A Student’s Guide to Special Relativity

This compact yet informative Guide presents an accessible route through Special
Relativity, taking a modern axiomatic and geometrical approach. It begins by
explaining key concepts and introducing Einstein’s postulates. The consequences of
the postulates – length contraction and time dilation – are unravelled qualitatively and
then quantitatively. These strands are then tied together using the mathematical
framework of the Lorentz transformation, before applying these ideas to kinematics
and dynamics. This volume demonstrates the essential simplicity of the core ideas of
Special Relativity, while acknowledging the challenges of developing new intuitions
and dealing with the apparent paradoxes that arise. A valuable supplementary resource
for intermediate undergraduates, as well as independent learners with some technical
background, the Guide includes numerous exercises with hints and notes provided
online. It lays the foundations for further study in General Relativity, which is
introduced briefly in an appendix.

n o r m a n g r ay is in the School of Physics and Astronomy, University of Glasgow,

where he has, over the last 20 years, taught courses covering both special and general
relativity. Since his PhD in particle theory, he has had a varied research career, most
recently concentrating on the interface between astronomy and computing. He is
author of A Student’s Guide to General Relativity (2019), which has been a successful
addition to Cambridge University Press’s growing series of Student’s Guides.
Other books in the Student’s Guide series:

A Student’s Guide to Laplace Transforms, Daniel A. Fleisch

A Student’s Guide to Newton’s Laws of Motion, Sanjoy Mahajan

A Student’s Guide to the Schrödinger Equation, Daniel A. Fleisch

A Student’s Guide to General Relativity, Norman Gray

A Student’s Guide to Analytical Mechanics, John L. Bohn

A Student’s Guide to Atomic Physics, Mark Fox

A Student’s Guide to Infinite Series and Sequences, Bernhard W. Bach Jr.

A Student’s Guide to Dimensional Analysis, Don S. Lemons

A Student’s Guide to Numerical Methods, Ian H. Hutchinson

A Student’s Guide to Waves, Daniel A. Fleisch, Laura Kinnaman

A Student’s Guide to Langrangians and Hamiltonians, Patrick Hamill

A Student’s Guide to Entropy, Don S. Lemons

A Student’s Guide to the Mathematics of Astronomy, Daniel A. Fleisch, Julia

Kregenow

A Student’s Guide to Vectors and Tensors, Daniel A. Fleisch

A Student’s Guide to Data and Error Analysis, Herman J. C. Berendsen

A Student’s Guide to Fourier Transforms, J. F. James

A Student’s Guide to Maxwell’s Equations, Daniel A. Fleisch
A Student’s Guide to Special Relativity

N O R M A N G R AY
University of Glasgow
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I do not know what I may appear to the world, but to
myself I seem to have been only like a boy playing on the
sea-shore, and diverting myself in now and then finding a
smoother pebble or a prettier shell than ordinary, whilst
the great ocean of truth lay all undiscovered before me.
Isaac Newton, as quoted in Brewster, Memoirs of
the Life, Writings, and Discoveries of Sir Isaac Newton,
Vol. 2, p. 407.

The eternally incomprehensible thing about the universe

is its comprehensibility. Albert Einstein (1936a)
 Contents

Preface page xi
Acknowledgements xv
Aims xvi
1 Introduction 1
1.1 The Basic Ideas 1
1.2 Events 3
1.3 Inertial Reference Frames 3
1.4 Simultaneity: Measuring Times 7
1.5 Simultaneity: Measuring Lengths 9
1.6 The Clock Hypothesis 10
1.7 Standard Configuration 12
1.8 Further Reading 13
Exercises 16
2 The Axioms 18
2.1 The First Postulate: the Principle of Relativity 18
2.2 The Second Postulate: the Constancy of the Speed of Light 27
Exercises 29
3 Length Contraction and Time Dilation 32
3.1 Simultaneity 32
3.2 Length Contraction and Time Dilation, Qualitatively 34
3.3 The Light Clock 37
3.4 The Horizontal Light Clock: Length Contraction 40
3.5 Is There Anything I Can Hold on To? 42
Exercises 43
4 Spacetime and Geometry 45
4.1 Natural Units 46
4.2 The Minkowski Diagram 50

vii
viii

4.3 Plane Rotations 55

4.4 The Invariant Interval 56
4.5 Changes of Frame, and Perspective 61
4.6 Length and Time in the Minkowski Diagram 63
4.7 Worldlines and Causality 66
Exercises 68
5 The Lorentz Transformation 72
5.1 The Derivation of the Lorentz Transformation 72
5.2 Addition of Velocities 75
5.3 The Invariant Interval and the Geometry of Spacetime 77
5.4 Proper Time and the Invariant Interval 78
5.5 Applications of the Lorentz Transformation 79
5.6 The Equations of Special Relativity in Physical Units 83
5.7 Paradoxes 84
5.8 Some Comments on the Lorentz Transformation 93
Exercises 98
6 Vectors and Kinematics 103
6.1 Three-Vectors 103
6.2 Four-Vectors 105
6.3 Velocity and Acceleration 110
6.4 Velocities and Tangent Vectors 114
6.5 The Frequency Vector, and the Doppler Shift 115
Exercises 118
7 Dynamics 125
7.1 Energy and Momentum 125
7.2 Photons 131
7.3 Relativistic Collisions and the Centre-of-Momentum Frame 132
7.4 But Where’s This Mass Coming From? 136
7.5 More Unit Fun: an Aside on Electron-volts 137
7.6 Relativistic Force 138
7.7 An Example: Compton Scattering 139
7.8 Not The End 140
Exercises 141
Appendix A An Overview of General Relativity 145
A.1 Some Thought Experiments on Gravitation 146
A.2 Geometry 153
A.3 Gravity 160
A.4 Solutions of Einstein’s Equation 163
 ix

Appendix B Relativity’s Contact with Experimental Fact 177

B.1 Special Relativity 178
B.2 General Relativity: Classical and Post-classical Tests 183
B.3 A Closer Look at the 1919 Eclipse Observations 186
Appendix C Maths Revision 194
C.1 Complex Numbers 194
C.2 The Hyperbolic Functions 196
C.3 Linear Algebra 198
Appendix D How to Do Calculations: a Recipe 200
D.1 Key Things to Remember 200
D.2 A Checklist for Relativity Problems 200
D.3 Which Equation When? 202
D.4 Rest, Moving and Stationary Frames: Be Careful! 203
D.5 Length Contraction, Time Dilation, and ‘Rest Frames’ 204

References 205
Index 211
 Preface

This book is an introduction to Special Relativity (SR), which intends to be

both comprehensive, in the sense of covering most of what a physicist will
be presumed to know about relativity, and accessible to those at a reasonably
early stage of their physics education. These aims are not incompatible.
Learning about relativity does not require a huge volume of material to be
learned, and the maths is not elaborate (the most advanced maths we use is
basic calculus).
The very significant challenge to approaching this area of physics is that it
is probably the first area we encounter where we cannot rely on our intuition,
and where we must rely on abstract arguments to navigate past our intuitions
to somewhere we could not otherwise reach. I fully embrace this in the
axiomatic approach of this book: we start with the two axioms or postulates
of SR, and then follow their consequences.
But SR is not just an abstraction. It is instead a foundational theory of
how our physical world is constructed, and the consequences we deduce
must match the physical world we find ourselves in. Making the link from
abstraction to nature is of course a little tricky – there are a lot of thought
experiments, and a large number of trains moving at implausibly high speeds,
in the pages ahead of you – but our destination is physics, not maths.
It is also possible to go too far the other way, and over-stress the continuity
with familiar ideas, downplaying the dislocation that the second axiom
necessarily implies. I don’t think this is a useful approach, because we cannot
really evade the disorientation which arises from the loss of simultaneity,
nor evade the necessity of thinking of time as something other than a rope
along which a sequence of snapshots of ‘the present’ are strung.
The nature of the middle path I wish to plot out in this book is therefore,
I hope, clear. I want you as the reader to appreciate SR as an account of

xi
xii Preface

the geometrical structure of our universe, and also see that structure as
something with concrete physical consequences.
I developed the material here over a number of years, in the course of
teaching an early-undergraduate course in SR at the University of Glasgow.
It has therefore been read, puzzled over, commented on, and corrected by
multiple cohorts of students, and the distribution of exercises in each chapter
reflects both their questions and difficulties, and the inevitable requirement
that I formally examine their understanding. That is, this book is not an
armchair exercise in finding ‘interesting topics in SR’, but an attempt to
bring students to the understanding of SR that they will need in their future
work, and which will expand their intellectual horizons. And of course
those lectures were not just a discussion group, but a course with an exam at
the end of it; the presence of that exam influenced the selection and shape
of material in a way which may also be of use to you.
The text here is therefore suitable to support a course: I covered the
content, without appendices, in ten fairly busy lectures (and did not declare
all of it examinable). But in converting the text to a book I have also had
in mind both the independent reader working their way into the subject,
and the student of another course who wants light shone on the landscape
from a different direction. I resisted adding new core material in this pro-
cess, but have expanded explanations here and there, and added various
supplementary comments, either historical, mostly in discursive footnotes,
or technical, in ‘dangerous-bend’ sections.
Although I believe my students’ experience validates this text as one good
route to SR, I insist that it is not the only route, nor even the only good
one. Amongst the unusual things about relativity is that there exists no
single royal road to an understanding of it, nor even a single obvious way of
mapping the territory. Relativity always makes more sense the second time
you read about it (and makes still more sense the first time you explain it to
someone else), and so it is always useful to expect to read more than one
introduction; to that end, I point to a number of other books in Chapter 1.
My goal for this book is not to disperse, but to join, this swarm of alternatives.
Throughout, I have freely referred you to other textbooks, to review articles,
and to original research articles. It is not necessary to follow each of these
outward-pointing links to get an understanding of the subject, but I hope
they indicate the richness of the subject’s connections with the rest of physics,
and with the history of physics and astronomy.
It’s easy for me to talk about the simplicity and minimalism of SR: you
may find the claim a little surprising, if you look ahead and see a lot of solid
text, including a dense undergrowth of primes and subscripts. A lot of this
 Preface xiii

is re-explanation, however. In Chapter 5, for example, I discuss some of the

‘paradoxes’ of SR, deliberately pointing to results of relativistic arguments
which are so unexpected as to seem surely wrong, and then going on to
discuss how those results are not only consistent, but merely a different
way of looking at ideas we have already encountered, so that we acquire a
more textured understanding of ideas that seemed insubstantial the first
time around. There is a lot of internal cross-reference knitting the book
together – pointing to another version of an idea introduced previously, or
presenting a preview of something done more carefully later – but I think of
it as having a single short thread running from the first chapter to the last,
and when you have undone and re-done these chapters in your head, along
with any other broader reading, I hope you will see that thread, too.

Each chapter starts with a few high-level aims. These are collected on
page xvi, and I think of them as the ‘intellectual table of contents’ of the
book.
The exercises at the end of each chapter are, I think, important in solidify-
ing your understanding of SR. A number of them are annotated with 𝑑+ , 𝑑−
or 𝑢+ to indicate ones which are more or less difficult, or more particularly
useful, than those around them.
I have included an appendix on the link to General Relativity and gravi-
tation. Although it is of course auxiliary to a book on Special Relativity, I
believe it is useful to show that the gap between the two areas is narrower
than it may at first appear, and that we can build a bridge which lands a
small but significant way into the new territory. The approach I have taken
to SR is designed to give this bridge as firm a foundation as possible, on the
SR side.
Similarly, the appendix on relativity’s contact with experiment is auxiliary
but, I hope, both useful and interesting. It it not concerned with the question
‘is relativity right?’ – the answer to which it takes as obviously ‘yes’ – but
instead steps back and discusses the nature of the relationship between
relativity, both special and general, and experimental corroboration, and
indeed has something to say about the relationship between corroboration,
science in general, and scientists as a community.
Throughout the book, I have included a number of historical asides. Spe-
cial Relativity does not have an unusually intricate history, but these asides
are present partly because they add an extra dimension to our knowledge
of the topic, but also because, by hinting at an alternative intellectual path
xiv Preface

not followed, they can colour in our understanding of the ideas as they have
developed in fact. This also seems a good point at which to mention my
deliberate habit of styling scientists’ names, when they become adjectives,
in lowercase: thus ‘Newton’s laws’, but ‘newtonian physics’. This is partly
because, by the time a topic acquires an adjective like this, it has absorbed
the work of a multitude of people beyond any original creator (see also
‘Stigler’s law’), and evading the ownership question with a lowercase letter
seems both fairer and less cumbersome than a hyphenated list of all an
idea’s retrospectively discovered co-discoverers. Even Newton would have
to go back to school if presented with a book on contemporary classical
mechanics.

Throughout these notes, there are occasional sidenotes, and one

or two complete sections, which are marked with a symbol like
this. These ‘dangerous bend’ paragraphs provide supplementary detail or
precision, or discuss extra subtleties, which are tangential but interesting,
or which comprise extra discussion of ideas which students have in the past
found confusing or easy to misunderstand. You will typically want to ignore
these on a first reading, and I will sometimes presume you are re-reading,
here, by referring in passing to ideas which are introduced only later in the
book.
 Acknowledgements

These notes have benefitted from very thoughtful comments, criticism and
error-checking, received from both colleagues and students, over the years
this course has been presented, for which I am very grateful. In particular I
would like to thank Richard Barrett, Andrew Conway, and Susan Stuart for
detailed criticial comments when turning the notes into a book.
Thank you also to Cleon Teunissen for help with the history of the term
‘Invariantz-Theorie’ (p. 98, note 9); to the contributors to Wikimedia Com-
mons for the image in Figure 1.3 (and so many resources elsewhere); and to
the Historical Naval Ships Association for the scan of Figure 1.4. And thank
you, finally, to the editorial staff at CUP, for their encouragement, precision,
and patience.

xv
 Aims

My goal is that you should:

1.1. understand the importance of events within Special Relativity, and
the distinction between events and their coordinates in a particular
frame;
1.2. appreciate why we have to define very carefully the process of mea-
suring distances and times, and how we go about this;
2.1. understand the two axioms of SR;
2.2. appreciate the significance and inevitability of the immediate conse-
quences of those axioms;
2.3. understand the ideas of a coordinate transformation, and of the co-
variance of an equation under a coordinate transformation;
3.1. understand why the concept of simultaneity is problematic in the
context of SR, and how we resolve these problems;
4.1. appreciate the role of geometry in understanding spacetime, specif-
ically the importance of the invariant interval and the Minkowski
diagram;
4.2. internalise the utility of units where 𝑐 = 1, as the natural units for
discussing events in spacetime;
5.1. understand the derivation of the Lorentz transformation, and recog-
nise its significance;
6.1. understand the concept of a 4-vector as a geometrical object, and the
distinction between a vector and its components;
7.1. understand relativistic energy and momentum, the concept of energy-
momentum as the magnitude of the momentum 4-vector, and conser-
vation of the momentum 4-vector; and
7.2. understand the distinction between invariant, conserved and constant
quantities.

xvi
 1
Introduction

Aims: you should:

1.1. understand the importance of events within Special Relativity, and the
distinction between events and their coordinates in a particular frame;
and
1.2. appreciate why we have to define very carefully the process of measuring
distances and times, and how we go about this.

1.1 The Basic Ideas

Relativity is simple. Essentially the only new physics which will be intro-
duced in this text boils down to two axioms:

1. All inertial reference frames are equivalent for the performance of all
physical experiments;
2. The speed of light has the same constant value when measured in any
inertial frame.

The work of understanding relativity consists of (i) appreciating what

these two axioms really mean, (ii) examining their direct consequences, and
(iii) thus discovering the way that we have to adjust the physics we already
know. Each of these steps presents challenges.
So when I say that ‘relativity is simple’, I do not mean that it is easy, simply
that this list of axioms is a short one. We will discover that these axioms
are more subtle than may at first appear, and that they lead to conclusions
which go against our usual intuitions. It is here that the difficulties arise:
the maths isn’t particularly hard, but we have to put a lot of effort into

1
2 1 Introduction

understanding ideas we thought were already clear, and try to think precisely
about processes we thought were intuitive: what do we mean when we talk
about ‘the length of a stick’?
Our first step is to understand the axioms, and we’ll start on that in the
next chapter. What does it mean to talk about a ‘transformation between
frames’, and why should the speed of light have such a significant place in
this story?
Chapters 3 and 4 are about the reasonably direct consequences of the
axioms. It’s in this pair of chapters that we discover the most surprising
features of SR – length contraction and time dilation – first qualitatively then
quantitatively. This pair of chapters is where the most profound conceptual
challenges are.
The various strands here are tied together by the main calculational tool
of SR, the ‘Lorentz transformation’, which we derive and study in Chapter 5.
This chapter is quite a long one, and detailed, so that it would be fairly easy to
get lost in it. However it is really only Section 5.1 that has the new material,
and the rest of the chapter is, again, an exploration of the consequences.
Those consequences are often surprising, and will frequently, I think, cause
you to re-read and re-think Chapters 3 or 4.
In Chapters 6 and 7 we look outwards to the rest of physics, and learn
how the principles of SR oblige us to recast familiar ideas such as velocity,
acceleration, frequency, momentum, mass, and energy.
Finally, in Appendix A, we look at how we can apply these ideas more
generally, and discover Einstein’s theory of gravity: General Relativity (GR).
We can’t dive too far beneath the surface here, without more advanced
mathematics, but our understanding of SR will allow us to at least see the
connection to the main structures of GR, and how they link to gravity.
Prior to all that, however, there are a couple of bits of terminology that
it’s useful to clarify in this first chapter, to avoid breaking the flow of the
argument in Chapter 2. In particular:

Events These are things like a light-flash, or a bang, which happen at a

particular place and time.
Reference frames These are the coordinate systems that we use to de-
scribe where and when events happen. We are concerned, in SR, with
the multiple reference frames that may be relevant in our analysis of
the physical world, and how the measurements in these frames relate
to each other. We need to pay particular attention to the special case
of the ‘inertial frame’.
Measuring lengths and times We have to be quite careful about how
 1.2 Events 3

we measure distances, in space or time. The obvious ways of doing

this contain ambiguities which can easily lead to confusion.

I expand on each of these points below. The following sections are rather
short (and possibly a little dry), but we will use the ideas in them again and
again and again.
It’s probably a good idea to re-read these sections repeatedly as you work
through the rest of the text. It possibly follows that these remarks might
not make perfect sense first time; they may even at first seem absurdly over-
precise, since they are making distinctions which may appear unnecessary
until you have understood some of the rest of the material.

1.2 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 a banger or firecracker exploding, or two things colliding.
There is nothing relative about an event: if two cars crash and metal
is bent, there is no ‘point of view’ from which the crash did not happen.
Although this may seem at this point to be too obvious to be worth stating,
we will discover that more things than we may expect are relative to our
‘point of view’, and we will use events as our way of navigating through the
puzzles this produces.
We will soon discover that, although we can all agree that a particular
event did happen, we might well have different answers to the questions
‘where?’ and ‘when?’ These are questions that we can answer using a
reference frame. [Exercise 1.1]

1.3 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, and I will use the two terms almost interchangeably.
The coordinate systems that spring first to mind are possibly the (𝑥, 𝑦, 𝑧)
or (𝑟, 𝜃, 𝜙) of physics problems. Reference frames need not be fixed to a
stationary body: a train driver most naturally sees the world in terms of
4 1 Introduction

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.
In mechanics problems, we are used to thinking of time as a free variable:
often, the point of a physics problem is to work out how the position of a
thrown ball, for example, varies in time – what is 𝑥(𝑡)? When thinking of
events, however, it can be useful to think of them as being located using four
coordinate numbers, (𝑡, 𝑥, 𝑦, 𝑧). We will come back to this point of view in
Chapter 4.
You can generate any number of reference frames, associated with various
things moving in various ways. In the context of SR, 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 circumstance – that is, in which frames – will this not be true?
Suppose you’re sitting in a train which is accelerating out of a station.1
A ball placed on a table in front of you will start to roll towards the rear of
the train, rather than staying put in the way that Newton’s first law seems to
say it should. This observation makes perfect 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. But in a reference frame attached to
the train, where ‘position’ is perhaps measured as the distance, 𝑥(𝑡), from
the rear of the train, this position will change without any force acting on
the ball. We refer to the station as an ‘inertial frame’, and the accelerating
train carriage, with respect to which Newton’s law appears not to hold, as
non-inertial. Similarly, if you are perched on a spinning children’s round-
about, and toss a ball to someone on the opposite side, it veers off to one
side (interpreting this as either ‘it appears to veer off to one side, from your
point of view’ or, more formally, ‘it will be measured to veer off, as observed
by someone using the rotating reference frame which is fixed to the round-
about’). This motion, again, is 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 catcher rotate out of the way. The playpark

1 We’re going to hear an awful lot more about this train. Although I will occasionally vary the
examples by talking about rockets or boats, a train going past a station platform presents
such an immediate picture of two reference frames, in constant relative motion, that it will
be hard to avoid. I see no reason for wanton innovation with respect to this particular aspect
of the subject.
 1.3 Inertial Reference Frames 5

is an inertial frame, the spinning roundabout is not. In both cases, you can
tell whether 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, it’s only
your grip on the roundabout that stops you flying off, pulled towards the
outside by centrifugal ‘force’.
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, 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, even though, to the person
watching the performance from the platform, the balls you’re juggling with
are moving at a hundred kilometres an hour, or so.
What we have concluded here is that, although different observers may
reasonably ascribe different coordinates to events, and different speeds,
there is no ambiguity about who is accelerating or not. If you are on a train
picking up speed as it leaves a station, you can feel the pressure of the seat
on your back, and be under no illusion that you are not moving, and there is
no point of view from which the drink on the table in front of you does not
look likely to spill. We will have more to say about this point in Section 2.1.
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 = 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 first law holds.
All this being said: don’t over-think this. An inertial frame is one that isn’t
accelerating.
Note that, in the context of SR, inertial frames are infinite in extent; also,
since all inertial frames move with constant velocity, it follows that no pair
of such frames mutually accelerate. [Exercise 1.2]

1.3.1 Further Remarks on Frames

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.
6 1 Introduction

In brief, in GR the definition of an inertial frame is one which is in free

fall, meaning one which is moving freely in a gravitational field, or freely
floating, unaccelerated in interstellar space. As in SR, this is a frame in
which Newton’s first law still holds. This definition is locally consistent with
the definition in SR, but allows us to start to discuss inertial frames which
are mutually accelerating (in the specific sense that the second derivative
of the separation is non-zero), such as two free-falling objects on opposite
sides of the Earth. I discuss this at greater length in Appendix A.
If we acknowledge the existence of GR then, to be nit-pickingly 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 the
curved surface of a rotating planet; we can ignore these in the huge majority
of cases. Secondly, we should be careful when talking about throwing balls
or juggling 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 kilometres 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
flying past asteroids, with some suitably baroque arrangement of air jets or
rockets, to supply the forces when necessary.
Also, this section is one of the few places in this text where I mention
‘acceleration’ (another is coming up in Section 1.6, when I talk of the ‘clock
hypothesis’). This is not because SR ‘cannot deal with acceleration’ as you
might see mistakenly claimed, but simply because the novel and counter-
intuitive relativistic effects, that we are going to discover in the chapters
to come, are not a result of accelerating frames or observers, but instead
of non-accelerating (that is, inertial) observers in mutual motion at large
relative velocity. Similarly, when we (implicitly) talk of motion under gravity
– such as when we talk of throwing a ball – it is perfectly reasonable for us
to do so using a newtonian theory of gravity (𝐹 = 𝑚𝑔) rather than anything
distractingly exotic.
The mass which is the constant of proportionality in Newton’s second law
is what defines inertial mass – it indicates, roughly speaking, how resistant
an object is to being accelerated by an applied force. This is distinct from
the gravitational mass of an object, which describes how much gravitational
field the object generates – in Newton’s law of gravitation, 𝐹 = 𝐺𝑀𝑚∕𝑟2 ,
 1.4 Simultaneity: Measuring Times 7

both masses are gravitational masses. It turns out, however, that whatever
the composition or construction of an object, its gravitational and inertial
masses, though logically completely distinct, are always measured to be
equal. This fact is more surprising than it may at first appear; we will
examine some of its consequences in Appendix A.

1.4 Simultaneity: Measuring Times

How do we measure times? In SR, we repeatedly wish to talk about the times
at which events happen, and more particularly the time intervals between
events. In Chapter 3, we will discover that two observers in relative motion,
who observe a pair of events, will not only ascribe different time coordinates
to those events (unsusprisingly, because they are in different coordinate
systems), but may also disagree about which event happens first, or whether
they are simultaneous. We must therefore be careful just what we mean by
‘the time of an event’.
One of the things we can hold onto in the rest of this text is that, if
two events at the same spatial position happen at the same time, they are
simultaneous for everybody. Einstein made this particularly clear, when he
talked about what it means to assign a ‘time’ to an event:
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.’ (Einstein 1905)

If, however, the event happens some distance away (answering a question
such as ‘what time did the train pass the next signal box?’), or if we want to
know what time was measured by someone in a moving frame (answering,
for example, ‘what was the time on the train-driver’s watch as the train
passed the signal box?’), things are not so simple, as most of the rest of this
text makes clear. Special Relativity 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
8 1 Introduction

Figure 1.1 Our observers, equipped with their clocks and surveyors’ wheels.

measured in that observer’s frame.2

What we never do is have an observer in one location measure and report
the time of an event at another location. To do so, we’d have to concern
ourselves with a number of complications, most prominently correcting for
the time of flight of the light from the event to our observer’s eye, which
depends on the distance between them, and so on. It would be possible to
be clever and correct for these, but we simply avoid doing so by exclusively
using local observers.
We suppose that a frame has a plentiful supply of observers, and that
we can position them, and their clocks, wherever we need to make an
observation (we’re going to take ‘observation’ and ‘measurement’ to be
synonyms here). Or you can imagine observers positioned everywhere in a
frame, ready to take note of any events which happen next to them. Our
observers – see Figure 1.1 – are both lazy and very short-sighted: once they
have established their location they never move, and they will only ever
observe events which happen right in front of them. When an event happens,
they note the time on their clock, ready to report it along with their measured
position when required.
Since the observers never move, the frame they are standing in is special
to them, it is the rest frame of the observer. At the risk of belabouring the
point, if we have two observers, one on a train and one on a station platform,
then the rest frame for one observer is the frame attached to the station
(within which neither the station nor the observer are moving), and the rest
frame for the other is attached to the train (within which the train is not
moving, but the station is).
The observers in a particular frame have one further property of impor-

2 Note that it makes no sense to talk of being ‘stationary with respect to an event’ or to talk of
‘the rest frame of an event’: since an event is instantaneous, it cannot be said to be moving in
any frame. This also means that there is no observer who is ‘special’ with respect to an event.
 1.5 Simultaneity: Measuring Lengths 9

Figure 1.2 Measuring the length of a train, using a ruler painted on the edge of
the station platform. The observers are standing on the platform. You should
imagine observers all along the platform; I’ve shown only the two who happen
to be next to the ends of the train at the observation time.

tance: as well as being stationary in the frame, we presume that they have
arranged things so that the clocks they carry are, and remain, mutually
synchronised. I have a little more to say about this in Section 2.2.1, but I
mention it here only to reassure you that this is not on the list of unexpectedly
complicated things.
Although an event might have a number of adjacent observers, in different
frames, this doesn’t imply that all observers report the same time. The
observers’ watches may differ for trivial reasons – perhaps their watches
are set to different time zones, or designed to run at different rates. Or, less
trivially, they may be running at different rates for relativistic reasons that
we’ll come to later. We assume that, despite these complications, all of the
clocks tick out a time – produce a number – which is linearly related to the
passage of time. [Exercise 1.3]

1.5 Simultaneity: Measuring Lengths

How should we measure the length of a moving object such as the train
carriage of Figure 1.2? The obvious method is some variant of laying out
a metre stick or measuring tape, and looking at how the markings line up
with the moving carriage. But doing so raises some awkward questions.
I said above that we want to avoid making any non-local measurements,
and make all observations with co-located observers. How do we go about
that in this situation? You obviously have to measure both ends of the
object simultaneously: does that ‘simultaneously’ depend on where you’re
standing relative to the object? There are questions here which are only
apparently straightforward, and to which SR provides surprising answers.
The way we measure lengths and times in SR is therefore as follows. We
position observers at strategic points in the reference frames of interest. We
can know these observers’ coordinates in one frame or another (I find it
useful to imagine the 𝑥-axis painted on the platform edge). The observers
10 1 Introduction

make records of events which happen at their location, and afterwards

compare notes and draw conclusions. Specifically, you would measure
the ‘length of a train’ by subtracting the coordinates of the two observers
who observed opposite ends of the train at a pre-arranged time (remember
that we are assuming that all of the observers in a particular frame have
pre-synchronised clocks). We return to this in Chapter 3.
If these two observers are inside the train carriage, stationary at opposite
ends, then they get the value you would intuit: this is merely a complicated
way of measuring the length they could obtain by stretching a tape-measure
between them. For the case of the observers on the platform measuring the
moving train, this does initially seem a complicated way of organising things,
but it has the virtue of being unambiguous about when things happen and
where.
To summarise, this approach relies on three things.
1. It requires a specific procedure for synchronising clocks, which is de-
scribed briefly in Section 2.2.1.
2. It assumes that there is no ambiguity about two events at the same posi-
tion 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.
3. We assume that moving clocks measure the passage of time accurately –
that they are ‘good clocks’ in the specific sense of the ‘clock hypothesis’
(see Section 1.6).
The second point in this list does need saying, possibly surprisingly: we
will later discover that the simultaneity of two events not at the same position
does depend on your point of view.

1.6 The Clock Hypothesis

The clock hypothesis is the assumption that there is nothing intrinsic to
motion, or to acceleration, which stops a clock being a reliable measure of
the passage of time.
The first (rather trivial) part of this is the presumption that, whatever the
technology we’re using for our clocks, it’s appropriate for the situation, so
that we’re not, for example, using a pendulum clock in space or on board
ship, or a clock which will snap if we accelerate it. A very concrete example
of a ‘good clock’ is an atomic clock, which depends on fundamental physics
 1.6 The Clock Hypothesis 11

Figure 1.3 A Han-dynasty horse-drawn odometer. With bells on. (Image:

stone rubbing of an ad 125 tomb carving; Wikimedia commons.)

to mark out a timescale.

Secondly, and much more significantly, the hypothesis asserts that there
is nothing about acceleration which necessarily affects the way a clock
measures the passage of time; an accelerating clock will ‘tick’ at the same
rate as an unaccelerated clock moving at the same instantaneous speed.3
Put another way, this is the hypothesis that there are no additional com-
plications we need worry about here, and that as far as we are concerned,
time is the thing showing on the face of a clock. We will occasionally talk of
a good clock – we are invoking the clock hypothesis when we do so.
An odometer, or a surveyor’s measuring wheel, is a device which records
how far a vehicle, or a surveyor, has travelled by recording how much the
device’s wheel has turned. This is a very direct measurement of the distance
travelled through space (though not quite as direct as the idea of laying out
a measuring rod end-to-end). These need be no more complicated than a
wheel on a stick, but it’s possible to make them more elaborate (see Figure 1.3
for a spectacular example).
A useful corresponding image, I find, is to imagine a clock to be like an
old-fashioned ‘taffrail log’ (Figure 1.4), which is type of propellor towed
behind a boat, which records how much water the boat has moved through.
It’s like an odometer for water. Analogously, a clock is a device which records
how much time the clock has been dragged through.

A further way of putting this, which will mean more to you after you
have looked at Chapter 5, is to say that proper time is the measure of
the length of a worldline, and that this length is a property of the worldline, as

3 In a famous experiment, Bailey et al. (1977) measured the decay rates (which mark the
passage of time, and thus count as a clock for this purpose) of muons in a storage ring at
CERN; because the particles were moving in a circle, they were subject to very high
centripetal accelerations. The muons decayed at the same rate as they would if they were
moving at the same speed in a straight line.
12 1 Introduction

Figure 1.4 A ‘taffrail log’: the propellor is towed through the water, and the
number of turns is recorded on the dials (from Plate 6 of S. B. Luce, Textbook
of Seamanship (1891); image courtesy Historical Naval Ships Association).

a path through spacetime, rather than being a property of any physical system
within spacetime, such as a clock. In these terms, the clock hypothesis is the idea
that a ‘good clock’ is a clock which, though it is a physical thing within spacetime,
faithfully records this worldline length.

1.7 Standard Configuration

Finally, a bit of conventional terminology to do with reference frames. Two
(inertial reference) frames 𝑆 and 𝑆 ′ , with spatial coordinates (𝑥, 𝑦, 𝑧) and
(𝑥 ′ , 𝑦 ′ , 𝑧′ ), respectively, and time coordinates 𝑡 and 𝑡′ are said to be in stan-
dard configuration if:

1. they are aligned so that the (𝑥, 𝑦, 𝑧) and (𝑥 ′ , 𝑦 ′ , 𝑧′ ) axes are parallel;
2. the frame 𝑆 ′ is moving along the 𝑥-axis with velocity 𝑣; and
3. we set the zero of the time coordinates so that the origins coincide at
𝑡 = 𝑡 ′ = 0; this means that the origin of the 𝑆 ′ frame (which of course
remains at position 𝑥 ′ = 0 by definition) is always at position 𝑥𝑆′ = 𝑣𝑡 in
frame 𝑆.

