when v and w have the same
direction, by compounding two transformations in accordance with § 3. If in addition to the
systems K and k figuring in § 3 we introduce still another system of coordinates k 0 moving
parallel to k, its initial point moving on the axis of Ξ30 with the velocity w, we obtain equations
between the quantities x, y, z, t and the corresponding quantities of k 0 , which differ from the
equations found in § 3 only in that the place of “v” is taken by the quantity
1 + vw/c2
from which we see that such parallel transformations—necessarily—form a group.
This equation was incorrectly given in Einstein’s original paper and the 1923 English translation as a =
tan−1 wy /wx .—JW
“X” in the 1923 English translation.—JW
Chapter 9
Gravitational waves
We have now deduced the requisite laws of the theory of kinematics corresponding to our
two principles, and we proceed to show their application to electrodynamics.31
The remainder of the paper is not given here, but can be obtained from John Walker’s web site at www.—BC
Does the inertia of a body depend upon its energy content?
A. Einstein, Annalen der Physik. 18 (1905) 639.
The results of the previous investigation lead to a very interesting conclusion, which is here
to be deduced.
I based that investigation on the Maxwell-Hertz equations for empty space, together with
the Maxwellian expression for the electromagnetic energy of space, and in addition the principle
The laws by which the states of physical systems alter are independent of the alternative, to
which of two systems of coordinates, in uniform motion of parallel translation relative to each
other, these alterations of state are referred (principle of relativity).
With these principles32 as my basis I deduced inter alia the following result (§ 8):—
Let a system of plane waves of light, referred to the system of coordinates (x, y, z), possess
the energy l; let the direction of the ray (the wave-normal) make an angle φ with the axis of x
of the system. If we introduce a new system of coordinates (ξ, η, ζ) moving in uniform parallel
translation with respect to the system (x, y, z), and having its origin of coordinates in motion
along the axis of x with the velocity v, then this quantity of light—measured in the system
(ξ, η, ζ)—possesses the energy33
1 − v cosφ
l∗ = l p c
1 − v 2 /c2
where c denotes the velocity of light. We shall make use of this result in what follows.
Let there be a stationary body in the system (x, y, z), and let its energy—referred to the
system (x, y, z) be E0 . Let the energy of the body relative to the system (ξ, η, ζ) moving as
above with the velocity v, be H0 .
Let this body send out, in a direction making an angle φ with the axis of x, plane waves
of light, of energy 12 L measured relative to (x, y, z), and simultaneously an equal quantity of
light in the opposite direction. Meanwhile the body remains at rest with respect to the system
(x, y, z). The principle of energy must apply to this process, and in fact (by the principle of
relativity) with respect to both systems of coordinates. If we call the energy of the body after
the emission of light E1 or H1 respectively, measured relative to the system (x, y, z) or (ξ, η, ζ)
respectively, then by employing the relation given above we obtain
E0 = E1 + L + L,
1 1 − vc cosφ
1 1 + v cosφ
H0 = H 1 + L p
+ Lp c
1 − v 2 /c2 2
1 − v 2 /c2
= H1 + p
1 − v 2 /c2
By subtraction we obtain from these equations
The principle of the constancy of the velocity of light is of course contained in Maxwell’s equations.—AS
See homework problem 11, p. 156.—BC
Chapter 9
Gravitational waves
H0 − E0 − (H1 − E1 ) = L
−1 .
1 − v 2 /c2
The two differences of the form H − E occurring in this expression have simple physical significations. H and E are energy values of the same body referred to two systems of coordinates
which are in motion relative to each other, the body being at rest in one of the two systems
(system (x, y, z)). Thus it is clear that the difference H − E can differ from the kinetic energy
K of the body, with respect to the other system (ξ, η, ζ), only by an additive constant C, which
depends on the choice of the arbitrary additive constants34 of the energies H and E. Thus we
may place
H0 − E0 = K0 + C,
H1 − E1 = K1 + C,
since C does not change during the emission of light. So we have
K0 − K1 = L
−1 .
1 − v 2 /c2
The kinetic energy of the body with respect to (ξ, η, ζ) diminishes as a result of the emission
of light, and the amount of diminution is independent of the properties of the body. Moreover,
the difference K0 − K1 , like the kinetic energy of the electron (§ 10), depends on the velocity.
Neglecting magnitudes of fourth and higher orders35 we may place
K0 − K1 =
1L 2
v .
2 c2
From this equation it directly follows36 that:—
If a body gives off the energy L in the form of radiation, its mass diminishes by L/c2 . The
fact that the energy withdrawn from the body becomes energy of radiation evidently makes no
difference, so that we are led to the more general conclusion that
The mass of a body is a measure of its energy-content; if the energy changes by L, the mass
changes in the same sense by L/9 × 1020 , the energy being measured in ergs, and the mass in
It is not impossible that with bodies whose energy-content is variable to a high degree (e.g.
with radium salts) the theory may be successfully put to the test.
If the theory corresponds to the facts, radiation conveys inertia between the emitting and
absorbing bodies.
A potential energy U is only defined up to an additive constant. If, for example, U depends on the distance
between particles, and the distance undergoes a Lorentz contraction, there is no reason to imagine that the
constant will stay the same.—BC
The purpose of making the approximation is to show that under realistic lab conditions, the effect exactly
mimics a change in Newtonian mass.
The object has the same velocity v before and after emission of the light, so this reduction in kinetic energy
has to be attributed to a change in mass.—BC
The foundation of the general theory of relativity
A. Einstein, Annalen der Physik 49 (1916) 769.
[A one-page introduction relating to history and personalities is omitted.—BC]
§1. Observations on the Special Theory of Relativity
The special theory of relativity is based on the following postulate, which is also satisfied by
the mechanics of Galileo and Newton. If a system of coordinates K is chosen so that, in relation
to it, physical laws hold good in their simplest form, the same laws also hold good in relation to
any other system of coordinates K 0 moving in uniform translation relative to K. This postulate
we call the “special principle of relativity.” The word “special” is meant to intimate that the
principle is restricted to the case when K 0 has a motion of uniform translation37 relative to K,
but that the equivalence of K 0 and K does not extend to the case of non-uniform motion of K 0
relative to K.
Thus the special theory of relativity does not depart from classical mechanics through the
postulate of relativity, but through the postulate of the constancy of the velocity of light in
vacuo, from which, in combination with the special principle of relativity, there follow, in the
well-known way, the relativity of simultaneity, the Lorentzian transformation and the related
laws for the behaviour of moving bodies and clocks.
The modification to which the special theory of relativity has subjected the theory of space
and time is indeed far-reaching, but one important point has remained unaffected. For the laws of
geometry, even according to the special theory of relativity, are to be interpreted directly as laws
relating to the possible relative positions of solid bodies at rest; and, in a more general way, the
laws of kinematics are to be interpreted as laws which describe the relations of measuring bodies
and clocks. To two selected material points of a stationary rigid body there always corresponds
a distance of quite definite length, which is independent of the locality and orientation of the
body, and is also independent of the time. To two selected positions of the hands of a clock at
rest relative to the privileged system of reference there always corresponds an interval of time
of a definite length, which is independent of place and time. We shall soon see that the general
theory of relativity cannot adhere to this simple physical interpretation of space and time.38
§2. The Need for an Extension of the Postulate of Relativity
In classical mechanics, and no less in the special theory of relativity, there is an inherent
epistemological defect which was, perhaps for the first time, clearly pointed out by Ernst Mach.
We will elucidate it by the following example:39 — Two fluid bodies of the same size and nature
Here Einstein defines the distinction between special and general relativity according to whether accelerated
frames of reference are allowed. The modern tendency is to pose this distinction in terms of flat versus curved
spacetime, so that accelerated frames of reference in flat spacetime are considered to be part of special relativity.
None of this has anything to do with the ability to describe accelerated objects. For example, special relativity is
perfectly capable of describing the twin paradox.—BC
Einstein is just starting to lay out his argument, and has not yet made clear in what sense these statements
about location-independence of clocks and rulers could be empirically tested. It becomes more clear later that
he means something like this. We could try to fill spacetime with a lattice of clocks and rulers, to synchronize
the clocks, and to construct the lattice so that it consisted of right angles and equal-length line segments. This
succeeds in special relativity, so that the geometry of spacetime is compatible with frames of reference that split
up spacetime into 3+1 dimensions, where the three dimensions are Euclidean. The same prescription fails in
general relativity.—BC
This example was described on p. 115.—BC
Chapter 9
Gravitational waves
hover freely in space at so great a distance from each other and from all other masses that only
those gravitational forces need be taken into account which arise from the interaction of different
parts of the same body. Let the distance between the two bodies be invariable, and in neither
of the bodies let there be any relative movements of the parts with respect to one another.
But let either mass, as judged by an observer at rest relative to the other mass, rotate
with constant angular velocity about the line joining the masses. This is a verifiable relative
motion of the two bodies. Now let us imagine that each of the bodies has been surveyed by
means of measuring instruments at rest relative to itself, and let the surface of S1 prove to be a
sphere, and that of S2 an ellipsoid of revolution. Thereupon we put the question — What is the
reason for this difference in the two bodies? No answer can be admitted as epistemologically
satisfactory,40 unless the reason given is an observable fact of experience. The law of causality
has not the significance of a statement as to the world of experience, except when observable
facts ultimately appear as causes and effects.
Newtonian mechanics does not give a satisfactory answer to this question. It pronounces as
follows: — The laws of mechanics apply to the space R1 , in respect to which the body S1 is at
rest, but not to the space R2 , in respect to which the body S2 is at rest. But the privileged
space R1 of Galileo, thus introduced, is a merely factitious 41 cause, and not a thing that can be
observed. It is therefore clear that Newton’s mechanics does not really satisfy the requirement
of causality in the case under consideration but only apparently does so, since it makes the
factitious cause R1 responsible for the observable difference in the bodies S1 and S2 .
The only satisfactory answer must be that the physical system consisting of S1 and S2
reveals within itself no imaginable cause to which the differing behaviour of S1 and S2 can be
referred. The cause must therefore lie outside this system. We have to take it that the general
laws of motion, which in particular determine the shapes of S1 and S2 , must be such that the
mechanical behaviour of S1 and S2 is partly conditioned in quite essential respects, by distant
masses which we have not included in the system under consideration. These distant masses
and their motions relative to S1 and S2 must then be regarded as the seat of the causes (which
must be susceptible to observation) of the different behaviour of our two bodies S1 and S2 . They
take over the rôle of the factitious cause R1 . Of all imaginable spaces R1 , R2 , etc., in any kind
of motion relative to one another there is none which we may look upon as privileged a priori
without reviving the above-mentioned epistemological objection. The laws of physics must be
of such a nature that they apply to systems reference in any kind of motion.42 Along this road
we arrive at an extension at the postulate of relativity.
