Astronomy 2 – Special Relativity – Part 1 Introduction Norman Gray Autumn 2013

```Astronomy 2 – Special Relativity – Part 1
Introduction
Norman Gray
Autumn 2013
Structure
The structure of the Special Relativity course is as follows.
Part 1: Introduction Central ideas and definitions. Inertial frames. How to measure
lengths and times, and how to synchronise clocks. [One lecture]
Part 2: The postulates Introducing and justifying the two postulates which SR rests
on. [One lecture]
Part 3: Spacetime and the Lorentz Transformation The central mathematical tool.
The Minkowski diagram and the idea of spacetime. The invariant interval. [Four
lectures]
Part 4: Relativistic kinematics Momentum, energy and force. [Four lectures]
Web pages:
http://physci.moodle.gla.ac.uk/course/view.php?id=15
the A2 moodle.
http://www.astro.gla.ac.uk/users/norman/lectures/A2SR/
the main web pages for this SR lecture course.
I’m going to assume that you have at least looked at these notes before the lecture
– not that you’ve necessarily understood them, but that you have a broad idea what I’m
going to be talking about, so that there will be a structure to the way to listen to the
next lecture.
I’ve also taken pains to include a number of exercises, which are generally, but
not always, keyed to particular objectives. You should make an effort to attempt these
exercises: more so than with other subjects, special relativity can seem to be intelligible
right up to the point where you’re required to solve a problem, and you don’t want to
discover this for the first time in the week before the exam.
A2 – Special Relativity
Aims and objectives for Part 1
The point of Aims and Objectives is twofold. They help me keep on track by reminding
me what things it’s important I cover; and they help you follow the course, by reminding
you of the motivation for the material I’m covering. The distinction between the two,
as far as I’m concerned, is simple.
The aims are the point of the course – why you’re doing the course, and why I’m
teaching it. These are the insights you’ll have, and the ideas you’ll understand,
long after the point where you’ve forgotten most of the details. Unfortunately, it’s
easy to claim, but difficult to show, you have this understanding. So. . . .
The objectives are the detailed skills, mastery of which demonstrates that you
have in fact achieved the aims of the course. Hint: it is a short step from objectives
to exam questions.
Aims: You should
1. understand the primacy of events within Special Relativity, and the distinction
between events and their coordinates in a particular frame.
2. appreciate why we have to define very carefully the process of measuring distances
Objectives: You should be able to demonstrate that you can
1. provide concise descriptions of the terms ‘event’, ‘reference frame’, ‘inertial
frame’ and ‘standard configuration’.
2. describe how an observer ascribes a position and time to an event.
3. describe how we might use a network of observers to measure lengths and times.
4. cast a described problem into standard configuration.
Throughout these notes, there are occasional sidenotes, marked like this.
These ‘dangerous bend’ paragraphs provide extra detail or precision, or discuss extra subtleties, which are supplementary to the text surrounding them. You
will typically want to ignore these on a first reading. None of the material in these
paragraphs, nor the material in the occasional ‘dangerous bend sections’, is examinable.

1-2
A2 – Special Relativity
1
The basic ideas
Relativity is simple. Essentially the only new physics which will be introduced in this
course boils down to just:
1. All inertial reference frames are equivalent for the performance of all physical
experiments (the Equivalence Principle);
2. The speed of light has the same constant value when measured in any inertial
frame.
The work you will do in this course consists of (a) understanding what these two
axioms really mean, and (b) examining both their direct consequences, and the way
that we have to adjust the physics we already know.
We will examine these axioms in part 2, study their direct consequences in part 3,
and their consequences for dynamics in part 4, but before we can start on any of this, we
have to understand what the axioms actually mean. For example, what is a ‘reference
frame’, and what is special about ‘inertial’ frames? It turns out that we also have to
be particularly careful about how we use terms like ‘measurement’: we have to ask
precisely what we mean when we talk of measuring distances, in space or time, or how
we would go about synchronising two clocks. It is this that makes SR challenging: the
maths isn’t particularly hard, but we have to put a lot of effort into understanding ideas
we thought were already clear, and try to think precisely about processes we thought
were intuitive, such as measuring the length of a stick.