This arrangement, shown in Figure 1.5, makes a lot of example problems

somewhat easier, and is the convention assumed by the ‘Lorentz transfor-
mation’ which we will meet in Chapter 5.
 1.8 Further Reading 13


y y v S y S  y

xS  = vt
t = 0 t > 0
t=0 t>0

x x
x x

Figure 1.5 Standard configuration: two frames 𝑆 and 𝑆 ′ , where frame 𝑆 ′ is

moving at speed 𝑣 with respect to frame 𝑆; the axes coincide at time 𝑡 = 𝑡′ = 0.

In Figure 1.5, both of the observers are at positions which can be given
coordinates in both frames, (𝑡, 𝑥, 𝑦, 𝑧) and (𝑡′ , 𝑥 ′ , 𝑦 ′ , 𝑧′ ). The left-hand ob-
server in each figure is at rest in frame 𝑆, meaning that they stay stationary at
position (𝑥, 𝑦, 𝑧), but of course move through time 𝑡, as shown on their clock;
their position in terms of coordinates (𝑥′ , 𝑦 ′ , 𝑧′ ) changes, with d𝑥′∕d𝑡′ = −𝑣.
The other observer, at rest in frame 𝑆 ′ , stays at coordinates (𝑥′ , 𝑦 ′ , 𝑧′ ), but
with changing 𝑥-coordinate: d𝑥∕d𝑡 = 𝑣.
When we refer to ‘frame 𝑆’ and ‘frame 𝑆 ′ ’, we will interchangeably be re-
ferring either to the frames themselves, or to the sets of coordinates (𝑡, 𝑥, 𝑦, 𝑧)
or (𝑡′ , 𝑥 ′ , 𝑦 ′ , 𝑧′ ). It’s also worth mentioning in passing that, in standard con-
figuration, we can presume that the corresponding coordinates in the two
frames, such as 𝑡 and 𝑡′ , or 𝑥 and 𝑥′ , will always be in the same units,
whether these be nanoseconds or parsecs.

In
standard
configuration,
the
motion
of
the
frames
is
always
along
the
𝑥-
and
𝑥 ′ -axes,
and
in
the
examples
we
use
we
can
always
choose
the
axes
so
that
this
is
so.
This
is
not
a
fundamental
restriction,
and
we
can
relax
it
at

the
cost
of
mere
algebra.
There
is
a
little
more
to
say
about
this
in
Section
5.2.1

1.8 Further Reading

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 the ‘physics’ section in
the library, looking through the SR books there, and finding one which
14 1 Introduction

makes sense to you.4

I recommend in particular the books listed below, which excel in different
ways.
r Taylor & Wheeler (1992) is an excellent account of Special Relativity,
written in a style which is simultaneously conversational and rigorous. It
introduces the subject from a geometrical perspective from the very be-
ginning, which makes it quite natural to make links to General Relativity.
r Rindler (2006) contains a clear account of the subject, using a broadly sim-
ilar tack to my own. Rindler takes great pains to confront and explain the
subtleties involved in the subject, and explains things both carefully and
lucidly; he is also much less geometrical in tone than Taylor & Wheeler.
I think that both Rindler and Taylor & Wheeler see SR essentially as a
prologomenon to GR, and are consequently admirably careful in their
treatment of SR, to the extent that in Rindler’s book the distinction be-
tween them is rather porous. Because of this, I don’t think Rindler is a
perfect first introduction to SR, but he is invaluable as a second source,
to turn to after a first source leaves you uncertain of a subtlety, or simply
wanting more detail. The second half of this book is about GR, and goes
substantially beyond the scope of this text (and he takes what is now
rather an old-fashioned approach).5
r French (1968) is a justly well-known account, which has been in print
since it was first published. It takes a rather traditional approach to the
subject. Here, the motivation for SR is particle physics, so it talks a lot
about dynamics, and is not much interested in geometry. In terms of style
and approach, I think of it as the polar opposite of Gourgoulhon (2013),
and enjoy both.
r Barton (1999) is another book with in some ways a rather traditional
approach, though more modern than French. Its particular strength is
that, like Rindler, it explains things very carefully and with scrupulous
reference to background material. This makes it slightly hard work as an
introduction to SR, but very valuable to me when it allows me to direct
you towards supporting details, when they might break the flow below.
r Takeuchi (2012) is a charming book which introduces the key ideas of
SR purely through diagrams, without any equations. It overlaps comfort-

4 There is an annotated and updated list of recommendable texts on relativity at

https://math.ucr.edu/home/baez/physics/Administrivia/rel_booklist.html.
5 Note that Rindler (1977) and Rindler (2006) are different books, even though they are by the
same author, and have almost identical titles. They are so similar that the latter almost
seems like a (significantly) revised edition of the former. I will refer to the 2006 book below,
but if you happen to have access only to the 1977 book, it is as rich in insight.
 1.8 Further Reading 15

ably with the material up to Chapter 4, and touches on the paradoxes in

Chapter 5.

Notation: If you refer to a range of books (and, to be clear, I think you

should), then you should be aware that there are a few notational variants
between texts. In particular, see the notes regarding this in Section 4.4 and
Section 6.2.
More advanced textbooks, on General Relativity, often include a very
compact account of Special Relativity in their initial chapters. It can be
useful to look at these once you have a basic understanding of the topic, as
this can often highlight the key features, and show you how ‘simple’ SR is,
in the slightly specialised sense I mentioned at the beginning of this chapter.
r Schutz (2009) is a textbook on General Relativity, and is therefore generally
beyond the scope of this text. However the first chapter gives a breakneck
account of SR, and the second a similarly abstract account of vector
analysis, which were influential on Chapter 6.
r Landau & Lifshitz (1975) covers a daunting range of classical physics,
but Chapter 1 covers SR in a particularly compact way, which might
help solidify your understanding after you have broadly understood the
material.
r Taylor et al. (2019) is a book on GR by the same authors as Taylor &
Wheeler (1992). It has an unusual style, but is just as vivid. As of 2021
the book is still only in draft, and available online.
r I should also mention the book Gray (2019).

There are also some books which might provide some historical insight
and context. It’s worthwhile taking at least a look at Einstein’s famous 1905
paper introducing the ideas which were later termed ‘Special Relativity’
(Einstein 1905). The first few sections are worth reading for their clarity and
directness (and I’m going to quote from this paper in the next chapter). That
said, this paper is of historical interest rather than being a great introduction
because, amongst other things, the notation is somewhat different from
what we would use today. It is also unusual, amongst foundational papers
in physics, for being at least intelligible; most analogous papers are for
historical specialists. Einstein’s own popular account of relativity (Einstein
1920) 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 (Lorentz et al. 1952) is a collection
of (translations of) original papers on the Special and General theories,
including Einstein’s 1905 paper, but also some earlier papers by Lorentz
16 1 Introduction

suggesting interpretations of the Michelson–Morley experiment.

Other books which have interesting notes on SR, but which should be
regarded as strictly supplemental, include: David Mermin’s book (1990) is
a (very good) collection of essays, covering a wide variety of topics from
the practice of physics, through quantum mechanics, to relativity; Chap-
ters 19 to 21 are about unusual approaches to the teaching of relativity.
Another book to mention, just because I like it, is Malcolm Longair’s collec-
tion of lecture notes and case studies, covering several areas in theoretical
physics (Longair 2020): it pulls no mathematical punches, but is full of
insights, including chapters on SR and GR.
There are several popular science books which are about, or which men-
tion, 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
careful detail (so they won’t get you through an exam), but in exchange
they spend their efforts on the underlying ideas. Those underlying ideas,
and the development of your intuition about relativity, are things that can
sometimes be forgotten in more formal texts. I’ve always liked Schwartz &
McGuinness (2012), which is a cartoon book but very clear, and which has a
special place in my affections as the book which first introduced schoolboy
me to the ideas of Special Relativity.

Exercises
Exercise 1.1 (§1.2)
Which of the following are events?

1. A supernova explosion.
2. A concert.
3. The whole country clapping hands at once.
4. A collision between two particles in the LHC. [ 𝑑− ]

Exercise 1.2 (§1.3)

Which of the following represent inertial frames?

1. A plane, while cruising at altitude.

2. A plane, while landing.
3. The surface of the Earth.
4. A stationary car.
 Exercises 17

5. A car moving at a straight line at a constant speed.

6. A car cornering at a constant speed.
7. A stationary lift.
8. A free-falling lift (this last one is rather subtle and relates to the dangerous-
bend paragraphs at the end of Section 1.3; don’t spend too much time
thinking about this, at least until you’ve looked at Appendix A). [ 𝑑− ]

Exercise 1.3 (§1.4)

Imagine you’re standing on a bridge over a motorway, looking at a traffic
cop standing beside the road. A car driver sneezes precisely at the point
when they’re passing the police officer. Whose watch do you consult to find
out the time of this sneeze in your frame (presuming all this is happening at
relativistic speed)?
1. Your own watch.
2. The police officer’s watch.
3. The driver’s watch. [ 𝑑− ]
 2
The Axioms

In this chapter, I am going to introduce the two axioms of Special Relativity.

These axioms are, to an extent, the only new physics introduced in this text:
once I have introduced them and made them plausible, the rest of our work
is devoted to examining their consequences, and the way in which they
change the physics we are already familiar with.

Aims: you should:

2.1. understand the two axioms of SR;

2.2. appreciate the significance and inevitability of the immediate conse-
quences of those axioms; and
2.3. understand the ideas of a coordinate transformation, and of the covari-
ance of an equation under a coordinate transformation.

2.1 The First Postulate: the Principle of Relativity

A form of the Principle of Relativity (or Relativity Principle, RP) was described
very clearly by Galileo, and his account is both clear and charming enough
to quote at length:
salvatius: Shut yourself up with some friend in the main cabin below
decks on some large ship, and have with you there some flies, butterflies,
and other small flying animals. Have a large bowl of water with some
fish in it; hang up a bottle that empties drop by drop into a wide vessel
beneath it. With the ship standing still, observe carefully how the little
animals fly with equal speed to all sides of the cabin. The fish swim
indifferently in all directions; the drops fall into the vessel beneath; and, in
throwing something to your friend, you need throw it no more strongly in
one direction than another, the distances being equal; jumping with your

18
 2.1 The First Postulate: the Principle of Relativity 19

feet together, you pass equal spaces in every direction. When you have
observed all these things carefully (though there is no doubt that when the
ship is standing still everything must happen in this way), have the ship
proceed with any speed you like, so long as the motion is uniform and not
fluctuating this way and that. You will discover not the least change in all
the effects named, nor could you tell from any of them whether the ship
was moving or standing still. In jumping, you will pass on the floor the
same spaces as before, nor will you make larger jumps toward the stern
than toward the prow even though the ship is moving quite rapidly, despite
the fact that during the time that you are in the air the floor under you will
be going in a direction opposite to your jump. In throwing something to
your companion, you will need no more force to get it to him whether he
is in the direction of the bow or the stern, with yourself situated opposite.
The droplets will fall as before into the vessel beneath without dropping
toward the stern, although while the drops are in the air the ship runs
many spans. The fish in their water will swim toward the front of their
bowl with no more effort than toward the back, and will go with equal
ease to bait placed anywhere around the edges of the bowl. Finally the
butterflies and flies will continue their flights indifferently toward every
side, nor will it ever happen that they are concentrated toward the stern,
as if tired out from keeping up with the course of the ship, from which
they will have been separated during long intervals by keeping themselves
in the air.
Galileo Galilei (1632), quoted in Taylor & Wheeler (1992, §3.1)

This is a very vivid account of the Relativity Principle (RP), which I shall
state more precisely at the end of this section. It’s also an illustration of the
idea of an inertial frame, which we discussed in Section 1.3. Another way
of phrasing the principle is that ‘you can’t tell if you’re moving’ – there’s
no experiment you can do which would allow you to distinguish between a
moving and a stationary frame.

The Relativity Principle as quoted here, and discussed below, is also

sometimes referred to as the Equivalence Principle (for SR). However,
when you go on to study GR, you will discover that it has an Equivalence Principle
of its own (and if one is reading a careful account, one discovers even weak, strong,
and semistrong variants of it). To avoid confusion, it seems best to leave the term
‘The Equivalence Principle’ to GR: the RP is deeply linked to the Equivalence
Principle, so the issue here is more one of terminology than physics. The various
equivalence principles are discussed at illuminating length in Rindler (2006,
ch. 1).

Galileo’s Relativity Principle implicitly refers only to mechanics. How-

ever, given that you need mechanical components to do any electromagnetic
experiment, and given that all mechanical objects are held together by
(atomic) electromagnetic forces, it would seem unavoidable that it must
20 2 The Axioms

x x = x − V t

Figure 2.1 An observer in a boat, at position 𝑥 ′ in the moving frame, and

position 𝑥 in the harbour frame.

apply to electromagnetism as well. And since we are made up from the

same atoms as everything else, it seems inevitable that the principle also
extends to biology. See also Einstein’s remarks quoted later in this section,
and Barton (1999, ch. 3); I have more to say about this in Section 5.8.2.
From the RP, one can show that, with certain obvious (but, as we will
discover, wrong) assumptions about the nature of space and time, one could
derive the galilean transformation (GT), which relates two frames in standard
configuration (see Section 1.7 or Figure 2.1):
𝑥 ′ = 𝑥 − 𝑉𝑡
𝑦′ = 𝑦
(2.1a)
𝑧′ = 𝑧
𝑡 ′ = 𝑡.
You have possibly seen this expression before, though probably not in this
form. This transformation relates the coordinates of an event (𝑡, 𝑥, 𝑦, 𝑧),
measured in frame 𝑆, to the coordinates of the same event (𝑡 ′ , 𝑥 ′ , 𝑦 ′ , 𝑧′ ) in
frame 𝑆 ′ . Differentiating these, we find that
𝑣𝑥′ = 𝑣𝑥 − 𝑉
𝑣𝑦′ = 𝑣𝑦 (2.1b)
𝑣𝑧′ = 𝑣𝑧 ,
and differentiating again
𝑎′ = 𝑎, (2.1c)
where 𝑣𝑥 = d𝑥∕d𝑡 is the 𝑥-component of velocity, and so on.
If we take the RP seriously (and we should, of course), it tells us that any
putative law of mechanics which does appear to allow you to distinguish
between reference frames cannot in fact be a law of physics. This means that
 2.1 The First Postulate: the Principle of Relativity 21

numbers might change from frame to frame – if you walk along a moving
train, you are moving faster with respect to the nearby platform than you
are moving with respect to the other passengers – but the physics doesn’t
change. If you can juggle on the platform, you can do so on the (smoothly)
moving train as well. If you throw a ball whilst on a moving train, the usual
constant-acceleration equations tell us that it will follow a parabola, with
certain parameters of maximum height, angle, and so on; someone watching
this ball from a station platform also sees a parabola: the ball is moving at a
different speed and a different angle, and it moves a greater distance – it is
a different parabola – but it remains a parabola nonetheless. The physics
has led us to solutions for the path of the ball which are different in the two
cases, but only to the extent of them being merely two variants of the same
shape.
How do we phrase this last paragraph in equation form? Consider New-
ton’s second law, in the alternative form 𝐹 = d𝑝∕d𝑡 – that is, we conceive of
force as a thing which changes the momentum of an object it acts on, which
only secondarily results in acceleration. Specifically, consider motion under
gravity, so that 𝐹 = 𝑚𝑔. Writing 𝑝 = 𝑚𝑣 = 𝑚d𝑥∕d𝑡, and 𝑝′ = 𝑚𝑣 ′ in the
primed frame,

d𝑥′
𝑝′ = 𝑚𝑣 ′ = 𝑚
d𝑡 ′
d
= 𝑚 (𝑥 − 𝑉𝑡) using Eq. (2.1a)
d𝑡
= 𝑝 − 𝑚𝑉

(that is, we are differentiating the 𝑥 ′ -coordinate with respect to the time-
coordinate in that frame, and not merely sticking primes on the expression).
Thus

𝑚𝑔 = d𝑝∕d𝑡 ⇔ 𝑚𝑔 = d𝑝′∕d𝑡′ .

It doesn’t matter what coordinates we use – i.e., it doesn’t matter which

frame we’re in – the laws of physics are the same.
Fundamental physical laws take the same form in different frames, which
means that their predicted consequences take the same form as well. Con-
sider the constant-acceleration equation

1
𝑥 = 𝑣0 𝑡 + 𝑎𝑡 2 . (2.2)
2
22 2 The Axioms

Transform this to the moving frame by replacing unprimed quantities by

primed ones using Eq. (2.1), and we find

1 2
𝑥 ′ = 𝑣0′ 𝑡′ + 𝑎′ 𝑡′ . (2.3)
2

That is, we find exactly the same relation, as if we had simply put primes
on each of the quantities in Eq. (2.2). This is known as ‘form invariance’,
or sometimes covariance, and indicates that the expression Eq. (2.3) has
exactly the same form as Eq. (2.2), with the only difference being that we
have different numerical values for the coefficients and coordinates (in
general, though, 𝑎′ = 𝑎 and 𝑡′ = 𝑡 according to the GT). Barton (1999, §2.3.3)
discusses this usefully; see also Exercise 2.4.
Going further, if all frames are equivalent, in the sense of the RP, then
there is no frame that is special, and in particular this means that we cannot
identify any frame corresponding to a state of absolute rest. But that in turn
means that the very idea of such a frame is redundant.
We can generalise this, and say that the RP, in classical mechanics, de-
mands that all laws of mechanics, and by obvious extension all other physical
laws, be covariant under the galilean transformation. Though you probably
wouldn’t naturally phrase things like this, this is entirely in accordance with
your (and my) physical intuition, and it seems amply corroborated by the
majority of our experience. Writing down Eq. (2.1) seems little more than
an exercise in notation.
Note carefully that, although we are talking in this section about ‘the
Principle of Relativity’, we are not yet talking about ‘Special Relativity’. The
RP is consistent with both newtonian physics and SR: it is the second axiom,
in Section 2.2 below, which distinguishes between them (I have more to say
about this in Section 5.8.1, which is a ‘dangerous bend’ section).
This first axiom is consistent with our intuition, with our mathematical
tastes, and also, it seems, with experiment.
Everything, therefore, seems to be rosy. [Exercises 2.1–2.3]

2.1.1 Electromagnetism
Everything, in fact, was rosy, until the end of the nineteenth century. Around
then, physicists were investigating Maxwell’s equations, one of the highpoints
of nineteenth-century physics, which unified all of the phenomena of elec-
tricity and magnetism into a single formalism of tremendously insightful
power and overwhelmingly successful application. For concreteness, let’s
 2.1 The First Postulate: the Principle of Relativity 23

remind ourselves what Maxwell’s equations look like:1

𝜌
∇⋅𝐄=
𝜖0
∇⋅𝐁=0
𝜕𝐁 (2.4)
∇×𝐄=−
𝜕𝑡
𝜕𝐄
∇ × 𝐁 = 𝜇0 (𝐉 + 𝜖0 ).
𝜕𝑡
These four equations are essentially Gauss’s laws for the electric and mag-
netic fields 𝐄 and 𝐁, Faraday’s law, and Ampère’s law, in terms of the charge
density 𝜌 and current 𝐉. Maxwell’s contribution (amongst many others)
was to realise that these four different things, extensively studied in the
nineteenth century, were facets of a single underlying structure. As such,
Maxwell’s equations were as amply corroborated as the various precursor
laws, and were used to show that electromagnetic phenomena from radio
waves to visible light were merely different views of a single thing, and that
they propagate at a single speed, the speed of light.
Maxwell’s equations work.
Unfortunately, they are not invariant under a galilean transformation.
The wave equation, and Maxwell’s equations, do not transform into them-
selves under a GT.

This is fairly easy to show for the wave equation, slightly more involved
for Maxwell’s equations. We have more to say about this in Section 5.8.2.
More advanced textbooks on electromagnetic theory also tend to have sections
on SR, which make this point more or less emphatically.
It appeared that Maxwell’s equations had their simplest form – that is,
Eq. (2.4) – only in a frame which was not moving. The fact that the equations
of electromagnetism are not invariant under the GT appeared to indicate
that, whenever you watched an electromagnetic experiment (such as an
ammeter, or a microwave oven) in a moving frame, it should work differently
from that same experiment in a stationary frame. Specifically, it suggests
that there actually exists such a unique absolutely stationary frame, which
is otherwise rendered unnecessary by the RP. [Exercise 2.4]

1 If you haven’t encountered Maxwell’s equations yet, or if this notation is unfamiliar to you,
don’t worry – the point here is to show that, excepting some notational complexities (OK,
quite a few complexities), they are as elegant and simple as Newton’s laws. The ∇ symbol on
the left hand side is a differential operator, so that this is a set of differential equations
relating spatial gradients in the electric and magnetic fields to distributions of charge and
current, and temporal changes in the magnetic and electric fields.
24 2 The Axioms

2.1.2 Einstein, Electromagnetism, and the Foundations of

Special Relativity
Einstein noted that electrodynamics appeared to be concerned only with
relative motion, and did not take a different form when viewed in a moving
frame. His 1905 paper is very clear on this point, and begins:
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. 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. (Einstein 1905)

The puzzle that Einstein is drawing attention to is that, although there are
two significantly different explanations of what is happening, when either
the magnet or the conductor is in motion, the observable current is identical.
The thing that is special about Einstein’s approach here is that he sees this,
not as a curiosity, but as a massive problem: why is there this unexplained
symmetry? What is it telling us?
Another, linked, problem was that of the aether. Since light is an electro-
magnetic wave, it seems obvious that, like water waves or sound waves,
there must be something that light waves propagate in. This ‘light medium’
was named the aether, and had the apparently contradictory properties of
being both very rigid (so that it could sustain the very high frequencies of
light) and very tenuous (so that objects such as planets could move through
it freely). The aether is an obvious candidate for the frame of absolute rest.
The Earth moves around the Sun in its orbit, with a constantly changing
velocity. It followed, therefore, that there was some point in its orbit at
which it had a maximum, and another point at which it had its minimum,
speed with respect to the putative aether. Although this speed is rather slow
compared to the speed of light, it should have been possible to measure
the change in the velocity of the Earth with respect to the aether or the
 2.1 The First Postulate: the Principle of Relativity 25

absolute rest frame. There was therefore a series of experiments in the late
nineteenth and early twentieth centuries which attempted to measure this
phenomenon: the Michelson–Morley aether-wind experiment attempted to
measure the different light-travel times for beams directed along and across
the flow of the aether; the Fizeau experiment and Lodge’s experiments
attempted to detect the extent to which the aether could be dragged along by
fast-moving objects on Earth. All of them failed: no-one was able to detect
the Earth’s movement through the aether, or the movement relative to the
absolute rest frame, which the galilean transformation and the apparently
necessary properties of an apparently necessary aether demanded.

The aether-wind experiments are discussed in most relativity textbooks.

French (1968), for example, gives clear accounts. I have quite a lot more
to say about this in Appendix B.

Thus Maxwell’s equations appear to be inconsistent with the galilean

transformation (in the sense that Eq. (2.4) is not form-invariant under the
GT), or else the equations are inconsistent with the relativity principle (in the
sense that they appear to point towards the existence of a special reference
frame). At this stage there were a number of options.
(i) Perhaps Maxwell’s equations were wrong. Although the unification
that Maxwell’s equations represent is intellectually satisfying, and although
they appear to precisely match experiment, perhaps there was some further
careful experiment which could be devised which would uncover some
contradiction between theory and experiment.
(ii) Perhaps the Relativity Principle was wrong, although one possible
result of the repeated failures to measure the Earth’s motion through the
aether could, I imagine, have been the weakening of the idea of an absolute
rest-frame.
(iii) Perhaps the GT was wrong, though this transformation seems so
obvious, and so bound up with our other preconceptions that it would be
difficult to see how this would be possible.
(iv) Perhaps there was some further physics at work. There were sugges-
tions that an experimental apparatus, or even the Earth itself, might be able
to drag the aether along with it, enough to wipe out any detection from the
Michelson–Morley experiment. Lorentz managed to find an alternative to
the GT – now known as the Lorentz or FitzGerald–Lorentz transformation2
– under which Maxwell’s equations are covariant, but he and the rest of the

2 For more about Lorentz, FitzGerald, and this transformation, see Section 5.8.2.
26 2 The Axioms

community were then left with the problem of explaining why electromag-
netism was apparently uniquely subject to a different transformation law
from everything else. The Lorentz transformation appeared to indicate that
objects would change their lengths, and time be distorted, when moving
head-on into the aether, without there being any clear physical mechanism
for this. It would have been clear that something was very wrong.3
This is a fascinating episode in the history of science, but the resolution
was that (i) Maxwell’s equations are right, (ii) the Relativity Principle is
right, (iii) the galilean transformation is only approximate, and (iv) Special
Relativity is the new physics to come out of this.
By saying that the GT is ‘approximate’ I mean that, although the transfor-
mation clearly works in our normal experience, we can find circumstances
(namely when we are moving quickly, at ‘relativistic speed’) where it pro-
duces wrong predictions.
Einstein explains this as clearly as anyone. In his 1905 paper which
introduced SR (‘On the Electrodynamics of Moving Bodies’), he opens with
the paragraph above commenting that only relative motion is important in
Maxwell’s equations, and then goes on to say, with magisterial finality:
Examples of this sort, together with the unsuccessful attempts to discover
any motion of the Earth relatively to the ‘light medium’, suggest that
the phenomena of electrodynamics as well as of mechanics possess no
properties corresponding to the idea of absolute rest. They suggest rather
that, as has already been shown to the first order of small quantities, the
same laws of electrodynamics and optics will be valid for all frames of
reference for which the equations of mechanics hold good. We will raise
this conjecture (the purport of which will hereafter be called the ‘Principle
of Relativity’) to the status of a postulate. . . (Einstein 1905)

In other words, he is saying that Galileo’s Relativity Principle, which had

really only been a statement about mechanical experiments, now applied to

3 There are some interesting historical details in Einstein’s ‘Autobiographical Notes’ (1991)
and Pais’s scientific biography of Einstein (2005, ch. 6). The apparent non-covariance of
Maxwell’s equations, and the problems with the aether theory, were well-recognised
problems by the end of the nineteenth century, and the historically interesting thing here is
the extent to which Einstein was aware of the prior work but didn’t feel he needed to build
on it, since his motivations for the 1905 paper were largely philosophical. Lorentz’s
discussion of the Michelson–Morley experiment describes how one might reconcile it with
the aether theory, by assuming inter-molecular forces are modified by the aether in a
particular way ‘though to be sure, there is no reason for doing so’ (Lorentz 1895, §4); and
Bell (2004, ch. 9) has described a potential way of teaching relativity, by considering the
electrostatic field of a point charge, which would now be regarded as extremely eccentric,
but which would have made a lot of sense to Einstein’s contemporaries, and which is
illuminating therefore (and see Section 5.8.2 below). It was clear in 1905 that the physics of
mechanics and of electromagnetism were intimately related to one another, so that the fact
they seemed to observe incompatible transformation laws was a major anomaly.
 2.2 The Second Postulate: the Constancy of the Speed of Light 27

all of physics.
We can recast the Principle of Relativity (RP), the first postulate of Special
Relativity, as follows:

The Principle of Relativity: All inertial frames are equivalent for

the performance of all physical experiments.

There is no physical (or chemical or biological or sociological or musical)

experiment I can do which will have a different result when I’m moving
uniformly from when I’m stationary. There is therefore no need for even
the idea of a standard of absolute rest. And further (but importantly for
Einstein’s approach) any theory which appears to require such a standard,
must be wrong in principle.
Put another way, although Newton devised his laws of mechanics in an
‘absolute rest frame’, and although Maxwell devised electromagnetism in an
‘aether frame’, the RP says that the same physical explanation will be good
in any inertial frame. It may not be obvious, but the demand that this be so
puts a severe constraint on the theories in question, or on the relationships
we must find between inertial frames. [Exercise 2.5]

2.2 The Second Postulate: the Constancy of the Speed

of Light
The passage I quoted above, from Einstein’s 1905 paper, goes on to say:
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. (Einstein 1905)

This is the second postulate of Special Relativity.

This doesn’t seem particularly remarkable at first reading; after all, we
know that ‘the speed of light’ is one of nature’s fundamental constants, at
𝑐 = 299 792 458 m s−1 . The sting is in the final remark, ‘independent of the
state of motion of the emitting body’. At first thought, there would seem to
be three things that ‘the speed of light’ could mean:

1. the speed relative to the emitter (like a projectile);

2. the speed relative to the transmitting medium (like water or sound); or
3. the speed relative to the detector.
28 2 The Axioms

Option 2 is ruled out by the first postulate: if this were true then the frame
in which light had this special value would be picked out as special; the RP
also incidentally excludes the notion of the aether.
Option 1 also turns out not to be the case, if the statement here is taken to
mean that light behaves just like a classical projectile. While light is always
emitted at the speed 𝑐, option 1 suggests that it may potentially arrive at a
different speed at a moving detector. This is what we would intuit from the
velocity addition part of the GT, Eq. (2.1b), which says that velocities add in
a straightforward way; this turns out not to be true for light, or indeed any
object moving at a significant fraction of the speed of light.
No, option 3 is the case, so that, no matter what sort of experiment you are
doing, whether you are directly observing the travel-time of a flash of light,
or doing some interferometric experiment, the speed of light relative to your
apparatus will always have the same numerical value. This is perfectly inde-
pendent of how fast you are moving relative to the source: it is independent
of whichever inertial frame you are in, so that another observer, measuring
the same flash of light from their moving laboratory, will measure the speed
of light relative to their detectors to have exactly the same value.
Constancy of 𝑐: There exists a finite constant speed
𝑐 = 299 792 458 m s−1 ,
such that anything which moves at this speed in one inertial frame is
measured to move at that speed in all other inertial frames.
Barton (1999, §3.1) gives a wonderfully careful expression of this. There is
no real way of justifying this postulate: it is simply a truth of our universe,
and we can do nothing more than simply demonstrate its truth through
experiment.
This experimental corroboration might take the form of a measurement
of the speed of light emitted from an orbiting body, at the phases in its orbit
when it is moving directly towards or away from us. The orbiting body can
be a particle in an accelerator, or a binary star orbiting its companion, but
in either case the measured light speed is determined to be independent of
the speed of the emitter, to impressively high accuracy.
For further discussion of this experimental support, and references to
further reading, see Appendix B, French (1968, chs. 2 and 3), and Barton
(1999, §3.4). [Exercise 2.6]
 Exercises 29

2.2.1 Synchronising Clocks

The second axiom states that the speed of light is a universal constant, and
it is this that allows us to define a procedure for synchronising clocks.
Suppose that you and I are some distance apart from one another, and
relatively at rest. We start off with two clocks which are going at the same
rate, but which are not initially synchronised. I send a flash of light in your
direction, and note the time when I do so, 𝑡1 . You hold up a mirror to reflect
you
the light back at me, and also note the time at which the flash arrived, 𝑡mid .
I note the time of the returned flash, 𝑡2 .
From the round-trip time, and knowing the speed of light, I can work out
how far you are from me (it’s simply 𝑐(𝑡2 − 𝑡1 )∕2). Since the light moved
at the same speed in both directions, I know that the time of the reflection
me
was 𝑡mid = (𝑡1 + 𝑡2 )∕2, and I can communicate this time to you in some
you
convenient way. You will compare that time to your 𝑡mid , and adjust the zero
of your clock to match. From that point on, our clocks are synchronised.
I describe this procedure in detail, not because it is complicated or partic-
ularly surprising, but to illustrate the extent to which we are here proceeding
extremely cautiously from the axioms, careful to avoid smuggling in any
extra assumptions. If we accept the axioms as true of our world, then we
are forced to accept SR.

There are a few further subtleties to this procedure which are not impor-
tant for our purposes; both Rindler (2006, §2.6) and Taylor & Wheeler
(1992, §2.6), for example, give details. One subtlety is that you may notice that
the above procedure assumes that the speed of light is the same in both directions:
although I doubt anyone seriously thinks that the speed of light is different, out
and back, it is surprisingly difficult (indeed, impossible to date) to devise a proce-
dure which avoids this assuption, or which is capable of measuring the ‘one-way’
speed of light.

Exercises
Exercise 2.1 (§2.1)
Consider a rocket at rest (𝑣0 = 0) at the origin of a frame 𝑆. At time 𝑡 = 0, it
starts to fire its rockets so that it moves along the 𝑥-axis, and at time 𝑡 = 𝑡1 we
find the rocket moving at speed 𝑣 = 𝑣1 . Consider a second frame 𝑆 ′ , moving
at speed 𝑉 along the 𝑥-axis, such that frames 𝑆 and 𝑆 ′ are in standard
configuration (so that 𝑥 = 𝑥 ′ = 0 when 𝑡 = 𝑡 ′ = 0; the rocket of course has
speed 𝑣 ′ and 𝑣1′ in frame 𝑆 ′ ).
30 2 The Axioms

Presume that both the rocket and the primed frame are moving slowly
enough that we can reasonably use the galilean transformation.
Work out the momentum of the rocket at 𝑡 = 0 and 𝑡 = 𝑡1 in the two
frames (that is, work out 𝑝0 = 𝑚𝑣0 , 𝑝1 = 𝑚𝑣1 , 𝑝0′ = 𝑚𝑣0′ and 𝑝1′ = 𝑚𝑣1′ ). Is
momentum frame-invariant?
Work out the change in momentum in the two frames: is this frame-
invariant?
Work out the kinetic energy and the change in kinetic energy in the two
frames: are these frame-invariant?