In addition to this weighty argument from the theory of knowledge, there is a well-known
physical fact which favours an extension of the theory of relativity. Let K be a Galilean system
of reference, i.e., a system relative to which (at least in the four-dimensional region under
consideration) a mass, sufficiently distant from other masses, is moving with uniform motion
in a straight line. Let K 0 be a second system of reference which is moving relative to K in
uniformly accelerated translation. Then, relative to K 0 , a mass sufficiently distant from other
Of course an answer may be satisfactory from the point of view of epistemology, and yet be unsound hysically,
if it is in conflict with other experiences. —AS
i.e., artificial —BC
At this time, Einstein had high hopes that his theory would be fully Machian. He was already aware of the
Schwarzschild solution (he refers to it near the end of the paper), which offended his Machian sensibilities because
it imputed properties to spacetime in a universe containing only a single point-mass. In the present example of
the bodies S1 and S2 , general relativity actually turns out to give the non-Machian result which Einstein here
says would be unsatisfactory.—BC
masses would have an accelerated motion such that its acceleration and direction of acceleration
are independent of the material composition and physical state of the mass.
Does this permit an observer at rest relative to K 0 to infer that he is on a “really” accelerated
system of reference? The answer is in the negative; for the above-mentioned relation of freely
movable masses to K 0 may be interpreted equally well in the following way. The system of
reference K 0 is unaccelerated, but the space-time territory in question is under the sway of a
gravitational field, which generates the accelerated motion of the bodies relative to K 0 .
This view is made possible for us by the teaching of experience as to the existence of a field
of force, namely, the gravitational field, which possesses the remarkable property of imparting
the same acceleration to all bodies.43 The mechanical behaviour of bodies relative to K 0 is
the same as presents itself to experience in the case of systems which we are wont to regard
as “stationary” or as “privileged.” Therefore, from the physical standpoint, the assumption
readily suggests itself that the systems K and K 0 may both with equal right be looked upon
as “stationary” that is to say, they have an equal title as systems of reference for the physical
description of phenomena.
It will be seen from these reflections that in pursuing the general theory of relativity we shall
be led to a theory of gravitation, since we are able to “produce” a gravitational field merely by
changing the system of coordinates. It will also be obvious that the principle of the constancy
of the velocity of light in vacuo must be modified, since we easily recognize that the path of
a ray of light with respect to K 0 must in general be curvilinear, if with respect to K light is
propagated in a straight line with a definite constant velocity.
§3. The Space-Time Continuum. Requirement of General Covariance for the Equations
Expressing General Laws of Nature
In classical mechanics, as well as in the special theory of relativity, the coordinates of space
and time have a direct physical meaning. To say that a point-event has the X1 coordinate x1
means that the projection of the point-event on the axis of X1 , determined by rigid rods and
in accordance with the rules of Euclidean geometry, is obtained by measuring off a given rod
(the unit of length) x1 times from the origin of coordinates along the axis of X1 . To say that
a point-event has the X4 coordinate x4 = t, means that a standard clock, made to measure
time in a definite unit period, and which is stationary relative to the system of coordinates and
practically coincident in space with the point-event,44 will have measured off x4 = t periods at
the occurrence of the event.
This view of space and time has always been in the minds of physicists, even if, as a rule, they
have been unconscious of it. This is clear from the part which these concepts play in physical
measurements; it must also have underlain the reader’s reflections on the preceding paragraph
for him to connect any meaning with what he there read. But we shall now show that we must
put it aside and replace it by a more general view, in order to be able to carry through the
postulate of general relativity, if the special theory of relativity applies to the special case of the
absence of a gravitational field.
In a space which is free of gravitational fields we introduce a Galilean system of reference
Eötvös has proved experimentally that the gravitational field has this property in great accuracy.—AS
We assume the possibility of verifying “simultaneity” for events immediately proximate in space, or — to
speak more precisely — for immediate proximity or coincidence in space-time, without giving a definition of this
fundamental concept.—AS
Chapter 9
Gravitational waves
K (x, y, z, t), and also a system of coordinates K 0 (x0 , y 0 , z 0 , t0 ) in uniform rotation45 relative to
K. Let the origins of both systems, as well as their axes of Z, permanently coincide. We shall
show that for a space-time measurement in the system K 0 the above definition of the physical
meaning of lengths and times cannot be maintained. For reasons of symmetry it is clear that a
circle around the origin in the X, Y plane of K may at the same time be regarded as a circle
in the X 0 , Y 0 plane of K 0 . We suppose that the circumference and diameter of this circle have
been measured with a unit measure infinitely small compared with the radius, and that we have
the quotient of the two results. If this experiment were performed with a measuring-rod46 at
rest relative to the Galilean system K, the quotient would be π. With a measuring-rod at rest
relative to K 0 , the quotient would be greater than π. This is readily understood if we envisage
the whole process of measuring from the “stationary” system K, and take into consideration
that the measuring-rod applied to the periphery undergoes a Lorentzian contraction, while the
one applied along the radius does not.47 Hence Euclidean geometry does not apply to K 0 .
The notion of coordinates defined above, which presupposes the validity of Euclidean geometry,
therefore breaks down in relation to the system K 0 . So, too, we are unable to introduce a
time corresponding to physical requirements in K 0 , indicated by clocks at rest relative to K 0 .
To convince ourselves of this impossibility, let us imagine two clocks of identical constitution
placed, one at the origin of coordinates, and the other at the circumference of the circle, and
both envisaged from the “stationary” system K. By a familiar result of the special theory of
relativity, the clock at the circumference — judged from K — goes more slowly than the other,
because the former is in motion and the latter at rest. An observer at the common origin
of coordinates, capable of observing the clock at the circumference by means of light, would
therefore see it lagging behind the clock beside him. As he will not make up his mind to let the
velocity of light along the path in question depend explicitly on the time, he will interpret his
observations as showing that the clock at the circumference “really” goes more slowly than the
clock at the origin. So he will be obliged to define time in such a way that the rate of a clock
depends upon where the clock may be.
We therefore reach this result: — In the general theory of relativity, space and time cannot
be defined in such a way that differences of the spatial coordinates can be directly measured by
the unit measuring-rod, or differences in the time coordinate by a standard clock.
The method hitherto employed for laying coordinates into the space-time continuum in a
definite manner thus breaks down, and there seems to be no other way which would allow us
to adapt systems of coordinates to the four-dimensional universe so that we might expect from
their application a particularly simple formulation of the laws of nature. So there is nothing for
it but to regard all imaginable systems of coordinates, on principle, as equally suitable for the
description of nature.48 This comes to requiring that: —
This example of a rotating frame of reference was discussed on p. 108.—BC
Einstein implicitly assumes that the measuring rods are perfectly rigid, but it is not obvious that this is
possible. This issue is discussed on p. 113.—BC
As described on p. 109, Ehrenfest originally imagined that the circumference of the disk would be reduced
by its rotation. His argument was incorrect, because it assumed the ability to start the disk rotating when it
had originally been at rest. The present paper marks the first time that Einstein asserted the opposite, that the
circumference is increased.—BC
This is a conceptual leap, not a direct inference from the argument about the rotating frame. Einstein started
thinking about this argument in 1912, and concluded from it that he should base a theory of gravity on nonEuclidean geometry. Influenced by Levi-Civita, he tried to carry out this project in a coordinate-independent way,
but he failed at first, and for a while explored a theory that was not coordinate-independent. Only later did he
return to coordinate-independence. It should be clear, then, that the link between the rotating-frame argument
and coordinate-independence was not as clearcut as Einstein makes out here, since he himself lost faith in it for
The general laws of nature are to be expressed by equations which hold good for all the
systems of coordinates, that is, are covariant with respect to any substitutions whatever (generally
It is clear that a physical theory which satisfies this postulate will also be suitable for the
general postulate of relativity.50 For the sum of all substitutions in any case includes those
which correspond to all relative motions of three-dimensional systems of coordinates. That
this requirement of general covariance, which takes away from space and time the last remnant
of physical objectivity,51 is a natural one, will be seen from the following reflection. All our
space-time verifications invariably amount to a determination of space-time coincidences.52 If,
for example, events consisted merely in the motion of material points, then ultimately nothing
would be observable but the meetings of two or more of these points. Moreover, the results
of our measurings are nothing but verifications of such meetings of the material points of our
measuring instruments with other material points, coincidences between the hands of a clock
and points on the clock-dial, and observed point-events happening at the same place at the same
The introduction of a system of reference serves no other purpose than to facilitate the
description of the totality of such coincidences. We allot to the universe four space-time variables
x1 , x2 , x3 , x4 in such a way that for every point-event there is a corresponding system of values
of the variables x1 . . . x4 . To two coincident point-events there corresponds one system of values
of the variables x1 . . . x4 , i.e., coincidence is characterized by the identity of the coordinates.
If, in place of the variables x1 . . . x4 , we introduce functions of them, x01 , x02 , x03 , x04 , as a new
system of coordinates, so that the systems of values are made to correspond to one another
without ambiguity, the equality of all four coordinates in the new system will also serve as an
expression for the space-time coincidence of the two point-events. As all our physical experience
can be ultimately reduced to such coincidences, there is no immediate reason for preferring
certain systems of coordinates to others, that is to say, we arrive at the requirement of general
a while.—BC
In this book I’ve used the more transparent terminology “coordinate independence” rather than “general
For more on this point, see p. 117.—BC
This is an extreme interpretation of general covariance, and one that Einstein himself didn’t hew closely to
later on. He presented an almost diametrically opposed interpretation in a philosophical paper, “On the aether,”
Schweizerische naturforschende Gesellschaft 105 (1924) 85.—BC
i.e., what this book refers to as incidence measurements (p. 96)—BC
Chapter 9
Gravitational waves
Appendix 2: Hints and solutions
Hints for Chapter 1
Page 38, problem 5: Apply the equivalence principle.
Solutions to Selected Homework Problems
Solutions for Chapter 1
Page 38, problem 3:
Pick two points P1 and P2. By O2, there is another point P3 that is distinct from P1 and
P2. (Recall that the notation [ABC] was defined so that all three points must be distinct.)