1.1
Events
An ‘event’ in SR is something that happens at a particular place, at a particular instant
of time. The standard examples of events are a flashbulb going off, or an explosion, or
two things colliding.
Note that it is events, and not the reference frames that we are about to mention,
that are primary. Events are real things that happen in the real world (we omit a mass
of philosophical detail, here); the separations between events are also real (we return to
this in more detail later); reference frames are a construct we add to events to allow
us to give them numbers, and to allow us to manipulate and understand them. That
is, events are not relative – everyone agrees that an event happens. SR is about how
we reconcile the different measurements of an event, that different, relatively moving,
observers make.
1.2
Inertial reference frames
We need to understand first what a reference frame is, and then what is special about
an inertial (reference) frame.
A reference frame is simply a method of assigning a position, as a set of numbers,
to events. Whenever you have a coordinate system, you have a reference frame. The
coordinate systems that spring first to mind are possibly the .x; y; z/ or .r; ; / of
physics problems. Reference frames need not be fixed to a stationary body, though.
A train driver most naturally sees the world in terms of distances in front of the train.
An approaching station can quite legitimately be said to be moving – speeding up and
slowing down – in the driver’s reference frame.
You can generate an indefinite number of reference frames, fixed to various things
moving in various ways. However, we can pick out some frames as special, namely
those frames which are not accelerating.
Imagine placing a ball at rest on a table: you’d expect it to stay in place. Similarly,
if you roll a ball across a table, you’d expect it to move in a straight line. This is merely
the expression of Newton’s first law: ‘bodies move in straight lines at constant velocity,
unless acted on by an external force’. In what circumstances will this not be true?
1-3
A2 – Special Relativity
Suppose you’re sitting in a train which is accelerating out of a station. A ball
placed on a table in front of you will start to roll towards the back of the train, rather
than staying put, as Newton’s first law says it should. This observation only makes
sense from the point of view of someone on the station platform, who sees the ball as
stationary, and the train being pulled from under it. The station is an inertial frame,
and the accelerating train carriage, where Newton’s law appears not to hold, is not.
Similarly, if you are perched on a spinning children’s roundabout, and toss a ball to
someone on the opposite side, it’ll veer off to one side (interpreting this as either ‘it’ll
appear to veer off to one side, from your point of view’ or, more formally, ‘it’ll be
measured to veer off, as observed by someone using the rotating reference frame which
is fixed to the roundabout’). This motion, again, is only immediately intelligible from
the point of view of someone standing watching all this go on, who sees the ball go
exactly where it should, but the person it’s aimed at turn out of the way. The playpark
is an inertial frame, the spinning roundabout is not. In both cases, you can tell if you’re
the one in the non-inertial frame: in the first case you feel yourself pushed back into
the train seat, and in the second case, your perch on the roundabout stops you flying
off, and you feel yourself thrown towards the outside by centrifugal ‘force’, and held in
your place only by the force exerted on you by your seat.
Acceleration and force are intimately connected with the notion of inertial frames
– an inertial frame is one which isn’t accelerated in any way. From that, you would
be correct to conclude that once the train has stopped accelerating, and is speeding
smoothly on its way (we imagine a perfectly smooth track), it becomes an inertial
frame again; if you closed your eyes, you wouldn’t be able to tell if you were on a
moving train or at rest in the station. Anything you can do whilst standing on a station
platform (such as juggling, perhaps), you can also do whilst racing through that station
on a train, irrespective of the fact that, to the person watching the performance from the
platform, the balls you’re juggling with are moving at a hundred miles an hour, or so.
Newton’s second law is more quantitative, since it relates the amount of force
applied to an object, the amount it is accelerated, and the body’s inertial mass, through
the well-known relation F D ma.
We can therefore define an ‘inertial frame’ as follows:
Definition of Inertial Frames: An inertial reference frame is a reference
frame, with respect to which Newton’s second law holds, to an adequate
approximation.