Exercise 2.2 (§2.1)

Suppose I have a fancy new cosmological theory that says that there’s a
special point in the universe – say, half-way between here and Andromeda –
which is such that the gravitational constant 𝐺 changes depending on how
far away from that point you are. Does this theory have a chance of being
right? Why? [ 𝑑− ]

Exercise 2.3 (§2.1)

From the constant-acceleration equations, you learned, early in your educa-
tion, how to analyse projectile motion: you discovered that, for a projectile
launched at speed 𝑢 at an angle 𝜃 to the horizontal, the time aloft was
𝑡 = 2𝑢 sin 𝜃∕𝑔 and the range was 𝑥 = 𝑢2 sin 2𝜃∕𝑔 (just to be clear, in this
exercise we’re still talking about galilean relativity, and the galilean transfor-
mation – we still haven’t got to Special Relativity).
(i) Consider a train moving through a station at a speed 𝑈. Draw a
diagram of this situation, indicating the two reference frames 𝑆 and 𝑆 ′ in
standard configuration.
(ii) Someone on the train (frame 𝑆 ′ ) throws a ball vertically into the air at
a speed 𝑢′ (i.e., 𝑢𝑥′ = 0 and 𝑢𝑦′ = 𝑢′ ). Calculate its time aloft, 𝑡′ , and range, 𝑥 ′ ,
measured in the moving frame, and reassure yourself that the latter is zero
(i.e., the ball comes vertically downwards and the person catches it again).
(iii) Do the same calculation from the point of view of a person standing
on the platform (i.e., work out the initial 𝑢𝑥 and 𝑢𝑦 for the ball, and work
out the consequent projectile motion).
(iv) Finally, take your answer from part (ii), and use the GT in Eq. (2.1)
to substitute the primed quantities. Confirm that you get the same answers
as in part (iii).
 Exercises 31

Exercise 2.4 (§2.1.1)

Consider the wave equation
𝜕2 𝜙 𝜕2 𝜙 𝜕2 𝜙 1 𝜕2 𝜙
+ + − = 0. (i)
𝜕𝑥2 𝜕𝑦 2 𝜕𝑧2 𝑐2 𝜕𝑡 2
Take
𝜕 𝜕𝑥′ 𝜕 𝜕𝑡′ 𝜕
= ′
+
𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑡 ′
𝜕 ′
𝜕𝑥 𝜕 𝜕𝑡′ 𝜕
= ′
+
𝜕𝑡 𝜕𝑡 𝜕𝑥 𝜕𝑡 𝜕𝑡 ′
𝜕 𝜕 𝜕 𝜕
= ′ = ′.
𝜕𝑦 𝜕𝑦 𝜕𝑧 𝜕𝑧
Using the GT, Eq. (2.1), show that Eq. (i) does not transform into the same
form under a GT. [ 𝑑+ ]

Exercise 2.5 (§2.1.2)

I have a friend moving past me in a rocket at a relativistic speed, and I
observe her watch to be ticking slower than mine (as we will discover later).
She examines my watch as I do this: is it ticking faster or slower than hers?
[ 𝑑− ]

Exercise 2.6 (§2.2)

You are on a train moving through a station at 50 m s−1 , and you throw a ball
forwards at 10 m s−1 : what is the speed of the ball as measured by someone
on the platform?
Now you are on a train moving at half the speed of light, and you shine a
torch forwards: what is the speed of the light from the torch as measured by
someone on the platform?
1. 0.5𝑐
2. 𝑐
3. 1.5𝑐
4. 2𝑐 [ 𝑑− ]
 3
Length Contraction and Time Dilation

Before thir eyes in sudden view appear

The secrets of the hoarie deep, a dark
Illimitable Ocean without bound,
Without dimension, where length, breadth, & highth,
And time and place are lost

John Milton, Paradise Lost, II, 890–894

In this chapter we will examine some immediate consequences of the axioms

of Chapter 2, and develop some qualitative understanding of these before
we mathematise things in the next chapter.

Aims: you should:

3.1. understand why the concept of simultaneity is problematic in the con-
text of SR, and how we resolve these problems.

3.1 Simultaneity
Imagine standing in the centre of a train carriage,1 with suitably agile friends
at either end: Fred (at the Front) and Barbara (at the Back). At a prearranged
time, say time ‘0’ on your carefully synchronised watches, you fire off a
flashbulb and your friends note down the time showing on their watches
when the flash reaches them (Figure 3.1). Since you are standing in the

1 The argument below ultimately originates from Einstein’s popular book about
relativity (Einstein 1920), first published in English in 1920. It is clearly ancestral to the
multiple versions, involving planes, trains, automobiles and rockets, in both popular and
professional books on relativity. The particular variant described here is most immediately
descended from Rindler’s version (2006).

32
 3.1 Simultaneity 33

3 3

Figure 3.1 A flashbulb in a stationary train.

3 1

Figure 3.2 Passing trains.

middle of the carriage, Fred’s and Barbara’s times must be the same as each
other. Comparing notes afterwards, you all find that it took some time for
the flash to travel from the middle of the carriage to the end, and that your
friends have noted down the same arrival time on their watches, time ‘3’,
say. In other words, Fred’s watch reading ‘3’, and Barbara’s watch reading ‘3’,
are simultaneous events in the frame of the carriage.

These watches are obviously not calibrated in seconds, but these could
be sensible values if the watches are telling time in the natural relativistic
time unit of light-metres. See Section 4.1 below.

Suppose now that this train is moving through a station as all this goes
on, and you look from the platform into the carriage – what would you see
from this point of view? You would see the light from the flash move both
forwards towards Fred and backwards towards Barbara, but remember that
you would not see the light moving forwards faster than the speed of light –
its speed would not be enhanced by the motion of the train – nor would you
see the light moving backwards at less than 𝑐. Since the back of the train is
rushing towards where the light was emitted, the flash would naturally get
to Barbara first, as illustrated in Figure 3.2. At that point Barbara’s watch
must read ‘3’, since the flash meeting her and her watch reading ‘3’ are
simultaneous at the same point in space at exactly the same time, and so
34 3 Length Contraction and Time Dilation

B 3 1 F

Y 1 3 Z

Figure 3.3 Two trains passing each other, with the observers’ locations indi-
cated.

must be simultaneous for observers in any frame. But at this point, the
light moving towards Fred cannot yet have caught up with him: since the
light reaches Fred when his watch reads ‘3’, his watch must still be reading
something less than that, ‘1’, say. In other words, Barbara’s watch reading ‘3’
and Fred’s watch reading ‘1’ are simultaneous events in your inertial frame
on the platform.
What is going on here? Are these events simultaneous or not? What this
tells us is that our notion of simultaneity is initially rather naïve, and that we
have to be very careful exactly what we mean when we talk of events as being
simultaneous. The only case where two events are quite unambiguously
simultaneous is if they take place at exactly the same point in space.

3.2 Length Contraction and Time Dilation,

Qualitatively
We’re not finished with the trains, yet. Imagine now we’re standing on the
platform and see, this time, two trains go past in opposite directions, at the
same speed as in the previous section. As well as Barbara and Fred, we have
Yvette and Zebedee stationed at the very front and back of the lower carriage,
also with clocks and well-sharpened pencils. We’ve cunningly arranged the
speeds, timetable and flashbulbs so that we can get the set of observations
represented in Figure 3.3, where the light has reached both rear observers
and neither front one. This time, because the lower carriage is moving in
the opposite direction, the same argument as in Section 3.1 has the clocks
showing times ‘1’ and ‘3’ the opposite way around.
We are going to presume that the trains pass close enough to each other,
and that the various observers have their noses pressed firmly enough against
the windows, that observers in the two trains can be taken to be in the same
 3.2 Length Contraction and Time Dilation, Qualitatively 35

L

H 3

Z ?

Figure 3.4 Two trains, a little later.

B 11 ?

? 11 Z

Figure 3.5 The two trains, a little later still.

location, and thus can make ‘local’ observations of each other’s clocks. In
Figure 3.3, for example, Barbara can at this instant legitimately read the
time on her watch and the time on Yvette’s. She cannot read the time on
Fred’s watch, nor Yvette’s watch at any other time, because those would not
be ‘local’ observations.
Now pause a moment, and take another set of observations, shown in
Figure 3.4 (we’ll come back to this image in a moment). And then take a
final set of observations when the two rear observers are beside each other,
this time getting Figure 3.5.

The images are intended to represent a suitable set of observers lined

up along the platform edge. You can if you wish think of these two
figures as ‘photographs’ of the scene, if that would be helpful. Note, however, that
these ‘photographs’ are emphatically not what you would see if you used a real
camera. A real such photo would include optical effects such as aberration and
the Doppler effect, along with effects arising from the propagation time between
different points in the scene, which would hopelessly confuse the issue here.

After this, the various observers calm down, amble together, and compare
notes. Barbara (standing at the back of the top carriage) could remark ‘I
saw the front of the other carriage pass me when my clock was reading “3” ’
36 3 Length Contraction and Time Dilation

(this is perfectly correct, as you can confirm by looking at the positions

in Figure 3.3). To which Fred would reply ‘But Zebedee passed me when
my clock showed time “1” – he must have been well past me at time “3” ’
(also true, from the same figure, and since Fred and Barbara’s clocks are
synchronised in their frame). Someone other than Fred was beside Zebedee
at time ‘3’. From this they, and we, can quite correctly remark that the
carriage they observed moving past them was measured to be shorter than
their own. They have measured the length of a moving carriage using the
procedure of Section 1.4, and found that it is shorter, by some amount which
we can’t calculate just yet, than a similar carriage (their own) which they
can measure at leisure when stationary. This is length contraction.
In particular, this intermediate observation is what we see in Figure 3.4,
where we see the clock being held by Hilary. Hilary is the observer who,
at time ‘3’, is adjacent to Zebedee, at the back of the lower carriage. Hilary
and Barbara can conclude that, since they saw respectively the back and
the front of the lower carriage at time ‘3’, the length of that carriage in their
frame is the distance between Barbara and Hilary in their frame, namely 𝐿′ .
Fred then says ‘I looked through the window at Zebedee’s clock, and I
noticed that it was reading “3”, when mine was reading “1” – it was two units
fast’ (this also is true, as you can see from Figure 3.3). Barbara says ‘Well, I
also saw Zebedee’s clock a bit later [in Figure 3.5], and it was reading “11”,
just like mine – it wasn’t fast at all.’ That is, there were 11 − 1 = 10 units
of time between Zebedee meeting Fred and Zebedee meeting Barbara, as
measured on Fred and Barbara’s clocks, but only 11 − 3 = 8 units of time
between those same two events, as recorded on Zebedee’s watch.
Fred and Barbara know that their own clocks were synchronised through-
out the encounter (they can make sure that their clocks are synchronised
at some point, and they know that they both go at the same rate), so they
can only conclude (correctly) that the clock they both saw was going more
slowly than theirs were. Zebedee’s clock has measured a smaller amount of
time passing, between encountering Fred and encountering Barbara, than
has been measured to pass in Fred and Barbara’s frame. If we say ‘time in
the other carriage is passing more slowly’, this is what we mean (note, by
the way, that we have nothing to say about Yvette’s clock since, in Figure 3.3
and Figure 3.5, it is only Barbara who is briefly adjacent to Yvette’s clock,
whereas both make successive local observations of Zebedee’s).
The startling thing is that Yvette and Zebedee would come to precisely
the same conclusions. Because this setup is perfectly symmetrical, they
would measure Barbara’s clock to be moving slowly, and Barbara and Fred’s
carriage to be length contracted. There is no sense in which one of the
 3.3 The Light Clock 37

carriages is absolutely shorter than the other.

We are not at this point able to calculate how much the lengths of the
carriages have contracted. We will come back to this shortly, in Section 3.4
(and see also Exercise 5.6). [Exercises 3.1 & 3.2]

3.3 The Light Clock

Having persuaded ourselves (I hope) of the existence of time dilation and
length contraction, it is easy to put numbers to the effects.
Imagine you’re on a train which is, again, moving through a station at
some non-relativistic speed. You throw a ball into the air with one hand,
and then catch it with the other: how would you describe this? You’d say
that the ball started off in your left hand, followed a parabolic path (like
anything thrown), landed in your right hand, and that it took one second
(say) to do it all.
Now imagine you’re on the platform watching this go on: how would
you describe it now? You’d say that it started off at the start of the platform,
landed a good way down the platform, and took one second to do it. These
two perceptions agree that the ball follows a parabolic path (different parabo-
lae, yes, but parabolae nonetheless – this is the Relativity Principle at work),
but they disagree on how far the ball travelled in flight. That disagreement
is easily explained: from the point of view of the observer on the platform,
the ball was travelling very quickly, since it had the train’s speed as well as
its own; so of course it covered more ground before it landed again.
None of this is mysterious. I’ve possibly made it sound mysterious by the
elaborate way I’ve described it. But I’ve described it that way to pull this
perfectly normal situation into line with the next step, the light clock.
The light clock (see Figure 3.6) is an idealised timekeeper, in which a
flash of light leaves a bulb, bounces off a mirror, and returns – this is one
‘tick’ of the clock.
We set things up so that the clock’s mirrors are arranged perpendicular
to the clock’s motion. This means that both the stationary and the moving
observer measure the same separation between them – there is no length con-
traction perpendicular to the motion. To see that this must be so, consider
the following reductio ad absurdum. Imagine there were a perpendicular
length contraction. Then observers measuring a train moving along a par-
allel railway track would see the train getting narrower; specifically, the
train’s axles would get shorter, so that at some speed the train would be
derailed with its wheels lying between the rails. However, from the point
38 3 Length Contraction and Time Dilation

Δt

Figure 3.6 Light clock.

of view of the observers in the train, it is the rails which would be moving;
therefore, if there were a perpendicular length contraction, the measured
distance between the rails would become shorter, and at some speed the
train would be derailed with its wheels lying outside the rails. These state-
ments must be either both true or both false; they contradict one another,
so they cannot both be true, and they must therefore both be false; they
are both consequences of the supposition that there exists a perpendicular
length contraction; so that statement must in turn be false, demonstrating
that there can be no such contraction.
If the mirror and the flashbulb are a distance 𝐿 apart, and I, standing by
the light clock in Figure 3.6, time the round trip as Δ𝑡 ′ seconds, then, since
the speed of light is the constant 𝑐,

2𝐿 = 𝑐Δ𝑡 ′ . (3.1)

Note that Δ𝑡 ′ here is the time interval on my watch, standing alongside and
moving with the clock.
Now observe the light clock, stationary on the train going through the
station, as you watch it from the platform (see Figure 3.7). The clock is in
motion, at a speed 𝑣, so that the flash of light is reflected by the opposite
mirror when it is a little way down the platform, and detected when it is still
further on. Here, one tick is timed as Δ𝑡 seconds, during which time the
clock will have moved a distance 𝑣Δ𝑡 down the platform.
How far has the light travelled? We know the light travelled at a speed 𝑐,
from the second axiom, and we timed its round trip at Δ𝑡 seconds, so the
light beam must have travelled a distance 𝑐Δ𝑡 in the time that the clock itself
travelled a distance 𝑣Δ𝑡. But from the figure,
2 2
𝑐Δ𝑡 𝑣Δ𝑡
( ) = 𝐿2 + ( ) . (3.2)
2 2
 3.3 The Light Clock 39

cΔt/2
L
vΔt/2
Δt

Figure 3.7 Light clock: the observer is standing on the station platform, and
times the round-trip time as Δ𝑡 on their clock.

Substituting 𝐿 from Eq. (3.1), and rearranging, we find

1∕2
𝑣2 Δ𝑡
Δ𝑡 ′ = (1 − 2 ) Δ𝑡 = , (3.3)
𝑐 𝛾

where the factor 𝛾 = 𝛾(𝑣) is defined as

−1∕2
𝑣2
𝛾 = (1 − 2 ) . (3.4)
𝑐

Now, the important thing about Eq. (3.3) is that it involves Δ𝑡 ′ , the time for
the clock to ‘tick’ as measured by me, standing next to it on the train, and it
involves Δ𝑡, the time as measured by you on the platform, and they are not
the same.
How can this possibly be? Why is this different from the perfectly reason-
able behaviour of the ball thrown down the carriage, in the non-relativistic
example at the beginning of this section? The difference is that when you
watched the ball from the platform, you saw it move with the speed it was
given plus the speed of the train – in other words, the person on the platform
and the person on the train had a perfectly reasonable disagreement about
the speed of the ball, which resulted in them agreeing on the time the ball
was in flight. In the relativistic example, however, both of them agree on
the speed of the light in the light clock, as the second axiom says they must.
Something has to give, and the result is that the two observers disagree on
how long the light takes for a circuit.
So at least one of the clocks is broken? They’re both in perfect working
order (this is again the ‘clock hypothesis’ which was briefly mentioned in
Section 1.6). They only work properly when they’re stationary? No, the
40 3 Length Contraction and Time Dilation

10

6
γ(v)
4

0.5 1
v/c

Figure 3.8 The gamma function, or Lorentz factor, 𝛾 = (1 − 𝑣 2 ∕𝑐2 )−1∕2 , plot-
ted as a function of 𝑣∕𝑐.

Relativity Principle tells us that there’s no sense in which either of them is

‘more stationary’ than the other, so that the clocks work in exactly the same
way whether they’re moving or not. No. . .
Both clocks are perfectly accurately measuring the passage of time; time is
flowing differently for the two observers.

Think back to the ‘taffrail log’ of Section 1.6: the clocks carried by the
moving and by the stationary observer have been dragged through different
distances in time.
The factor 𝛾 = (1 − 𝑣2 ∕𝑐2 )−1∕2 is known by a couple of different names,
including the ‘Lorentz factor’; you will become very familiar with this expres-
sion. As you can see from Eq. (3.3) or Eq. (3.5), it indicates how significant
the relativistic effects are, for a given velocity 𝑣. At 𝑣 = 0, it has the value
𝛾(0) = 1, showing that there is no time dilation for a stationary frame. As
you can see in Figure 3.8, the factor stays very close to one√ for much of its
range, and even at nearly 90% of the speed of light (𝑣∕𝑐 = 3∕2 = 0.87) it
is still only 𝛾 = 2; as 𝑣 increases beyond this, however, the Lorentz factor
grows very rapidly, becoming infinite at 𝑣 = 𝑐. Thus there is no point at
which relativistic effects suddenly switch on – we are always in a relativistic
universe – but they are ignorable at lower speeds. [Exercises 3.3 & 3.4]

3.4 The Horizontal Light Clock: Length Contraction

In Section 3.3 we saw how to use the light clock to obtain the time-dilation
formula, Eq. (3.3). Just for an encore, and as another illustration of how we
can do calculations using just the ideas we have so far, let’s consider a variant
 3.4 The Horizontal Light Clock: Length Contraction 41

of this light clock, which lets us derive the length-contraction formula (we
obtain this result again, in a rather more compact way, in Section 5.5.1, and
that version may be a little easier to follow).
Consider a horizontal light clock, such as a train carriage of length 𝐿0
with a mirror at the front. Take the carriage to be at rest in frame 𝑆 ′ , with
the rear of the carriage at the origin of 𝑆 ′ . The carriage is measured to be
length 𝐿 in the frame of the station; we know from Section 3.2 that the
distance 𝐿 will be less than 𝐿0 , as a result of length contraction, but we do
not know, at this point, just how much shorter it will be. This length, 𝐿0
(which we could write 𝐿′ if we preferred), is sometimes referred to as the
proper length of the carriage, meaning its length in a frame in which it is
stationary.
Imagine a light flash at 𝑥 ′ = 0 at time 𝑡′ = 0, which is reflected from the
′
mirror at the front, which we will call event ○ 1 , at times denoted 𝑡1 and 𝑡 in
1
the two frames, and detected at the back of the carriage again, in event ○ 2 ,

at times 𝑡2 and 𝑡2′ . By considering the round-trip time in the two frames, we
can find expressions for 𝐿 and 𝐿0 in terms of 𝑣 and 𝑐 (you might want to try
deriving this yourself, before reading on).
In frame 𝑆 ′ , the analysis is simple. The light travels a distance 2𝐿0 in
time 𝑡2′ , so

2𝐿0 = 𝑐𝑡2′ .

Because the original flash and event ○ 2 happen at the same location in 𝑆 ′ –

we are reading the same clock twice – we can use the time-dilation formula,
Eq. (3.3), to relate 𝑡2 and 𝑡2′ , and discover that 𝑡2 = 𝛾𝑡2′ (the time-dilation
formula relates time intervals, not time coordinates, so we are here relating
the intervals between time 𝑡 = 𝑡 ′ = 0, when the light flashed, and times 𝑡2
and 𝑡2′ when the reflection returned).
In frame 𝑆, we examine the light’s travel from the origin to event ○ 1 ,

and separately its travel between events ○ and ○. The light travels at 𝑐, so
1 2

𝑥1 = 𝑐𝑡1 . In doing so, it has travelled the length of the carriage (as measured
in frame 𝑆), plus the distance the carriage has travelled in that time, so that
𝑥1 = 𝐿 + 𝑣𝑡1 . Thus

𝐿
𝑡1 = .
𝑐−𝑣

On the return leg, similarly, we find 𝑥1 − 𝑥2 = 𝑐(𝑡2 − 𝑡1 ) and 𝑥1 − 𝑥2 =

42 3 Length Contraction and Time Dilation

𝐿 − 𝑣(𝑡2 − 𝑡1 ). Adding 𝑡1 + (𝑡2 − 𝑡1 ) we find

𝐿 𝐿 2𝐿𝑐 2𝐿 2𝐿
𝑡2 = + = 2 = = 𝛾2
𝑐−𝑣 𝑐+𝑣 𝑐 − 𝑣2 𝑐(1 − 𝑣 2 ∕𝑐2 ) 𝑐
2𝐿 0
= 𝛾𝑡2′ = 𝛾 ,
𝑐
and thus, after a little rearrangement, that
𝐿0
𝐿= . (3.5)
𝛾
The length of the carriage in the platform frame, 𝐿, is less than its length in
the carriage frame.
I’ve added a few extra remarks about length contraction and time dilation
in Appendix D. [Exercise 3.5]

3.5 Is There Anything I Can Hold on To?

At this point you may be feeling rather seasick. People tend to find relativity
rather disorienting, as more and more pillars of their dynamical intuition
are kicked away. You can end up in the situation where you trust none of
your steps in any direction, and find yourself unable to move at all, for fear
that the whole edifice will come tumbling down.
Distinguish between what you know, and what you intuit. In fact, rather
little of what you know has changed: it comes down to not much more
than the relativity of simultaneity (‘not everyone agrees that two events are
simultaneous’), which leads to length contraction and time dilation as fairly
direct consequences of the second axiom. Unfortunately, both of these eat
away at our intuition of how moving objects behave.
Look at what has not changed, however. For example, in our discussion
of the trains in Figure 3.2, it is still true that the backwards-moving light
flash hit the rear of the carriage before the forwards-moving one hit the front
because, reasonably enough, the carriage rear was moving into the flash. It
is still true that the order of events at a single point in space is absolutely
fixed. It is still true that when Fred and Barbara looked at their own watches
those watches told them the local time accurately, and when they looked
at the nearby watches on the other train (which are, in principle, at the
same point in space and time as Fred and Barbara), they could reasonably
measure, or observe, what they saw there.
The focus on local measurements can help us understand what is going
on. If we concentrate on working out what the participants would measure
 Exercises 43

– that is, on what they would see happening at their own point in space and
time – rather than on what we intuitively expect them to see, then when we
find ourselves puzzled, saying ‘that couldn’t possibly happen, because. . . ’,
we have someplace to start.2

Exercises
Exercise 3.1 (§3.2)
Which of the following statements are true, referring to the observers at the
front and back of the train carriage, in the discussion of Section 3.2?

1. Fred and Barbara’s watches stay synchronised with each other.

2. Fred and Barbara’s watches stay synchronised with the clocks in the other
carriage.
3. Fred and Barbara measure their carriage to get shorter when they’re
moving. [ 𝑑− ]

Exercise 3.2 (§3.2)

Which of the following statements are true, referring to the observers at the
front and back of the train carriage, in specifically Figure 3.2?

1. Fred and Barbara measure the other carriage to get shorter when it’s
moving relative to them.
2. Fred and Barbara measure the speed of light in the other carriage to be
less than 𝑐.
3. By measuring the Doppler shift of a light signal sent from the back of
the carriage to the front, the two observers can determine the carriage’s
velocity to any desired accuracy.

2 In this focus on (local) measurement, we can see the influence of the philosophical
‘positivism’ which influenced Einstein in his development of SR. This is the claim that it is
observations of the external world that give us knowledge, rather than a priori suppositions.
There is some apparent tension with the rather axiomatic approach to SR which Einstein
then develops, but while the approach is rather abstract, the contact between the theory and
the world is via tangible events or measurements, rather than abstractions such as ‘absolute
space’.
44 3 Length Contraction and Time Dilation

Exercise 3.3 (§3.3)

When discussing the light clock, just before Eq. (3.2), we we saw the phrase
‘one tick is timed as Δ𝑡 seconds’. This is Δ𝑡 on whose clock?
1. The watch of a person on the train.
2. The watch of a person on the platform edge.
3. The station clock.
4. The time attached to the photon of light. [ 𝑑− ]

Exercise 3.4 (§3.3)

Repeat the analysis of Section 3.3 for a ‘tennis-ball clock’. Balls are fired
horizontally across a carriage (moving at a non-relativistic speed): the time
it takes them to bounce back to the starting point is one ‘tick’ of the clock.
What is the crucial point at which the two analyses diverge?

Exercise 3.5 (§3.4)

Consider the following argument.
Consider two frames 𝑆 and 𝑆 ′ in standard configuration; there is a rod
of length 𝐿0 laid along the 𝑥 ′ -axis with one end at the origin. What is the
length of this rod, 𝐿, as measured in the 𝑆 frame?
Imagine our line of observers stretching along the 𝑥-axis, all with syn-
chronised clocks (showing 𝑡). At time 𝑡 = 0, when the origins of the two
frames coincide (so that one end of the rod is at coordinate 𝑥 = 0), precisely
one of the observers in the 𝑆 frame will be standing opposite the far end
of the rod, located at coordinate 𝑥 ′ = 𝐿0 in the 𝑆 ′ frame; this observer has
coordinate 𝑥 = 𝐿.
Let us arrange for a firework to go off at the origin at time 𝑡 = 𝑡 ′ = 0:
travelling at speed 𝑐, the light from this will reach our observers at coordi-
nates 𝑥 = 𝐿 and 𝑥 ′ = 𝐿0 at times 𝑡 = 𝐿∕𝑐 and 𝑡′ = 𝐿0 ∕𝑐. But, from Eq. (3.3),
we know that any two events separated by time 𝑡′ in the moving frame would
be measured to be separated by time 𝑡 = 𝛾𝑡′ in the stationary one. Thus, the
time intervals 𝐿∕𝑐 and 𝐿0 ∕𝑐 are related, and so
𝐿 = 𝛾𝐿0 [WRONG!].
This conclusion is wrong, as comparison with Eq. (3.5) shows, so the argu-
ment is fallacious. What’s wrong with it? [ 𝑑+ ]
 4
Spacetime and Geometry

But in the dynamic space of the living Rocket, the double integral has a differ-
ent meaning. To integrate here is to operate on a rate of change so that time
falls away: change is stilled. . . . ‘Meters per second’ will integrate to ‘meters.’
The moving vehicle is frozen, in space, to become architecture, and timeless.
It was never launched. It will never fall.
Thomas Pynchon, Gravity’s Rainbow

Chapter 3 has shown us that the axioms of SR take us into unexpected

territory. Now it is time to explore the new landscape, and discover that
geometry can help orient us.

Aims: you should:

4.1. appreciate the role of geometry in understanding spacetime, specifically
the importance of the invariant interval and the Minkowski diagram;
and
4.2. internalise the utility of units where 𝑐 = 1, as the natural units for
discussing events in spacetime.
In Chapter 3 we saw that if a set of observers, in motion relative to each
other, separately observe a common pair of events, they will make separate
measurements of the events’ coordinates in space and time, and produce
different values for those positions. This is not simply due to insignificant
differences such as having differing origins – having the observers’ frames
in standard configuration (Section 1.7) deals with that. Nor is it due to using
different measurement scales – numbers can be trivially converted from one
unit to another, and in any case we are about to discuss (Section 4.1) the
way we systematically handle this problem. It is also not due to any effect
of light-travel time: all the observers make measurements only of events
immediately local to them, and the observations we have of ‘distant’ events

45
46 4 Spacetime and Geometry

are relayed from observers local, in space and time, to those events. Instead,
we have discovered that not only will different observers disagree about the
order in which separated events are observed to occur (Section 3.1), but
(partly as a consequence, and as we saw in Section 3.2), observers in different
frames will not agree about the separation in time, or the separation in space,
between two events. Such separations are things we would think are solidly
established (once we have put aside the trivial complications we have just
discussed); it is disturbing to find that they are not.
The relationship between the different measurements is not random
– the coordinates obtained by one observer are systematically related to
those obtained by another. You are already familiar with this general idea:
the space between objects in your environment is not changed when we
change the units we use to measure it (such as kilometres versus inches), and
there are all sorts of regularities in the network of such distances, such as
Pythagoras’s theorem, or that the internal angles of a triangle add up to 180°.
That is, we are familiar with the idea of geometry. We also learn to become
comfortable with the idea that the rules of geometry in our immediate
environment (Pythagoras, and all that) have to be adjusted when we consider
Earth-sized distances, or distances on the sky: the geometry of the surface
of a sphere has different rules from those that Euclid wrote down. We are
about to learn a further set of geometrical rules, and learn that we can make
sense of the things we encountered in the last chapter by seeing them as the
consequences of a set of geometrical rules that we have not known about
before now.
This chapter takes a first look at these ideas, before I pull them together
in a geometrical approach to the Lorentz transformation (LT, Eq. (5.6)), in
the next chapter. This way of approaching Special Relativity is, I think,
tremendously powerful, and (a separate advantage) creates a natural bridge
to General Relativity, which we will explore a little more in Appendix A.
Before we can properly embark on this, however, we must look more
carefully at the way in which time and space are interrelated.
And before we do that, it is convenient to get our units of measurement
straight.

4.1 Natural Units

We tend to choose units appropriate to our topic of conversation – hours
for appointments, days for calendars, metres for our height, kilometres for
our maps, and parsecs for galaxies. What units should we use when we are
 4.1 Natural Units 47

talking about moving near the speed of light? The fact that the speed of light
is 3 × 108 m s−1 tells us that if we persist in using metres and seconds, we’ll
get a lot of very large or very small numbers, and lose all hope of developing
much in the way of intuition.
If the problem were merely one of large numbers, we could settle on the
gigametre as our usual unit, give it a handy name, and be done with the
question. However we will soon discover that space and time are not as
distinct as might at first appear, and that having different names for the
separations in these directions can obscure this. It’s as if we had decided
to measure distances east–west in kilometres and distances north–south in
inches. Much better is to measure the two in the same units.
One possibility is to use time as a measure of distance. We do this naturally
when we talk of the Earth being about 8 light-minutes from the Sun, or the
nearest star being a little more than 4 light-years away. We can also talk
of the light-second, of 1 s = 3 × 108 m (the Sun has a diameter of 4.6 s), or
the light-nanosecond of 30 cm. In these units, light moves at a speed of one
light-second per second, 𝑐 = 1 s s−1 , or one light-year per year, 𝑐 = 1 yr yr−1 ;
that is, 𝑐 = 1, a unitless number.1
There’s nothing wrong with this in principle, but a more common con-
vention in this context is to instead use space as a measure of time, and use
the light-metre as our time unit, with a light-metre being the time it takes
for light to travel one metre; that is, a little more than 3.3 ns. In these units,
light travels a distance of one metre in a time of one light-metre, so again
𝑐 = 1 m m−1 as a unitless number (we did this, in fact, in Section 3.1). These
are referred to as natural units. Talking of ‘metres of time’ feels initially
unnatural, but it’s not any odder than talking about (light-)years as a dis-
tance, and we’re generally quite comfortable with that. When we need to
distinguish them, we refer to SI units – that is, metres and seconds – as
physical units.
In fact, since 1983, the International Standard definition of the metre is
that it is the distance light travels in 1/299 792 458 seconds; that is, the speed
of light is 299 792 458 m s−1 by definition, without measurement uncertainty,
and so 𝑐 is therefore demoted to being merely a conversion factor between
two different units of time. The ‘second’ being used here is the SI second,
which is defined so that a particular atomic transition in caesium has a
specific defined frequency in hertz (for further documentation of the various
resolutions here, see the ‘SI Brochure’ (BIPM 2019)). International Atomic

1 It is not dimensionless, since it still has the dimensions LT−1 , but since both dimensions use
the same units, they cancel.
48 4 Spacetime and Geometry

Time (TAI) consists of a count of such seconds from a particular epoch in

1977.
In exactly the same sense, the inch is defined as 25.4 mm, and this figure of
25.4 is merely a conversion factor between two different, and only historically
distinct, units of length.2 It does look odd to see the unit conversion written
as 1 = 3 × 108 m s−1 ; it would be similarly odd, but similarly rational, to
write the definition of the inch as 1 = 25.4 mm in−1 .
It is the fact that the speed of light is a universal frame-independent
constant – one of the axioms of SR – that allows us to use this speed as
a conversion factor in this way. This procedure would make no sense in
newtonian physics, where Eq. (2.1b) asserts that the speed of light is a frame-
dependent quantity.
There are several advantages to all this. (i) As we will learn below, space
and time are not as distinct as our intuition might suggest, but having
different units for the two ‘directions’ can obscure this; and (ii) if we measure
time in light-metres, then we no longer need the conversion factor 𝑐 in our
equations, which are consequently simpler (from here on, we’ll refer to
the time units as simply ‘metres’, and I’ll let you mentally insert the ‘light-’
prefix yourself). We also quote other speeds in these units of metres per
metre, so that all speeds are unitless and less than one. For the rest of this
text, unless otherwise noted, all dimensions and speeds will be quoted
in natural units: 𝑐 = 1. Most relativity textbooks do not do this, and quote
equations in physical units instead.3 You can find a table of these later on,
in Section 5.6.
It is easy, once you have a little practice, to convert values and equations
between the different systems of units (or at least, it is no more confusing
than unit conversions normally are).
For example, to convert 10 J = 10 kg m2 s−2 to natural units, we could
proceed in two ways. Since 𝑐 = 1, we have 1 s = 3 × 108 m, and so

2 If there really were some convention that east–west distances were in kilometres and
north–south ones in inches, then land surveyors would all be familiar with the conversion
factor 𝑘 = 2.54 × 10−5 km in−1 , and the equations in their textbooks would be littered with
factors of 𝑘 and 𝑘2 and so on. Instead, surveyors would surely write their textbook equations
in ‘natural units’ where 𝑘 = 1, and when they want to calculate ‘real-world’ numbers, they
would have to learn the techniques for either converting those measurements to natural
units, or re-inserting the ‘missing’ factors of 𝑘 into their equations, to re-obtain ‘physical
units’. But of course land surveyors would amongst themselves avoid this km/in conversion;
and of course we as spacetime-surveyors want to avoid the analogous convention of
heterogeneous units.
3 Natural units are not a modern innovation, cruelly designed to confuse students. Eddington
used natural units throughout his ‘Report’ (1920), which was the first book-length
description in English of Special and General Relativity.
 4.1 Natural Units 49

1 s−2 = (9 × 1016 )−1 m−2 . So 10 kg m2 s−2 = 10 kg m2 × (9 × 1016 )−1 m−2 =

1.1 × 10−16 kg.
Alternatively (saying the same thing in a slightly different way), we can
write 1 = 3 × 108 m s−1 , or 1 = (3 × 108 )−1 s m−1 . Thus

10 J = 10 kg m2 s−2 × (1)2
= 10 kg m2 s−2 × (3 × 108 )−2 s2 m−2
= 1.1 × 10−16 kg.

To convert from natural units to physical ones, it is useful to consider

the dimensions of the quantities in question. Thus to convert an energy
1.1 × 10−16 kg to physical units, we recall that the dimensions of energy are
ML2 T−2 , and thus write

𝐸 = 1.1 × 10−16 kg
= 1.1 × 10−16 kg m2 m−2 , recalling our time units are m
= 1.1 × 10−16 kg m2 × (3 × 108 s−1 )2 , unit conversion
= 10 kg m2 s−2 = 10 J.