Applying O2 again, there must be a further point P4 out beyond P3, and by O3 this can’t be
the same as P1. Continuing in this way, we can produce as many points as there are integers.
Page 38, problem 4:
(a) If the violation of (1) is tiny, then of course Kip won’t really have any practical way to
violate (2), but the idea here is just to illustrate the idea, so to make things easy, let’s imagine an
unrealistically large violation of (1). Suppose that neutrons have about the same inertial mass as
protons, but zero gravitational mass, in extreme violation of (1). This implies that neutron-rich
elements like uranium would have a much lower gravitational acceleration on earth than ones
like oxygen that are roughly 50-50 mixtures of neutrons and protons. Let’s also simplify by
making a second unrealistically extreme assumption: let’s say that Kip has a keychain in his
pocket made of neutronium, a substance composed of pure neutrons. On earth, the keychain
hovers in mid-air. Now he can release his keychain in the prison cell. If he’s on a planet, it
will hover. If he’s in an accelerating spaceship, then the keychain will follow Newton’s first law
(its tendency to do so being measured by its nonzero inertial mass), while the deck of the ship
accelerates up to hit it.
(b) It violates O1. O1 says that objects prepared in identical inertial states (as defined by
two successive events in their motion) are predicted to have identical motion in the future. This
fails in the case where Kip releases the neutronium keychain side by side with a penny.
Page 38, problem 5: By the equivalence principle, we can adopt a frame tied to the tossed
clock, B, and in this frame there is no gravitational field. We see a desk and clock A go by.
The desk applies a force to clock A, decelerating it and then reaccelerating it so that it comes
back. We’ve already established that the effect of motion is to slow down time, so clock A reads
a smaller time interval.
Page 38, problem 6: (a) Generalizing the expression gy/c2 for the fractional time dilation to
the case of a nonuniform field, we find Φ/c2 , where Φ is the Newtonian gravitational potential,
i.e., the gravitational energy per unit mass. The shell theorem gives a gravitational field g =
M r/R3 . Integration shows Φ = M r2 /2R3 . The difference in the gravitational potential between
these two points, divided by c2 , is Φ/c2 = M/2c2 R, which comes out to be 3.5×10−10 . This is the
fractional difference in clock rates. (b) The probe’s velocity is on the same order of magnitude
as escape velocity from the inner solar system, so very roughly we can say that v ∼ |∆Φ|,
where ∆Φ is the difference between
pthe gravitational potential at the earth’s orbit and infinity.
This gives a Doppler shift v/c ∼ |∆Φ|/c. We saw in part a that the gravitational Doppler
shift was ∆Φ/c2 , which is the square of this quantity, and therefore much smaller.
Page 39, problem 7: (a) In case 1 there is no source of energy, so the particle cannot radiate.
In case 2-4, the particle radiates, because there are sources of energy (loss of gravitational energy
in 2 and 3, the rocket fuel in 4).
(b) In 1, Newton says the object is subject to zero net force, so its motion is inertial. In
2-4, he says the object is subject to a nonvanishing net force, so its motion is noninertial. This
matches up with the results of the energy analysis.
(c) The equivalence principle, as discussed on page 39, is vague, and is particularly difficult
to apply successful and unambiguously to situations involving electrically charged objects, due
to the difficulty of defining locality. Applying the equivalence principle in the most naive way,
we predict that there can be no radiation in cases 2 and 3 (because the object is following a
geodesic, minding its own business). In case 4, everyone agrees that there will be radiation
observable back on earth (although it’s possible that it would not be observable to an observer
momentarily matching velocities with the rocket). The naive equivalence principle says that 1
and 4 must give the same result, so we should have radiation in 1 as well. These predictions are
wrong in two out of the four equations, which tells us that we had better either not apply the
equivalence principle to charged objects, or not apply it in such a naive way.
Page 39, problem 8:
(a) The dominant form of radiation from the orbiting charge will be the lowest-order nonvanishing multipole, which in this case is a dipole. The power radiated from a dipole scales like
d2 ω 4 , where d is the dipole moment. For an orbit of radius r, this becomes q 2 r2 ω 4 . To find
the reaction force on the charged particle, we can use the relation p = E/c for electromagnetic
waves (section 1.5.5), which tells us that the force is equal to the power, up to a proportionality
constant c. Therefore ar ∝ q 2 r2 ω 4 /m. The gravitational acceleration is ag = ω 2 r, so we have
ar /ag ∝ (q 2 /m)ω 2 r, or ar /ag ∝ (q 2 /m)ag , where the ag on the right can be taken as an orbital
parameter, and for a low-earth orbit is very nearly equal to the usual acceleration of gravity at
the earth’s surface.
(b) In SI units, ar /ag ∼ (k/c4 )(q 2 /m)ag , where k is the Coulomb constant.
(c) The result is 10−34 . If one tried to do this experiment in reality, the effect would be
impossible to detect, because the proton would be affected much more strongly by ambient
electric and magnetic fields than by the effect we’ve calculated.
Remark: It is odd that the result depends on q 2 /m, rather than on the charge-to-mass ratio
q/m, as is usually the case for a test particle’s trajectory. This means that we get a different
answer if we take two identical objects, place them side by side, and consider them as one big
object! This is not as unphysical as it sounds. The two side-by-side objects radiate coherently,
so the field they radiate is doubled, and the radiated power is quadrupled. Each object’s rate
of orbital decay is doubled, with the extra effect coming from electromagnetic interactions with
the other object’s fields.
Solutions for Chapter 2
Chapter 9
Gravitational waves
Page 84, problem 1:
(a) Let t be the time taken in the lab frame for the light to go from one mirror to the other,
and t0 the corresponding interval in the clock’s frame. Then t0 = L, and (vt)2 +L2 = t2 , where the
use of the same L in both equations makes use of our prior knowledge that there is no transverse
length contraction. Eliminating L, we find the expected expression for γ, which is independent
of L (b) If the result of a were independent of L, then the relativistic time dilation would depend
on the details of the construction of the clock measuring the time dilation. We would be forced
to abandon the geometrical interpretation of special relativity. (c) The effect is to replace vt with
vt+at2 /2 as the quantity inside the parentheses in the expression (. . .)2 +L2 = t2 . The resulting
correction terms are of higher order in t than the ones appearing in the original expression, and
can therefore be made as small in relative size as desired by shortening the time t. But this is
exactly what happens when we make the clock sufficiently small.
Page 85, problem 2:
Since gravitational redshifts can be interpreted as gravitational time dilations, the gravitational time dilation is given by the difference in gravitational potential gdr (in units where
c = 1). The kinematic effect is given by dγ = d(v 2 )/2 = ω 2 rdr. The ratio of the two effects is
ω 2 R cos λ/g, where R is the radius of the Earth and λ is the latitude. Tokyo is at 36 degrees
latitude, and plugging this in gives the claimed result.
Page 85, problem 3:
(a) Reinterpret figure j on p. 82 as a picture of a Sagnac ring interferometer. Let light waves
1 and 2 move around the loop in opposite senses. Wave 1 takes time t1i to move inward along
the crack, and time t1o to come back out. Wave 2 takes times t2i and t2o . But t1i = t2i (since
the two world-lines are identical), and similarly t1o = t2o . Therefore creating the crack has no
effect on the interference between 1 and 2, and splitting the big loop into two smaller loops
merely splits the total phase shift between them. (b) For a circular loop of radius r, the time
of flight of each wave is proportional to r, and in this time, each point on the circumference
of the rotating interferometer travels a distance v(time) = (ωr)(time) ∝ r2 . (c) The effect is
proportional to area, and the area is zero. (d) The light clock in c has its two ends synchronized
according to the Einstein prescription, and the success of this synchronization verifies Einstein’s
assumption of commutativity in this particular case. If we make a Sagnac interferometer in the
shape of a triangle, then the Sagnac effect measures the failure of Einstein’s assumption that all
three corners can be synchronized with one another.
Page 85, problem 5:
Here is the program:
The diagonal components of the result are both 1 + η12 /2 + η22 /2 + η1 η2 + . . . Everything after
the 1 is nonclassical. The off-diagonal components are η1 + η2 + η1 η22 /2 + η2 η12 /2 + . . ., with the
third-order terms being nonclassical.
Solutions for Chapter 3
Page 119, problem 1:
(a) As discussed in example 4 on page 95, a cylinder has local, intrinsic properties identical
to those of flat space. The cylindrical model therefore has the same properties L1-L5 as our
standard model of Lorentzian space, provided that L1-L5 are taken as purely local statements.
(b) The cylindrical model does violate L3. In this model, the doubly-intersecting world-lines
described by property G will not occur if the world-lines are oriented exactly parallel to the
cylinder. This picks out a preferred direction in space, violating L3 if L3 is interpreted globally.
Frames moving parallel to the axis have different properties from frames moving perpendicular
to the axis.
But just because this particular model violates the global interpretation of L3, that doesn’t
mean that all models of G violate it. We could instead construct a model in which space wraps
around in every direction. In the 2+1-dimensional case, we can visualize the spatial part of such
a model as the surface of a doughnut embedded in three-space, with the caveat that we don’t
want to think of the doughnut hole’s circumference as being shorter than the doughnut’s outer
radius. Giving up the idea of a visualizable model embedded in a higher-dimensional space,
we can simply take a three-dimensional cube and identify its opposite faces. Does this model
violate L3? It’s not quite as obvious, but actually it does. The spacelike great-circle geodesics of
this model come in different circumferences, with the shortest being those parallel to the cube’s
We can’t prove by constructing a finite number of models that every possible model of G
violates L3. The two models we’ve found, however, can make us suspect that this is true, and
can give us insight into how to prove it. For any pair of world-lines that provide an example of
G, we can fix a coordinate system K in which the two particles started out at A by flying off
back-to-back. In this coordinate system, we can measure the sum of the distances traversed by
the two particles from A to B. (If homegeneity, L1, holds, then they make equal contributions
to this sum.) The fact that the world-lines were traversed by material particles means that we
can, at least in principle, visit every point on them and measure the total distance using rigid
rulers. We call this the circumference of the great circle AB, as measured in a particular frame.
The set of all such circumferences has some greatest lower bound. If this bound is zero, then
such geodesics can exist locally, and this would violate even the local interpretation of L1-L5.