That is, an intertial frame is one which is not accelerating. It follows that inertial frames
are, in the context of SR, infinite in extent; also, that all inertial frames move with
constant velocity, so that no pair of such frames mutually accelerate.
Of course, there is rather more to it than that. This definition suffices for Special
Relativity, but once we consider General Relativity (GR) we have both the
need, and the mathematical tools, for a more fundamental definition. In brief, in GR the
definition of an inertial frame is one which is in free fall, meaning moving freely in a
gravitational field, or freely floating, unaccelerated in interstellar space. This definition is
locally consistent with the definition in SR, but allows us to start to discuss frames which
are mutually accelerating.

To be precise, I shouldn’t really talk of train carriages and station platforms as
inertial frames. Firstly, there are tiny corrections due to the fact that we are on a
rotating, curved, planet; we can ignore these. Secondly, we should be careful when talking
about throwing balls or juggling (as I do repeatedly) within an inertial frame, since, because
of the presence of the force of gravity, a frame sitting on earth is not inertial according
to GR’s stricter definition. However, as long as we are talking about SR rather than GR,
as long as all the relevant motion (of inertial frames) is horizontal, and as long as no-one
throws the ball further than a hundred miles or so (!), denying ourselves any mention of
projectile motion would achieve nothing beyond removing a vivid and natural example to
focus on. If you really want to, you can remove gravity from the examples by imagining
the events taking place not in train carriages going through stations, but in space capsules

1-4
A2 – Special Relativity
flying past asteroids, with some suitably baroque arrangement of air jets or rockets, to
supply the forces when necessary.
The mass which is the constant of proportionality in Newton’s second law is
what defines inertial mass. It is distinct from gravitational mass, which describes
how closely bound the object is to the gravitational field, and is the proportionality constant
in Newton’s law of gravitation – it is this that you measure when you weigh something.
In fact, these two quantities, though logically completely distinct, are always measured to
be equal, and it is the examination of this astonishing observation that leads us to General
Relativity.

1.3
See example 1.1
Measuring lengths and times: simultaneity
How do we measure times? In SR, we repeatedly wish to talk about the time at which
an event happens. If the event happens in front of our nose, we can just look at our own
watch. This is an important point. One of the things we can hold onto in the rest of this
course, is that if two events at the same spatial position happen at the same time, they
are simultaneous for everybody. This is why a time measurement by a local observer
is always reliable. Observers at that same spatial position but moving in different
frames may produce different numbers, but their measurement is, by definition, the
measurement of the time of that event in that frame. Einstein made this particularly
clear:
We must 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.” [1]
If the event happens some distance away (answering a question such as “what
time does the train pass the next signal box?”), however, or if we want to know what
time was measured by someone in a moving frame (answering, for example, “what is
the time on the train-driver’s watch as the train passes through the station?”), things
are not so simple, as most of the rest of this lecture course makes clear. SR is very
clear about what we mean by ‘the time of an event’: when we talk about the time of
an event, we always mean the time of the event as measured on a clock carried by
a local observer, that is, an observer at the same spatial position as the event (which
is rather unfortunate if the event in question is an explosion of some type – but what
are friends for?), who is stationary with respect to the frame they represent. We will
typically imagine more than one observer at an event; indeed we imagine one local
observer per frame of interest, stationary in that frame, and responsible for reporting
the space and time coordinates of the event ‘as measured in that frame’.
We suppose that a frame has a limitless supply of observers, scattered throughout
space. These ‘same-frame’ observers have two properties of importance: as well as
being at rest in their frame, the clocks they carry are synchronised with all others in
their frame. This implies a specific procedure for synchronising clocks: this isn’t too
difficult to define, within the context of the two axioms mentioned at the start of Sect. 1
and discussed in greater detail in the next part, and both Rindler [2, §2.5] and Taylor
and Wheeler [3, §2.6], for example, give details. When an observer is asked for the
coordinates of an event which happened near them (in their frame, only ever in their
frame), they can respond with the spatial coordinates of their fixed station, and the time
the event happened as noted from their local clock.