It helps to recall that, with 1 m = (3 × 108 )−1 s, a metre is a very small unit
of time, or that with 1 m−1 = 3 × 108 Hz something that repeats once per
metre-of-time is happening at a very high frequency. Similarly, it might be
useful to think that when we write an energy as 𝐸 = 1 kg, what we mean
is 𝐸 = 1 kg m2 m−2 , to be dimensionally correct, but we’ve skipped writing
the cancelling units.
It’s also possible to write equations with and without explicit factors
of 𝑐. We have seen the Lorentz factor 𝛾 = (1 − 𝑣2 ∕𝑐2 )−1∕2 . In units where
𝑐 = 1, this simplifies to 𝛾 = (1 − 𝑣2 )−1∕2 . This expression is dimensionally
consistent because, in these units, 𝑣 has no units (note: it still has the
dimensions of LT−1 , but our choice of time units means that these cancel).
To convert this expression back to units where 𝑣 and 𝑐 have physical units,
we simply have to add enough factors of 𝑐 inside the expression to make
it dimensionally consistent. Another way of thinking about this is to take
the various factors of 𝑐 to be still present in the equations, but ‘invisible’
because they have numerical value 𝑐 = 1 m m−1 . Resurfacing these factors,
by deducing what power of 𝑐 must be present, then allows us to put in the
physical-units value of 𝑐.

In General Relativity, people tend to work in units where the gravita-

tional constant 𝐺 is also 1, so that mass has the same units as distance
and time; the effect is that the mass of an object has the same numerical value
50 4 Spacetime and Geometry

t a c
b


1
t1
x
x1

Figure 4.1 A Minkowski diagram: this shows the sequence of events corre-
sponding to a flashing bulb which is (a) at rest at 𝑥 = 0, (b) accelerating from
rest, and (c) moving at (nearly) the speed of light.

as the radius of the black hole that would result if the object were sufficiently
compressed (in these terms, the Sun has a mass of 3 km). In relativistic quantum
mechanics and particle physics, likewise, units are chosen so that Planck’s con-
stant ℏ = 𝑐 = 1, and everything is quoted in energy units (usually electron-volts;
see Section 7.5). As of 2018, all of the base units of SI are defined via conversion
factors, in the same way as the metre – for example, the kilogramme is defined so
that Planck’s constant has a specific exact value (BIPM 2019).

[Exercises 4.1–4.3]

4.2 The Minkowski Diagram

How can we visualise motion?
Any event (say event ○ 1 ) has a set of four coordinates (𝑡1 , 𝑥1 , 𝑦1 , 𝑧1 ) in

frame 𝑆. Since we will almost always restrict our attention to frames in stan-
dard configuration, and since there is no transverse contraction, the 𝑦 and 𝑧
coordinates are uninteresting. We can plot the remaining two coordinates
(𝑡1 , 𝑥1 ) on a diagram, to obtain something like Figure 4.1.
Imagine a flashing bulb which is stationary in frame 𝑆, at the origin. Each
flash is a separate event: each event will have 𝑥 = 0 and successively increas-
ing 𝑡. We can plot these points to obtain the line marked ‘a’ in Figure 4.1;
the line connecting these is the worldline of the bulb. If instead the bulb
accelerates along the 𝑥-axis, then we obtain the line ‘b’. As the bulb moves
faster, the distance in space between two successive flashes, Δ𝑥, and their
separation in time, Δ𝑡, are related as Δ𝑥 = 𝑣Δ𝑡, giving the worldline’s gradi-
ent on the diagram as Δ𝑡∕Δ𝑥 = 1∕𝑣. Since 𝑣 < 1 for any physical object, an
object’s worldline always has a gradient larger than 45°.
If an object moves at speed 𝑐, then in 1 m of time it will travel 1 m along
the 𝑥-axis, so its worldline will be a line at 45° to the axes, giving line ‘c’.
 4.2 The Minkowski Diagram 51

Figure 4.2 A flashbulb in the middle of a train carriage.

Notice that the worldline (b) approaches a 45° angle.

This diagram, characteristically with the 𝑥-axis horizontal and the 𝑡-axis
vertical, is a Minkowski diagram, and the space it describes is referred to as
Minkowski space or Minkowski spacetime.4
Consider now a train moving along at speed 𝑣, with a flashbulb in the
centre, and a mirror at each end, as illustrated in Figure 4.2. Call the train’s
frame 𝑆 ′ (so positions in this frame will be described using coordinates 𝑥 ′
and 𝑡 ′ ), and locate its spatial origin (that is, coordinate 𝑥1′ = 0) at the flash-
bulb; take the carriage to be 6 m long. We can identify a number of events
which take place here:
r The bulb goes off at time 𝑡′ = −3 m; call this event ○ 1 .
1
r Events ○ 2 and ○ 3 are that flash being reflected from mirrors on the 𝑥 ′ -

axis, at positions 𝑥2′ = +3 m and 𝑥3′ = −3 m (these reflections will happen

simultaneously in frame 𝑆 ′ , at 𝑡2′ = 𝑡3′ = 0).
r Event ○ ′
4 is these reflected flashes reaching the centre again, at 𝑡 = +3 m
4
and 𝑥4′ = 0 (note that this discussion has taken place entirely within
frame 𝑆 ′ ; all these times are obtained simply from ‘distance is speed times
time’).

The worldline of the flashbulb is simple, and is a straight line lying along
the 𝑡′ -axis, like worldline ‘a’ in Figure 4.1. We can plot these events, the
worldline of the flashbulb, and the worldlines of the light flashes, on a
Minkowski diagram, to obtain Figure 4.3(a). What does this sequence of
events look like on a Minkowski diagram for frame 𝑆?
r Firstly, the worldline of the moving flashbulb is a slanted line in this
frame; we know from the previous paragraph that this worldline lies
along the 𝑡′ -axis, so we can draw that axis in immediately. From 𝑥 = 𝑣𝑡,
or rather 𝑡 = 𝑥∕𝑣, we can see that this line has gradient 1∕𝑣.

4 Hermann Minkowski (1864–1909) introduced the idea in a talk in 1908 (see Section 5.3),
where he emphasised the importance and fertility of this geometrical approach. The
diagram is less exotic than it may at first appear: if it were drawn with 𝑥 vertical and 𝑡 as the
horizontal axis, there would be nothing odd about it at all.
52 4 Spacetime and Geometry

t t

4
t

4
x


3 
2 
2

x 
3
x


1

1

(a) (b)

Figure 4.3 A light flash, at event ○ 1 , reflected at events ○

2 and ○ 3 , and re-
′
observed at ○ 4 , as shown (a) in frame 𝑆 , in which the flashbulb is stationary,

and (b) in 𝑆, in which the flashbulb is moving.

r The events ○ 1 and ○ 4 happen at the location of the flashbulb, therefore

they are located on the worldline of the flashbulb an equal distance either
side of the origin. We’ll shortly be able to work out just where on this
worldline the events appear.
r So where are the events ○ 2 and ○3 ? The light which travels from event ○ 1

to events ○ 2 and ○ 3 must have a worldline which is angled at 45° (the fact

that this is true in all frames is another statement of the second axiom);
and the reflected light which travels from events ○ 2 and ○
3 back to event ○ 4

must also be angled at 45°.

r The point where these light worldlines ‘turn around’ is the point where
they are reflected, that is, at events ○ 2 and ○ 3 . We therefore see that the

events ○ 2 and ○ 3 , which are simultaneous in the frame 𝑆 ′ (they have the

same 𝑡 ′ -coordinate), are not simultaneous in the frame 𝑆; this is yet another
illustration of the relativity of simultaneity discussed in Section 3.1).
r The 𝑡 ′ -axis joins the events ○ 1 and ○ 4 in Figure 4.3, and the 𝑥 ′ -axis joins

events ○ 2 and ○ 3 . This must also be true for the newly discovered locations

of these events in the 𝑆 frame.

This means that we can show, in Figure 4.3, the positions of the four events ○
1

to ○ as they would be measured in frame 𝑆, as well as the positions of the

4

axes of the 𝑆 ′ frame (the 𝑡′ -axis is the worldline of the moving flashbulb,
and the 𝑥′ -axis is the line of simultaneity in the moving frame).
In Figure 4.3(a), events which happen at the same time in frame 𝑆 ′ – that
is, which have the same 𝑡′ -coordinate – are connected by a line parallel to
the 𝑥 ′ -axis. This remains true in Figure 4.3(b). Therefore in any Minkowski
diagram, we can indicate the coordinates of a marked event, in each of
 4.2 The Minkowski Diagram 53

t
t
slope 1/v

1
t1

t1 x1 x

x
x1

Figure 4.4 The coordinates of an event projected onto the (𝑡, 𝑥) and (𝑡 ′ , 𝑥 ′ )
axes of a Minkowski diagram.

B t F


1 
2

x
x = 3

Figure 4.5 The Minkowski diagram of the flashes in Figure 3.1, shown in the
frame of the carriage.

the frames displayed in the diagram. We can see this in Figure 4.4, where
the event ○
1 has coordinates (𝑡1 , 𝑥1 ) in frame 𝑆, and the same event has

coordinates (𝑡1′ , 𝑥1′ ) in frame 𝑆 ′ . [Exercise 4.4]

4.2.1 Example Minkowski Diagram: the Moving Trains

We can draw on the Minkowski diagram the motions of the trains in Fig-
ure 3.1 and Figure 3.2. Call the frame fixed to the train carriage 𝑆 ′ , with
its origin at the centre of the carriage. In this frame, the worldlines of the
back and front of the carriage (marked 𝐵 and 𝐹 respectively) are stationary,
so they run vertically on the diagram; the light flash from the centre runs
forwards and back at speed 𝑐 = 1, so is diagonal. We can therefore deduce
the location of the two events ○ 1 , ‘the flash reaches the back of the carriage’,

and ○ 2 , ‘the flash reaches the front of the carriage’, as shown in Figure 4.5.

How does this appear if we draw the Minkowski diagram of frame 𝑆, the
station platform that the carriage is moving past? In this frame, the front and
back are moving from left to right, so their worldlines slope upwards to the
right (𝑐 = 1 in both frames, remember). The light flashes move diagonally as
54 4 Spacetime and Geometry

B F
t


2

1

Figure 4.6 The Minkowski diagram of the flashes in Figure 3.2.

B F
t t


2

1
x

Figure 4.7 The Minkowski diagram for Section 4.2.1, completed.

before. The events are the same as before: event ○1 is on both worldline 𝐵 and

the worldline of the rearward-moving flash; event ○ 2 is on worldline 𝐹 and

that of the forwards-moving flash. We thus obtain Figure 4.6, and we can
immediately see that, although events ○ 2 are simultaneous in 𝑆 ′ (in
1 and ○

Figure 4.5), they are not simultaneous in frame 𝑆, as we saw in Section 3.1.
We know that events ○ 1 and ○2 are simultaneous in 𝑆 ′ , so that the 𝑥 ′ -axis

must be parallel to the line joining them, and that the worldlines 𝐵 and 𝐹 are
parallel to the 𝑡′ -axis, so we can go on to complete the Minkowski diagram
in Figure 4.7 (it’s unfortunately the case that Minkowski diagrams end up
looking a lot more messy and confusing when complete, than they look
when you’re building them up by thinking through the description of a
problem).
It may by now be clear that the geometry of the Minkowski diagram – the
relationships between lengths and angles and what is and isn’t perpendicular
– is very different from the geometry we’re used to. We need to find out more
about that.
 4.3 Plane Rotations 55

t
t

x

Figure 4.8 The dots represent the events where the flashes from an offshore
lighthouse beam are observed on a linear shoreline.

4.2.2 Example Minkowski Diagram: a Lighthouse

The Minkowski diagram can help us see what is going on in a given SR
problem: if we plot the relevant events on the diagram, we can see their
relationship more clearly. However, it can also support some arguments
directly. For example (this argument is taken from Rindler (2006, §2.9)),
imagine a flashing lighthouse beam being swept across a distant shore. If
the shore is far enough away and the beam is turned quickly enough, the
illuminated points can be made to travel arbitrarily fast – faster than the
speed of light. We can plot such a path on the Minkowski diagram as the
dotted line in Figure 4.8. Because it is travelling faster than light, it has a
shallower gradient than the light beam at 45°. However, this means that
there can be some frame in which the 𝑥 ′ -axis lies along this dotted line –
that is, some frame in which all the light flashes happen simultaneously. In
any faster frame – the primed frame in Figure 4.8 – the illuminated points
will be measured to travel in the opposite direction. This indicates that no
causal influence – no light flash or anything travelling slower than it – can
travel from one dot to the next.
Even when you cannot entirely solve a problem using a Minkowski dia-
gram, they are extremely useful when you are trying to visualise a problem,
prior, perhaps, to turning it into a set of events to manipulate using the
Lorentz transformation we will learn about in the next chapter. Although it
may seem a rather abstract way of noting down the information in a problem,
with practice, you can often qualitatively solve the problem before plugging
in any numbers at all.

4.3 Plane Rotations

Before the next step, we need a quick review of plane geometry – the geom-
etry we’re used to (that is, there’s no Special Relativity in this section).
56 4 Spacetime and Geometry

C
C
y P

r
y

α x
θ
x

Figure 4.9 Rotation in the plane: the point 𝑃 has different coordinates in two
frames.

In Figure 4.9 we see frames 𝐶 and 𝐶 ′ , where the latter has been rotated
by an angle 𝜃 with respect to the former. The point 𝑃 has coordinates (𝑥, 𝑦)
in the frame 𝐶 (here, as elsewhere, I am taking ‘frame’ and ‘coordinate
system’ to be synonyms). The same point 𝑃 has coordinates (𝑥 ′ , 𝑦 ′ ) in the
frame 𝐶 ′ ; what is the relationship between the two sets of coordinates (𝑥, 𝑦)
and (𝑥′ , 𝑦 ′ )?
A little geometry (observe that the distance 𝑟 is the same in both frames,
express 𝑥, 𝑦, 𝑥 ′ and 𝑦 ′ in terms of 𝛼, 𝜃 and 𝑟, and eliminate 𝑟 and 𝛼) gives

𝑥 ′ = 𝑥 cos 𝜃 + 𝑦 sin 𝜃 (4.1a)

𝑦′ = −𝑥 sin 𝜃 + 𝑦 cos 𝜃. (4.1b)

The point 𝑃 is distance 𝑟 from the origin. Pythagoras’s theorem tells us

that the coordinates in 𝑆 are such that 𝑟2 = 𝑥 2 + 𝑦 2 ; the transformation in
Eq. (4.1) additionally implies that 𝑟2 = 𝑥 ′2 + 𝑦 ′2 (this can be seen directly
from the figure, and is consistent with the result of explicitly calculating
𝑥 ′2 + 𝑦 ′2 using Eq. (4.1)). That is, although the coordinates of the point 𝑃 in
the two frames are different, the distance 𝑟 is the same in both frames – it is
an invariant of the transformation:

𝑥 2 + 𝑦 2 = 𝑟2 = 𝑥 ′2 + 𝑦 ′2 . (4.2)

4.4 The Invariant Interval

Let us look again at Figures 4.4 and 4.9, and place them side-by-side in
Figure 4.10.
These are starting to look alike. In both cases, we have a single position 𝑃,
or event ○1 , described by two sets of coordinates in two frames. These
 4.4 The Invariant Interval 57

C t
t
C
y P slope 1/v

1
r t1

y
t1 x1 x
α x
θ x
x x1
(a) (b)

Figure 4.10 (a) A length-preserving rotation in the euclidean plane (Fig-

ure 4.9); and (b) a length-preserving transformation in Minkowski space (Fig-
ure 4.4).

coordinates can be systematically related to each other. Given any pair

of points (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ), in the euclidean plane of Figure 4.10(a), we
can define the separations Δ𝑥 = 𝑥2 − 𝑥1 and Δ𝑦 = 𝑦2 − 𝑦1 , and use the
constraint that the distance; note, by the way, that we read Δ𝑥2 as (Δ𝑥)2 , and
not Δ(𝑥 2 ). The quantity Δ𝑥2 + Δ𝑦 2 is an invariant under the transformation
to derive Eq. (4.1). We can do the same for the distance 𝑟2 = 𝑥 2 + 𝑦 2 =
𝑥′2 + 𝑦 ′2 . Can we find a similar invariant for the transformation between
frames which is represented by the Minkowski diagram, Figure 4.10(b)?
Look back at the light clock, illustrated in Figure 3.6 on p. 38: how ‘far
apart’ are event ○ 1 , the light flashing, and event ○2 , the reflected light being

received? In between these two events, the light has travelled a total dis-
tance 2𝐿, so the temporal separation between events ○ 1 and ○ 2 is Δ𝑡 ′ = 2𝐿

(in units where the speed is 𝑐 = 1), even though the spatial separation
between the two events, as measured in this frame, is Δ𝑥′ = 0. Inspired
by the pythagorean distance above, let’s write that separation-squared as
𝑠′2 = Δ𝑡 ′2 = (2𝐿)2 . That interval involves only the time between the two
events, but the two events happened at the same location (Δ𝑥′ = 0), so
any contribution from that spatial separation will not be present in this
expression.
Now look at Figure 3.7, showing the light clock as viewed from the frame
it is moving through, and again ask how far apart are the same two events,
as measured in this frame. This time, the separation between the events
includes both a temporal separation Δ𝑡, and a spatial separation Δ𝑥 = 𝑣Δ𝑡.
Taking further pythagorean inspiration, let’s suppose that there is a quasi-
pythagorean expression which represents the distance here, and write
𝑠2 = Δ𝑡 2 + 𝑎Δ𝑥2 ,
58 4 Spacetime and Geometry

for some constant 𝑎 we don’t yet know (we are at this point taking a very
non-axiomatic approach to thinking about this, in contrast to the approach
we took in Chapter 2). We rewrite Eq. (3.2) using our new convention of
𝑐 = 1, and find

Δ𝑡 2 = (2𝐿)2 + (𝑣Δ𝑡)2 = Δ𝑡 ′2 + Δ𝑥 2 .

Combining these, we conclude that

𝑠2 = Δ𝑡 ′2 + Δ𝑥 2 + 𝑎Δ𝑥2 = 𝑠′2 + (1 + 𝑎)Δ𝑥 2 .

Looking at this, we can see that we will have 𝑠2 = 𝑠′2 – that is, it will be an
invariant of the transformation – if 𝑎 = −1, or

𝑠2 = Δ𝑡 2 − Δ𝑥 2 . (4.3)

This is our ‘distance function’, which has the same property of frame invari-
ance in Minkowski space, that 𝑑2 = Δ𝑥 2 + Δ𝑦 2 has in euclidean space. It
is a difference of squares rather than a sum of squares, and this is the first
clue that the geometry of Minkowski space – that is, the rules of distance
there – is systematically different from euclidean space. The argument in
this paragraph, leading up to Eq. (4.3), is founded on the requirement, in
Section 3.3, as re-examined here, that the speed of light is measured to be
the same in both frames.
This quantity 𝑠2 is a frame-invariant separation between two events,
where Δ𝑡 and Δ𝑥 are the differences in coordinates of the two events. Since
the orientation of our axes is arbitrary, we can immediately generalise
Eq. (4.3) into a (3 + 1)-dimensional version, where

𝑠2 = Δ𝑡 2 − (Δ𝑥 2 + Δ𝑦 2 + Δ𝑧2 ). (4.4)

This quantity is referred to as the interval, or sometimes, interchangeably,

as the squared interval or the invariant interval. It is also sometimes written
as Δ𝑠2 , but since the interval is always a difference, the Δ is somewhat
redundant. It is also the first appearance of the metric of General Relativity,
which we will learn a little more about in Appendix A.

I have chosen to introduce the interval, here, by observing that this

particular combination of space and time intervals is frame-invariant.
This follows from the axioms we have chosen, and their consequences in terms of
time dilation and length contraction. An alternative approach is to simply assert
this invariance as an axiom, in place of the constancy of 𝑐, and deduce the rest
of SR from that, in the same way that we could imagine asserting Pythagoras’s
theorem, and deducing euclidean geometry from there.
 4.4 The Invariant Interval 59

Although the interval is what we will mean by ‘distance’ in Minkowski

space, the negative sign in Eq. (4.3) means that the ‘distance’ has some rather
unexpected properties.
Firstly, consider two events which are separated by a time Δ𝑡, but which
happen at the same location (so that Δ𝑥 = 0; the light clock is an example
of this). According to Eq. (4.3) the ‘distance’ between these events (I’ll drop
the scare-quotes from now on) is simply 𝑠2 = Δ𝑡 2 , and as you can see, this
distance will always be positive.
Now consider a flash of light emitted at one event, and received at another
a distance Δ𝑥 away. Since the light moves at speed 𝑐 = 1, the second event
will take place a time Δ𝑡 = Δ𝑥 later than the first. This means, according to
Eq. (4.3), that the distance between the two events is 𝑠2 = 0. In other words,
any two events which are separated by a light flash are, in this geometry,
zero distance apart.
Finally, consider two events which are further separated in space than
they are in time, so that Δ𝑥2 is greater than Δ𝑡2 . In this case, the quantity 𝑠2
will be negative. It’s rather unexpected to find that a distance function can be
negative, and we will look at some of the consequences of this below. Note
that it is the quantity written ‘𝑠2 ’ which is the distance function: there is
no straightforward interpretation of the quantity ‘𝑠’; instead, we might find
ourselves reasoning that, if two events are separated by a distance 𝑠2 < 0,
then we can conclude that there ′
√ is a frame 𝑆 in which they are separated
′
by Δ𝑡 = 0, in which |Δ𝑥 | = −𝑠2 .′2

Let’s make this a little more concrete, and link it to what we learned in
Chapter 3. Specifically, is this consistent with Eq. (3.3)? Let event ○ 1 be the

light being emitted from the flashbulb at the bottom of the light clock, and
event ○ 2 the light being received there again, after its round trip. As before,

we take the primed frame, 𝑆 ′ , to be fixed to the clock, and we can decide
to put its space origin at the location of the bulb and its time origin at the
instant the bulb flashes. This means that the coordinates of the two events
are, respectively, (𝑡1′ , 𝑥1′ ) = (0, 0) (by the definition of our frame’s origin) and
(𝑡2′ , 𝑥2′ ) = (2𝐿, 0) (this is a restatement of Eq. (3.1) with 𝑐 = 1, combined with
the statement that ○ 2 happens at the spatial origin). The interval between

these two events is therefore

′2
𝑠12 = Δ𝑡 ′2 − Δ𝑥 ′2
( )2 ( )2
= 𝑡2′ − 𝑡1′ − 𝑥2′ − 𝑥1′
= (2𝐿)2 . (4.5)
What are the coordinates of these same events in the frame 𝑆, through
60 4 Spacetime and Geometry

which frame 𝑆 ′ is moving? Because we assume that the two frames are in
standard configuration (Section 1.7), we know immediately that an event
which happened at the (spatial and temporal) origin of frame 𝑆 ′ happened
at the origin of frame 𝑆 also, so the coordinates in frame 𝑆 of event ○ 1 are

(𝑡1 , 𝑥1 ) = (0, 0). For definiteness, let us suppose that the light clock has
′2
size 𝐿 = 1 m (and so 𝑠12 = 4 m2 ) and is moving at speed 𝑣 = 3∕5, so that 𝛾 =
′2
5∕4. Thus the interval between the two events is, according to Eq. (4.5), 𝑠12 =
2
4 m (we shouldn’t attempt to interpret this as an ‘area’, of course, but we
will be well-behaved and include the units in numerical expressions below).
Equation (3.3) tells us that 𝑡2 = 2.5 m (since 𝑡2′ = 2 m and 𝛾 = 5∕4), and
Figure 3.7 reminds us that 𝑥2 = 𝑣𝑡2 , or 𝑥2 = 1.5 m. Putting these together,
we find
2
𝑠12 = Δ𝑡 2 − Δ𝑥 2
= (𝑡2 − 𝑡1 )2 − (𝑥2 − 𝑥1 )2
= (2.5 m)2 − (1.5 m)2
= 4 m2 ,
′2 2
equal to 𝑠12 . Thus we see explicitly that the interval 𝑠12 has the same value
when worked out using the 𝑆 frame’s coordinates, as it has when worked
out using the 𝑆 ′ frame’s coordinates.
More generally, we can see that, in this scenario, Δ𝑡 = 𝛾Δ𝑡′ = 𝛾2𝐿 and
Δ𝑥 = 𝑣Δ𝑡 (compare Section 3.3). Thus

Δ𝑡′2 − Δ𝑥 ′2 = (2𝐿)2
Δ𝑡 2 − Δ𝑥 2 = 𝛾2 (2𝐿)2 − 𝑣 2 𝛾2 (2𝐿)2
= 𝛾2 (1 − 𝑣 2 )(2𝐿)2
= (2𝐿)2 .

Note: In the discussion in this section, we could with equal justice have
started off by defining 𝑠′2 = −Δ𝑡′2 , and we would have ended up with an
interval 𝑠2 = Δ𝑥 2 − Δ𝑡2 or, more generally, 𝑠2 = −Δ𝑡 2 + Δ𝑥 2 + Δ𝑦 2 + Δ𝑧2 .
Choosing one or the other is purely a matter of convention. I slightly prefer
the convention 𝑠2 = Δ𝑡 2 − Δ𝑥 2 , because it means that the interval is the
same as the ‘proper time’, which we come to later; but there’s very little in
it. Arbitrary though it may be, the convention determines the signs of a
number of important equations in the text, and different texts choose different
conventions here, so you should be aware of this if you read around the
subject. Most significantly, the definitions of ‘timelike’ and ‘spacelike’ in
Section 4.7 swap signs. The sum of the signs of the terms is known as the
 4.5 Changes of Frame, and Perspective 61

y y (1m)2 = Δx2 + Δy 2

Δy = 1m Δy

x Δx
x
(a) (b)

Figure 4.11 A metre stick in front of you: (a) perpendicular to the line of sight;
and (b) at an angle.

signature, so that Eq. (4.4), with signs (+, −, −, −), has signature −2, and
the above alternative definition, with signs (−, +, +, +), has signature +2.
[Exercises 4.5 & 4.6]

4.5 Changes of Frame, and Perspective

In Figure 4.10 I hoped to illustrate both the arbitrariness of the coordinates of
a point, or an event, in two different reference frames, and the commonalities
between a rotation and a change of frame in a Minkowski diagram. The
closeness of the analogy between these two things is, for me, one of the key
insights of this part of our path through SR.
Imagine holding a metre stick in front of you, directly perpendicular
to your line of sight (Figure 4.11a). The two ends are, obviously, a metre
apart from each other in the 𝑦-direction. Now partly rotate the stick away
from you (Figure 4.11b): the ends of the stick are now closer together in the
𝑦-direction, but separated in the 𝑥-direction by exactly the right amount to
make the pythagorean squared-distance between the two ends of the metre
stick, Δ𝑥2 + Δ𝑦 2 = 1 m2 , as before. You are not in the least bit surprised
at this, because you have an intuitive understanding of the invariants of
euclidean geometry (although I agree that may not be how you phrase this
to yourself at this point). We know that Δ𝑥 and Δ𝑦 here are not frame-
independent properties of the measuring rod, but instead a joint property of
the rod and our choice of coordinate system.
We can see a very similar thing happening when we look at the separa-
tion between two events in Minkowski space, in two frames, as shown in
62 4 Spacetime and Geometry

t
t


1

x
Δt Δt Δx

x

2
Δx

Figure 4.12 Two events in Minkowski space, and their separations in two ref-
erence frames. Compare Figure 4.4.

Figure 4.12. In one frame, the events are separated by Δ𝑥 in space and Δ𝑡 in
2
time, and thus by an interval of 𝑠12 = Δ𝑡 2 − Δ𝑥 2 . Viewed from a different
frame, however, these two events are separated by different amounts of space
and time, purely because of the change of frame, but the interval between
2
them, 𝑠12 = Δ𝑡 ′2 − Δ𝑥 ′2 , is the same.
Exactly as in Figure 4.11(b), the separations along any one coordinate
axis are frame-dependent and physically meaningless. It is only the interval
Δ𝑥2 + Δ𝑦 2 , or Δ𝑡2 − Δ𝑥 2 , that is significant, depending on whether we are
talking about the geometry of, respectively, euclidean space (our usual
intuition) or Minkowski space (the intuition we must develop for SR).
The correspondence between Figures 4.10(a) and 4.10(b) is not intended
to be some vague handwaving analogy. On the contrary, it is a very precise
analogy: the same thing is happening in both cases. The only thing that is
different between the two cases is the geometry of the space in question. In
the first case, the transformation from one set of coordinates to the other
(Eq. (4.1)) is such that it preserves euclidean distance (Eq. (4.2)). In the sec-
ond, the coordinate transformation is the one which preserves ‘Minkowski
distance’, Eq. (4.3), with a specific transformation – the expression corre-
sponding to Eq. (4.1) – which we are going to discover in the next chapter. If
you think of moving from one inertial frame to another as a ‘rotation’ (keep
the scare-quotes), you will not go far wrong. If you can find a place in your
head, where Figure 4.12 looks as natural as Figure 4.11(b), you will have
reached some relativistic nirvana.5

5 I haven’t got there yet.

 4.6 Length and Time in the Minkowski Diagram 63

t
s2 = 1

t=1

s2 = −1
x=1 x

Figure 4.13 All the points on the two hyperbolae are the same Minkowski-
distance from the origin.

4.6 Length Contraction and Time Dilation in the

Minkowski Diagram
At the cost of some admittedly rather forbidding-looking Minkowski dia-
grams, it is possible to illustrate time dilation and length contraction graphi-
cally. You might want to come back to this section a little later, once you have
become more comfortable with Minkowski diagrams, in the next chapter.
In Figure 4.9, we could draw an arc indicating the locus of points (𝑥, 𝑦)
with the same value of the invariant 𝑟 (this locus is known, outside of a maths
class, as that exotic object the ‘Circle’). Can we do a similar thing with the
invariant 𝑠2 on the Minkowski diagram? Yes: in Figure 4.13 we have drawn
the locus of points for which 𝑡 2 − 𝑥 2 = +1 (upper line) and −1 (lower). This
makes very clear how different is the geometry of a Minkowski diagram
from that on the plane in Figure 4.9; all the points on each hyperbola in the
figure are the same distance 𝑠2 from the origin, either 𝑠2 = 1 (upper line) or
𝑠2 = −1 (lower line).
Imagine a set of clocks which tick every 10 m of time (i.e., every 33 ns in
physical units). We put one on the station platform (at 𝑥 = 0) and another
in a moving train carriage (at 𝑥 ′ = 0), and set things up so that there is a
tick at 𝑡 = 𝑡 ′ = 0. The platform clock will tick again at 𝑡 = 10 m, and the
train clock at 𝑡′ = 10 m. The interval between ticks of a particular clock will
always be 𝑠2 = (10 m)2 − 02 . This is a quite general property: if an inertially
moving clock is present at two events (in this case, successive ticks of the
clock), then the interval between them is governed entirely by the separation
in time in the clock’s frame, since in this frame the two events are separated
by Δ𝑥 = 0; that time interval is what is showing on the clock’s face (this idea
that the clock face shows the ‘distance through spacetime’ is what motivates
the image of the taffrail log in Section 1.6). This time interval – which is
64 4 Spacetime and Geometry

t t


2 x
10m

1

Figure 4.14 Two clock-ticks on clocks at rest in frames 𝑆 and 𝑆 ′ . In each case,
the previous tick was at time 𝑡 = 𝑡′ = 0. The null line is shown dotted.

t
nt
t fro

x


4

3

10m x

Figure 4.15 A 10 m long train (at rest in 𝑆 ′ ) moving through a station (at rest
in 𝑆) with a 10 m long platform, also showing (dot-dashed) the worldline of
the front of the train.

very immediately related to the Minkowski interval – is known as the proper

time (we have a little more to say about this in Section 5.4).
Now look at Figure 4.14. The second tick of the platform clock is at
time 10 m, at the point marked ○ 1 , and the second tick of the train clock is

2 (how do we know that? Because that event is on the 𝑡 ′ -axis, and the
at ○
hyperbola marks out the locus of points which are an interval of 𝑠2 = (10 m)2
from the origin). But the 𝑡-coordinate of this event ○ 2 is larger than 10 m –

this is time dilation.