If the bound is nonzero, then let’s fix a circle that has this minimum circumference. Mark the
spatial points this circle passes through, in the frame of reference defined above. This set of
points is a spacelike circle of minimum radius. Near a given point on the circle, the circle looks
like a perfectly straight axis, whose orientation is presumably random. Now let some observer
K0 travel around this circle at a velocity v relative to K, measuring the circumference with a
Lorentz-contracted ruler. The circumference is greater than the minimal one measured by K.
Therefore for any axis with a randomly chosen orientation, we have a preferred rest frame in
which the corresponding great circle has minimum circumference. This violates L3. Thanks to
physicsforums user atyy for suggesting this argument.
More detailed discussions of these issues are given in Bansal et al.,
0503070v1, and Barrow and Levin,
Page 119, problem 2:
In these Cartesian coordinates, the metric is diagonal and has elements with opposite signs.
Due to the SI units, it is not possible for the two nonzero elements of the metric to have the
same units. Let’s arbitrarily fix gtt = 1. Then we must have gxx = c−2 . Using the metric to
Chapter 9
Gravitational waves
lower the index on dsa , we find dsa = (dst , c−2 dsx ).
Page 120, problem 7:
According to the Einstein summation convention, the repeated index implies a sum, so the
result is a scalar. As shown in example 14 on p. 106, each term in the sum equals 1, so the
result is unitless and simply equals the number of dimensions.
Page 119, problem 3:
(a) The first two violate the rule that summation only occurs over up-down pairs of indices.
The third expression would result in a quantity that couldn’t be classified as either contravariant
or covariant. (b) In differential geometry, different elements of the same tensor can have different
units. Since, as remarked in the problem, Uaa were to be interpreted as a sum, this mean
adding things that had different units. In the expression pa − qa , even if we suppose that
p and q both represent the same type of physical quantity, e.g., force, their covariant and
contravariant versions would not necessarily have the same units unless we happened to be
working in coordinates such that the metric was unitless.
Page 119, problem 4:
Assuming the mountaineer uses radians and the metric system, the coordinates have units
1, 1, and m (where 1 means a unitless quantity and m means meters — radians are not really
units). Therefore the units of an infinitesimal difference in coordinates dsa are also (1, 1, m).
Because the coordinates are orthogonal, the metric is diagonal. If we want gab dsa dsb to have
units of m2 , then its diagonal elements must have units of (m2 , m2 , 1). The upper-index metric
g ab is the inverse of its lower-index version gab , so its units are (m−2 , m−2 , 1). Mechanical work
has units of N · m, so given dW = Fa dsa , the units of Fa must be (N · m, N · m, N). Raising the
index on the force using g ab gives (N/m, N/m, N).
Page 119, problem 5:
The only aspect of the geometrical representation that needs to be changed is that instead of
representing an upper-index vector using a pair of parallel lines, we should use a pair of parallel
Page 120, problem 8:
The coordinate T would have a discontinuity of 2πωr2 /(1 − ω 2 r2 ). Reinserting factors of c
to make it work out in SI units, we have 2πωr2 c−2 /(1 − ω 2 r2 c−2 ) ≈ 207 ns. The exact error in
position that would result is dependent on the geometry of the current position of the satellites,
but it would be on the order of c∆T , which is ∼ 100 m. This is considerably worse than civilian
GPS’s 20-meter error bars.
Page 120, problem 9:
The process that led from the Euclidean metric of example 7 on page 102 to the nonEuclidean one of equation [3] on page 111 was not just a series of coordinate transformations.
At the final step, we got rid of the variable t, reducing the number of dimensions by one.
Similarly, we could take a Euclidean three-dimensional space and eliminate all the points except
for the ones on the surface of the unit sphere; the geometry of the embedded sphere is nonEuclidean, because we’ve redefined geodesics to be lines that are “as straight as they can be”
(i.e., have minimum length) while restricted to the sphere. In the example of the carousel, the
final step effectively redefines geodesics so that they have minimal length as determined by a
chain of radar measurements.
Page 120, problem 10:
(a) No. The track is straight in the lab frame, but curved in the rotating frame. Since the
spatial metric in the rotating frame is symmetric with respect to clockwise and counterclockwise,
the metric can never result in geodesics with a specific handedness. (b) The dθ02 term of the
metric blows up here. A geodesic connecting point A, at r = 1/ω, with point B, at r < 1/ω,
must have minimum length. This requires that the geodesic be directly radial at A, so that
dθ0 = 0; for if not, then we could vary the curve slightly so as to reduce |dθ0 |, and the resulting
increase in the dr2 term would be negligible compared to the decrease in the dθ02 term. (c) No.
As we found in part a, laser beams can’t be used to form geodesics.
Page 120, problem 11: A and B are equivalent under a Lorentz transformation, so the Penrose
result clearly includes B. The outline of the sphere is still spherical. C is also equivalent to A
and B, because there are only two effects (Lorentz contraction and optical aberration), and both
of them depend only on the observer’s instantaneous velocity, not on his history of motion. D
is not a well-defined question. When asking this question, we’re implicitly assuming that the
sphere has some “real” shape, which appears different because the sphere has been set into
motion. But you can’t impart an angular acceleration to a perfectly rigid body in relativity.
Page 121, problem 12: Applying the de Broglie relations to the relativistic identity m2 =
p − p , we find the dispersion relation to be m = ω − k . The group velocity is dω/dk =
1 − (m/ω)2 . Applying
p the de Broglie relations to this, and associating the group velocity
with v, we have v = 1 − (m/E)2 , which is equivalent to E = mγ. Since E = mγ has been
established, and m2 = E 2 − p2 was assumed, it follows immediately that p = mγvp
holds as well.
All hell breaks loose if we try to associate v with the phase velocity, which is ω/k = 1 + (m/k)2 .
For example, the phase velocity is always greater than c(= 1) for m > 0.
Solutions for Chapter 4
Page 156, problem 1:
The four-velocity of a photon (or of any massless particle) is undefined. One way to see this
is that dτ = 0 for a massless particle, so v i = dxi /dτ involves division by zero. Alternatively,
pi = mv i would always give an energy and momentum of zero if v i were well defined, yet we
know that massless particles can have both energy and momentum.
Page 156, problem 2:
To avoid loss of precision in numerical operations like subtracting v from 1, it’s better
p derive an ultrarelativistic approximation. The velocity corresponding to a given γ is v =
1 − γ −2 ≈ 1 − 1/2γ 2 , so 1 − v ≈ 1/2γ 2 = (m/E)2 /2. Reinserting factors of c so as to make
the units come out right in the SI system, this becomes (mc2 /E)2 /2 = 9 × 10−9 .
Page 156, problem 8:
The time on the clock is given by s = ds, where the integral is over the clock’s world-line.
The quantity ds is our prototypical Lorentz scalar, so it’s frame-independent. An integral is
just a sum, and the tensor transformation laws are linear, so the integral of a Lorentz scalar is
still a Lorentz scalar. Therefore s is frame-independent. There is no requirement that we use an
inertial frame. It would also work fine, for example, in a frame rotating with the earth. We don’t
even need to have a frame of reference. All of the above applies equally well to any coordinate
system at all, even one that doesn’t have any sensible interpretation as some observer’s frame
Chapter 9
Gravitational waves
of reference.
Page 156, problem 10:
Such a transformation would take an energy-momentum four-vector (E, p), with E > 0, to
a different four-vector (E 0 , p0 ), with E 0 < 0. That transformation would also have the effect of
transforming a timelike displacement vector from the future light cone to the past light cone. But
the Lorentz transformations were specifically constructed so as to preserve causality (property
L5 on p. 52), so this can’t happen.
Page 156, problem 11:
A spatial plane is determined by the light’s direction of propagation and the relative velocity of the source and observer, so the 3+1 case reduces without loss of generality to 2+1
dimensions. The frequency four-vector must be lightlike, so its most general possible form
is (f , f cos θ, f sin θ), where θ is interpreted as the angle between the direction of propagation and the relative velocity. Putting this through a Lorentz boost along the x axis, we find
f 0 = γf (1 + v cos θ), which agrees with Einstein’s equation on page 358, except for the arbitrary
convention involved in defining the sign of v.
Page 157, problem 12:
The exact result depends on how one assumes the charge is distributed, so this can’t be any
more than a rough estimate.
The energy
density is (1/8πk)E 2 ∼ ke2 /r4 , so the total energy
R −4
R −2
is an integral of the form r dV ∼ r dr, which diverges like 1/r as the lower limit of
integration approaches zero. This tells us that most of the energy is at small values of r, so to
a rough approximation we can just take the volume of integration to be r3 and multiply by a
fixed energy density of ke2 /r4 . This gives an energy of ∼ ke2 /r. Setting this equal to mc2 and
solving for r, we find r ∼ ke2 /mc2 ∼ 10−15 m.
Remark: Since experiments have shown that electrons do not have internal structure on this
scale, we conclude that quantum-mechanical effects must prevent the energy from blowing up
as r → 0.
Page 157, problem 17:
Doing a transformation first by u and then by v results in E00 = E−v×(u×E)+(u+v)×B.
This is not of the same form, because if B = 0, we can have E00 6= E.
Solutions for Chapter 5
Page 199, problem 1:
The answer to this is a little subtle, since it depends on how we take the limit. Suppose we
join two planes with a section of a cylinder having radius ρ, and let ρ go to zero. The Gaussian
curvature of a cylinder is zero, so in this limit we fail to reproduce the correct result. On the
other hand, suppose we take a discus of radius ρ1 whose edge has a curve of radius ρ2 . in the
limit ρ1 → +∞, ρ2 → 0+ , we can get either K = 1/(ρ1 ρ2 ) → 0 or K → +∞, depending on how
quickly ρ1 and ρ2 approach their limits.
Page 199, problem 6:
(a) Expanding in a Taylor series, they both have gtt = 1 + 2gz + . . .
(b) This property holds for [2] automatically because of the way it was constructed. In [1],
the nonvanishing Christoffel symbols (ignoring permutations of the lower indices) are Γtzt = g
and Γz tt = ge2gz . We can apply the geodesic equation with the affine parameter taken to be the
proper time, and this gives z̈ = −ge2gz ṫ2 , where dots represent differentiation with respect to
proper time. For a particle instantaneously at rest, ṫ = 1/ gtt = e−2gz , so z̈ = −g.
(c) [2] was constructed by performing a change of coordinates on a flat-space metric, so it
is flat. The Riemann tensor of [1] has Rtztz = −g 2 , so [1] isn’t flat. Therefore the two can’t be
the same under a change of coordinates.