If we find a number of observers located immediately adjacent to an event, the
above doesn’t imply that all observers report the same time. The observers may have
watches set to different time zones, their watches may be designed to run at different
rates (perhaps someone has an exotic watch that ticks out 1000ths of a day), someone’s
computer might be ticking out seconds since the start of 1970, or the watches may be
1-5
A2 – Special Relativity
running at different rates for relativistic reasons that we’ll come to later. We assume
that despite these complications, all of the clocks are good clocks: whatever they tick
out, they do so in a linear way.

The key mathematical observation is that there is a set of linear transformations
between the times being shown on the different observers’ clocks and watches.
For example, imagine a trainful of world leaders speeding through a busy station
when a protestor throws a paint-bomb at the train. Questioned afterwards, a white-witha-hint-of-puce commuter explains that it happened exactly when the 7.29 express was
due, and that he is the 117th commuter along from the coffee-stall. Describing the
same incident, the member of the entourage nearest the impact describes it happening
three hours before the conference starts (the only timescale which matters to the jetlagged passengers on the train), and two carriages up from the president. The two local
observers have different origins of space and of time, and the two sets of observers, on
the train and on the platform, have clocks which are synchronised with respect to the
others in their own frame, but have little relation to the clocks in the other.
So much for coordinates; how should we measure the length of a moving object?
The obvious methods – some variant of laying a ruler along the object to be measured
and examining the marks on the ruler – make too many assumptions about how the
world is. To measure something accurately this way, you have to correct for the time it
takes for light to travel from the ruler and the (moving) object to you. You obviously
have to measure both ends of the object simultaneously: does that ‘simultaneously’
depend on where you’re standing relative to the object? The obvious approaches to this
problem beg too many questions to which SR provides surprising answers.
The way we measure lengths and times in SR is therefore as follows. We station
observers at strategic points in the reference frames of interest (in principle, we have
at our disposal an infinite array of observers scattered throughout space). We can
know these observers’ coordinates in one frame or another. The observers make
measurements of events which happen at their location, and afterwards compare notes
and draw conclusions. For example, you would measure the ‘length of a rod’ by
subtracting the coordinates of the two observers who observed opposite ends of the rod
at a prearranged time.
We can summarise what we have discovered so far. This approach relies on three
things.
1. It requires a specific procedure for synchronising clocks.
2. It assumes that there is no ambiguity about two events at the same position and
time being regarded as simultaneous. This has to be true: the fact that two cars
crash – because they were in the same position at the same time, and so are
attempting to occupy the same space simultaneously – cannot depend in any way
on your point of view.
y
3. We assume that moving clocks measure the passage of time accurately. We do
not assume anything about accelerating clocks, because in SR we largely avoid
discussion of acceleration, but we do assume that there is nothing magical about
motion which causes clocks to go wrong. This is known as the clock hypothesis,
and is related to the Equivalence Principle which is one of the axioms of SR.
y0
v
1.4
z0
z
x0
x
Figure 1: Standard config.
1-6
Standard configuration
Finally, a bit of terminology to do with reference frames. Two frames S and S 0 , with
spatial coordinates .x; y; z/ and .x 0 ; y 0 ; z 0 / and time coordinates t and t 0 are said to be
in standard configuration if:
1. they are aligned so that the .x; y; z/ and .x 0 ; y 0 ; z 0 / axes are parallel;
A2 – Special Relativity
2. the frame S 0 is moving along the x axis with velocity V ;
3. we set the zero of the time coordinates so that the origins coincide at t D t 0 D 0
(which means that the origin of the S 0 frame is always at position x D V t).
This arrangement, shown in Fig. 1, makes a lot of example problems somewhat easier,
and is the arrangement assumed by the ‘Lorentz Transformation’ which we will meet
later.
When we refer to ‘frame S ’ and ‘frame S 0 ’, we will interchangably be referring
either to the frames themselves, or to the sets of coordinates .t; x; y; z/ or .t 0 ; x 0 ; y 0 ; z 0 /.
1.5
Rest, moving and stationary frames – be careful!