Now measure 10 m along the station platform, and mark this point in
Figure 4.15. Have a 10 m long train go through the station, timetabled so that
the back of the moving carriage is at 𝑥 = 𝑥 ′ = 0 at time 𝑡 = 𝑡 ′ = 0 (standard
configuration again). The worldline of the back of the carriage lies along
the 𝑡 ′ -axis, and the worldline of the front is parallel to this.
Suppose that an observer at the front of the carriage sets off a firework
 4.6 Length and Time in the Minkowski Diagram 65

t
t

1
2

10 m

x
x

Figure 4.16 The same events as shown in Figure 4.14, but this time shown in
the frame 𝑆 ′ .

at time 𝑡′ = 0 (that is, simultaneously, in 𝑆 ′ , with the back of the carriage

being at 𝑥 = 𝑥 ′ = 0). That front firework will be positioned at event ○ 3 . The

hyperbola again marks out the set of points which are the same distance –
10 m in this case – from the origin, and simultaneity in 𝑆 ′ implies that the
event ○ 3 is on the 𝑥 ′ -axis. How long does the platform party measure this

train carriage to be? The way they measure that is to identify where the
front of the carriage was at some instant (choosing for example event ○ 3 , at

the position and time of the firework explosion), and find another event at
the back of the carriage at the same time in 𝑆, namely event ○ 4 . Directly from

the diagram, you can see that the spatial distance, in 𝑆, between events ○ 3

and ○ 4 , is less than 10 m – this is length contraction.

Now let’s look at those events in a Minkowski diagram drawn in the

train’s frame; see Figure 4.16. As before, the worldline of the zero marker
on the platform scale lies along the 𝑡-axis. Events ○ 1 and ○ 2 are, as before,

at positions (𝑡, 𝑥) = (10 m, 0) and (𝑡 ′ , 𝑥 ′ ) = (10 m, 0), respectively. The ‘cal-

ibration’ hyperbola again shows that the two events are the same interval
away from the origin. In this frame, and unlike Figure 4.14, event ○ 1 is later
′ ′
than event ○ 2 (in the sense that 𝑡 > 𝑡 ), so it is the platform’s clock which is
1 2
time-dilated in the train’s frame.
If we do the same thing with events ○ 3 and ○ 4 , we get Figure 4.17. As
′
before, event ○ is at the front of the train, 𝑥 = 10 m, and the point which
3

the platform observers measure as simultaneous with it, ○ 4 , is also shown

for reference (just as in Figure 4.15, the line joining ○ 4 and ○ 5 is parallel to

the 𝑥-axis). I have also shown a new event ○ 5 , which is at the 10 m on the

platform (as we can see from the calibration hyperbola), and an event ○ 6 ,

which is at the zero-end of the station platform (it’s on the worldline of the
end of the platform, overlaid on the 𝑡-axis), which happens simultaneously
66 4 Spacetime and Geometry

t

front
t

4


3

x

5

6

Figure 4.17 The same events as Figure 4.15, in 𝑆 ′ .

′ ′
with ○ 5 in the train frame (i.e., 𝑡 = 𝑡 ); it is clear that the separation between
5 6
events ○ 5 and ○ 6 – that is, between the zero and 10 m marks on the platform

– is less than 10 m: the platform is length contracted in the train’s frame.

Just as we saw at the end of Section 3.2, both length contraction and time
dilation are symmetrical. The train is contracted in the platform’s frame,
and the platform in the train’s frame; and the train’s clocks are dilated or
slow in the platform’s frame, and the platform’s clocks in the train’s, all by
the same factor, 𝛾.
We are now ready to derive the Lorentz transformation, but before we
do that, there are a couple of other remarks we can make about Minkowski
space. [Exercise 4.7]

4.7 Worldlines and Causality

In Figure 4.4, back on p. 53, we saw how we ascribe (𝑡, 𝑥) and (𝑡 ′ , 𝑥 ′ ) coordi-
nates to an event. In this figure, the 𝑡′ -axis indicates the locus of positions of
a particle at the origin of the 𝑆 ′ frame, as that frame moves through 𝑆. This
line, which is straight in this case but would be curved for a more general
motion of a point, is the worldline of a particle at rest at 𝑥 ′ = 0. As suggested
above, the worldline of a photon, or anything else moving at the speed of
light, always appears as a 45° diagonal line on a Minkowski diagram.
Since the interval is just 𝑠2 = Δ𝑡 2 − Δ𝑥 2 , we can see that all the points
on a photon’s worldline have 𝑠2 = 0, all the points below that diagonal have
𝑠2 < 0, and all those above it have 𝑠2 > 0. Such separations are termed,
 4.7 Worldlines and Causality 67

t 
5

s
2
future s2 > 0

=

4

0

3
elsewhere
s2 < 0 x

0


0

2
=

past s2 > 0
s2


1

Figure 4.18 Light cones.

respectively, null (or lightlike), spacelike and timelike separations.

We can illustrate these terms in Figure 4.18. Events ○ 5 and ○1 have time-

like separations from an event ○ 0 , at the origin, events ○4 and ○2 have null

separations from ○ 0 , and event ○ 3 has a spacelike separation from ○ 0 . We

can immediately categorise the five events in terms of the type of separation
between the event and an event at the origin.
Event ○5 timelike separation from ○ 0 : a slower-than-light signal could

travel from the origin to ○.

5

Event ○4 null separation: only a speed-of-light signal could make it

from ○ 0 to ○
4 .

Event ○3 spacelike separation: no signal could travel between ○ 0 and ○

3 ,

but there is a frame in which ○ and ○ are simultaneous.

0 3

Event ○2 null separation: a speed-of-light signal could travel from ○ 2

to ○
0 .

Event ○1 timelike separation: a slower-than-light signal could get

from ○ 1 to ○
0 .

Although we are familiar with time being simply divided into a future and
a past, we can see from this Minkowski diagram that spacetime is divided
into three regions, the familiar ‘future’ and ‘past’, plus a region indicated by
‘elsewhere’, consisting of events which cannot interact causally with events
at the origin.
Figure 4.18 shows only the 𝑥- and 𝑡-dimensions. If we showed the 𝑦-
dimension as well, and rotated the figure around the 𝑡-axis, the dashed
lines would turn into a cone defined by 𝑠2 = 𝑡 2 − (𝑥2 + 𝑦 2 ) = 0. This null
cone demarcates areas of spacetime in the same way as the diagonals in the
68 4 Spacetime and Geometry

figure. Every event has a null cone, pointing both forwards and backwards
along the time axis: events ‘inside’ the null cone are timelike separated from
it, with 𝑠2 > 0, and are in the possible future of the event or its possible
past, depending on whether 𝑡 > 0 or 𝑡 < 0; events ‘outside’ the null cone
are spacelike separated from it, and are ‘elsewhere’; events on the null cone
have a null separation from it, and are reachable only by a light signal.
Although it is harder to visualise, the full null cone is the surface 𝑠2 =
𝑡2 − (𝑥 2 + 𝑦 2 + 𝑧2 ) = 0 in spacetime. [Exercises 4.8 & 4.9]

Exercises
Exercise 4.1 (§4.1)
Convert the following to units in which 𝑐 = 1:
1. 1J
2. 1 ns
3. light-bulb power, 100 W
4. Planck’s constant, ℎ = 6.626 × 10−34 J s
5. jogging pace, 3 m s−1
6. momentum of a jogger, 300 kg m s−1
7. power station output, 1 GW
Convert the following to physical units:
1. velocity 10−2
2. time 9.46 × 1015 m
3. acceleration 1.11 × 10−16 m−1
4. 𝑡 ′ = 𝛾(𝑡 − 𝑣𝑥) (this is Eq. (5.6), from the next chapter)
5. the ‘mass-shell’ equation 𝐸 2 = 𝑝2 + 𝑚2
If you’re having trouble working out whether to multiply or divide, I find
it helpful to remember that, just as an inch is a small number of kilometres, a
light-metre is a small number of seconds, or a light-second is a large number
of metres.

Exercise 4.2 (§4.1)

What is 9 N = 9 kg m s−2 in natural units?
1. 10−16 kg
2. 10−16 kg m−1
 Exercises 69

t
t

4
x

2


3
x


1

Figure 4.19 Reflections in the light clock, in the stationary frame.

3. 10−16 kg m s−2

You can also do this without doing any arithmetic – think of the dimen-
sions. [ 𝑑− ]

Exercise 4.3 (§4.1)

From Exercise 4.2, just above, 9 N is 10−16 kg m−1 in natural units. Given
that, work out the acceleration of a mass of 0.1 kg in natural units (you
should end up with something in units of m−1 ). What does it mean, in a
point of view where speeds are fractions of the speed of light, to have an
acceleration in units of per-metre?

Exercise 4.4 (§4.2)

Which of the following is true in frame 𝑆 in Figure 4.19?

1. Event ○
3 happens before event ○2

2. Event ○ happens at the same time as ○

3 2

3. Event ○
3 happens after ○
2

And in frame 𝑆 ′ ? [ 𝑑− ]

Exercise 4.5 (§4.4)

The discussion around Eq. (4.5) showed that Δ𝑡2 − Δ𝑥 2 was conserved in
that case. Demonstrate that the quantity Δ𝑡2 + Δ𝑥 2 is not conserved; that is
that Δ𝑡2 + Δ𝑥 2 ≠ Δ𝑡′2 + Δ𝑥 ′2 .
70 4 Spacetime and Geometry

Exercise 4.6 (§4.4)

Re-do the calculation at the end of Section 4.4 with a few different speeds,
such as 𝑣 = 4∕5, or 𝑣 = 12∕13, and observe that you get the same value
for 𝑠2 each time. [ 𝑑− 𝑢+ ]

Exercise 4.7 (§4.6)

Cosmic rays entering the top of the Earth’s atmosphere collide with it, and
produce showers of elementary particles. Of these showers, almost the
only ‘exotic’ particles to reach the ground are muons. These muons have
a half-life of 𝑡1∕2 = 2.2 μs, and are travelling at relativistic speeds with, on
average, a 𝛾 of 40. The majority of the produced muons reach the ground,
where they can be detected.
(a) Supposing that the muons are created at a nominal height of 15 000 m,
calculate how long it takes them to reach the ground, in the Earth frame,
giving the answer in seconds and in half-lives.
(b) Calculate how long it takes the muons to reach Earth, in the muons’
frame, giving the answer, again, in seconds and half-lives.
(c) Can this be regarded as a test of Special Relativity?
(d) In the frame of the muons, the Earth is moving towards them at
(approximately) the speed of light. Calculate the distance between the
altitude at which the muons are created, and the surface of the Earth, in
the muons’ frame, and thus how long it takes for the muons to traverse it,
giving your answer, again, in seconds and half-lives.
(e) Comment on the equality or inequality of your answers to parts (b)
and (d).

Exercise 4.8 (§4.7)

The usual train travels through the usual station at a speed 𝑣 = 1∕5, in
units where 𝑐 = 1. The train guard sets off a firecracker 500 m in time after
the train has passed the station, when the train is 100 m further down the
track (both distances measured in the station’s frame). Draw a Minkowski
diagram, indicating on it the worldlines of the station and the train, and the
explosion event. At what time is the explosion visible at the station?

Exercise 4.9 (§4.7)

Consider three events, with coordinates (𝑡1 , 𝑥1 ) = (1, 1), (𝑡2 , 𝑥2 ) = (6, 4) and
 Exercises 71

2 2 2
(𝑡3 , 𝑥3 ) = (4, 6) (take 𝑐 = 1). Calculate the intervals 𝑠12 , 𝑠13 and 𝑠23 between
the three pairs of events, state whether each is timelike, spacelike or null,
and in each case whether the earlier event could influence the later event
through a signal travelling no faster than light.
 5
The Lorentz Transformation

In Section 3.2 we saw how observers could make measurements of lengths

and times in frames which are in relative motion, and reasonably disagree
about the results – the phenomena of length contraction and time dilation.
In Section 3.3, we were able to put numbers to this and derive a quantitative
relation, Eq. (3.3), between the duration of a ‘tick’ of the light clock as
measured in two frames. We want to do better than this, and find a way
to relate the coordinates of any event, as measured in any pair of frames
in relative motion. That relation – a transformation from one coordinate
system to another – is the Lorentz transformation (LT). The derivation in
Section 5.1 has a lot in common with the account given in Rindler (2006).
Most of the work of this chapter is in Section 5.1. The rest is, to a greater
or lesser extent, commentary on that, with several sections being marked
with dangerous bends, and thus omissible.

Aims: you should:

5.1. understand the derivation of the Lorentz transformation, and recognise

its significance.

5.1 The Derivation of the Lorentz Transformation

Consider two frames in standard configuration, and imagine an event such
as a flashbulb going off; observers in each of the two frames will be able
to measure the coordinates of this event. Those observers will of course
produce different numbers for those coordinates – they will disagree about
the precise time and location of the event – with those in frame 𝑆 producing
coordinates (𝑡, 𝑥, 𝑦, 𝑧), and those in 𝑆 ′ producing (𝑡 ′ , 𝑥 ′ , 𝑦 ′ , 𝑧′ ). It is our task

72
 5.1 The Derivation of the Lorentz Transformation 73

now to calculate the relationship between those two sets of numbers.

First of all, we can note that 𝑦 ′ = 𝑦 and 𝑧′ = 𝑧, since this is just a re-
statement of the lack of a perpendicular length contraction, as discussed
in Section 3.3. Therefore we can unproblematically make things easier
for ourselves by supposing that the event takes place on the 𝑥-axis, so that
𝑦 = 𝑧 = 0 (problem: construct for yourself the argument that there is no loss
of generality here). Therefore our problem has reduced to that of obtaining
(𝑡 ′ , 𝑥 ′ ) for this event, given (𝑡, 𝑥) (or vice versa).
Now imagine a second event, located at the origin with coordinates (0, 0)
in frame 𝑆; since the frames are in standard configuration, we can imme-
diately deduce that this event has coordinates (0, 0) in frame 𝑆 ′ also. Since
we have two events, we have an interval between them, with the value
𝑠2 = (𝑡 − 0)2 − (𝑥 − 0)2 = 𝑡 2 − 𝑥 2 in frame 𝑆. We will now assume that the
interval is frame-independent (recall Section 4.4): the calculation of this
interval done by the observer in the primed frame will produce the same
value, restating Eq. (4.3):

𝑡 2 − 𝑥 2 = 𝑠2 = 𝑡 ′2 − 𝑥 ′2 . (5.1)

Thus the relationship between (𝑡, 𝑥) and (𝑡 ′ , 𝑥 ′ ) must be one for which
Eq. (5.1) is true.
Equation (5.1) is strongly reminiscent of Eq. (4.2), and we can make it
more so by writing 𝑙 = i𝑡 and 𝑙 ′ = i𝑡 ′ , so that Eq. (5.1) becomes1

𝑙 2 + 𝑥 2 = −𝑠2 = 𝑙 ′2 + 𝑥 ′2 . (5.2)

But we already know a transformation which satisfies this constraint, and so

can conclude that the pairs (𝑙, 𝑥) and (𝑙′ , 𝑥 ′ ) can be related via the analogue
of Eq. (4.1), and thus that

𝑥 ′ = 𝑥 cos 𝜃 + 𝑙 sin 𝜃 (5.3a)

𝑙′ = −𝑥 sin 𝜃 + 𝑙 cos 𝜃, (5.3b)

for some angle 𝜃, which depends on 𝑣, the relative speed of frame 𝑆 ′ in

frame 𝑆. That is, this specifies a linear relation for which Eq. (5.1) is true.
If we finally write 𝜃 = i𝜙,2 and recall the trigonometric identities sin i𝜙 =

1 You may wish to refer to Appendix C, to refresh your memory on complex numbers, and
trigonometric functions using them.
2 If 𝜃 is pure imaginary then, from the Taylor expansions, cos 𝜃 is real and sin 𝜃 is pure
imaginary, so that 𝑥′ and 𝑙′ are real and pure imaginary, respectively, as they should be.
74 5 The Lorentz Transformation

i sinh 𝜙 and cos i𝜙 = cosh 𝜙, then

𝑡 ′ = 𝑡 cosh 𝜙 − 𝑥 sinh 𝜙 (5.4a)

𝑥′ = −𝑡 sinh 𝜙 + 𝑥 cosh 𝜙 (5.4b)

(I have swapped the order of the two equations, relative to Eq. (5.3), and
swapped 𝑥 and 𝑡 in each row, for later notational convenience; you might
want to take a look at Exercise 6.5 for an alternative way of arriving at these
equations.)
Now consider an event at the spatial origin of the moving frame, that is,
at 𝑥 ′ = 0 for some unknown 𝑡 ′ . What are the coordinates of this event in the
unprimed frame? That’s easy: if it happens at time 𝑡 in the unprimed frame
then it happens at position 𝑥 = 𝑣𝑡 in that frame (because the two frames are
in standard configuration), in which case Eq. (5.4b) can be rewritten as

tanh 𝜙(𝑣) = 𝑣. (5.5)

The quantity 𝜙(𝑣) is sometimes called the ‘rapidity’, and the application of
an LT to a frame is sometimes called a ‘boost’ of that frame.
Since we now have 𝜙 as a function of 𝑣, we also have, in Eq. (5.4), the full
transformation between the two frames; but combining Eq. (5.4) and Eq. (5.5)
2 2
with a little hyperbolic trigonometry (remember cosh 𝜙 − sinh 𝜙 = 1), we
can rewrite Eq. (5.4) in the more usual form

𝑡 ′ = 𝛾(𝑡 − 𝑣𝑥) (5.6a)

𝑥′ = 𝛾(𝑥 − 𝑣𝑡). (5.6b)
𝑦′ =𝑦 (5.6c)
𝑧′ = 𝑧, (5.6d)

where we have included the trivial transformations for 𝑦 and 𝑧, which

we deduced above. Here 𝛾 is (as in Eq. (3.4) but now with 𝑐 = 1; see also
Section C.2)
( )−1∕2
𝛾(𝑣) = cosh 𝜙 = 1 − 𝑣2 . (5.7)

If frame 𝑆 ′ is moving with speed 𝑣 relative to 𝑆, then 𝑆 must have a

speed −𝑣 relative to 𝑆 ′ . Swapping the roles of the primed and unprimed
frames, the transformation from frame 𝑆 ′ to frame 𝑆 is exactly the same as
Eq. (5.6), but with the opposite sign for 𝑣:

𝑡 = 𝛾(𝑡 ′ + 𝑣𝑥 ′ ) (5.8a)
𝑥= 𝛾(𝑥 ′ + 𝑣𝑡 ′ ), (5.8b)
 5.2 Addition of Velocities 75

which can be verified by direct solution of Eq. (5.6) for the unprimed coordi-
nates. [Exercises 5.1 & 5.2]

5.2 Addition of Velocities

Adding and subtracting the expressions in Eq. (5.4), and recalling that e±𝜙 =
cosh 𝜙 ± sinh 𝜙, we find
𝑡′ − 𝑥 ′ = e𝜙 (𝑡 − 𝑥) (5.9a)
𝑡′ + 𝑥 ′ = e−𝜙 (𝑡 + 𝑥), (5.9b)
as yet another form (once we add 𝑦 ′ = 𝑦 and 𝑧′ = 𝑧) of the LT.
Consider now three frames, 𝑆, 𝑆 ′ and 𝑆 ′′ . If 𝑆 and 𝑆 ′ are in standard
configuration with relative velocity 𝑣1 , and 𝑆 ′ and 𝑆 ′′ are in standard config-
uration with relative velocity 𝑣2 (implying that the direction of velocity 𝑣1
is parallel to that of 𝑣2 ), then frames 𝑆 and 𝑆 ′′ will also be in standard
configuration with some third velocity 𝑣; we obviously cannot presume
that 𝑣 = 𝑣1 + 𝑣2 , as it would be under a galilean transformation. However,
applying Eq. (5.9a) twice shows us that
𝑡 ′′ − 𝑥 ′′ = e𝜙 (𝑡 − 𝑥) = e𝜙1 +𝜙2 (𝑡 − 𝑥), (5.10)
where 𝜙, 𝜙1 and 𝜙2 are the rapidity parameters corresponding to 𝑣, 𝑣1 and 𝑣2 .
This shows us how to add velocities: Eq. (5.5) plus a little more hyperbolic
trigonometry (tanh(𝜙1 + 𝜙2 ) = (tanh 𝜙1 + tanh 𝜙2 )∕(1 + tanh 𝜙1 tanh 𝜙2 ))
produces
𝑣 1 + 𝑣2
𝑣= . (5.11)
1 + 𝑣1 𝑣2
We used Eq. (5.10) as a stepping-stone on our way to the velocity-addition
formula. However it also demonstrates the important point that, if frames 𝑆
and 𝑆 ′ are related by a LT, and frame 𝑆 ′ is related to a further frame 𝑆 ′′ by
another LT, then there is also necessarily a single LT which takes us from 𝑆
to 𝑆 ′′ . This is another hint that length contraction and time dilation are not
simply isolated consequences of a change of speed, but glimpses of a larger
underlying structure. [Exercises 5.3 & 5.4]

5.2.1 The Lorentz Transformation as a Group

The form of the LT shown in Eq. (5.9), and the addition law in Eq. (5.10),
together conveniently indicate three interesting things about the LT: (i) for
76 5 The Lorentz Transformation

any two transformations performed one after the other, there exists a third
with the same net effect (i.e., the LT is ‘transitive’); (ii) there exists a trans-
formation (with 𝜙 = 0) which maps (𝑡, 𝑥) to themselves (i.e., there exists
an identity transformation); and (iii) for every transformation (with 𝜙 = 𝜙1 ,
say) there exists another transformation (with 𝜙 = −𝜙1 ) which results in
the identity transformation (i.e., there exists an inverse). If we add to these
three properties the observation that, for three Lorentz transformations 𝐿1 ,
𝐿2 and 𝐿3 , the associative property holds, 𝐿3 ◦(𝐿2 ◦𝐿1 ) = (𝐿3 ◦𝐿2 )◦𝐿1 , then
these are enough to indicate that the LT is an example of a mathematical
‘group’, known as the ‘Lorentz group’.
The Lorentz group consists of all those transformations which leave
invariant the interval in Eq. (4.4) (or, as we shall see in the next chapter,
which leave the lengths of 4-vectors invariant). The LT of Eq. (5.6) describes
transformations where the motion is along the 𝑥-axis (i.e., standard con-
figuration). It is possible to generalise this transformation to boosts in any
spatial direction, and to include spatial rotations (it’s usual to use one of a few
different notations when discussing this, to avoid the discussion becoming
unbearably messy); this is referred to as the group of restricted (homogeneous)
Lorentz transformations. The ‘restriction’ is that this excludes reflections in
space or time, and the full Lorentz group consists of the restricted group
extended with transformations which flip the signs of the coordinates. If
we further add translations – transformations which change the origin of
the coordinate system – we obtain the ‘inhomogeneous Lorentz transfor-
mations’, also known as the Poincaré transformations, which represent the
Poincaré group.
Groups are very closely related to the mathematical study of symmetry (to
say a sphere is ‘symmetric’ is to say that there is a mathematically analysable
set of rotations you can perform, which leave the sphere looking unchanged).
Einstein was repeatedly motivated by ideas of symmetry, and invariance,
and they are of profound importance in modern physics.
Poincaré has a lot to do with the mathematical history of relativity. You
might also be interested in a discussion (Adlam 2011) of how much Poincaré
understood of what we now know as relativity, and how his approach to
relativity is linked to a particular philosophical position on science (this is
firmly in the category of dangerous-bend remarks).
 5.3 The Invariant Interval and the Geometry of Spacetime 77

5.3 The Invariant Interval and the Geometry of

Spacetime
Is the interval merely a mathematical curiosity? No; it is the key to a pro-
found reappraisal of our picture of space and time.
The LT as represented by Eq. (5.4) looks like a rotation – indeed like the
rotation which Eq. (5.3) would represent were 𝑙 and 𝜃 real. The hyperbolic
functions and the pattern of signs tell us that it is not exactly the same, but
the similarities are instructive.
A crucial – indeed the defining – feature of a rotation in a plane is that
it preserves euclidean distance. If you have a point (𝑥, 𝑦) in the plane, and
you rotate the axes so that the same point now has coordinates (𝑥 ′ , 𝑦 ′ ) =
(𝑥 cos 𝜃 + 𝑦 sin 𝜃, −𝑥 sin 𝜃 + 𝑦 cos 𝜃), then you know that the distance 𝑥 2 +
𝑦 2 will be equal to 𝑥 ′2 + 𝑦 ′2 . There’s nothing which marks out one pair (𝑥, 𝑦)
as more fundamental in any sense than another pair (𝑥′ , 𝑦 ′ ), nor even any
important difference between the 𝑥- and 𝑦-axes – they’re just a pair of
directions on the plane. The only fixed thing here is the euclidean distance.
In the same way, the crucial feature of the LT is that it preserves the inter-
val 𝑠2 . Just as in the previous paragraph, there is nothing fundamental about
the particular pair of coordinates (𝑡, 𝑥) which our clocks and measuring
rods pick out for us; the coordinates (𝑡 ′ , 𝑥 ′ ) which are natural for another,
moving, observer, are just as fundamental. Similarly, and surprisingly, there
is no real difference between the 𝑡- and 𝑥-axes – they’re just different di-
rections in the relevant geometrical space. This is really an astonishingly
radical view of space and time, and drove Minkowski to remark, famously,
that
The views of space and time which I wish to lay before you have sprung
from the soil of experimental physics, and therein lies their strength. They
are radical. Henceforth space by itself, and time by itself, are doomed to
fade away into mere shadows, and only a kind of union of the two will
preserve an independent reality. (Minkowski 1908)

In the newtonian world-view, Space was a three-dimensional geometrical

space, which obeyed Euclid’s geometrical axioms, and specifically preserved
the euclidean length. Time entered in a simple way: history consisted of
this three-dimensional space moving gracefully through Time, with each
instant of history having its ‘own’ space associated with it. Special Rela-
tivity teaches us that the world is not like this after all: instead we live in
a four-dimensional universe, with three space dimensions and one time
dimension – Spacetime. Spacetime does not obey euclidean geometry, since
a ‘rotation’ in spacetime (i.e., a Lorentz transformation) preserves, not eu-
78 5 The Lorentz Transformation

clidean distance, but the interval 𝑠2 . The geometry of spacetime contains a

great deal more structure than this, which you will learn much more about
if you study General Relativity.
A useful picture, I find, is to imagine two cars driving roughly northwards
in the desert, at the same speed: 𝐴 is moving north-northwest and 𝐵 is
moving north-northeast; thus 𝐴 is moving ‘forwards’ (i.e., NNW) faster
than 𝐵 is moving towards the NNW, and 𝐵 is moving ‘forwards’ (i.e., NNE)
faster than 𝐴 is moving towards the NNE. That is, each of the two cars is
moving forwards steadily and continuously both faster and slower than the
other – this apparent paradox evaporates, and is revealed as the result of
careless language, as soon as one realises that the two cars have different
ideas of ‘forward’, because of the different orientations of their reference
frames – one reference frame is rotated with respect to the other. Exactly
analogously, remember that at the end of Section 3.2 we realised that the
observers in the two trains each thought that the other was moving through
time more slowly than they were: just as with the cars in the desert, this is
perfectly true, because the two sets of observers have different notions of
‘forwards in the desert’ or ‘forwards in time’ as a result of their systematically
‘rotated’ reference frames. This picture seems to me to be the most concrete
illustration of the relationship between the passage of time, and the geometry
of spacetime.

5.4 Proper Time and the Invariant Interval

In some frame 𝑆, consider two events, one at the origin, and one at coor-
dinates (𝑡, 𝑥). The frame is arbitrary – there is nothing special about the
numbers 𝑥 and 𝑡 obtained in this frame.
Consider now a clock moving at a constant velocity 𝑣 in 𝑆, chosen so that
it is present at both events (we assume for the moment that the two events
are close enough that this is possible for a clock moving at a sub-light speed).
Of all the time and space coordinates which could be used to describe the
motion of this clock, there is one we can pick out as special: the frame, 𝑆 ′ ,
in which the clock is at rest at the origin. In this frame, the clock’s position
is constant, 𝑥 ′ = 0, because both events happen at the origin, and the time
coordinate 𝑡 ′ is the time which the clock actually shows on its face. We write
this particular time coordinate as 𝜏, and call it the proper time. Starting from
the frame 𝑆 in which the clock has coordinates (𝑡, 𝑥), we can use Eq. (5.6a)
 5.5 Applications of the Lorentz Transformation 79

to transform to the clock’s rest frame and write

𝜏 = 𝛾(𝑡 − 𝑣𝑥). (5.12)
The frame 𝑆 ′ is special – it’s the one where the clock is at rest – but the
frame 𝑆 is arbitrary, and no matter how frame 𝑆 is moving with respect to
the clock, it would always be possible to calculate this number 𝜏, showing
on the face of the clock. That proper time is invariant under a LT.
Noting that the velocity required for this scenario is simply 𝑣 = 𝑥∕𝑡, we
can put this velocity into Eq. (5.12) to find 𝜏2 = 𝑡 2 − 𝑥 2 . At this point we
realise that we have discovered another version of Eq. (4.4), and that the
idea that the proper time is an easily graspable proxy for the interval is what
is behind much of the discussion in Section 4.6.

5.5 Applications of the Lorentz Transformation

The applications of the LT are basically all the same: given a set of coordi-
nates in one frame, what are the coordinates of the same event or events in
another, moving, frame? I provide a ‘recipe’ for doing relativity problems in
Section D.2.

5.5.1 Length Contraction

A rod of length 𝐿0 lies along the 𝑥 ′ -axis of frame 𝑆 ′ , with one end at the
origin; what is its length as measured in the 𝑆 frame? Frames 𝑆 and 𝑆 ′ are
of course in standard configuration.
At time 𝑡 = 0, there will be two observers in 𝑆 who are adjacent to the
ends of the moving rod: they let off two bangers, or firecrackers; call these
events ○ 1 and ○ 2 . We can find the length of the rod by considering the

coordinates of these two events in the two frames (this is the definition of
the length of the rod, as measured in the ‘stationary’ frame – see Section 1.4).
In frame 𝑆 ′ , there is no complication: any events which happen at the
ends of the rod will be a distance 𝐿0 apart, so that 𝑥2′ = 𝐿0 .
In 𝑆, the two events happen at 𝑡-coordinates 𝑡1 = 𝑡2 = 0, and 𝑥-
coordinates 𝑥1 = 0 and 𝑥2 = 𝐿. Since the origins were coincident at 𝑡 =
𝑡′ = 0, we know that 𝑥1 = 𝑥1′ = 0 (standard configuration again). From
Eq. (5.8), we can write down that
( )
𝑥2 = 𝛾 𝑥2′ + 𝑣𝑡2′ (5.13a)
(′ ′
)
0 = 𝑡2 = 𝛾 𝑡2 + 𝑣𝑥2 . (5.13b)
80 5 The Lorentz Transformation

t

1

x = L = 3 m

Figure 5.1 Events in the carriage frame, 𝑆 ′ .

Equation (5.13b) tells us that 𝑡2′ = −𝑣𝑥2′ ; substituting this into Eq. (5.13a)
and writing 𝑥2 = 𝐿 and 𝑥2′ = 𝐿0 , we have
( ) 𝐿0
𝐿 = 𝛾 1 − 𝑣2 𝐿0 = , (5.14)
𝛾
showing that the moving rod is measured, by observers in the stationary
frame, to be shorter than its ‘rest length’ 𝐿0 .
Compare Section 3.4. The only difference between this section and that
one is that here, we were able to short-circuit the rather elaborate argument
of Section 3.4, since the LT in a sense provides the machinery for that
argument in a ready-to-use form. [Exercise 5.5]

5.5.2 A Worked Example: The Trains Again

As in Section 3.1, consider a train moving past a station platform, this time
at a specific speed 𝑣 = 0.5 (this is the same setup as in Figure 3.2, but going
at a different speed); the carriage is 6 m long, and a flashbulb is set off in the
centre of the carriage at time 𝑡 ′ = 0. (i) What time does the flash reach the
back of the carriage as measured in the carriage? (ii) And as measured on the
platform? (iii) At the instant when the flash reaches the back of the carriage,
what is the position of the forwards-moving wavefront, as measured on the
platform? (iv) And as measured in the carriage? (v) Where is the front of
the carriage at this instant (measured in the platform’s frame)? (vi) So what
is the reading, at this instant, on the clock attached to the front of the train?
We will work through this using the recipe of Section D.2.
Firstly, we have to identify the frames we want to use, and draw the
Minkowski diagram. Let the platform’s frame be 𝑆 and the carriage’s be 𝑆 ′ ,
with the two frames in standard configuration, and choose the origin of the 𝑆 ′
frame to be at the centre of the carriage (the origin of 𝑆, on the platform, is
arbitrary). In drawing the Minkowski diagram(s) you will naturally identify
events. The worldlines of the ends of the carriage are drawn in Figure 5.1.
Event ○ 1 is the flash reaching the back of the carriage, and so sits on both
 5.5 Applications of the Lorentz Transformation 81


1

2 
3
x

Figure 5.2 Events in the platform frame, 𝑆.

that worldline and the worldline of the light flash from the origin, and thus
at the intersection of those two worldlines.
In the platform frame, we draw the worldlines of the ends of the carriage,
in motion to the right, and the worldlines of the same light flashes, again at
slope 1; see Figure 5.2. Exactly as before, event ○ 1 is on the intersection of

the same two worldlines.