(d) [2] is flat, so its curvature is constant. [1] has the property that under the transformation
z → z + c, where c is a constant, the only change is a rescaling of the time coordinate; by
coordinate invariance, such a rescaling is unobservable.
Page 200, problem 7: (a) 0 ≤ x ≤ 1
(b) 0 ≤ x < 1
(c) x2 ≤ 2
Page 200, problem 8: The double cone fails to satisfy axiom M2, because the apex has
properties that differ topologically from those of other points: deleting it chops the space into
two disconnected pieces.
Page 200, problem 9: When we use a word like “torus,” there is some hidden ambiguity. We
could mean something strange like the following. Suppose we construct the three-dimensional
space of coordinates (x, y, z) in which all three coordinates are rational numbers. Then let a
torus be the set of all such points lying at a distance of 1/2 from the nearest point on a unit
circle. This is in some sense a torus, but it doesn’t have the topological properties one usually
assumes. For example, two continuous curves on its surface can cross without having a point of
intersection. We can’t get anywhere without assuming that the word “torus” refers to a surface
that has the usual topological properties.
Now let’s prove that it’s a manifold using both definitions.
Using the topological definition, M1 is satisfied with n = 2, because every point on the surface
lives in a two-dimensional neighborhood. M2 holds because the only differences between points
are those that are not topological, e.g., Gaussian curvature. M3 holds due to the interpretation
outlined in the first paragraph.
Alternatively, we can use the local-coordinate definition. We have already shown that a
circle is a 1-manifold, which can be coordinatized in two patches by an angle φ. The torus can
therefore be coordinatized by a pair of such angles, (φ1 , φ2 ), in four patches. Again we need to
assume the interpretation given above, since otherwise real-number pairs like (φ1 , φ2 ) wouldn’t
have the same topology as points on the rational-number torus.
Page 200, problem 10: In the torus, we can construct a closed curve C that encircles the
hole. If we have a homeomorphism, C must have an image C0 under that homeomorphism
that is a closed curve in the sphere. C0 can then be contracted continuously to a point, and
since the inverse of the homeomorphism is also continuous, it would be possible to contract C
continuously to a point. But this is impossible because C encircles the hole.
Page 200, problem 11: (a) The Christoffel symbols are (assuming I didn’t make a mistake
in calculating them by hand) Γtxx = (1/2)ptp−1 and Γxxt = Γxtx = (1/2)pt−1 . (b) After that, I
resorted to a computer algebra system (Maxima), which told me that, for example, the Ricci
tensor has Rtt = (p/2 − p2 /4)t−2 .
Chapter 9
Gravitational waves
Solutions for Chapter 6
Page 238, problem 3: (a) In the center of mass frame, symmetry guarantees that the test
particle exits with a speed equal to the speed with which it entered, and the entry and exit
velocities are v and −v. Now let’s switch to the sun’s frame. This involves adding u to all
velocities, so the entry and exit velocities become v + u and −v + u. The difference in speed is
(b) The derivation assumed that velocities add linearly when you change frames of reference,
which is a nonrelativistic approximation. Relativistically, velocities combine not like u + v but
like (u + v)/(1 + uv). If you put in v = 1, the result for the combined velocity is always 1.
This is a funny case where we can get the answer to a gravitational problem purely through
special relativity. We might worry that the SR-based answer is wrong, because we really need
GR for gravity. But we can get the same answer from GR, since GR says that a test particle
always follows a geodesic, and a lightlike geodesic always remains lightlike. The reason SR
worked is that an observer could watch a patch of flat space far away from the black hole,
observe a wave-packet of light passing through that patch on the way to the black hole, and
then observe it again on the way back out. Since the patch is flat, SR works.
238,√problem 4: (a) For a displacement with dφ = 0, we have ds2 = gtt dt2 , so gtt =
ds/dt = 3 sin t cos t. For an azimuthal displacement, ds = ydφ, so gφφ = y = sin3/2 t.
(b) At places on the surface of revolution corresponding to the cusps of the astroid, one or
both of the lower-index elements of the metric go to zero, which means that the corresponding
upper-index elements blow up. These are the sharp points of the surface at the x axis and the
sharp edge at its waist. There are at least coordinate singularities there, but the question is
whether they are intrinsic. The only intrinsic measure of curvature in two dimensions is the
Gaussian curvature, which can be interpreted as (minus) the product of the curvatures along
the two principal axes, here k1 = −(2/3) csc 2t and k2 = 1/y = sin−3 t. At the waist, both
factors blow up, so the Gaussian curvature, which is intrinsic, blows up, and this is not just a
coordinate singularity. The same thing happens at the tips. Interestingly, a geodesic that hits
one of these singularities can still be traced through in a continuous way and extended onward
such that its arc length remains finite. This property is called geodesic completeness.
Page 238, problem 5: (a) There are singularities at r = 0, where gθ0 θ0 = 0, and r = 1/ω,
where gtt = 0. These are considered singularities because the inverse of the metric blows up.
They’re coordinate singularities, because they can be removed by a change of coordinates back
to the original non-rotating frame.
(b) This one has singularities in the same places. The one at r = 0 is a coordinate singularity,
because at small r the ω dependence is negligible, and the metric is simply that of ordinary
plane polar coordinates in flat space. The one at r = 1/ω is not a coordinate singularity. The
following Maxima code calculates its scalar curvature R = Raa , which is esentially just the
Gaussian curvature, since this is a two-dimensional space.
The result is R = 6ω 2 /(1 − 2ω 2 r2 + ω 4 r4 ). This blows up at r = 1/ω, which shows that this is
not a coordinate singularity. The fact that R does not blow up at r = 0 is consistent with our
earlier conclusion that r = 0 is a coordinate singularity, but would not have been sufficient to
prove that conclusion.
(c) The argument is incorrect. The Gaussian curvature is not just proportional to the angular
deficit , it is proportional to the limit of /A, where A is the area of the triangle. The area of
the triangle can be small, so there is no upper bound on the ratio /A. Debunking the argument
restores consistency with the answer to part b.
Page 239, problem 9: The only nonvanishing Christoffel symbol is Γttt = −1/2t. The antisymmetric treatment of the indices in Rabcd = ∂c Γadb − ∂d Γacb + Γace Γedb − Γade Γecb guarantees
that the Riemann tensor must vanish when there is only one nonvanishing Christoffel symbol.
Page 239, problem 10: The first thing one notices is that the equation Rab = k isn’t written
according to the usual rules of grammar for tensor equations. The left-hand side has two lower
indices, but the right-hand side has none. In the language of freshman physics, this is like
setting a vector equal to a scalar. Suppose we interpret it as meaning that each of R’s 16
components should equal k in a vacuum. But this still isn’t satisfactory, because it violates
coordinate-independence. For example, suppose we are initially working with some coordinates
xµ , and we then rescale all four of them according to xµ = 2xµ . Then the components of Rab
all scale down by a factor of 4. But this would violate the proposed field equation.
Page 239, problem 11: The following Maxima code calculates the Ricci tensor for a metric
with gtt = h and grr = k.
Inspecting the output (not reproduced here), we see that Rφφ = 0 requires k 0 /k = h0 /h. Since
the logarithmic derivatives of h and k are the same, the two functions can differ by at most a
constant factor c. So now we do a second iteration of the calculation:
Chapter 9
Gravitational waves
The result for Rrr is independent of c. Since h is essentially the gravitational potential, we have
the requirements h0 > 0 (because gravity is attractive) and h00 < 0 (because gravity weakens
with distance). Therefore we find that Rrr is positive, and we do not obtain a vacuum solution.
Page 239, problem 12: This idea is not well defined because it implicitly assumes that we
can fix a global frame of reference. The notion of reversing velocity vectors (i.e., reversing
the spacelike components of 4-velocities) implies that there are some velocity vectors whose
spacelike parts are zero, so that they aren’t changed by a flip. This amounts to choosing a
frame of reference. To be able to do the flip globally, you’d have to have some sensible notion
of a global frame of reference, but we don’t necessarily have that. (In a spacetime with closed
timelike curves, there is also the issue that we don’t have complete freedom to choose initial
conditions on a spacelike surface, because these conditions might end up not being consistent
with themselves when evolved around a CTC.)
Solutions for Chapter 7
Page 260, problem 2: (a) If she makes herself stationary relative to the sun, she will still
experience local geometrical changes because of the planets. (b) If it was to be impossible
for her to prove the universe’s nonstationarity, then any world-line she picked would have to
experience constant local geometrical conditions. A counterexample is any world-line extending
back to the Big Bang, which is a singularity with drastically different conditions than any other
region of spacetime. (c) To maintain a constant local geometry, she would have to “surf” the
wave, but she can’t do that, because it propagates at the speed of light. (d) There are places
where the local mass-energy density is increasing, and the field equations link this to a change
in the local geometry.
Page 260, problem 4:
Under these special conditions, the geodesic equations become r̈ = Γrtt ṫ2 , φ̈ = 0, ẗ = 0,
where the dots can in principle represent differentation with respect to any affine parameter we
like, but we intend to use the proper time s. By symmetry,
there will
√ be no motion in the z
direction. The Christoffel symbol equals −(1/2)e (cos 3r − 3 sin 3r). At a location where
the cosine equals 1, this is simply −er /2. For ṫ, we have dt/ds = 1/ gtt = e−r/2 . The result of
the calculation is simply r̈ = −1/2, which is independent of r.
Page 260, problem 5:
The Petrov metric is one example. The metric has no singularities anywhere, so the r
coordinate can be extended from −∞ to +∞, and there is no point that can be considered the
center. The existence of a dφdt term in the metric shows that it is not static.
A simpler example is a spacetime made by taking a flat Lorentzian space and making it wrap
around topologically into a cylinder, as in problem 1 on p. 119. As discussed in the solution
to that problem, this spacetime has a preferred state of rest in the azimuthal direction. In
a frame that is moving azimuthally relative to this state of rest, the Lorentz transformation
requires that the phase of clocks be adjusted linearly as a function of the azimuthal coordinate
φ. As described in section 3.5.4, this will cause a discontinuity once we wrap around by 2π, and
therefore clock synchronization fails, and this frame is not static.
Solutions for Chapter 8
Page 329, problem 2: No. General relativity only allows coordinate transformations that are
smooth and one-to-one (see p. 97). This transformation is not smooth at t = 0.
Page 329, problem 5: (a) The Friedmann equations are
= Λ−
(ρ + 3P )
= Λ+
ρ − ka−2
The first equation is time-reversal invariant because the second derivative stays the same under
time reversal. The second equation is also time-reversal invariant, because although the first
derivative flips its sign under time reversal, it is squared.