Frame S 0 will often be the rest frame; however, it should always be the rest frame
of something. Yes, it does seem a little counterintuitive that it’s the ‘moving frame’
that’s the rest frame, but it’s called the rest frame because it’s the frame in which the
thing we’re interested in – be it a train carriage or an electron – is at rest. It’s in the
rest frame of the carriage that the carriage is measured to have its rest length or proper
length. In general, talking of the ‘rest’, ‘moving’ and ‘stationary’ frames is a bit sloppy,
because (and this is part of the point) what’s ‘moving’ and what’s ‘stationary’ is relative
to who’s doing the observing: the carriage is moving in the observer’s rest frame,
and the observer is moving in the carriage’s rest frame. For this reason it’s generally
better to talk of ‘the frame of the station platform’ (or equivalently ‘the rest frame of
the platform’) or ‘the (rest) frame of the particle’, but to avoid sounding pointlessly
pedantic, we’ll stick to the more informal version most of the time, if the identity of the
frames is clear from context. But whenever you see or write ‘rest frame’ or ‘stationary
frame’, you should always ask the question ‘whose frame?’
Similarly, it never makes sense to talk of ‘the primed frame’ or ‘the unprimed
frame’ to refer to ‘the moving frame’ unless you’ve previously made clear which frame
is which, through a diagram or through a textual explanation. Usually, for the sake of
consistency, we draw diagrams so that the ‘moving’ frame is the primed one, but that’s
just a neat convention, and not anything you can start a calculation with.
You should be extremely precise here, not just for the sake of your reader (though
remember that that reader might be your exam marker!), but because being fussily
precise here helps you organise a problem in your head, and so make a good start on
solving it. Quite a lot of relativity problems are actually quite simple if (and only if)
people are clear in their own head what they’re talking about.
2
See example 1.2
When learning relativity, even more than with other subjects, you benefit from hearing
or reading things multiple times, from different authors, and from different points of
view. I mention a couple of good introductions below, but there is really no substitute
for going to section ‘Physics C25’ in the library, looking through the SR books there,
and finding one which makes sense to you.
I recommend in particular the books listed below, in rough order of preference.
Taylor & Wheeler [3] is an excellent account of Special Relativity, written in
a style which is simultaneously conversational and rigourous. It introduces the
subject from a geometrical perspective from the very beginning, which makes it
quite natural to make links to General Relativity.
Rindler [2] contains a clear account of the subject, using a rather similar tack to my
own in this course. Rindler takes great pains to confront and explain the subtleties
involved in the subject, and explains things sparsely but precisely. I think that both
Rindler and Taylor & Wheeler see SR essentially as a prologomenon to GR, and
1-7
A2 – Special Relativity
are consequently admirably careful in their treatment of SR. Only the first half
or so of this book covers SR: the remainder is about GR, and goes substantially
beyond the scope of this course.
French [4] is a justly well-known account. It takes a rather traditional approach
to the subject: it’s good on the experiments, such as the Michelson-Morley
experiment, but lacks excitement for me.
Moore [5] is another book which takes a fairly geometrical approach. It takes
pains to explain the ideas of inertial frames clearly.
Carroll & Ostlie [6] is the set book for Astronomy 2, and has a chapter on
relativity. However while their introduction to SR is good, their approach and
general notation are significantly different from mine (I think they underemphasise
the geometrical aspects), enough that it might be confusing to use this text as a
support for the lectures.
Schutz [7] is a textbook on General Relativity (in fact it’s the set book for the
honours/masters course on that subject), and is therefore generally beyond the
scope of this course. However the first chapter gives a breakneck account of
SR, and the second a similarly abstract account of vector analysis, which were
influential on parts 3 and 4 of this course. This book is available as an electronic
resource via the university library.
Einstein’s own popular account of relativity [8] is very readable, though it’s
naturally a little old-fashioned in places. You can see the influence of this book,
and its examples, in many later SR textbooks.
The Principle of Relativity [9] is a collection of (translations of) original papers
on the Special and General theories, including Einstein’s paper of 1905 [8], but
also some earlier papers by Lorentz suggesting interpretations of the MichelsonMorley experiment. The first few sections, at least, of the 1905 paper are worth
reading as a current introduction to SR.