Question (i): This one’s easy: we’re looking for the time coordinate of
′
event ○ 1 in the moving frame, thus 𝑡 . We’ve implicitly been told the position
1
of this event, and if we take the length of the carriage, in its own frame,
to be 2𝐿′ = 6 m (thus defining 𝐿′ in a temporarily convenient way), then
𝑥1′ = −𝐿′ = −3 m. The flash goes off at time 𝑡′ = 0 and moves backwards
and forwards at speed 𝑐 = ±1. The back of the carriage is at coordinate
𝑥1′ = −3 m, so the flash reaches the back at time 𝑡1′ = −𝑥1′ = 3 m. At this
point, we’ve successfully written down most of what the question is telling
us (Section D.2 tells us to ‘write down what you know’).
(ii) This question translates into a request for (𝑡1 , 𝑥1 ), given that we now
know (𝑡1′ , 𝑥1′ ). The inverse LT is, from Eq. (5.8), 𝑡 = 𝛾(𝑡 ′ + 𝑣𝑥 ′ ), 𝑥 = 𝛾(𝑥 ′ +
𝑣𝑡′ ), where
1 1 2
𝛾(𝑣 = 0.5) = √ =√ =√ .
1−𝑣 2 1 − 1∕4 3
√ √
Therefore we have 𝑥1 = − 3 m and 𝑡1 = + 3 m (we can spot that |𝑥1 | =
|𝑡1 |, indicating that the signal from the origin to event ○ 1 is travelling at

𝑐 = 1, exactly as it should be – that’s reassuring).

(iii) In order to associate an event with this part of the question, we
imagine one which happens on the worldline of the light flash, which also
happens at the same time 𝑡 as event ○ 1 (it’s hard to imagine a physical event

which happens just here, which is incidentally telling us that this is a rather
artificial question to ask, but the point is that we must synthesise such an
event in our heads, in order to have something to attach coordinates to).
82 5 The Lorentz Transformation

t t

x

1


3
x

Figure 5.3 The final diagram, showing both frames.

Call this event ○ 2 . Thus we can draw it onto Figure 5.2 on the forwards-

going light flash worldline, and simultaneous with event ○ 1 (i.e., at the same

height as ○ 1 on the diagram). The flash started at the origin 𝑥 = 0 of the 𝑆

√
frame at time 𝑡 = 0 and moves at speed√𝑐 = 1, so at time 𝑡2 = 𝑡1 = 3 m the
wavefront must be at position 𝑥2 = + 3 m.
(iv) We need the coordinates of event ○ 2 in the moving frame; that is, we

want (𝑡2 , 𝑥2 ) given that we now have (𝑡2 , 𝑥2 ). From Eq. (5.6), 𝑥2′ = 𝛾(𝑥 −
′ ′

𝑣𝑡) = 1 m and 𝑡2′ = 𝛾(𝑡 − 𝑣𝑥) = 1 m (since the flash is moving at speed 𝑐 = 1,
we must have 𝑥2′ = 𝑡2′ , so this result is reassuring).
(v) Saying ‘at this instant (measured in the platform’s frame)’ tells us that
we are looking for a 𝑡 and not a 𝑡 ′ . We’re looking for the coordinates of an
event ○ 3 which is such that 𝑡3 = 𝑡1 , and the position of which is at the front

of the carriage. Marking this in Figure 5.2 confirms to us that all three events
have been described as being simultaneous in the platform frame. With 𝐿′
as defined above, we can reason that the front of the carriage is at (constant)
coordinate 𝑥 ′ = 𝐿′ = 3 m in the 𝑆 ′ frame and so at coordinate 𝑥 = 𝐿 = 𝐿′ ∕𝛾
at time 𝑡 = 0 (by length contraction or from Eq. (5.6b)). Therefore the front
of the carriage, moving √ at speed 𝑣, is afterwards at coordinate 𝑥 = 𝐿 + 𝑣𝑡.
At time 𝑡3 √ = 𝑡2 = 3, √ the coordinate of the front will therefore be 𝑥3 =
(3 m)∕𝛾 + 3∕2 = 2 3 (alternatively, we can spot that we know 𝑥3′ and 𝑡3 ,
and want 𝑥3 , so we can use the LT 𝑥3′ = 𝛾(𝑥3 − 𝑣𝑡3 ) and rearrange for 𝑥3 ;
this is more direct, but you will often have to do simple speed-times-time
reasoning of this sort).
(vi) Using the result of part (v), we immediately obtain 𝑡3′ = 𝛾(𝑡3 − 𝑣𝑥3 ) =
√
0, from Eq. (5.6a). Alternatively, we have 𝑡3 = 3 m and 𝑥3′ = 3 m and want
𝑡3′ ; we can therefore rearrange Eq. (5.8a) to find 𝑡′ = 𝑡∕𝛾 − 𝑣𝑥 ′ and get the
same result without going via step (v).
I’ve drawn both frames in the same diagram, in Figure 5.3. Notice that
event ○ 3 is on the 𝑥 ′ -axis, reassuringly consistent with having coordinate
 5.6 The Equations of Special Relativity in Physical Units 83

𝑡3′ = 0. It happened at the same time, in 𝑆 ′ , as the light flash.

Note that I’ve drawn Figure 5.2 and Figure 5.3 in their final ‘fair-copy’
version. Working through this example on paper, I added the various events
to the diagram as I progressed through the various questions, and the result-
ing diagram was a bit of a mess. It’s only in retrospect that it becomes clear
how to draw the final tidy diagram.
This is quite an intricate calculation (made rather more intricate by my
indicating some alternative routes, here and there). Note, however, that we
used the LT only in parts (ii), (iv) and (vi). In parts (i), (iii) and (v) we used
nothing more sophisticated than ‘distance is speed times time’, plus a bit of
careful thought about what distance and what time we meant. Although
this seems fiddly, it ends up being quite routine with practice, as long as you
use a systematic approach such as that of Appendix D, and move step by
step, from what you know to what you are to calculate.
[Exercises 5.6 & 5.7]

5.6 The Equations of Special Relativity in Physical

Units
Back in Section 4.1 we introduced ‘natural units’, where we measure both
distance and time in units of metres. This makes a lot of SR’s equations look
both neater and more symmetrical. Here, however, we’ll repeat some of the
key equations of SR in physical units, partly (a) so that you’ve seen them at
least once, as you might see them in other texts, and (b) to perhaps persuade
you that they are unattractively messy in this form.
The interval (see Eq. (4.3)):

𝑠2 = Δ𝑡 2 − Δ𝑥 2 X→ 𝑠2 = 𝑐2 Δ𝑡2 − Δ𝑥 2 .

The Lorentz transformation (see Eq. (5.6)):

( 𝑣𝑥 )
𝑡′ = 𝛾(𝑡 − 𝑣𝑥) X→ 𝑐𝑡 ′ = 𝛾 𝑐𝑡 −
𝑐
𝑥′ = 𝛾(𝑥 − 𝑣𝑡) X→ 𝑥 ′ = 𝛾(𝑥 − 𝑣𝑡)
−1∕2
( )−1∕2 𝑣2
𝛾= 1 − 𝑣2 X→ 𝛾 = (1 − 2 ) .
𝑐
84 5 The Lorentz Transformation

And the addition of velocities (see Eq. (5.11)):

𝑣 1 + 𝑣2 𝑣 1 + 𝑣2
𝑣= X→ 𝑣= .
1 + 𝑣1 𝑣2 1 + 𝑣1 𝑣2 ∕𝑐2
Each of these expressions was obtained from the corresponding expression
in natural units by the simple method of adding enough powers of 𝑐 in each
term, that the expression ends up dimensionally correct. Another way is to
replace 𝑡 ↦→ 𝑐𝑡 and 𝑣 ↦→ 𝑣∕𝑐.

5.7 Paradoxes
The teaching of SR conventionally and usefully includes various paradoxes.
In this context, ‘paradox’ means something that seems wrong at first sight,
but isn’t. These so-called paradoxes are thought experiments where we
arrive at conclusions which are probably unexpected, but which are simply
correct deductions from ideas we at least partly understand. We look at
them in order to deepen our understanding of, and familiarity with, the
ideas of SR.

5.7.1 The Twins

This is a famous one.
Firstly, imagine Odysseus sailing (at a relaxed non-relativistic speed)
between two points, say from Troy to Ithaca. The direct route would follow
the 𝑦-axis in Figure 5.4(a). Contrast this with a route which takes a diversion
via Circe, producing the dog-leg route in the figure. Taking this to be a flat
euclidean surface, a straight line is the shortest distance between two points,
so that the direct route to Ithaca is, of course, shorter than the indirect one.
If, in order to measure the distance sailed beyond any doubt, Odysseus towed
a taffrail log behind him through the wine-dark sea, it would turn a smaller
number of times if he took the direct route, than if he took the indirect one.
So far, so obvious.
Now imagine two intrepid scientists, Odysseus and Penelope, who are
conveniently twins. Penelope leaves Earth in a spaceship to travel to a star
25 light-years away, and does so at a speed 𝑣 = 0.5; then turns round and
comes back. How long is she away for as timed by Odysseus on Earth, and
how long in her own frame? If she travels a total distance of 50 light-years at a
speed 𝑣 = 0.5, then the journey will take 100 years as measured by Odysseus.
However, we know that the time measured on board the spaceship will be
 5.7 Paradoxes 85

y t
Ithaca t = 100 yr

P
O

Circe
O O
x x
Troy 25 ly
(a) (b)

Figure 5.4 Travellers’ paths: (a) Odysseus can go from Troy to Ithaca by two
different routes; (b) Penelope travels outward at speed 𝑣 = 0.5 for 25 yr, and
then returns.

less, according to Eq. (3.3) above: for 𝑣 = 0.5, we have 𝛾 = 1.15, and only 87
years will pass for Penelope, who will consequently be substantially younger
than Odysseus when she returns to Earth.
That seems a little odd, but we’re used to peculiarity in SR, now. However,
some bright spark then points out that, relative to Penelope, it is Odysseus
who has been moving, so shouldn’t the whole situation be symmetrical, just
as with the trains in Section 3.1 above, forcing us to conclude that Odysseus
will be younger than Penelope? This is nonsensical – although two trains
can be mutually measured each to be shorter than the other, two clocks
(Penelope and Odysseus) can’t logically each be showing less time than the
other.3
The paradox is dissolved as soon as we point out that Penelope cannot
conclude that she is not moving (and thus that Odysseus is moving), since of
the two only Penelope experiences the change of velocity at the distant star,
with an absolutely observable acceleration – SR discusses the relationships
between inertial frames, and Penelope does not (indeed cannot) remain in a
single inertial frame for the entire round trip. This becomes clearer when
we consider the Minkowski diagram.
In Figure 5.4(b), we see the path which Penelope takes through spacetime,
on the way to the star and back. It appears that Penelope’s route is the longer,

3 The two situations are not as directly comparable as may at first appear. A measurement of
distance necessarily involves events separated in space, so we must deal with questions of
simultaneity. In contrast, observations of clocks represent a sequence of observations of a
single object as it moves through time and space.
86 5 The Lorentz Transformation

t t t

4

x
a

3

2 
1

x

Figure 5.5 The dog-leg in spacetime, annotated, showing Odysseus’s rest

frame 𝑆, and the rest frame of Penelope on the outward (𝑆 ′ ) and return (𝑆 ′′ )
journeys. The origin of the frame 𝑆 ′′ is at event ○ 1 , and the dashed line 𝑎 is

parallel to the 𝑥′ -axis. The dotted lines are the paths of light flashes moving in
the positive and negative 𝑥 direction.

but remember that the plane of the Minkowski diagram is not a euclidean
surface, so that our intuitions about lengths and angles are not reliable. It is
possible to show, in fact, that a straight line is the longest distance between
two points in Minkowski space (see Exercise 5.8), so that Odysseus, taking
the straight route, has travelled a greater distance through spacetime than
has Penelope. Since, √ for a pair of timelike-separated events, the distance
through spacetime, 𝑠2 , is the same as the proper time 𝜏 between the events
(Section 5.4), we can say that the proper time along Odysseus’s path is greater
than the proper time along Penelope’s, which is to say that a clock carried
with him will show a greater time elapsed than one carried by Penelope,
which is to say that Odysseus is older than Penelope when they re-meet.
It is at this point that I, for one, get the most immediate benefit from
the analogy of the clock as a spacetime version of the taffrail log. When
Odysseus sailed, in a boat, from Troy to Ithaca via Circe, the taffrail log
showed the amount of Mediterranean he moved through; it would have
turned fewer times if he’d gone home by the direct route. When Odysseus
and Penelope move from take-off to event ○ 4 , their clocks show how much

spacetime they have moved through. Clock-Penelope moved through less;

she is younger.
Another way of approaching the puzzle is to ask what events are simulta-
neous in the various different frames. In Figure 5.5 we see the same path
through spacetime as in Figure 5.4(b), but this time including the axes of
the 𝑆 ′ frame (in which Penelope is at rest, on the outward journey; she is
thus moving along the 𝑡 ′ -axis in that frame), and the 𝑆 ′′ frame (in which
 5.7 Paradoxes 87

she is at rest on the return journey, moving along the 𝑡 ′′ -axis; note that
frame 𝑆 ′′ is unusually not in standard configuration with respect to the
other two frames). This diagram makes it very clear that Penelope changes
inertial frame at the turnaround event, ○ 1 . We also see that, just before the
4
turnaround, Penelope sees event ○ 2 , which is at Odysseus’s location, 𝑥 = 0,

as being simultaneous with ○ 1 in 𝑆 ′ (the line 𝑎 is parallel to the 𝑥 ′ -axis). Im-

mediately after the turnaround, however, she sees event ○ 3 as simultaneous

with event ○; in this frame, event ○ happened in the past, with respect
1 2

to ○1 . If Penelope has skipped her relativity lectures, and is not aware of the

significance of the change of frame, she will simply ‘miss’ the time interval
between events ○ 2 and ○ 3 . During the two ‘cruise’ phases, before and after

the turnaround, the two observers’ clocks are indeed symmetrically slower
than each other, but the frame-independent difference between the two
observers’ elapsed time is attributable to the gap between ○ 2 and ○ 3 . This

gap, one might say, is where Odysseus’s extra ageing comes from.
There is an alluring blind alley here, prompted by the presence of the
acceleration at the turning-point, even prompting some folk to insist that
the ‘Twins Paradox’ needs GR to resolve it. It does not: there is no need to
talk about any actual acceleration: rather than actually turning round at
the star, Penelope could simply set the clock of another traveller she meets
there, who is already travelling at the right speed in the return direction, or
hand over her log-book to them. Our conclusion would then be to do with
the total elapsed time on the two legs of the journey, as calculated from the
log-books, rather than counting the grey hairs on one miraculously unaged
traveller; but this is exactly the same conclusion as above, merely in a less
vivid form.
As a final remark, a real Penelope would of course accelerate towards
the destination, and then slow to a halt before accelerating in the opposite
direction for the return journey. Gourgoulhon (2013, §2.6, where he refers
to the scenario by its alternative name of Langevin’s traveller) works through
this more realistic case in detail. This doesn’t change the logic of the argu-
ment, but firstly it illustrates the way that SR can comfortably deal with
acceleration (we touch on this also in Exercise 6.11); and secondly it may
reassure you that, when we introduce the simplification of an instantaneous
turnaround, above, we are not also introducing a subtle flaw in the argu-
ment. [Exercise 5.8]

4 More precisely, and to avoid the word ‘sees’, Penelope can arrange a network of friends at
′
rest in frame 𝑆 ′ , one of whom ends up local to event ○ 2 , and logs it as happening at time 𝑡 ,
2
′
2 in frame 𝑆 ′ , which is numerically equal to 𝑡 .
the time of event ○ 1
88 5 The Lorentz Transformation

5.7.2 The Pole in the Barn

A farmer with a 20 m pole holds it horizontally and runs towards a barn
which is 10 m deep.5 The farmer’s wife,√standing by the barn door, sees
him running at a speed (specifically 𝑣 = 3∕2) at which 𝛾 = 2. The pole is
therefore length contracted to have a measured length of 10 m in the barn’s
frame, so that the pole will fit entirely into the barn, and the farmer’s wife
can slam the door behind him, with the ‘20 m’ pole entirely (and briefly!)
within the 10 m barn (in other words, the farmer’s wife is ‘measuring the
length of the pole’ using yet another variant of the procedure described in
Section 1.4).
However, in the farmer’s frame, the barn is rushing towards him with
the same 𝛾-factor, so that the barn is measured to have a length of only 5 m,
making it even more outlandish that the pole should fit in the barn.
The reason this sounds so bizarre is that we are forgetting the central
insight of this chapter: spatially separate events which are simultaneous
in one frame will not be simultaneous in another. When we say ‘the pole
is entirely within the barn’ we mean something like ‘there is at least one
pair of events, located at each end of the pole while those ends are within
the barn, which are simultaneous in the frame of the barn’. However, this
pair of events are not simultaneous in the frame of the pole. In other words,
the paradox comes about because of the seemingly innocuous remark ‘the
pole will fit entirely into the barn’ which we smuggled into the description
above; this remark is meaningful only when it is associated with a particular
reference frame: it is true within the farmer’s wife’s frame, but is simply not
true within the farmer’s.
In Figure 5.6, we can see the worldlines of the barn (this is another of
those Minkowski diagrams which is a lot more intelligible when you build
it up gradually, than when you look at the final result). Here, the front and
back of the barn are labelled ‘FB’ and ‘BB’, and the front and back of the pole,
which is lying along the 𝑥 ′ -axis with one end at the origin, are ‘FP’ and ‘BP’.
Event ○ 1 is an event which happens at the place and time where the front of

the barn and the back of the pole are coincident (the place and time of the
slammed-shut door), and event ○ 2 is an event which happens at the place

and time where the back of the barn and the front of the pole are coincident.
We can see that events ○ 1 and ○ 2 are simultaneous in 𝑆 – they have the same

𝑡-coordinate – this is the situation where the pole is moving at a speed where

5 This is also known as the ‘ladder paradox’, and was first described, in a slightly different
form, by Rindler (1961). You can also think of it as the car-in-the-garage paradox, but do
wear a seat-belt.
 5.7 Paradoxes 89

t
t

BP FP

x

4

1
x

2

3

FB BB

Figure 5.6 The pole moving through the barn. Frame 𝑆 is the barn, frame 𝑆 ′
is attached to the pole. The worldlines FP, FB, BP, BB, are the front/back of
the pole/barn. For events, see the text.

it is length contracted just the right amount to fit exactly inside the barn, in
the sense of the previous paragraph. From this same diagram, you can see
that event ○ 4 , which is an event which happens at the front of the pole at

the same 𝑡 ′ as event ○ 1 , is beyond the back of the barn: at the instant of 𝑡 ′

when the back of the pole enters the barn in the farmer’s frame (at time
𝑡′ = 𝑡1′ = 𝑡4′ ), the front of the pole is well clear of the back of the barn, and
the event when the front of the pole hits the back of the barn (event ○ 2 ) is

simultaneous in the farmer’s frame with event ○ 3 , which is an event in front

of the barn – the back of the pole hasn’t entered the barn at this point. The
constant spatial distance between events ○ 1 and ○ 4 , or between events ○ 3

and ○ 2 , is the rest length of the pole, 20 m, and the distance between ○ 1 and ○ 2

is the rest length of the barn, 10 m. You can plainly see from this figure that
it is impossible to find a pair of events which have the same 𝑡′ -coordinate,
which are both between the FB and BB worldlines: the pole is never entirely
within the barn in the farmer’s frame.
Figure 5.7 is the same set of events, but drawn in the farmer’s frame, 𝑆 ′ .
Here, the worldlines of the barn show it to be moving in the negative 𝑥 ′ -
direction.
We can look at the problem another way, by asking the question ‘sup-
posing that the back of the barn were made of super-strong concrete, how
would the trailing end of the pole know when to stop?’ We cannot assume
that the pole is completely rigid; thus when the front of the pole hits the back
wall of the barn, the shock wave – the information that this has happened –
takes a finite time to make it to the back of the pole, so that the back of the
90 5 The Lorentz Transformation

t t
FB BB


1 
4
x

2

3
x

BP FP

Figure 5.7 The barn moving past the pole (this view is in the pole’s frame).
The events and worldlines are the same as in Figure 5.6.

pole keeps moving forwards into the barn even after the front of the pole has
halted at the back wall. The information about the front of the pole stopping
moves rearwards at the sound-speed in the pole: for the pole to be rigid
would require an infinite sound-speed. Very shortly after the barn door has
been slammed shut, the pole’s recoil will smash through the door (or more
likely be vaporised, but let’s not worry ourselves with physical practicalities
at this point).
What this example shows is that any conclusion which you correctly
reach in one frame must be reachable in any other frame, even though the
detailed mechanisms might be different.
Exercise 5.9, though it is framed in different language, works through
this problem in illuminating detail. [Exercise 5.9]

5.7.3 Bell’s Accelerating Spaceships

Consider a pair of identical rockets parked on neighbouring launch pads,
on the 𝑥-axis. They take off simultaneously (in the launch-pad frame) and
fly identical flight plans, accelerating continuously along the 𝑥-axis.6
We now suppose that the rockets are connected to each other by a piece of
string, which is taut when the rockets are on the launch pads. The question:
does the string break during the rockets’ acceleration to relativistic speeds?
Viewed in the launch-pad frame, the rockets and the thread are length
contracted. Since the thread is length contracted, and since, in the launch-
pad frame, the two spaceships remain the same distance apart from each

6 This paradox is discussed in Bell (1976), and has become known as ‘Bell’s spaceship paradox’,
although he quotes it as originating somewhat earlier (Dewan & Beran 1959, Dewan 1963).
 5.7 Paradoxes 91

t r2 r1

t

p3 x

p1 p2

Figure 5.8 Bell’s rockets: two spaceships moving along the 𝑥-axis at increas-
ingly relativistic speeds. The two rockets have worldlines 𝑟1 and 𝑟2 .

other, the string can no longer stretch from one rocket to the other, and
therefore must break at some point.
There is a contrary argument, however, which has it that, in the rockets’
frame, the two rockets are stationary, so the string between them is stationary,
so there’s no length contraction, so the string will still stretch between the
rockets, so it won’t break.
Which of these arguments holds up – does the string break or not? Since
the string does or does not break, one of these arguments is mistaken.
You may wish to think the arguments through before reading on. It’s
useful to consider how the two rockets’ accelerations will be observed by
the pilots of the ‘other’ rocket. When doing so, imagine the accelerations
happening as a sequence of instantaneous increments in speed (when do
these jumps happen?), rather than being continuous; also, drawing a well-
chosen Minkowski diagram makes the problem very simple.
In Figure 5.8 we can see the Minkowski diagram of the rockets’ motion,
plus the axes of a reference frame, 𝑆 ′ , which is instantaneously co-moving
with the rear rocket (that is, an inertial frame which, at a specific instant,
is briefly moving at the same velocity as the accelerating rocket, so that its
𝑡′ -axis is parallel to the 𝑟2 worldline at that point; 𝑆 and 𝑆 ′ are not in standard
configuration). The key to resolving the paradox is the insight that, once the
rockets are moving at speed, their clocks are no longer synchronised from
each other’s point of view.
Imagine one-year anniversary parties, events 𝑝1 and 𝑝2 , on board the
two spaceships, when a year of proper time has passed on board. Since the
rockets have identical acceleration schedules, these will be simultaneous in
the launch-pad frame (though not, of course, at 𝑡 = 1 yr).
92 5 The Lorentz Transformation

t r2 r1

t

x
p3

p1 p2

Figure 5.9 Bell’s rockets: the same as Figure 5.8, but with the axes of the front
rocket’s frame drawn in.

In the frame of the rear rocket, however, events 𝑝1 and 𝑝2 are not simul-
taneous; instead it is the events 𝑝1 and 𝑝3 which are simultaneous there.
But event 𝑝3 is after the front rocket’s one-year party, and thus later in the
rockets’ acceleration schedule, so the rear rocket’s pilot (presuming they’ve
forgotten their relativity lectures) will conclude that the front pilot is ‘cheat-
ing’ on the acceleration schedule, by both organising their party ahead of
time (𝑝2 ), and by applying more thrust than agreed, with the result that the
front rocket pulls away, causing the string to break.
The piece of string, which has never attended a relativity lecture in its
short and sorry life, and which is obliged to have one end at 𝑝1 and the other
simultaneously (in its frame7 ) at 𝑝3 , simply detects that the front rocket is
further from the rear rocket than the string can cope with, and breaks.
If you sketch in the axes of a frame 𝑆 ′′ which is instantaneously co-moving
with the front rocket at 𝑝3 (Figure 5.9), then you will see that the event 𝑝3 is
simultaneous in frame 𝑆 ′′ with an event on 𝑟2 with a smaller 𝑡 than 𝑝1 , that
is, at an earlier point in the acceleration schedule: the front rocket’s pilot
(equally negligent of their relativity lectures) believes that the rear rocket
has fallen behind schedule, so it is lagging behind, causing the string to
break (as usual, different frames’ observers have different explanations for
what happens, but they must agree on the results).
The error in the contrary argument above is the phrase ‘in the rockets’
frame. . . ’ – there is no single ‘rockets’ frame’ once they’ve started accelerat-
ing. Here, as in Figure 5.5, the key part of the resolution is to identify the

7 There is no complication about ‘the string’s frame’: even a modest continuous acceleration
produces a relativistic speed before long – see Exercise 6.11 – so we can take the string to be
always in equilibrium. And yes; there is such a thing as ‘string theory’; no, it’s not this.
 5.8 Some Comments on the Lorentz Transformation 93

frames in which things are or are not simultaneous. In Figure 5.5 Penelope
‘misses’ a chunk of time when she switches from frame 𝑆 ′ to 𝑆 ′′ ; in Figure 5.8,
the relativity-naïve pilots accuse each other of bad navigation, because they
have not realised that both rockets have changed frame between the launch
and the one-year party. [Exercise 5.10]

5.8 Some Comments on the Lorentz

Transformation
This section has a ‘dangerous bend’ symbol, not because it is particularly
hard, but because it is something of a tangent from the thread of the text.

5.8.1 The Derivation of the Lorentz Transformation

The route I have chosen here, to derive the LT, is to demand that the in-
terval Eq. (5.1) be invariant under the transformation, and then work out
what transformation provides this. This is motivated by the observation, in
Section 4.4, that the interval is invariant. This is partly justified by the fact
that it works, in the sense of obtaining the LT, which is observed to match
nature. But it also highlights the importance of the geometry of Minkowski
space, which arises from our choice of 𝑠2 as our definition of distance. This
also leads quite naturally to the geometrical approach to General Relativity
which we will meet in Appendix A.
Einstein’s 1905 paper (which is reproduced in Lorentz et al. (1952)) de-
rives the LT by what may now seem a rather circuitous route which starts
with the demand that clocks in both the stationary and the moving frame are
synchronised, using the procedure in Section 2.2.1, and works out what the
relationship must be, between the two frames’ space and time coordinates,
in order for this to be so. But even there, in a footnote in §3 of that paper,
he says (adjusting notation) ‘The equations of the Lorentz Transformation
may be more simply deduced directly from the condition that in virtue of
these the relation [Δ𝑥2 = (𝑐Δ𝑡)2 ] shall have as its consequence the second
relation [Δ𝑥 ′2 = (𝑐Δ𝑡 ′ )2 ].’ The ideas of symmetry, and of frame invariance,
were pervasive in Einstein’s work, and have become organising principles
for much of modern physics.
Taking Einstein’s hint, we could choose to take the invariance of the
interval as our starting point for relativity, instead of the second axiom of
Chapter 2, and deduce from that the geometry of Minkowski space, and
94 5 The Lorentz Transformation

eventually the Lorentz transformation. This is the approach taken by Gour-

goulhon (2013), for example: this is a powerful and captivating approach,
but a rather abstract one; it may appeal to you if your interests run towards
pure maths. Whether this approach is more or less physically illuminating
is probably a matter of taste.
An alternative mathematically direct, but similarly less physically illumi-
nating, route to the LT is to note that the transformation from the unprimed
to the primed coordinates must be linear, if the equations of physics are to be
invariant under a shift of origin. That is, we must have a transformation like
𝑡′ = 𝐴𝑥 + 𝐵𝑦 + 𝐶𝑧 + 𝐷𝑡, and similarly (with different coefficients) for 𝑥 ′ ,
𝑦 ′ and 𝑧′ . By using the Relativity Principle and the constancy of the speed
of light, one can deduce the transformation given in Eq. (5.6). There are
different versions of the same discussion in Landau & Lifshitz (1975, §2),
Taylor & Wheeler (1992, §§L.4–5), Barton (1999, §4.3), Rindler (2006, §2.6),
and Schutz (2009, §1.6).
Finally, Rindler (2006, §2.11) shows an even more powerful consequence
of the same ideas, and in particular observes that we can choose multiple
alternative second axioms. If we were to choose as our second axiom ‘there is
no upper limit to speeds in nature’, then we could examine the consequences
of these axioms and before too long deduce the galilean transformation. That
is, there is nothing mathematically wrong with this axiom – it leads to no
inconsistency – it simply fails to match our universe.
If on the other hand we chose as the second axiom ‘there is a finite
upper limit to speeds in nature’, then we deduce Einstein’s SR (and must
subsequently assert that light is a thing which travels at that upper speed).
There are no options other than these two: something which moves at that
upper speed-limit in one frame must be observed to move at that same limit
speed in every other frame; if this were not the case then, by virtue of the
first axiom, observers in one frame could deduce that they were moving
absolutely. The second axiom could in fact be anything that is unique to SR,
such as ‘coordinates in inertial frames are related by the LT’: the statement
‘light always travels at 𝑐’ is simply crisp and historically motivated, and shows
Einstein’s excellent taste in choosing a starting point for his argument.
The discussion in Section 5.8.2 shows a further alternative way of deriving
the LT which, to me, is much less clear than the geometrical approach of
this section. Yet again, your opinion may vary, depending on your tastes
and physics background.
If you think all of these are too long-winded, you might enjoy Rindler’s
‘World’s fastest way to get the relativistic time-dilation formula’ (1967).
 5.8 Some Comments on the Lorentz Transformation 95

5.8.2 The Lorentz Transformation and Maxwell’s

Equations
It can be shown that the electric field at position 𝐫 = (𝑥, 𝑦, 𝑧), of a charge 𝑞
moving with speed 𝑣 along the 𝑥-axis, has components8
𝑞
𝐸𝑥 = 𝑥′ (5.15a)
4𝜋𝜖0 𝑟3
𝑞
𝐸𝑦 = 𝛾𝑦 (5.15b)
4𝜋𝜖0 𝑟3
𝑞
𝐸𝑧 = 𝛾𝑧, (5.15c)
4𝜋𝜖0 𝑟3
where 𝛾 = (1 − 𝑣2 )−1∕2 , 𝑟2 = 𝑥 ′2 + 𝑦 2 + 𝑧2 , 𝑥 ′ = 𝛾(𝑥 − 𝑥(𝑡)),
̄ ̄ is the
and 𝑥(𝑡)
instantaneous position of the charge at time 𝑡.
As is evident from Eq. (5.15), the electric field is no longer spherically
symmetric, and will have a pattern which is squashed in the direction of
motion, along the 𝑥-axis. The field around a stationary charge is symmetric,
and by choosing the angular velocity carefully, we could set a charged test
particle in a circular orbit around the original charge, with period 𝑇. It can
be shown (Bell 1976) that the corresponding test particle orbiting the moving
charge, with the squashed electric field, has an orbit compressed along the
𝑥-axis, with a period of 𝛾𝑇, greater than the period of the original system.
Bell shows that, if the motion of the particle around the moving charge
is expressed using the change of variables (in units where 𝑐 = 1)
𝑥′ = 𝛾(𝑥 − 𝑣𝑡) (5.16a)
𝑦′ =𝑦 (5.16b)
𝑧′ =𝑧 (5.16c)
𝑡′ = 𝛾(𝑡 − 𝑣𝑥), (5.16d)
then the expressions for the field of the moving charge, and the motion of the