(b) We typically do not think of a singularity as being a point belonging to a manifold at all. If
we want to create this type of connected, symmetric back-to-back solution, then we need the Big
Bang singularity to be a point in the manifold. But this violates the definition of a manifold,
because then the Big Bang point would have topological characteristics different from those of
other points: deleting it separates the spacetime into two pieces.
Page 329, problem 4: Example 13 on page 303, the cosmic girdle, showed that a rope that
stretches over cosmological distances does expand significantly, unlike Brooklyn, nuclei, and
solar systems. Since the Milne universe is nothing but a flat spacetime described in funny
coordinates, something about that argument must fail. The argument used in that example
relied on the use of a closed cosmology, but the Milne universe is not closed. This is not a
completely satisfying resolution, however, because we expect that a rope in an open universe
will also expand, except in the special case of the Milne universe.
In a nontrivial open universe, every galaxy is accelerating relative to every other galaxy.
By the equivalence principle, these accelerations can also be seen as gravitational fields, and
tidal forces are what stretch the rope. In the special case of the Milne universe, there is no
acceleration of test particles relative to other test particles, so the rope doesn’t stretch.
Example 15 on page 306, the cosmic whip, resulted in the conclusion that the velocity of
the rope-end passing by cannot be interpreted as a measure of the velocity of the distant galaxy
to which the rope’s other end is hitched, which makes sense because cosmological solutions are
nonstationary, so there is no uniquely defined notion of the relative velocity of distant objects.
The Milne universe, however, is stationary, so such velocities are well defined. The key here is
that nothing is accelerating, so the time delays in the propagation of information do not lead to
ambiguities in extrapolating to a distant object’s velocity “now.”
The Milne case also avoids the paradox in which we could imagine that if the rope is sufficiently long, its end would be moving at more than the speed of light. Although there is no
limit to the length of a rope in the Milne universe (there being no tidal forces), the Hubble
law cannot be extrapolated arbitrarily, since the expanding cloud of test particles has an edge,
beyond which there is only vacuum.
Chapter 9
Gravitational waves
Page 329, problem 6: The cosmological constant is a scalar, so it doesn’t change under
reflection. The metric is also invariant under reflection of any coordinate. This follows because
we have assumed that the coordinates are locally Lorentzian, so that the metric is diagonal.
It can therefore be written as a line element in which the differentials are all squared. This
establishes that the Λgab is invariant under any spatial or temporal reflection.
The specialized form of the energy-momentum tensor diag(−ρ, P , P , P ) is also clearly invariant under any reflection, since both pressure and mass-energy density are scalars.
The form of the tensor transformation law for a rank-2 tensor guarantees that the diagonal
elements of such a tensor stay the same under a reflection. The off-diagonal elements will flip
sign, but since only the G and T terms in the field equation have off-diagonal terms, the field
equations remain valid under reflection.
In summary, the Einstein field equations retain the same form under reflection in any coordinate. This important symmetry property, which is part of the Poincaré group in special
relativity, is retained when we make the transition to general relativity. It’s a discrete symmetry, so it wasn’t guaranteed to exist simply because of general covariance, which relates to
continuous coordinate transformations.
Page 329, problem 7: (a) The Einstein field equations are Gab = 8πTab + Λgab . That means
that in a vacuum, where T = 0, a cosmological constant is equivalent to ρ = (1/8π)Λ and
P = −(1/8π)Λ. This gives ρ + 3P = (1/8π)(−2Λ), which violates the SEC for Λ > 0, since part
of the SEC is ρ + 3P ≥ 0.
(a) Since our universe appears to have a positive cosmological constant, and the paper by
Hawking and Ellis assumes the strong energy condition, doubts are raised about the conclusion
of the paper as applied to our universe. However, the theorem is being applied to the early
universe, which was not a vacuum. Both P and ρ were large and positive in the early, radiationdominated universe, and therefore the SEC was not violated.
Page 330, problem 8:
(a) The Ricci tensor is Rtt = g 2 e2gz , Rzz = −g 2 . The scalar curvature is 2g 2 , which is
constant, as expected.
(b) Both Gtt and Gzz vanish by a straightforward computation.
(c) The Einstein tensor is Gtt = 0, Gxx = Gyy = g 2 , Gzz = 0. It is unphysical because it has
a zero mass-energy density, but a nonvanishing pressure.
Page 330, problem 9:
This proposal is an ingenious attempt to propose a concrete method for getting around the
fact that in relativity, there is no unique way of defining the relative velocities of objects that
lie at cosmological distances from one another.
Because the Milne universe is a flat spacetime, there is nothing to prevent us from laying
out a chain of arbitrary length. The chain will not, for example, be subject to the kind of tidal
forces that would inevitably break a chain that was lowered through the event horizon of a
black hole. But this only guarantees us that we can have a chain of a certain length as measured
in the chain’s frame. An observer at rest with respect to the chain describes all the links of
the chain as existing simultaneously at a certain set of locations. But this is a description in
(T , R) coordinates. To an observer who prefers the FRW coordinates, the links do not exist
simultaneously at these locations. This observer says that the supposed locations of distant
points on the chain occurred far in the past, and suspects that the chain has broken since then.
The paradox can also be resolved from the point of view of the (T , R) coordinates. The
chain is long enough that its end hangs out beyond the edges of the expanding cloud of galaxies.
Since there are no galaxies beyond the edge, so there are no galaxies near the end of the chain
with respect to which the chain could be moving at > c.
Page 330, problem 10: Frames are local, not global. One of the things we have to specify in
order to define a frame of reference is a state of motion. To define the volume of the observable
universe, there end up being three spots in the definition at which we might need to pick a state
of motion. I’ve labeled these 1-2-3 below.
Observer O is in some state of motion [1] at event A. O’s past light-cone intersects the
surface of last scattering (or some other surface where some other physically well-defined thing
happens) in a spacelike two-surface S. S does not depend on O’s state of motion. At every
event P on S, we define a state of motion [2] that is at rest relative to the Hubble flow, and
we construct a world-line that starts out in this state of motion and extends forward in time
inertially. One of these world-lines intersects O’s world-line at A. Let the proper time interval
along this world-line be t. We extend all the other world-lines from all the other P by the same
interval of proper time t. The end-points of all these world-lines constitute a spacelike 2-surface
B that we can define as the boundary of the observable universe according to O. Let R be the
3-surface contained inside B. In order to define R, we need to define some notion of simultaneity,
which depends on one’s state of motion [3]. If we like, we can pick this state of motion to be
one at rest with respect to the Hubble flow. Given this choice, we can define the volume V of
R (e.g., by chopping R up into pieces and measuring those pieces using rulers that are in this
state of motion).
State of motion 1 had absolutely no effect on V , but states of motion 2 and 3 did. If O is
not at rest relative to the Hubble flow at A, then 2 and 3 do not match O’s state of motion at
A. This probably means that O will object that V is not the answer in his frame but in someone
else’s. However, there is no clear way to satisfy O by modifying the above definition. We can’t
just say that 2 and 3 should be chosen to be the same as O’s state of motion at A, because
frames are local things, so matching them to O’s motion at A isn’t the same as matching them
at points far from A. In a cosmological solution there is no well-defined notion of whether or
not two cosmologically distant objects are at rest relative to one another.
In particular, it is not meaningful to try to calculate a reduced value of V based on Lorentz
contraction for O’s velocity relative to the Hubble flow. Lorentz contractions can’t be applied
to a curved spacetime.
Page 330, problem 11: The Friedmann equations reduce to
= − (1 + 3w)ρ
Eliminating ρ, we find
= −β
where β = (1 + 3w)/2. For a solution of the form a ∝ tδ , calculation of the derivatives results in
δ = 1/(1 + β) = (2/3)/(1 + w). For dust, δ = 2/3, which checks out against the result on p. 313.
Chapter 9
Gravitational waves
For radiation, δ = 1/2. For a cosmological constant, w = −1 gives δ = ∞, so the solution has a
different form.
Page 330, problem 12: The integral is exactly the same as the one in example 18 on p. 314
for the dust case, except that the exponent 2/3 is generalized to δ = (2/3)/(1 + w), as shown
in the solution to problem 11. The result is L/t = 1/(1 − δ) = (w + 1)/(w + 1/3). In the
radiation-dominated case, we have L/t = 2.
Solutions for Chapter 9
Page 344, problem 1: (a) The members of the Hulse-Taylor system are spiraling toward one
another as they lose energy to gravitational radiation. If one of them were replaced with a
low-mass test particle, there would be negligible radiation, and the motion would no longer be
a spiral. This is similar to the issues encountered on pp. 39ff because the neutron stars in the
Hulse-Taylor system suffer a back-reaction from their own gravitational radiation.
(b) If this occurred, then the particle’s world-line would be displaced in space relative to a
geodesic of the spacetime that would have existed without the presence of the particle. What
would determine the direction of that displacement? It can’t be determined by properties of
this preexisting, ambient spacetime, because the Riemann tensor is that spacetime’s only local,
intrinsic, observable property. At a fixed point in spacetime, the Riemann tensor is even under
spatial reflection, so there’s no way it can distinguish a certain direction in space from the
opposite direction.
What else could determine this mysterious displacement? By assumption, it’s not determined by a preexisting, ambient electromagnetic field. If the particle had charge, the direction
could be one imposed by the back-reaction from the electromagnetic radiation it had emitted
in the past. If the particle had a lot of mass, then we could have something similar with gravitational radiation, or some other nonlinear interaction of the particle’s gravitational field with
the ambient field. But these nonlinear or back-reaction effects are proportional to q 2 and m2 ,
so they vanish when q = 0 and m → 0.
The only remaining possibility is that the result violates the symmetry of space expressed by
L1 on p. 52; the Lorentzian geometry is the result of L1-L5, so violating L1 should be considered
a violation of Lorentz invariance.
Photo Credits
Cover Galactic center: NASA, ESA, SSC, CXC, and STScI
15 Atomic clock on plane:
Copyright 1971, Associated press, used under U.S. fair use exception to copyright law.
Gravity Probe A: I believe this diagram to be public domain, due to its age and the improbability of its copyright having been renewed.
20 Stephen Hawking: unknown NASA photographer, 1999, public-domain product of NASA. 22 Eotvos: Unknown source. Since Eötvös
died in 1919, the painting itself would be public domain if done from life. Under U.S. law,
this makes photographic reproductions of the painting public domain.