All of these books, except Carroll & Ostlie, which you are supposed to have yourself,
are on restricted loan in the Library’s Short Loan collection.
Barton [10] is another book with a rather traditional approach. It goes into a
great deal of detail about experimental corroboration of the results of SR (a little more
modern than French), and explains its results with great care. These are both valuable
features, but it might be best to treat this book as a supporting resource, as it would be
easy to find the quantity of detail overwhelming.
Other books which have interesting notes on SR include: David Mermin’s book
[11] is a (very good) collection of essays, covering a wide variety of topics from the
practice of physics, through quantum mechanics, to relativity. Chapters 19 to 21 are
about unusual approaches to the teaching of relativity. John Bell’s book [12] is another
collection of (sometimes rather high-level) essays; they’re mostly illuminating essays
on quantum mechanics, but chapter 9 is about a rather classically-oriented approach to
relativity, which would be rather nicely partnered with Lorentz’s papers in [9]. Another
book to mention, just because I like it, is Malcolm Longair’s collection of lecture
notes and case studies, covering several areas in theoretical physics [13]: it pulls no
mathematical punches, but is full of insights, including a chapter on SR.
There are several popular science books which are about, or which mention,
relativity – these aren’t to be despised just because you’re now doing the subject
‘properly’. These books tend to ignore any maths, and skip more pedantic detail (so
they won’t get you through an exam), but in exchange they spend their effots on the
underlying ideas. Those underlying ideas, and developing your intuition about relativity,
are things that can sometimes be forgotten in more formal courses. I’ve always liked
[14], which is a cartoon book but very clear, and I’ve heard good things about Brian
Cox’s Why does E D mc 2 ?.
1-8
A2 – Special Relativity
References
[1] Albert Einstein. Zur Elektrodynamik bewegter Körper (on the electrodynamics
of moving bodies). Annalen der Physik, 17:891, 1905. Reprinted in [9].
[2] Wolfgang Rindler. Essential Relativity: Special, General and Cosmological.
Springer-Verlag, 2nd edition, 1977.
[3] Edwin F Taylor and John Archibald Wheeler. Spacetime Physics. W H Freeman,
2nd edition, 1992.
[4] A P French. Special Relativity. Chapman and Hall, 1971.
[5] Thomas A Moore. A Traveler’s Guide to Spacetime. McGraw-Hill, 1995.
[6] Bradley W Carroll and Dale A Ostlie. Modern Astrophysics. Addison Wesley,
1996.
[7] Bernard F Schutz. A First Course in General Relativity. Cambridge University
Press, second edition, 2009.
[8] Albert Einstein. Relativity: The Special and the General Theory. Routledge
Classics, 2001.
[9] H A Lorentz, A Einstein, H Minkowski, and H Weyl. The Principle of Relativity.
Dover, 1952.
[10] Gabriel Barton. Introduction to the Relativity Principle. John Wiley and Sons,
1999.
[11] N David Mermin. Boojums All the Way Through. Cambridge University Press,
1990.
[12] J S Bell. Speakable and Unspeakable in Quantum Mechanics. Cambridge
University Press, 1987.
[13] M S Longair. Theoretical Concepts in Physics. Cambridge University Press,
1984.
[14] Joseph Schwartz and Michael McGuinness. Introducing Einstein. Icon Books,
1999. Used to be known as ‘Einstein for Beginners’.
Examples
Example 1.1 (section 1.2)
Which of these are inertial frames? (i) A motorway bridge; (ii) a stationary car; (iii) a
car moving at a straight line at a constant speed; (iv) a car cornering at a constant speed;
(v) a stationary lift; (vi) a free-falling lift (the last one is rather subtle and relates to the
dangerous-bend paragraphs at the end of Sect. 1.2). (Objective 1)
Example 1.2 (section 1.5)
[Come back to this question after you’ve studied the length-contraction and timedilation sections of Part 3.] State the time dilation and length contraction formulae,
explaining precisely what the various symbols mean.
Revision c3fc251e64d6, 2013-10-24
1-9
```