8 This expression was first established by Heaviside in 1888 or 1889 (in different notation; see
Heaviside (1889)). Although the result looks reasonably simple, the calculation is famously
hard: there is a discussion of the various routes to the result in Jefimenko (1994), which
refers to the derivation as ‘one of the most complicated procedures in classical
electromagnetic theory’. The discussion in this section follows that in Bell (1976), which is
also the source for ‘Bell’s spaceship paradox’ of Section 5.7.3. Although this paper is now
most commonly cited (slightly erroneously) merely as the source of the paradox, Bell’s goal
in the paper was to claim that the argument in this section is a more intuitive way of arriving
at SR than the axiomatic approach. I think this is true only for those with a substantial
pre-existing familiarity with advanced EM theory, but that it is largely unintelligible for
those without this familiarity, as well as missing the fundamental insights mentioned at the
end of this section.
96 5 The Lorentz Transformation

test particle in orbit around it, reduce to the same form as the (spherically
symmetrical) expressions for a stationary charge, and the particle again
orbits with period 𝑇; and further, that Maxwell’s equations in these changed
variables have the same form as Maxwell’s equations in the original variables
(compare Exercise 2.4).
Equations (5.16) are of course now familiar. These and similar expres-
sions were well known to physicists for at least a decade prior to Einstein’s
1905 paper. The LT is also sometimes referred to as the FitzGerald–Lorentz
transformation, since it appears to have been George Francis FitzGerald who
first suggested (1889) that ‘the length of material bodies changes, according
as they are moving through the ether or across it’, which was elaborated as
an idea by Hendrik Antoon Lorentz (1895, 1904). It may be J. J. Thomson
who first put the change of variables Eq. (5.16a) into print (Thomson 1889),
as a way of recovering the spherical symmetry of Maxwell’s equations, in the
case of a particle in motion in an electric field, but it does seem that it was
Lorentz who came closest to writing down, what it now seems fair to call,
the Lorentz transformation. The papers I have mentioned here form part of
a network of dauntingly complicated attempts to discuss the consequences
of Maxwell’s equations in a frame moving with respect to the aether and –
although this is certainly not how these authors would have described their
work – to find a transformation which left form-invariant the equations
of electromagnetism. It’s easy to think, now, of SR as part of mechanics
(think of all these trains moving about at high speed and, in chapters to
come, particles colliding), and to be slightly puzzled at why it would occur
to Einstein to think of specifically light as having some special status. But
Einstein’s 1905 paper is, even looking only at its title, located within the
study of electromagnetism; and almost half of it, subtitled ‘Electrodynamical
part’ and beginning with ‘Transformation of the Maxwell–Hertz equations
for empty space’, is manifestly connected to the same network of concerns
as above. Despite thus advertising its connections with previous work, the
1905 paper is remarkable for the completeness and concision of its reboot of
physics as the study of symmetry in nature, which Minkowski could be said
to declare as complete in 1908.
Recall that, at this point in this section, we have not yet talked about
Special Relativity; so far, this is all (advanced) classical electromagnetism.
If the ‘moving charge’ described above is the nucleus of an atom, and the
orbiting charges are the atomic electrons, then the above analysis is telling
us that those atoms will be ‘squashed’ along the direction of motion by a
factor of 𝛾, and thus that a rod which is, for example, Avogadro’s number
of atoms long will be contracted in length by the same factor. If the ‘rod’
 5.8 Some Comments on the Lorentz Transformation 97

is instead your brain, in motion while sitting in a travelling train, then not
only will it be contracted along the direction of your motion, but all of
the electrons around all of the atoms in it will be orbiting with a period
which is longer than the period at rest, by another factor of 𝛾. The result,
presumably, would be that you would think slower by a factor of 𝛾, and
the watch in your hand – with its components also made of atoms – would
tick slower by a factor of 𝛾, ticking out 𝑡′ rather than 𝑡. Thus all of your
observations within the carriage would be in terms of (𝑡 ′ , 𝑥 ′ ) rather than the
(𝑡, 𝑥) appropriate to measurements on the stationary platform. But that in
turn means that all of the electromagnetic measurements you make, in the
carriage, would be indistinguishable from the corresponding measurements
made on the platform, even though they would be measured to go more
slowly by observers standing on that platform. It would be impossible,
based on experiments and observations made within the carriage, for you to
discover whether you were in motion.
It is possible to imagine, at this point, a counterfactual history of rela-
tivity without Einstein, in which all of the equations of SR are obtained as
consequences of Eq. (5.16), and we discover, as a consequence of these, the
invariance of the interval 𝑠2 . The measurable results would be largely the
same as what we have now, but the route and the foundations would be very
different.
We might end up concluding that an absolute rest frame was a physi-
cally real thing, even though, by a peculiarity of Maxwell’s equations, it was
impossible to detect (this appears to have been Lorentz’s position, just as,
looking further back, it may have been Newton’s position even though his
theory, similarly, contained nothing which would allow the rest frame to
be identified). If you believe in atoms, and in particular atoms composed
of electrons orbiting a nucleus (a belief which was not universal in the first
decades of the twentieth century), then this feature of Maxwell’s equations
transfers itself to material bodies, and thus to the rest of physics. The ar-
gument above, about the atoms in your brain, makes it plausible that you
would think slower whilst moving, but it doesn’t prove it. All together, this
argument from Maxwell’s equations explains why we don’t see an aether
and why ‘moving clocks run slow’, but leaves a large number of questions
still open, and in addition doesn’t address the question of why it should
be that Maxwell’s equations have this odd Lorentz-group symmetry, rather
than anything more intuitive.
Einstein’s axiomatic starting point, in contrast, focuses on the underlying
geometry of, and properties of, spacetime, rather than one particular physical
theory, and it therefore immediately has universal applicability.
98 5 The Lorentz Transformation

Einstein’s two axioms say nothing of electromagnetism, nor of mechanics:

the first axiom says that all inertial frames are equivalent for all physical
experiments (the absolute in Einstein’s work is not absolute rest, but absolute
symmetry). This axiom is consistent with the galilean transformation as
well as Lorentz’s, so the role of the second axiom is to pick out the latter from
the former: saying ‘light always moves at 𝑐’ is a fine second axiom, saying
‘Δ𝑡2 − Δ𝑥 2 is frame-invariant’ would be a perfectly adequate alternative
(as Einstein hints in a footnote in the 1905 paper, and as I discussed in
Section 5.8.1).
At the risk of taking a tangent to the tangent, this seems a good place to
observe that the ‘theory of relativity’ is something of a misnomer, since calling
it relativity muddles the problem (the frame dependence of coordinates) with
the solution (the invariance of the interval).9 That is, Einstein’s theory solves
the problem by identifying new absolutes, most specifically the invariance
of 𝑠2 between frames. In the nearest that Einstein came to an autobiography
(Einstein 1991), he repeatedly stresses the methodological and philosophical
importance of identifying invariance or symmetry under transformation.

Exercises
Exercise 5.1 (§5.1)
Look back at Figure 3.3 on p. 34, and consider the following two frames.
Frame 𝑆 is attached to the right-moving carriage and has its spatial origin 𝑥 =
0 at the centre of the top carriage; frame 𝑆 ′ is attached to the left-moving
carriage and has its spatial origin 𝑥 ′ = 0 at the centre of that carriage. Sketch
the position of these carriages/frames at time 𝑡 = 0. Are these frames in
standard configuration? Can we use the Lorentz transformation to relate
the coordinates of these frames? [ 𝑢+ ]

9 It’s a fairly common observation that Einstein disliked the term, but it’s unexpectedly hard
to find a specific source for it. The nearest I have been able to find is a 1921 letter to
Eberhard Zschwimmer (Einstein Papers, volume 12, document 250,
https://einsteinpapers.press.princeton.edu/vol12-doc/372), in which he says that the name
‘relativity theory’ can lead to philosophical misunderstandings, and agrees that ‘invariance
theory’ (Invariantz-Theorie) would better describe the method; he acknowledges, however,
that it was even then too late to change the term.
 Exercises 99

Exercise 5.2 (§5.1)

√
A spaceship cruises past the Earth travelling at speed 𝑣 = 3∕2. The ship
is carrying a beacon flashing 120 times per minute (that is, 0.5 s between
flashes).
Explain how you would use a network of observers to make direct mea-
surements of the length of the spaceship, and the interval between flashes,
in the Earth’s frame (that is, without using the LT).
Use the time dilation formula to obtain the interval between flashes, as
measured on Earth.
Earth traffic control sends a radio signal to the ship 1010 m (a little under
a minute) after the ship passes, as measured on Earth; the flashing stops
at a time 3 × 1010 m after the flyby, as measured on the ship. Sketch these
two events on a Minkowski diagram, obtain their coordinates, and calculate
the invariant interval between them. State whether the end of the flashing
could be attributed to the signal from traffic control, and why.

Exercise 5.3 (§5.2)

What happens to 𝛾 as we take the velocity 𝑣 towards zero in Eq. (5.7) – that
is, as we move to the non-relativistic velocities of our everyday experience?
The LT of Eq. (5.6), written in physical units, is
( 𝑣𝑥 )
𝑐𝑡 ′ = 𝛾 𝑐𝑡 −
𝑐
𝑥′ = 𝛾(𝑥 − 𝑣𝑡).

What is the form of the LT in this slow-speed limit? Do you recognise this?
From Eq. (5.11), how do velocities add in this limit, where either 𝑣1 or 𝑣2
is small compared with 𝑐 = 1? And what happens if one of the velocities is
already the speed of light?

Exercise 5.4 (§5.2)

In the notation of Eq. (5.10), show that 𝛾(𝑣) = cosh 𝜙, and that

𝛾(𝑣)
= 1 + 𝑣1 𝑣2 .
𝛾(𝑣1 )𝛾(𝑣2 )
100 5 The Lorentz Transformation

Exercise 5.5 (§5.5.1)

In Section 5.5.1 the LT was used to obtain the length-contraction formula
(Eq. (5.14)) directly. Use a similar argument to obtain the time-dilation
result of Eq. (3.3), using the LT of Eq. (5.6).

Exercise 5.6 (§5.5.2)

What is the speed of the trains in Figure 3.2 and Figure 3.3 (p. 34)? Remember
that the train carriage is 6 m long, and that the clocks are showing times in
units of metres.
Further, show that the times shown in Figure 3.5 are correct.
What is the length of the lower train, which Barbara and Hilary measure
(i.e., 𝐿′ in Figure 3.4)? [This question is a little tricky, not because it’s
conceptually hard, but because it’s a bit like a logic puzzle, in seeming to
have too little information; getting the right choice of frames is important.]
[ 𝑑+ ]

Exercise 5.7 (§5.5.2)

A train enters a tunnel travelling at a speed 3∕5 (in units where 𝑐 = 1). The
tunnel is 500 m long, and the train 100 m long, both as measured in their
rest frames. The train has clocks at the extreme front and back, which
are synchronised in the train’s frame. The rear clock is observed to show
time 0 m at the instant the rear of the train disappears into the tunnel. The
front clock is also observed when it emerges from the other end of the tunnel.
Draw a Minkowski diagram for the frame of the tunnel, showing the
train’s motion, including the worldlines of the clocks and the tunnel ends,
and the three events consisting of ○ 1 the rear clock disappearing into the

tunnel, an event ○ 1 in 𝑆 ′ , and

2 at the front of the train, simultaneous with ○

an event ○3 at the point where the front clock emerges from the tunnel.

By using the Lorentz transformation and/or the length-contraction for-

mula, or otherwise, deduce the time on the front clock when it emerges
from the tunnel.

Exercise 5.8 (§5.7.1)

In Figure 5.10, we see two alternative routes between events ○ 1 and ○ 3 ,

namely a direct route, and one via a second event ○. Take the interval
2

between events ○2 and ○3 to be null. By considering the spatial, temporal, and

invariant intervals between these three events, work out the total invariant
 Exercises 101

t

3


2

x

1

Figure 5.10 A ‘dog-leg’ path in Minkowski space.

interval along the two paths, and thus conclude that in Minkowski space a
straight line is the longest timelike interval between two points.
It is reasonable to assume that the total interval along a path is the sum
of the intervals along its segments. [ 𝑑+ ]

Exercise 5.9 (§5.7.2)

Take two cars, moving along the 𝑥-axis at a speed 𝑣, separated by 1 km in
their frame. The lead car passes first a checkpoint and then, 0.5 km further
on in the checkpoint frame, a traffic policeman.
Sketch a Minkowski diagram of these events, indicating at least the world-
lines of the cars, the checkpoint and the policeman, and marking the events:
r ○1 , the lead car passing the checkpoint;
r ○2 , the position of the second car at the same time as ○
1 in the cars’ frame;
r ○3 , the position of the second car at the same time as ○
1 in the checkpoint

frame;
r ○4 , the second car reaching the checkpoint; and
r ○5 , the first car reaching the policeman.

[It will probably help if you choose the frames such that event ○ 1 has coor-

dinates 𝑥1 = 𝑡1 = 𝑥1′ = 𝑡1′ = 0, but if you want to do it another way, that’s

fine.]
Give expressions for the coordinates of these events (in either 𝑆 or 𝑆 ′ as
appropriate), as expressions in terms of the speed 𝑣. This is the ‘write down
what you know’ step of the Appendix D recipe.
2
Give an expression for the interval 𝑠45 between events ○4 and ○5 , in terms

of 𝑣, and calculate numerical values for this in the cases (i) 𝑣 = 1∕2, (ii) 𝑣 =
3∕5 and (iii) 𝑣 = 4∕5. In each of the three cases, state, with an explanation,
whether it is possible for the traffic policeman to signal to the checkpoint to
102 5 The Lorentz Transformation

lower a barrier before the second car arrives.

[Optional extra: draw Minkowski diagrams for the three cases (i), (ii)
and (iii), which illustrate the answers you obtain above. This might illumi-
nate both the answer to this question, and the pole-in-the-barn problem in
the farmer’s frame.] [ 𝑢+ ]

Exercise 5.10 (§5.7.3)

A piezoelectric crystal is one which generates an electrical voltage when it is
squashed. Imagine that the ‘carriages’ in Figure 3.2 were in fact piezoelectric
crystals, and recall that we worked out in Section 3.2 that the observers on
the other carriages (and, though we did not work this out explicitly, the
observers on the station platform) would measure the top carriage/crystal
to be shorter than it would be when it was at rest. Would the crystal be
generating a voltage because it’s shorter? [ 𝑑+ ]
 6
Vectors and Kinematics

In Chapter 5, we used the axioms of Chapter 2 to obtain the Lorentz trans-

formation. That allowed us to describe events in two different frames in
relative motion. That part was rather mathematical in style. Now we are
going to return to the physics, and describe motion: velocity, acceleration,
momentum, energy and mass.

Aims: you should:

6.1. understand the concept of a 4-vector as a geometrical object, and the
distinction between a vector and its components.
Chapter 5 was concerned with static events as observed from moving
frames. In this part, we are concerned with particle motion.
Before we can explain motion, we must first be able to describe it. This
is the subject of kinematics. We will first have to define the vectors of four-
dimensional Minkowski space, and specifically the velocity and acceleration
vectors.

6.1 Three-Vectors
You are familiar with 3-vectors – the vectors of ordinary three-dimensional
euclidean space. To an extent, 3-vectors are merely an ordered triple of
numbers, but they are interesting to us as physicists because they represent a
more fundamental geometrical object: the three numbers are not just picked
at random, but are the vector’s components – the projections of the vector
onto three orthogonal axes (that the axes are orthogonal is not essential to
the definition of a vector, but it is almost always simpler than the alternative).
That is, the components of a vector are functions of both the vector and our

103
104 6 Vectors and Kinematics

y
y θ
A
Ay
Ay Ax x

x
Ax

Figure 6.1 A displacement vector in 3-d euclidean space.

choice of axes, and if we change the axes, then the components will change
in a systematic way.
For example, consider a prototype displacement vector (Δ𝑥, Δ𝑦, Δ𝑧).
These are the components of a vector with respect to the usual axes 𝐞𝑥 ,
𝐞𝑦 and 𝐞𝑧 . If we rotate these axes, say by an angle 𝜃 about the 𝑧-axis, to
obtain axes 𝐞′𝑥 , 𝐞′𝑦 and 𝐞′𝑧 , we obtain a new set of coordinates (Δ𝑥 ′ , Δ𝑦 ′ , Δ𝑧′ ),
related to the original coordinates by

⎛ Δ𝑥′ ⎞ ⎛ cos 𝜃 sin 𝜃 0 ⎞ ⎛ Δ𝑥 ⎞

⎜ Δ𝑦 ′ ⎟ = ⎜ − sin 𝜃 cos 𝜃 0 ⎟ ⎜ Δ𝑦 ⎟ . (6.1)
⎜ ′⎟ ⎜ ⎟⎜ ⎟
Δ𝑧 0 0 1 Δ𝑧
⎝ ⎠ ⎝ ⎠⎝ ⎠
These new components describe the same underlying vector 𝐀 =
(Δ𝑥, Δ𝑦, Δ𝑧), as shown in Figure 6.1 (omitting the 𝑧-axis). Although any
random triple of numbers (Δ𝑥, Δ𝑦, Δ𝑧) describes some vector, to some extent
what turns the number triple into a vector is the existence of this underly-
ing object, which implies all the other sets of coordinates (𝐴𝑥′ , 𝐴𝑦′ , 𝐴𝑧′ ) =
(Δ𝑥′ , Δ𝑦 ′ , Δ𝑧′ ), that have a particular functional relation to the origi-
nal (𝐴𝑥 , 𝐴𝑦 , 𝐴𝑧 ) = (Δ𝑥, Δ𝑦, Δ𝑧).
Note that this prototype vector is a displacement vector; the ‘position
vector’, in contrast, which goes from the origin to a point, is not a vector in
this sense.

Ideally, you should think of the vectors here as being defined as a length
plus direction, defined at a point, rather than an arrow spread out over
a finite amount of space (you might think of this as some sort of infinitesimal
displacement vector). If you think of an electric or magnetic field, which has a
size and a direction at every point, you’ll have a very good picture of a vector in
the sense we’re using it here.

There is a swift review of linear algebra in Section C.3; make sure you
are comfortable with the terms and definitions mentioned there.
 6.2 Four-Vectors 105

6.2 Four-Vectors
As we saw in Section 5.3, we can regard the events of SR taking place in a
4-dimensional space termed spacetime. Here, the prototype displacement
4-vector is (Δ𝑡, Δ𝑥, Δ𝑦, Δ𝑧), relative to the space axes and wristwatch of a
specific observer, and the transformation which takes one 4-vector into
another is the familiar LT of Eq. (5.6), or

⎛ Δ𝑡 ′ ⎞ ⎛ 𝛾 −𝛾𝑣 0 0⎞ ⎛ Δ𝑡 ⎞
⎜Δ𝑥′ ⎟ ⎜−𝛾𝑣 𝛾 0 0⎟ ⎜Δ𝑥 ⎟
⎜Δ𝑦 ′ ⎟ = ⎜ 0 0 1 0⎟ ⎜Δ𝑦 ⎟
(6.2a)
⎜ ′⎟ ⎜ ⎟⎜ ⎟
Δ𝑧 0 0 0 1 Δ𝑧
⎝ ⎠ ⎝ ⎠⎝ ⎠
for the ‘forward transformation’ and
⎛ Δ𝑡 ⎞ ⎛ 𝛾 +𝛾𝑣 0 0⎞ ⎛ Δ𝑡 ′ ⎞
⎜Δ𝑥⎟ ⎜+𝛾𝑣 𝛾 0 0⎟ ⎜Δ𝑥′ ⎟
⎜Δ𝑦 ⎟ = ⎜ 0 0 1 0⎟ ⎜Δ𝑦 ′ ⎟
(6.2b)
⎜ ⎟ ⎜ ⎟⎜ ⎟
Δ𝑧 0 0 0 1 Δ𝑧′
⎝ ⎠ ⎝ ⎠⎝ ⎠
for the inverse transformation (that the matrices are inverses of each other
can be verified by direct multiplication). These give the coordinates of
the same displacement as viewed by a second observer whose frame is in
standard configuration with respect to the first.
This displacement 4-vector Δ𝐑 = (Δ𝑡, Δ𝑥, Δ𝑦, Δ𝑧) we can take as the
prototype 4-vector, and recognise as a 4-vector anything which transforms
in the same way under the coordinate transformation of Eq. (6.2a) (this may
seem a rather abstract way of defining vectors, but we will see a concrete
example in Section 6.5).
We write the components of a general vector as 𝐀 = (𝐴0 , 𝐴1 , 𝐴2 , 𝐴3 )
(do note that the superscripts are indexes, not powers),1 or collectively 𝐴𝜇 ,
where the greek index 𝜇 runs from 0 to 3. We will also occasionally use latin
superscripts like 𝑖 or 𝑗: these should be taken to run over the ‘space’ indexes,
from 1 to 3.
An arbitrary vector 𝐀 has components 𝐴𝜇 in a frame 𝑆, as illustrated in
Figure 6.2. The components 𝐴1 , 𝐴2 and 𝐴3 (that is, 𝐴𝑖 ) are just as you would
expect, namely the projections of the vector 𝐀 onto the 𝑥-, 𝑦- and 𝑧-axes.
The component 𝐴0 is the projection of the vector onto the 𝑡-axis – it’s the
timelike component. In a frame 𝑆 ′ , the space and time axes will be different,

1 This is now, I think, the most common notation, but you can still find books which observe
the slightly more old-fashioned convention of labelling these as (𝐴1 , 𝐴2 , 𝐴3 , 𝐴4 ).
106 6 Vectors and Kinematics

t t
A
A0 A0

A1 x
x
A1

Figure 6.2 A displacement vector in 4-d Minkowski space.

and so the projections of the vector 𝐀 onto these axes will be different. Just
as in Figure 6.1, we find the projection of a point by moving it parallel to
one of the axes. For example, we find the projection onto the 𝑥-axis of the
end of the vector 𝐀, by moving that point parallel to the 𝑡-axis until it hits
the 𝑥-axis, and we find the projection of the same point onto the 𝑥 ′ -axis by
moving it parallel to the 𝑡 ′ -axis until it hits the 𝑥′ -axis.
Given the components of the vector in one frame, we want to be able
to work out the components of the same vector in another frame. We can
obtain this transformation by direct analogy with the transformation of the
displacement 4-vector, and take an arbitrary vector 𝐀 to transform in the
same way as the prototype 4-vector Δ𝐑.
That is, given an arbitrary vector 𝐀, the transformation of its compo-
nents 𝐴𝜇 in 𝑆 into its components 𝐴′𝜇 in 𝑆 ′ is exactly as given in Eq. (6.2a):

′
⎛𝐴0 ⎞ ⎛ 𝛾 −𝛾𝑣 0 0⎞ ⎛𝐴0 ⎞
⎜𝐴1′ ⎟ ⎜−𝛾𝑣 𝛾 0 0⎟ ⎜𝐴1 ⎟
⎜𝐴2′ ⎟ = ⎜ 0 0 1 0⎟ ⎜𝐴 2 ⎟
. (6.3a)
⎜ 3′ ⎟ ⎜ ⎟⎜ ⎟
𝐴 0 0 0 1 𝐴3
⎝ ⎠ ⎝ ⎠⎝ ⎠
Since this is a matrix equation, the inverse transformation is straightforward:
it is just the matrix inverse of this:
′
⎛𝐴0 ⎞ ⎛ 𝛾 𝛾𝑣 0 0⎞ ⎛𝐴0 ⎞
⎜𝐴1 ⎟ ⎜𝛾𝑣 𝛾 0 0⎟ ⎜𝐴1′ ⎟
⎜𝐴2 ⎟ = ⎜ 0 0 1 0⎟ ⎜𝐴2′ ⎟ . (6.3b)
⎜ 3⎟ ⎜ ⎟ ⎜ ′⎟
𝐴 0 0 0 1 𝐴3
⎝ ⎠ ⎝ ⎠⎝ ⎠
You may also see this written out in matrix form as

𝐀′ = Λ𝐀; 𝐀 = Λ−1 𝐀′ (6.4)

where the transformation matrix Λ is as in Eq. (6.3a). Written out explicitly,

 6.2 Four-Vectors 107

t t

A0 A
A0

A1
x
A1 x

Figure 6.3 The displacement vector of Figure 6.2 in the ‘other’ frame.

the expressions in Eq. (6.3a) are

′
𝐴0 = 𝛾(𝐴0 − 𝑣𝐴1 )
′
𝐴1 = 𝛾(𝐴1 − 𝑣𝐴0 )
′ (6.5)
𝐴2 = 𝐴2
′
𝐴3 = 𝐴3 ,
and, as you can see, these have the same form as the more familiar expression
for the Lorentz transformation of an event’s coordinates, as in Eq. (5.6). Note
also that the primed and unprimed frames are perfectly symmetrical. The
observers at rest in the unprimed frame 𝑆 regard the primed frame 𝑆 ′ as
moving at speed 𝑣 with respect to them. However, the observers in the
primed frame see the unprimed one moving at speed −𝑣; they would also
naturally swap the assignment of primed and unprimed frame, with their
frame, say 𝑃, being the unprimed one and the other, 𝑃′ , being the primed
one. In Figure 6.3 we can see the vector 𝐀 in Minkowski space, drawn from
the point of view of these ‘primed’ observers.
Now, these two sets of observers are describing the same events and
frames, but with opposite notation, and so with an opposite sign for 𝑣. We
can translate between the two sets of notations by simultaneously swapping
primed and unprimed quantities and swapping the sign of 𝑣. But all this
is, is a way of translating between a transformation and its inverse. This
is exactly what happens between, for example, Eq. (5.6) and Eq. (5.8), or
between Eq. (6.2a) and Eq. (6.2b).
I’ve talked repeatedly of events being frame-independent. Vectors are
also frame-independent, even though their components are not (this is what
distinguishes a vector from being just a collection of four numbers).
More linear algebra, directly echoing the corresponding results in the
previous section, and in Section C.3: Just as with 3-vectors, if 𝐀 and 𝐁
are 4-vectors, so is 𝑎𝐀 (for all 𝑎 ∈ ℝ), and so is 𝐀 + 𝐁, with components
108 6 Vectors and Kinematics

(𝐴0 + 𝐵0 , 𝐴1 + 𝐵1 , 𝐴2 + 𝐵2 , 𝐴3 + 𝐵3 ). Where the scalar product of ordinary

3-vectors is straightforward, the different geometry of spacetime means that
the useful scalar product for 4-vectors is defined as

𝐀 ⋅ 𝐁 = 𝐴0 𝐵 0 − 𝐴1 𝐵 1 − 𝐴2 𝐵 2 − 𝐴3 𝐵 3 (6.6a)
∑
= 𝜂𝜇𝜈 𝐴𝜇 𝐵𝜈 (6.6b)
𝜇,𝜈

= 𝐀𝑇 𝜂𝐁, (6.6c)

where the matrix 𝜂𝜇𝜈 is defined as

𝜂𝜇𝜈 = diag(1, −1, −1, −1) (6.7)

(denoting the matrix with components which are all zero except for
(1, −1, −1, −1) on its diagonal), and in the last line I have written the equiva-
lent matrix expression involving column vectors 𝐀 and 𝐁. The scalar product
of a vector with itself, 𝐀 ⋅ 𝐀, is its magnitude (or length-squared, written
|𝐀|2 or 𝐴2 ), and from this definition we can see that the magnitude of the
displacement vector is Δ𝐑 ⋅ Δ𝐑 = Δ𝑡 2 − Δ𝑥 2 − Δ𝑦 2 − Δ𝑧2 = 𝑠2 .
This matrix 𝜂𝜇𝜈 is known as the metric, and through its use in defining
the magnitude of the vector, it defines what ‘distance’ means in Minkowski
space; that is, we could take the above definition of Δ𝐑 ⋅ Δ𝐑 as the definition
of the invariant interval 𝑠2 (we will see a lot more of this when we look at
General Relativity). You can verify by explicit calculation that the matrix 𝜂
has the property that, when it is acted upon by the transformation matrix of
Eq. (6.3a),

Λ𝑇 𝜂Λ = 𝜂 (6.8)

(where Λ𝑇 is the transpose of the matrix Λ). That is, it transforms into itself,
telling us that the definition of distance in one coordinate system is the same
in every transformed coordinate system; this is another statement of the
frame-independence of the invariant interval, 𝑠2 .
The metric of three-dimensional euclidean space is diag(1, 1, 1), which
(for 3-vector 𝐚, and by analogy with Eq. (6.6)) gives us 𝐚 ⋅ 𝐚 = 𝑎𝑥2 + 𝑎𝑦2 + 𝑎𝑧2 ,
or Pythagoras’s theorem, as the definition of distance in euclidean space.
Just as we discussed in Section 4.7 for intervals, 4-vectors can be timelike,
spacelike or null, depending on whether their magnitude is positive, negative
or zero; note that, since the magnitude is not positive-definite (i.e., it can be
negative), even a non-zero vector can be null. Just as in Section 6.1, we say
that two vectors are orthogonal if their scalar product vanishes. For example,
the two vectors 𝐴 = (1, 2, 0, 0) and 𝐵 = (2, 1, 0, 0) have scalar product 1 ×
 6.2 Four-Vectors 109

2 − 2 × 1 = 0, and so are orthogonal. It follows that, in this geometry, a null

vector (with 𝐀 ⋅ 𝐀 = 0) is orthogonal to itself!
The magnitude of a 4-vector is frame-invariant. The magnitude of 𝐀 + 𝐁
is (𝐀 + 𝐁) ⋅ (𝐀 + 𝐁) = 𝐴2 + 𝐵2 + 2𝐀 ⋅ 𝐁, and since |𝐀 + 𝐁|2 , 𝐴2 , and 𝐵2 are
all three frame-invariant, the scalar product 𝐀 ⋅ 𝐁 must be frame-invariant
also.
We can see this explicitly by calculating the scalar product of 𝐀′ = Λ𝐀
and 𝐁′ = Λ𝐁:
𝐀′ ⋅ 𝐁′ = (Λ𝐀) ⋅ (Λ𝐁) using Eq. (6.4)
= (Λ𝐀)𝑇 𝜂(Λ𝐁) using Eq. (6.6)
= 𝐀𝑇 (Λ𝑇 𝜂Λ)𝐁
= 𝐀𝑇 𝜂𝐁 using Eq. (6.8)
= 𝐀 ⋅ 𝐁.
[Exercises 6.1–6.4]

6.2.1 Some Observations on 4-vectors

There’s a lot going on in this section, and a lot of rather profound maths is
being glanced at, to do with symmetry, invariance, and some of the mathe-
matical physics that underlies General Relativity. None of this is necessary
to understand the sections that follow, but the rather miscellaneous remarks
below address some questions of detail which may occur to those with a
mathematical background.
It’s no coincidence that the same expression appears in both Eq. (5.6) and
Eq. (6.5), but it’s important that you realise that slightly different things are
happening in the two cases. In Eq. (5.6), the LT relates the coordinates of a
single event in two reference frames; in Eq. (6.3), the LT relates the compo-
nents of a single vector in two frames. These are not the same thing, because
the coordinates of an event are not the components of a ‘position vector’. The
‘position vector’ is not a vector in the terms of this section, because the posi-
tion vector is not frame independent; since it stretches from the frame origin
to the position of something, it is obviously different in different frames.
This is why it is the displacement 4-vector that we take as the prototype
4-vector above; since the events (i.e., locations in spacetime) at each end of
the displacement vector are frame-independent, the displacement 4-vector,
taken as a geometrical object, is frame-independent also.
There’s an element of arbitrariness in the choice of the matrices in Eq. (6.1)
and Eq. (6.2a), but these choices have the crucial property of preserving the
110 6 Vectors and Kinematics

magnitude of the appropriate displacement vector (i.e., preserving euclidean

or lorentzian geometry). Similarly, the definition of 𝜂 in Eq. (6.7) provides a
scalar product which is physically meaningful.
The set of numbers 𝜂𝜇𝜈 are not just any old matrix, but in particular the
components of a tensor – a mathematical structure which is a generalisation
of a vector, and which is vital for the mathematical study of General Relativity.
It is not even an arbitrary tensor: it is the form taken in a non-accelerating
frame (ie, in SR) by the metric tensor which is a crucial object in General
Relativity. It is the metric tensor which encapsulates the notion of the
distance between two events in spacetime.
Related to that, the positions of the indexes in Eq. (6.6) are not arbitrary.
The details do not matter for us here, although they are crucial in a study of
General Relativity, but I have preserved the notational convention in order
to remain at least visually compatible with any more advanced texts you
may consult.
While we’re on the subject of mathematical niceties, I’ll observe that,
since the magnitude of a vector can be negative, we can’t in general take
its square root to find the quantity that we might want to call the ‘norm’ or
length of the vector; this is why the word ‘distance’ is in scare-quotes in this
section. Secondly, the scalar or ‘dot’ product is the only vector product in
Minkowski 4-space; there is no cross-product defined (that’s a peculiarity of
euclidean 3-space). Finally, and firmly in the tradition of physicists being
rather casual about mathematical niceties, you will sometimes see 𝐀 ⋅ 𝐁
being referred to as an ‘inner product’; a mathematician would insist that
an inner product must be positive semidefinite, which Eq. (6.6) indicates is
not true for the Minkowski scalar product.
‘When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it
means just what I choose it to mean – neither more nor less.’
‘The question is,’ said Alice, ‘whether you can make words mean so many
different things.’
‘The question is,’ said Humpty Dumpty, ‘which is to be master – that’s all.’
Lewis Carroll, Through the Looking Glass

[Exercise 6.5]

6.3 Velocity and Acceleration

Since the displacement 4-vector Δ𝐑 is a vector (in the sense that it transforms
properly according to Eq. (6.2a)), so also is the infinitesimal displacement
d𝐑. We can write the components of d𝐑 as d𝑥 𝜇 , so that the magnitude of this
 6.3 Velocity and Acceleration 111

infinitesimal displacement vector is d𝐑 ⋅ d𝐑 = (d𝑥 0 )2 − (d𝑥1 )2 − (d𝑥2 )2 −

(d𝑥 3 )2 , or d𝑡 2 − |d𝐫|2 (writing d𝐫 2 = d𝑥 2 + d𝑦 2 + d𝑧2 ). Since the proper
time 𝜏 (see Section 5.4) is a Lorentz scalar,2 we can divide each component
of this infinitesimal displacement by the proper time and still have a vector.
This latter vector is the 4-velocity:

d𝐑 d𝑥 0 d𝑥1 d𝑥2 d𝑥3

𝐔= =( , , , ), (6.9)
d𝜏 d𝜏 d𝜏 d𝜏 d𝜏

By the same argument, the 4-acceleration

d2 𝑥 0 d2 𝑥 1 d2 𝑥 2 d2 𝑥 3
𝐀=( , , , ) (6.10)
d𝜏2 d𝜏2 d𝜏2 d𝜏2

is a 4-vector, also. We can more naturally write these as

d𝑥𝜇
𝑈𝜇 =
d𝜏
and
d𝑈 𝜇 d2 𝑥 𝜇
𝐴𝜇 = = .
d𝜏 d𝜏2
Let us examine these components in more detail, taking d𝐑 to be the
infinitesimal displacement of a particle. In a frame (a coordinate system)
in which the particle is not moving, d𝐫 = 0, so that d𝐑 ⋅ d𝐑 = d𝜏2 , where 𝜏
is the proper time (remember that proper time is a ‘clock attached to the
particle’). Since this magnitude of d𝐑 is frame-invariant,

d𝜏2 = d𝑡 2 − |d𝐫|2 ,

so that
2
d𝜏 (d𝜏)2 |d𝐫|2 1
( ) = 2
= 1 − 2
= 1 − 𝑣2 = 2 .
d𝑡 (d𝑡) (d𝑡) 𝛾
Taking the square root and inverting, we immediately find that
d𝑡
= 𝛾, (6.11)
d𝜏

2 By ‘scalar’ I merely mean ‘a number’, in a context where we want to distinguish the quantity
from a vector. Note that a number is necessarily frame-invariant.
112 6 Vectors and Kinematics

and so
d𝑥0 d𝑡
𝑈0 = = =𝛾 (6.12a)
d𝜏 d𝜏
d𝑥𝑖 d𝑡 d𝑥 𝑖
𝑈𝑖 = = = 𝛾𝑣 𝑖 . (6.12b)
d𝜏 d𝜏 d𝑡
You can view Eq. (6.11) as yet another manifestation of time dilation. Thus
we can write
𝐔 ≡ (𝑈 0 , 𝑈 1 , 𝑈 2 , 𝑈 3 ) = (𝛾, 𝛾𝑣 𝑥 , 𝛾𝑣 𝑦 , 𝛾𝑣 𝑧 ) = 𝛾(1, 𝑣 𝑥 , 𝑣 𝑦 , 𝑣 𝑧 ). (6.13a)
We will generally write this, below, as
𝐔 = 𝛾(1, 𝐯), (6.13b)
using 𝐯 to represent the three (space) components of the (spatial) velocity
vector, but as notation this is perhaps a little ‘slangy’.
Note that I will consistently use upper-case letters for 4-vectors (either
as a vector 𝐀, written with bold-face, or referring to their components 𝐴𝜇 ),
and lower-case letters for 3-vectors (either 𝐚 or 𝑎𝑖 ).
In a frame which is co-moving with a particle, the particle’s velocity is
𝐔 = (1, 0, 0, 0), so that, from Eq. (6.6), 𝐔 ⋅ 𝐔 = 1; since the scalar product is
frame-invariant, it must have this same value in all frames, so that, quite
generally, we have the relation
𝐔 ⋅ 𝐔 = 1. (6.14)
You can confirm that this is indeed true by applying Eq. (6.6) to Eq. (6.13).
Here, we defined the 4-velocity by differentiating the displacement 4-
vector, and deduced its value in a frame co-moving with a particle. We can
now turn this on its head, and define the 4-velocity as a vector which has
magnitude 1 and which points along the 𝑡-axis of a co-moving frame (this
is known as a ‘tangent vector’, and is effectively a vector ‘pointing along’
the worldline). We have thus defined the 4-velocity of a particle as the
vector which has components (1, 𝟎) in the particle’s rest frame. Note that
the magnitude of the vector is always the same; the particle’s speed relative
to a frame 𝑆 is indicated not by the ‘length’ of the velocity – its magnitude,
which is always 1 – but by the direction of the vector in Minkowski space,
in the frame 𝑆. We can then deduce the form in Eq. (6.13) as the Lorentz-
transformed version of (1, 𝟎). Compare Section 6.4 and Section 7.1.1.