25 Earth: NASA,
Apollo 17. Public domain. 25 Orion: Wikipedia user Mouser, GFDL. 25 M100: European
Southern Observatory, CC-BY-SA. 25 Supercluster: Wikipedia user Azcolvin429, CC-BY-SA.
25 Artificial horizon: NASA, public domain. 26 Upsidasium: Copyright Jay Ward Productions, used under U.S. fair use exception to copyright law..
36 Pound and Rebka photo:
Harvard University. I presume this photo to be in the public domain, since it is unlikely to
have had its copyright renewed.
41 Lorentz: Jan Veth (1864-1925), public domain.
Muon storage ring at CERN: (c) 1974 by CERN; used here under the U.S. fair use doctrine.
65 Galaxies: Hubble Space Telescope. Hubble material is copyright-free and may be freely
used as in the public domain without fee, on the condition that NASA and ESA is credited
as the source of the material. The material was created for NASA by STScI under Contract
NAS5-26555 and for ESA by the Hubble European Space Agency Information Centre.
Gamma-Ray burst: NASA/Swift/Mary Pat Hrybyk-Keith and John Jones.
85 Graph from
Iijima paper: Used here under the U.S. fair use doctrine. 91 Levi-Civita: Believed to be public
domain. Source:
74 Ring laser gyroscope: Wikimedia commons user Nockson, CC-BY-SA licensed. 94 Einstein’s ring: I have lost the information about the source of the bitmapped image. I would
be grateful to anyone who could put me in touch with the copyright owners.
48 Map of
isotherms: J. Hanns, 1910, public domain.
49 Human arm: Gray’s Anatomy, 1918, public
domain. 118 SU Aurigae’s field lines: P. Petit, GFDL 1.2. 115 Galaxies: Hubble Space
Telescope. Hubble material is copyright-free and may be freely used as in the public domain
without fee, on the condition that NASA and ESA is credited as the source of the material.
The material was created for NASA by STScI under Contract NAS5-26555 and for ESA by
the Hubble European Space Agency Information Centre.
144 Chandrasekhar: University
of Chicago. I believe the use of this photo in this book falls under the fair use exception to
copyright in the U.S. 149 Relativistic jet: Biretta et al., NASA/ESA, public domain. 159
Rocks: Siim Sepp, CC-BY-SA 3.0. 160 Jupiter and comet: Hubble Space Telescope, NASA,
public domain.
161 Earth: NASA, Apollo 17. Public domain.
161 Moon: Luc Viatour, CC-BY-SA 3.0.
162 Heliotrope: ca. 1878, public domain.
162 Triangulation
survey: Otto Lueger, 1904, public domain. 166 Triangle in a space with negative curvature: Wikipedia user Kieff, public domain. 172 Eclipse: Eddington’s original 1919 photo,
public domain.
185 Torsion pendulum: University of Washington Eot-Wash group, http:
193 Asteroids: I believe the use of this photo in this book falls under the fair use exception to copyright
in the U.S. 193 Coffee cup to doughnut: Wikipedia user Kieff, public domain. 201 Coin:
Kurt Wirth, public-domain product of the Swiss government. 204 Bill Unruh: Wikipedia user
Childrenofthedragon, public domain.
226 Accretion disk: Public-domain product of NASA
and ESA. 241 Wilhelm Killing: I believe this to be public domain the US, since Killing died in
early 1923.. 241 Surface of revolution: Shaded rendering by Oleg Alexandrov, public domain.
267 Cavendish experiment: Based on a public-domain drawing by Wikimedia commons user
Chapter 9
Gravitational waves
Chris Burks. 268 Simplified diagram of Kreuzer experiment: Based on a public-domain drawing by Wikimedia commons user Chris Burks. 268 Kreuzer experiment: The diagram of the
apparatus is redrawn from the paper, and the two graphs are taken directly from the paper. I
believe the use of these images in this book falls under the fair use exception to copyright in the
270 Apollo 11 mirror: NASA, public domain.
278 Magnetic fipole: based on a figure by Wikimedia Commons user Geek3, CC-BY-SA licensed. 290 Penzias-Wilson antenna:
NASA, public domain. 295 Friedmann: Public domain. 297 Lemaı̂tre: Ca. 1933, public
domain.. 319 Cosmic microwave background image: NASA/WMAP Science Team, public domain. 327 Dicke’s apparatus: Dicke, 1967. Used under the US fair-use doctrine. 334 LIGO
and LISA sensitivities: NASA, public domain.
333 Graph of pulsar’s period: Weisberg and
aberration, 121
absolute geometry, 19
abstract index notation, 51, 102
equivalent to birdtracks, 51
acceleration four-vector, 126
Adams, W.S., 16
ADM mass, 336
affine geometry, 43
affine parameter, 44
Aharonov-Bohm effect, 191
angular defect, 163
angular momentum, 153, 155
antigravity, 26, 283
antisymmetrization, 102
Aristotelian logic, 68
Ashtekar formulation of relativity, 236
asymptotically flat, 249
atomic clocks, 15, 74
background independence, 117
baryon acoustic oscillations, 320
Bell, John, 66
spaceship paradox, 66, 200
Big Bang, 298
Big Crunch, 94
Big Rip, 316, 317
birdtracks, 47
birdtracks notation, 47
covariant derivative, 178
equivalent to abstract index notation, 51
metric omitted in, 106
rank-2 tensor, 105
Birkhoff’s theorem, 251, 254, 324
black body spectrum, 204
black hole, 223
definition, 253
event horizon, 224
formation, 147, 225
Newtonian, 65
no-hair theorem, 252
observational evidence, 225
radiation from, 230
singularity, 223
black string, 233, 254
Bohr model, 82
boost, 53
Brans-Dicke theory, 28, 33, 323, 332
Brown-Bethe scenario, 146
BTZ black hole, 233
cadabra, 188, 190
Cartan, 186
curved-spacetime theory of Newtonian gravity, 41, 116, 161
Casimir effect, 288
center of mass-energy, 264, 280
Cerenkov radiation, 143
Chandrasekhar limit, 144
charge inversion, 107
Chiao’s paradox, 30, 33
Christoffel symbol, 176
chronology protection conjecture, 20, 259
clock “postulate”, 84, 124
cloning of particles, 203
closed cosmology, 294
closed set, 200
closed timelike curves, 13, 20, 258
violate no-cloning theorem, 203
comoving cosmological coordinates, 292
geodesic, 373
Compton scattering, 157
conformal cosmological coordinates, 292
conformal geometry, 303
congruence, 284
conical singularity, 229
connection, 90
conservation laws, 148
from Killing vectors, 246
continuous function, 194
contravariant vector, 100
summarized, 389
coordinate independence, 97
coordinate singularity, 223
coordinate transformation, 97
correspondence principle, 33, 35, 57, 206
cosmic censorship, 227, 228
cosmic microwave background, 308
discovery of, 289
isotropy of, 291
cosmic rays, 16
cosmological constant, 64, 287
no variation of, 288
observation, 319
cosmological coordinates
comoving, 292
conformal, 292
standard, 292
covariant derivative, 173, 174
in electromagnetism, 173, 174
in relativity, 174
covariant vector, 100
summarized, 389
ctensor, 207
curvature, 159
Gaussian, 164
in two spacelike dimensions, 162
intrinsic versus extrinsic, 95
Kretchmann invariant, 223
none in one dimension, 162
of spacetime, 87
Ricci, 161
Ricci scalar, 223
Riemann tensor, 168
scalar, 223
sectional, 161
tensors, 168
tidal versus local sources, 160
Cvitanović, Predrag, 47
Cygnus X-1, 225
dark energy, 25
dark matter, 322
de Sitter, Willem, 82, 292
deflection of light, 171, 220
degeneracy, 234
Dehn twist, 97
covariant, 173, 174
in electromagnetism, 173
in relativity, 174
diffeomorphism, 97
Dirac sea, 130, 156
dominant energy condition, 276
Doppler shift, 133
dual vector, 47
summarized, 389
dust, 132
Eötvös experiments, 22
Eddington, 171
Ehrenfest’s paradox, 109
Einstein field equation, 263, 287
Einstein summation convention, 51
Einstein synchronization, 250, 347
Einstein tensor, 263
Einstein-Cartan theory, 186
electromagnetic fields
transformation properties of, 157
electromagnetic potential four-vector, 137, 141,
electromagnetic tensor, 141
electron capture, 145
elliptic geometry, 92
energy, see also conservation laws
of gravitational fields, 270
energy conditions, 275
violated by cosmological constant, 288
equiconsistency, 92
equivalence principle
accelerations and fields equivalent, 23
application to charged particles, 30, 33
no preferred field, 142
not mathematically well defined, 30, 276
spacetime locally Lorentzian, 28
Erlangen program, 108
ether, 70
event horizon, 201
expansion scalar, 284
extra dimensions, 232
extrinsic quantity, 95
Fermat’s principle, 137
field equation, Einstein, 263
fine structure constant, 82
foliation, 284
is a dual vector, 48
four-vector, 124
acceleration, 126
momentum, 126
velocity, 124
frame dragging, 151, 192, 257
frame of reference
inertial, 24
ambiguity in definition, 29
frequency vector, 50, 133
Friedmann equations, 295
Friedmann-Robertson-Walker cosmology, 295
Gödel metric, 291
Gödel’s theorem, 92
Gödel, Kurt, 92
gauge transformation, 118, 173
Gaussian curvature, 164
Gaussian normal coordinates, 164
Gell-Mann, Murray, 130
general covariance, 97
general relativity
defined, 29
geodesic, 22
as world-line of a test particle, 22, 344
differential equation for, 178
geodesic completeness, 373
geodesic equation, 178
geodesic incompleteness, 229
geodetic effect, 170, 212
geometrized units, 208
elliptic, 92
hyperbolic, 166
spherical, 93
Goudsmit, 82
frames of reference used in, 112
timing signals, 141
gravitational constant, 208
gravitational field
uniform, 199, 255, 330
gravitational mass, 21, 267
active, 267
passive, 267
gravitational potential, see potential
gravitational red-shift, see red-shift
gravitational shielding, 283
gravitational waves
empirical evidence for, 332
energy content, 334
propagation at c, 331
propagation at less than c, for high amplitudes, 331
rate of radiation, 340
transverse nature, 337
Gravity Probe A, 17
Gravity Probe B, 74, 142
frame dragging, 151
geodetic effect calculated, 212
geodetic effect estimated, 170
group, 107
group velocity, 51
Hafele-Keating experiment, 15, 74
Hawking radiation, 230
Hawking, Stephen, 20
hole argument, 114
homeomorphism, 194
Hoyle, Fred, 290
Hubble constant, 285, 295, 313
Hubble flow, 314
Hubble, Edwin, 289
Hulse, R.A., 220
Hulse-Taylor pulsar, 220, 342
hyperbolic geometry, 166
index gymnastics notation, 104
raising and lowering, 104
inertial frame, see frame, inertial
inertial mass, 21, 267
information paradox, 203, 228
inner product, 107
intrinsic quantity, 95
isometry, 107
Ives-Stilwell experiments, 134
Jacobian matrix, 152
Kasner metric, 256
Killing equation, 243
Killing vector, 241
orbit, 241
Kretchmann invariant, 223, 249, 260
Kreuzer experiment, 267
large extra dimensions, 232
Lemaı̂tre, Georges, 298
length contraction, 56
Lense-Thirring effect, 151, 257
Levi-Civita symbol, 153, 184
Levi-Civita, Tullio, 91, 116, 153
deflection by sun, 171, 220
light clock, 84
light cone, 64
lightlike, 64
Aristotelian, 68
loop quantum gravity, 69
Lorentz boost, 53
lowering an index, 104
lune, 94
Mössbauer effect, 35
Mach’s principle, 116, 261, 322
manifold, 193
geodesically complete, 373
active gravitational, 267
ADM, 336
gravitational, 21, 267
inertial, 21, 267
passive gravitational, 267
mass-energy, 128
ADM, 336
Maxima, 77, 207
orbit of, 210
metric, 99
none in Galilean spacetime, 100
Michelson-Morley experiment, 70
Milne universe, 299
Minkowski, 41
mathematical, 92
momentum four-vector, 126
muon, 16
neighborhood, 194
neutrino, 131
neutron star, 145, 220
no-cloning theorem, 203
no-hair theorems, 252
normal coordinates, 164
null energy condition, 276
observable universe, 314
size and age, 314
open cosmology, 294
open set, 194
optical effects, 121
Killing vector, 241
orientability, 152
in time, 211
orthogonality, 125
parallel postulate, 18
parallel transport, 89, 90
parity, 107
Pasch, Moritz, 19
patch, 197
graphical notation for tensors, 47
Penrose, Roger, 121, 228
Penrose-Hawking singularity theorems, 284, 298
Penzias, Arno, 289
Petrov classification, 337
Petrov metric, 257, 260
phase velocity, 51
mass, 131
Pioneer anomaly, 304
Planck mass, 186
Planck scale, 185
Playfair’s axiom, 18
Poincaré group, 107, 377
of gravitational waves, 337
of light, 130
potential, 32
Hansen’s, 254
not defined in arbitrary spacetimes, 254
relativistic vs. Newtonian, 254
Pound-Rebka experiment, 16, 34
Poynting vector, 279
principal group, 108
prior geometry, 117
projective geometry, 97
proper distance, 292
proper time, 123
pulsar, 145, 220
raising an idex, 104
rank of a tensor, 101
rapidity, 66
kinematic versus gravitational, 255, 306
gravitational, 16, 34
Ricci curvature, 161
defined, 169
Ricci scalar, 223
Riemann curvature tensor, 168
Riemann tensor
defined, 168
rigid-body rotation, 109
Rindler coordinates, 199
ring laser, 74
Abraham, 94
Robinson, Abraham, 88
rotating frame of reference, 108, 256
rigid, 109
Sagittarius A*, 226, 252
Sagnac effect, 110, 251
defined, 74
in GPS, 60
proportional to area, 85
defined, 46
scalar curvature, 223
Schwarzschild metric, 211
in d dimensions, 232
Schwarzschild, Karl, 205
gravitational, 283
change of, 234
defined as a list of signs, 206
defined as an integer, 234
singularity, 20, 227
conical, 229
coordinate, 223
formal definition, 229
naked, 230
timelike, 230
singularity theorems, 284
Sirius B, 16
spacelike, 64
spaceship paradox, 66, 200
special relativity
defined, 29
spherical geometry, 93
spherical symmetry, 247
spontaneous symmetry breaking, 314
standard cosmological coordinates, 292
static spacetime, 251
stationary, 248
asymptotically, 250
steady-state cosmology, 290
stress-energy tensor, 162, 263
divergence-free, 262
interpretation of, 266
of an electromagnetic wave, 277
symmetry of, 266
string theory, 185
strong energy condition, 276
surface of last scattering, 290
Susskind, Leonard, 232
Sylvester’s law of inertia, 234
symmetrization, 102
spherical, 247
symmetry breaking
spontaneous, 314
Einstein convention, 250, 347
tachyon, 158
tangent space, 242
Tarski, Alfred, 92
Taylor, J.H., 220
tensor, 101, 139
antisymmetric, 102
Penrose graphical notation, 47
rank, 101
symmetric, 102
transformation law, 139
tensor density, 152
tensor transformation laws, 138
Terrell, James, 121
Thomas precession, 73, 171, 213
Thomas, Llewellyn, 83
time dilation
gravitational, 15, 33
nonuniform field, 60
kinematic, 15, 56
time reversal, 107
of the Schwarzschild metric, 211
symmetry of general relativity, 211
time-orientable, 211
timelike, 64
Tolman-Oppenheimer-Volkoff limit, 146
topology, 193
torsion, 181
tensor, 183
trace energy condition, 276
transformation laws, 138
transition map, 197
transverse polarization
of gravitational waves, 337
of light, 130
trapped surface, 284
triangle inequality, 107
Type III solution, 337
Type N solution, 337
Uhlenbeck, 82
uniform gravitational field, 199, 255, 330
unitarity, 203, 228
geometrized, 208
observable, 314
size and age, 314
upsidasium, 26
defined, 46
dual, 47
Penrose graphical notation, 47
summarized, 389
vectors and dual vectors, 100
summarized, 389
velocity addition, 66
velocity four-vector, 124
velocity vector, 124
spacetime, 155
volume expansion, 284
Waage, Harold, 26
wavenumber, 133
gravitational, see gravitational waves
weak energy condition, 276
weight of a tensor density, 152
Wheeler, John, 26
white dwarf, 144
Wilson, Robert, 289
world-line, 21
Euclidean geometry (page 18):
E1 Two points determine a line.
E2 Line segments can be extended.
E3 A unique circle can be constructed given any point as its center and any line segment as
its radius.
E4 All right angles are equal to one another.
E5 Parallel postulate: Given a line and a point not on the line, exactly one line can be drawn
through the point and parallel to the given line.53
Ordered geometry (page 19):
O1 Two events determine a line.
O2 Line segments can be extended: given A and B, there is at least one event such that [ABC]
is true.
O3 Lines don’t wrap around: if [ABC] is true, then [BCA] is false.
O4 Betweenness: For any three distinct events A, B, and C lying on the same line, we can
determine whether or not B is between A and C (and by statement 3, this ordering is
unique except for a possible over-all reversal to form [CBA]).
Affine geometry (page 43):
In addition to O1-O4, postulate the following axioms:
A1 Constructibility of parallelograms: Given any P, Q, and R, there exists S such that [PQRS],
and if P, Q, and R are distinct then S is unique.
A2 Symmetric treatment of the sides of a parallelogram: If [PQRS], then [QRSP], [QPSR],
and [PRQS].
A3 Lines parallel to the same line are parallel to one another: If [ABCD] and [ABEF], then
Experimentally motivated statements about Lorentzian geometry (page 388):
L1 Spacetime is homogeneous and isotropic. No point has special properties that make it
distinguishable from other points, nor is one direction distinguishable from another.
L2 Inertial frames of reference exist. These are frames in which particles move at constant
velocity if not subject to any forces. We can construct such a frame by using a particular
particle, which is not subject to any forces, as a reference point.
This is a form known as Playfair’s axiom, rather than the version of the postulate originally given by Euclid.
L3 Equivalence of inertial frames: If a frame is in constant-velocity translational motion
relative to an inertial frame, then it is also an inertial frame. No experiment can distinguish
one inertial frame from another.
L4 Causality: There exist events 1 and 2 such that t1 < t2 in all frames.
L5 Relativity of time: There exist events 1 and 2 and frames of reference (t, x) and (t0 , x0 )
such that t1 < t2 , but t01 > t02 .
Statements of the equivalence principle:
Accelerations and gravitational fields are equivalent. There is no experiment that can
distinguish one from the other (page 23).
It is always possible to define a local Lorentz frame in a particular neighborhood of spacetime (page 28).
There is no way to associate a preferred tensor field with spacetime (page 142).
Coordinates cannot in general be added on a manifold, so they don’t form a vector space,
but infinitesimal coordinate differences can and do. The vector space in which the coordinate
differences exist is a different space at every point, referred to as the tangent space at that point
(see p. 242).
Vectors are written in abstract index notation with upper indices, xa , and are represented
by column vectors, arrows, or birdtracks with incoming arrows, → x.
Dual vectors, also known as covectors or 1-forms, are written in abstract index notation with
lower indices, xa , and are represented by row vectors, ordered pairs of parallel lines (see p. 48),
or birdtracks with outgoing arrows, ← x.
In concrete-index notation, the xµ are a list of numbers, referred to as the vector’s contravariant components, while xµ would be the covariant components of a dual vector.
Fundamentally the distinction between the two types of vectors is defined by the tensor
transformation laws, p. 138. For example, an odometer reading is contravariant because converting it from kilometers to meters increases it. A temperature gradient is covariant because
converting it from degrees/km to degrees/m decreases it.
In the absence of a metric, every physical quantity has a definite vector or dual vector
character. Infinitesimal coordinate differences dxa and velocities dxa /dτ are vectors, while
momentum pa and force Fa are dual (see p. 141). Many ordinary and interesting real-world
systems lack a metric (see p. 49). When a metric is present, we can raise and lower indices at
will. There is a perfect duality symmetry between the two types of vectors, but this symmetry
is broken by the convention that a measurement with a ruler is a ∆xa , not a ∆xa .
For consistency with the transformation laws, differentiation with respect to a quantity flips
the index, e.g., ∂µ = ∂/∂xµ . The operators ∂µ are often used as basis vectors for the tangent
plane. In general, expressing vectors in a basis using the Einstein notation convention results
in an ugly notational clash described on p. 245.
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