Equations (6.12) can lead us to some intuition about what the velocity
vector is telling us. When we say that the velocity vector in the particle’s
rest frame is (1, 𝟎), we are saying that, for each unit proper time 𝜏, the particle
 6.3 Velocity and Acceleration 113

moves the same amount through coordinate time 𝑡, and not at all through space 𝑥;
the particle ‘moves into the future’ directly along the 𝑡-axis. When we are talking
instead about a particle which is moving with respect to some frame, the equation
𝑈 0 = d𝑡∕d𝜏 = 𝛾 tells us that the particle moves through a greater amount of this
frame’s coordinate time, 𝑡, per unit proper time (where, again, the ‘proper time’
is the time showing on a clock attached to the particle) – ‘time dilation’ yet again.

Turning now to the acceleration 𝐀𝜇 , we have

d2 𝑥 0 d𝑈 0 d𝛾 d𝛾
𝐴0 = 2
= = =𝛾 ≡ 𝛾𝛾̇ (6.15a)
d𝜏 d𝜏 d𝜏 d𝑡
d2 𝑥 𝑖 d𝑈 𝑖 d ( 𝑖 )
𝐴𝑖 = = = 𝛾 (𝛾𝑣 𝑖 ) = 𝛾 𝛾𝑣
̇ + 𝛾𝑎 𝑖 (6.15b)
d𝜏2 d𝜏 d𝑡
(where 𝑣𝑖 and 𝑎𝑖 are the ordinary velocity and acceleration 𝑣𝑖 = d𝑥 𝑖 ∕d𝑡,
𝑎𝑖 = d2 𝑥 𝑖 ∕d𝑡2 , and 𝛾̇ = d𝛾∕d𝑡), or

̇ 𝛾𝐯
𝐀 = 𝛾 (𝛾, ̇ + 𝛾𝐚) . (6.16)

There is no inertial frame in which an accelerating particle is always at rest;

at any instant, however, such a particle has a definite velocity, and so there is
a frame – the instantaneously co-moving inertial frame (ICRF) – in which the
particle is briefly at rest.3 In this frame, where 𝐯 = 𝟎, we have 𝐔 = (1, 𝟎)
and 𝐀 = (𝛾, ̇ 𝐚) = (0, 𝐚) (since 𝛾̇ contains a factor of 𝐯), so that

𝐔⋅𝐀=0

in this co-moving frame, and therefore in all frames. From the result in this
co-moving frame we can deduce the magnitude of the 4-acceleration

𝐀 ⋅ 𝐀 = −𝑎2 ,

defining the proper acceleration 𝑎 as the magnitude of the acceleration in

the instantaneously co-moving inertial frame. See also Exercise 6.11.
Finally, given two particles with velocities 𝐔 and 𝐕, and given that the
second has velocity 𝑣 with respect to the first, then in the first particle’s rest
frame the velocity vectors have components 𝐔 = (1, 𝟎) and 𝐕 = 𝛾(𝑣)(1, 𝐯).
Thus

𝐔 ⋅ 𝐕 = 𝛾(𝑣),

and this scalar product is, again, frame-independent. [Exercise 6.6]

3 This is also referred to as the momentarily co-moving reference frame (MCRF) by some
authors.
114 6 Vectors and Kinematics

y
dr/dλ

r
λ
x

Figure 6.4 The derivative of a vector is the tangent to its curve.

6.4 Velocities and Tangent Vectors

An alternative route to the velocity vector is to use the idea of a tangent vector.
This is how vectors are defined in GR, and is a more fundamental approach
to vectors than the one described above. This section is itself something of a
tangent, which expands on the mention of ‘tangent vectors’ in the paragraph
following Eq. (6.14). The account here is still rather compressed, and I
offer it only to provide a hint of the more abstract, but more beautiful and
powerful, way in which these things are handled in GR.
Consider a vector 𝐫 = (𝑥, 𝑦) on the euclidean plane. If the components 𝑥
and 𝑦 are functions of some parameter 𝜆, then the vector function 𝐫(𝜆)
will trace out a path (𝑥(𝜆), 𝑦(𝜆)) on the plane. If we differentiate these
components with respect to the parameter 𝜆, then we will obtain an object
which obviously tells us something about the path. For example, if we
follow the path 𝐫(𝜆) = (cos 𝜆, sin 𝜆), parameterised by 𝜆, we find it traces
out a circle. Differentiating this straightforwardly, we find
d𝐫
= (− sin 𝜆, cos 𝜆),
d𝜆
which is a vector which, when plotted at the position 𝐫(𝜆), as in Figure 6.4,
can be clearly seen to be tangent to the path.
Now consider the spacetime vector 𝐑 = (𝑡(𝜆), 𝑥(𝜆), 𝑦(𝜆), 𝑧(𝜆)), which
draws out a path in spacetime. The path this traces out is the worldline –
the set of events which take place along a moving particle’s path through
spacetime – and a reasonable parameter to use is the particle’s proper time, 𝜏
– the time showing on the face of a clock attached to the moving particle.
( )
We therefore have a path 𝐑 = 𝑥0 (𝜏), 𝑥 1 (𝜏), 𝑥 2 (𝜏), 𝑥 3 (𝜏) and, exactly as we
did on the euclidean plane above, can differentiate it to obtain

d𝐑 d𝑥 0 d𝑥1 d𝑥2 d𝑥3

=( , , , ),
d𝜏 d𝜏 d𝜏 d𝜏 d𝜏
(compare Eq. (6.9)) as a 4-vector tangent to the worldline, which clearly
 6.5 The Frequency Vector, and the Doppler Shift 115


2
ΔR


1

λ
λ
uΔt

Figure 6.5 A wavetrain: waves of wavelength 𝜆 moving at speed 𝑢. In time Δ𝑡,

the wavecrests move from their initial position (dashed) to an advanced one
(solid).

contains information about the ‘speed’ of the particle, and which we can
define to be the velocity 4-vector. By the same argument that led up to
Eq. (6.14), we discover that this vector has magnitude 𝐔 ⋅ 𝐔 = 1, and that
its direction corresponds precisely to the 𝑡-axis of a frame co-moving with
the particle.
The point of this approach is that the idea of a path, and the idea of a
tangent vector to that path, are both geometrical ideas, existing at a level
beneath coordinates (which are more-or-less algebraic things), and so can
be defined and discussed without using coordinates, and so without having
any dependence on reference frames. They are therefore manifestly frame-
independent.

6.5 The Frequency Vector, and the Doppler Shift

In this section, we will examine a particular (non-obvious) 4-vector, and
exploit its properties to deduce the relativistic Doppler effect.
Imagine a series of waves of some type (not necessarily light waves),
moving in a direction 𝐧 = (𝑙, 𝑚, 𝑛) at a speed 𝑢, and imagine following a
point on the crest of one of these waves; take the vector 𝐧 to be a unit vector,
so 𝐧 ⋅ 𝐧 = 1. The point will have a displacement (Δ𝑥, Δ𝑦, Δ𝑧) in time Δ𝑡,
and so we will have
𝑙Δ𝑥 + 𝑚Δ𝑦 + 𝑛Δ𝑧 = 𝑢Δ𝑡,
for two events on the same wavecrest.
Now imagine a whole train of such waves, separated by wavelength 𝜆,
as illustrated in Figure 6.5. Consider the separation between one event on
116 6 Vectors and Kinematics

the crest of a wave and another on a wavecrest 𝑁 (integer) wavecrests away

(that is, a different wavecrest), and write this as Δ𝑅 = (Δ𝑡, Δ𝑥, Δ𝑦, Δ𝑧). We
can observe that the time separation between events 1 and 2 is still Δ𝑡, so
that the wavetrain will have shifted a distance 𝑢Δ𝑡, meaning that the spatial
separation between the two events is 𝑢Δ𝑡 + 𝑁𝜆. But this spatial separation
is also, as before, 𝑙Δ𝑥 + 𝑚Δ𝑦 + 𝑛Δ𝑧, so that we can write
𝑙Δ𝑥 + 𝑚Δ𝑦 + 𝑛Δ𝑧 = 𝑢Δ𝑡 + 𝑁𝜆
or
𝑢Δ𝑡 − 𝑙Δ𝑥 − 𝑚Δ𝑦 − 𝑛Δ𝑧 = −𝑁𝜆. (6.17)
Defining the frequency 𝑓 = 𝑢∕𝜆, we can rewrite this as
𝐋 ⋅ Δ𝐑 = −𝑁, (6.18)
where the dot represents the scalar product of Eq. (6.6), and the frequency
4-vector is
𝐧 𝑙 𝑚 𝑛
𝐋 = (𝑓, ) = (𝑓, , , ) . (6.19)
𝜆 𝜆 𝜆 𝜆
We have written this as a vector, but what is there to say that this is really
a vector – that is, that it is the components in one frame of an underlying
geometrical object – and isn’t merely four numbers in a row? We know that
Δ𝐑 is a vector – it is the prototype 4-vector – so we know that its components
transform according to Eq. (6.2b): Δ𝑡 = 𝛾(Δ𝑡 ′ + 𝑣Δ𝑥 ′ ), Δ𝑥 = 𝛾(Δ𝑥 ′ + 𝑣Δ𝑡′ ),
Δ𝑦 = Δ𝑦 ′ and Δ𝑧 = Δ𝑧′ . We can substitute this into Eq. (6.18), rearrange to
gather terms Δ𝑡′ , Δ𝑥′ , Δ𝑦 ′ , Δ𝑧′ and, just as we did before Eq. (6.18), rewrite
to find
𝑙 𝑙 𝑚 𝑛
𝐋 ⋅ Δ𝐑 = [𝛾 (𝑓 − 𝑣 ) , 𝛾 ( − 𝑣𝑓) , , ] ⋅ Δ𝐑′ .
𝜆 𝜆 𝜆 𝜆
But the vector in the square brackets is exactly the vector 𝐋′ we would obtain
if we transformed the frequency vector 𝐋 according to the transformation
matrix Eq. (6.3). That is, we have spotted that 𝐋′ = Λ𝐋, and thus established
that
𝐋′ ⋅ Δ𝐑′ = 𝐋 ⋅ Δ𝐑 = −𝑁,
or in other words that it is a frame-invariant quantity. We have therefore
established that the object 𝐋 defined by Eq. (6.19) really is a 4-vector, since it
transforms in the same manner as the prototype 4-vector Δ𝐑 (there is no
significance to the value being negative: the important thing is that this is a
frame-invariant scalar).
 6.5 The Frequency Vector, and the Doppler Shift 117

y

θ
x

Figure 6.6 A wavetrain: moving at an angle 𝜃 ′ to the 𝑥′ -axis.

Is this not inevitable? Not quite: imagine if we had naïvely defined the
frequency 4-vector as a vector whose space components were 𝐧∕𝜆 and whose
time component was defined to be zero. On transformation by either of
the routes in the previous paragraph, the vector would acquire a non-zero
time component, so that the transformed vector would have a different form
from the untransformed one. The components of such a ‘vector’ would not
transform in the same way as Δ𝐑, so it would not be a proper 4-vector, so
that we would not be able to identify an underlying geometrical object of
which these were the components.
Can we use the frequency 4-vector for anything? Yes. Imagine that the
wavetrain is moving at an angle 𝜃 ′ in the (𝑥′ , 𝑦 ′ ) plane, so that its direction
is 𝐧 = (cos 𝜃′ , sin 𝜃 ′ , 0) for some angle 𝜃 ′ in frame 𝑆 ′ (Figure 6.6). In that
case we have

cos 𝜃′ sin 𝜃′
𝐋′ = [𝑓 ′ , , , 0] (6.20)
𝜆′ 𝜆′
cos 𝜃′ cos 𝜃′ sin 𝜃′
𝐋 = [𝛾 (𝑓 ′ + 𝑣 ′ ) , 𝛾 ( ′ + 𝑣𝑓 ′ ) , , 0] . (6.21)
𝜆 𝜆 𝜆′

Now, by direct analogy with Eq. (6.20), write

cos 𝜃 sin 𝜃
𝐋 = [𝑓, , , 0] , (6.22)
𝜆 𝜆

and
compare
Eq.
(6.21)
and
(6.22)
component
by
component.
After
a
bit
of

rearrangement,
we
find

𝑣
𝑓 = 𝑓 ′ 𝛾 (1 + cos 𝜃 ′ ) (6.23)
𝑢′
cos 𝜃 ′ + 𝑣𝑢′
cos 𝜃 = . (6.24)
1 + 𝑣𝑢′ cos 𝜃′

Or, for the case of light, where 𝑢′ = 1, we have the simpler and well-known
118 6 Vectors and Kinematics

versions
𝑓 = 𝑓 ′ 𝛾(1 + 𝑣 cos 𝜃′ ) (6.25)
cos 𝜃 ′
+𝑣
cos 𝜃 = . (6.26)
1 + 𝑣 cos 𝜃 ′
Equation (6.23) is the relativistic Doppler effect, and describes the change
in frequency of a wave, as measured in a frame moving with respect to the
frame in which it was emitted. This applies for everything from water waves
(for which the effect would be exceedingly small) all the way up to light, for
which 𝑢 = 1.
Equation (6.24) shows that a wave travelling at an angle 𝜃 ′ in the moving
frame 𝑆 ′ is measured to be moving at a different angle 𝜃 in a frame 𝑆, with
respect to which 𝑆 ′ is moving with speed 𝑣. To calculate the change in the
speed of the wave, we could laboriously eliminate variables from Eq. (6.21)
and (6.22), but much more directly, we can make use of the fact that the
magnitudes of vectors are conserved under Lorentz transformation; thus
𝐋 ⋅ 𝐋 = 𝐋′ ⋅ 𝐋′ or, again using Eq. (6.5) and 𝑓 = 𝑢∕𝜆,
1 1
𝑓 2 (1 − 2
) = 𝑓 ′2 (1 − ′2 ) .
𝑢 𝑢
We could rewrite this to obtain an expression for 𝑢′ , but simply from this
form we can see that if 𝑢 = 1, the fact that neither 𝑓 nor 𝑓 ′ is zero implies
that 𝑢′ = 1 also (as the second postulate says).

Note that there are multiple possible ways to set up this problem, from
a notational point of view, so that if you are comparing the discussion
here with another text, make sure the various symbols mean what you expect.

[Exercises 6.7–6.17]

Exercises
Exercise 6.1 (§6.2)
Verify that the transformation matrices of Eq. (6.3a) and Eq. (6.3b) are in-
verses. You will probably find it convenient to omit the (trivial) 𝑦 and 𝑧
components, and instead write

1 −𝑣
Λ = 𝛾( ).
−𝑣 1

Observe additionally that Λ is obtained from Λ−1 by negating 𝑣. [ 𝑑− ]

 Exercises 119

Exercise 6.2 (§6.2)

Verify Eq. (6.8). As in Exercise 6.1, ignore the trivial components and write

1 0
𝜂=( ).
0 −1

You can write this as a matrix expression, or as a sum over components.

[ 𝑑− ]

Exercise 6.3 (§6.2)

Consider the vectors:
r 𝐀 = (1, 2, 0, 0)
r 𝐁 = (2, 1, 0, 0)

Calculate 𝐀 ⋅ 𝐁, and the magnitudes of 𝐀 and 𝐁. [ 𝑑− ]

Exercise 6.4 (§6.2)

Consider the vectors 𝐀 = (5, 3, 4, 0), 𝐁 = (1, 1, −1, −1) and 𝐂 = (3, 2, −1, 0).
Calculate 𝐀 ⋅ 𝐁, 𝐀 ⋅ 𝐂 and 𝐁 ⋅ 𝐂: which pair is orthogonal? Calculate also
the magnitudes of 𝐀, 𝐁 and 𝐂, and indicate which vectors are null, timelike
and spacelike.

Exercise 6.5 (§6.2.1)

Consider, instead of Eq. (6.2a), the transformation

⎛ Δ𝑡 ′ ⎞ ⎛𝑎 𝑏 0 0⎞ ⎛ Δ𝑡 ⎞
⎜Δ𝑥′ ⎟ ⎜ 𝑐 𝑑 0 0⎟ ⎜Δ𝑥⎟
⎜Δ𝑦 ′ ⎟ = ⎜ 0 0 1 0⎟ ⎜Δ𝑦 ⎟
(i)
⎜ ′⎟ ⎜ ⎟⎜ ⎟
Δ𝑧 0 0 0 1 Δ𝑧
⎝ ⎠ ⎝ ⎠⎝ ⎠
which is the simplest transformation which ‘mixes’ the 𝑥- and 𝑡-coordinates.
2
By requiring that Δ𝑠′ = Δ𝑠2 = Δ𝑡 2 − Δ𝑥 2 after this transformation, find
constraints on the parameters 𝑎, 𝑏, 𝑐, 𝑑 (you can freely add the constraint 𝑏 =
𝑐; why?), and by setting 𝑏 proportional to 𝑎 deduce the matrix Eq. (6.2a).
Bonus: by instead parameterising 𝑎, 𝑏, 𝑐, 𝑑 with suitable hyperbolic
functions, recover Eq. (5.3). [ 𝑑+ ]
120 6 Vectors and Kinematics

Exercise 6.6 (§6.3)

I throw a ball vertically into the air. At the top of its path, what is the
instantaneously comoving reference frame?

r The frame of the room.

r A frame moving upwards with the same speed as I threw the ball.
r A frame moving downwards with the ball’s terminal velocity. [ 𝑑− ]

Exercise 6.7 (§6.5)

Consider a water wave moving along the 𝑥-axis at speed 10 m s−1 , with
wavelength 5 m. What is its frequency 4-vector?

r (2, 2, 0, 0)
r (2, 0.2, 0, 0)
r (10, 5, 0, 0) [ 𝑑− ]

Exercise 6.8 (§6.5)

One might naïvely consider defining a 4-vector representing the electric
field as (0, 𝐸 𝑥 , 𝐸 𝑦 , 𝐸 𝑧 ), where the 𝐸 𝑖 are the components of the electric field,
and the time component is defined to be identically zero. Explain briefly
why this cannot be a 4-vector. [ 𝑢++ ]

Exercise 6.9 (§6.5)

Obtain the inverse to Eq. (6.25) and Eq. (6.26) by applying Eq. (6.3) to
Eq. (6.22) and comparing with Eq. (6.20). Verify that the resulting expres-
sion matches that which you obtain through the procedure described in
Section 6.2: swapping primed with unprimed quantities and changing the
sign of 𝑣.

Exercise 6.10 (§6.5)

Obtain Eq. (6.24) by observing that tan 𝜃 = 𝐿2 ∕𝐿1 (the 𝑥 and 𝑦 components
of 𝐋) and doing the trigonometry. You should recall that sec2 𝑥 = 1 + tan2 𝑥,
and it will be useful to notice that 1 = 𝛾2 (1 − 𝑣 2 ).
 Exercises 121

Exercise 6.11 (§6.5)

A rocket starts at rest at a space station, and accelerates along the 𝑥-axis at a
constant rate 𝛼 in its own frame 𝑆 ′ (i.e., in the ICRF). The rocket’s (proper)
acceleration 4-vector is therefore 𝐴′ = (0, 𝛼, 0, 0).
(a) Using the (inverse) Lorentz transformation equations, to transform
from an inertial frame instantaneously co-moving with the rocket into the
space station frame, calculate the components of the rocket’s acceleration
vector in the space station’s frame, and confirm that 𝐀 ⋅ 𝐀 = −𝛼 2 in this
frame also. Write down the rocket’s velocity 4-vector in this frame, and
confirm that 𝐔 ⋅ 𝐀 = 0. Differentiate this with respect to the proper time, to
obtain

d𝐔 d𝛾 d
= 𝛾 ( , (𝛾𝑣), 0, 0) .
d𝜏 d𝑡 d𝑡

Hence deduce that 𝛾𝑣 = 𝛼𝑡, and thus that

1 1
= 2 2 + 1. (i)
𝑣2 𝛼 𝑡

(b) If, at time 𝑡, the rocket sets off a flashbulb which has frequency 𝑓 ′ in
its frame, use the Doppler formula to show that the light is observed, at the
space station, to have frequency
(√ )
𝑓= 1 + 𝛼2 𝑡2 − 𝛼𝑡 𝑓 ′ .

What is this factor if the flashbulb is set off at time 𝑡 = 3∕(4𝛼)?

(c) Rearrange Eq. (i) to produce an expression for 𝑣, write 𝑣 = d𝑥∕d𝑡, and
integrate to obtain the constant-acceleration equation for SR,

2
1 1
(𝑥 + ) − 𝑡2 = 2
𝛼 𝛼

(you may or may not find the substitution 𝛼𝑡 = sinh 𝜃 useful or obvious
here). This is the equation for a hyperbola. By sketching this on a Minkowski
diagram and considering the point at which the asymptote intersects the
𝑡-axis, demonstrate that it is impossible for the space station to signal to the
retreating rocket after a time 1∕𝛼.
(d) How long does it take, from launch at 𝑣 = 0, for a rocket accelerating
at 𝑔 = 10 m s−2 to be moving at 𝛾 = 2?
[This question is a little algebra-heavy, but instructive.] [ 𝑑+ 𝑢+ ]
122 6 Vectors and Kinematics

Exercise 6.12 (§6.5)

Consider an ambulance passing you with its siren sounding. At the instant
the ambulance is level with you, its siren has its ‘rest’ frequency.
If the ambulance were moving at a relativistic speed, what would the
siren sound like as it passed you? Would it be:
r different from its rest frequency?
r the same as its rest frequency? [ 𝑑− ]

Exercise 6.13 (§6.5)

Consider (again) an ambulance passing you with its blue light flashing.
If the ambulance were moving at a relativistic speed, what would be
the measured frequency of the ambulance’s blue light, as a function of its
frequency in its rest frame, 𝑓 ′ ? If we put 𝜃 ′ = 𝜋∕2 into Eq. (6.25) (examining
the light which is emitted perpendicular to the ambulance’s motion), then
we obtain 𝑓 = 𝛾𝑓 ′ . But this suggests that the frequency in our frame is
measured to be higher than the frequency in the rest frame, and appears to
directly contradict the calculations in Exercise 5.2. What is wrong with this
argument? [ 𝑑+ ]

Exercise 6.14 (§6.5)

The traditional traffic-lights example: You are driving towards some traffic
lights showing red (take wavelength 𝜆 = 600 nm). How fast do you have
to be driving so that they are Doppler-shifted enough that they appear
green (wavelength 𝜆 = 500 nm)? [Hint: rewrite Eq. (6.23) in terms of 𝜌 =
𝑓 ′ ∕𝑓, set 𝑢 = 1 and 𝜃 = 0, and rearrange to make 𝑣 the subject.] Does this
sound like a reasonable defence when you’re had up in court on a charge of
dangerous driving?

Exercise 6.15 (§6.5)

Light emitted from a distant star may be Doppler-shifted, so that we detect it
at a frequency which is different from the frequency at which it was emitted
(for example, the star may be a member of a binary system). You may have
come across this Doppler shift expressed as
( 𝑣) ′
𝑓 = 1+ 𝑓.
𝑐
 Exercises 123

This is not the same as Eq. (6.25), even for 𝜃′ = 0. Why not? Derive this
expression from Eq. (6.25).

Exercise 6.16 (§6.5)

In 1725, James Bradley measured an aberration in the altitude of a star
that was dependent on the Earth’s motion at a given point in its orbit. The
observational result of Bradley’s investigation is

Δ𝜃 = 𝜅 sin 𝜃

where Δ𝜃 is the aberration in the star’s altitude 𝜃, and 𝜅 = 20′′. 496 is the
constant of aberration.
Consider pointing a laser at a star which has altitude 𝜃 in a frame in
which the Earth is at rest, and altitude 𝜃 ′ in a frame which is moving at the
Earth’s orbital velocity of approximately 𝑣 = 29.8 km s−1 (the two frames are
equivalent to making an observation where the Earth’s motion is orthogonal
to the line-of-sight to the star, and a frame, three months later, which is
moving towards the star; also, it’s easier to solve this variant of the problem
rather than the equivalent problem of the light coming from the star, since
it avoids lots of minus signs in the algebra). Show that the two altitudes are
related by:
cos 𝜃 ′ + 𝑣
cos 𝜃 = . (i)
1 + 𝑣 cos 𝜃 ′
Given this, it is merely tedious to work out that sin 𝜃 = sin 𝜃′ ∕(𝛾(1 +
𝑣 cos 𝜃′ )).
Deduce that, for small 𝑣,

Δ𝜃 = 𝜃 ′ − 𝜃 ≈ 𝑣 sin 𝜃 ′ ,

and verify that this is consistent with Bradley’s observational result.

The main challenge to this question is keeping track of the trigonometry,
but it’s useful drill nonetheless. [Recall that tan2 𝜃 = 1∕ cos2 𝜃 − 1, that
𝛼 ≈ sin 𝛼 for small 𝛼, and that sin(𝛼 − 𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽.]
[ 𝑑+ ]

Exercise 6.17 (§6.5)

Recall the expression for cos 𝜃 in Exercise 6.16.
A source in the moving frame 𝑆 ′ emits light isotropically. Deduce that
′
light emitted into the forward half-sphere (that is, with half-angle 𝜃 ′ ≤ 𝜃+ =
124 6 Vectors and Kinematics

𝜋∕2) is observed, in the frame 𝑆, to be emitted into a cone with half-angle 𝜃+ ,

where cos 𝜃+ = 𝑣. This is known as the headlight effect.
 7
Dynamics

In the previous chapter, we have learned how to describe motion; we now

want to explain it. In newtonian mechanics, we do this by defining quantities
such as momentum, energy, force and so on. To what extent can we do this
in the context of relativity, with our new 4-vector tools?

Aims: you should:

7.1. understand relativistic energy and momentum, the concept of energy-

momentum as the magnitude of the momentum 4-vector, and conser-
vation of the momentum 4-vector; and
7.2. understand the distinction between invariant, conserved and constant
quantities.

7.1 Energy and Momentum

We can start with momentum. We know that in newtonian mechanics,
momentum is defined as mass times velocity. We have a velocity, so we
can try defining a momentum 4-vector, for a particle of mass 𝑚 moving at
velocity 𝐯, as

𝐏 = 𝑚𝐔 = 𝑚𝛾(1, 𝐯). (7.1)

Since 𝑚 is a scalar, and 𝐔 is a 4-vector, 𝐏 must be a 4-vector also. Remember

also that 𝛾 is a function of the 3-vector 𝐯: 𝛾(𝐯).
In the rest frame
√ of the particle, this becomes 𝐏 = 𝑚(1, 𝟎): it is a 4-vector
whose length ( 𝐏 ⋅ 𝐏) is 𝑚, and which points along the particle’s worldline.
That is, it points in the direction of the particle’s movement in spacetime.
Since this is a vector, its magnitude and its direction are frame-independent

125
126 7 Dynamics

P1 P3

P2 P4

Figure 7.1 The momenta involved in a collision.

quantities, so a particle’s 4-momentum vector always points in the direction

of the particle’s worldline, and the 4-momentum vector’s length is always 𝑚.
We’ll call this vector the momentum (4-)vector, but it’s also called the 4-
momentum or the energy-momentum vector, and Taylor & Wheeler (coining
the word in an excellent chapter on it) call it the momenergy vector in order
to stress that it is not the same thing as the energy or momentum (or mass)
that you are used to.

Note that here, and throughout, the symbol 𝑚 denotes the mass as
measured in a particle’s rest frame. The reason I mention this is that
some treatments of relativity, particularly older ones, introduce the concept of
the ‘relativistic mass’ 𝑚(𝑣) = 𝛾(𝑣)𝑚0 , distinct from the ‘rest mass’, 𝑚0 . The only
(dubious) benefit of this is that it makes a factor of 𝛾 disappear from a few equa-
tions, making them look a little more like their newtonian counterparts; the cost
is that of introducing one more new concept to worry about, which doesn’t help
much in the long term, and which can obscure aspects of the energy-momentum
vector. Rindler (2006, §6.2), for example, introduces the relativistic mass, but
his subsequent discussion of relativistic force is sufficiently confusing, from our
notational point of view (it obliges one to introduce the notions of ‘longitudi-
nal’ and ‘transverse relativistic mass’!) that I feel it quite amply illustrates the
unhelpfulness of the whole idea.

The discussion of dynamics, in this chapter, also draws on chapter 2 of

Schutz (2009). This is a textbook on General Relativity, but chapter 2 de-
scribes Special Relativity using the beautiful and powerful geometrical language
which is used extensively in GR. I have toned this down for this section, but the
motivation is the same. It also draws on the insightful discussion in Taylor &
Wheeler (1992). ‘Insightful’ doesn’t really do justice to Taylor & Wheeler – you
should read chapter 7 of the book if you possibly can.

Now consider a pair of incoming particles 𝐏1 and 𝐏2 which collide and

produce a set of outgoing particles 𝐏3 and 𝐏4 (see Figure 7.1; this can be triv-
ially extended to more than two particles). Suppose that the total momentum
 7.1 Energy and Momentum 127

is conserved:
𝐏1 + 𝐏 2 = 𝐏 3 + 𝐏 4 . (7.2a)
This is an equation between 4-vectors. Equating the time and space coordi-
nates separately, recalling Eq. (7.1), and writing 𝐩 ≡ 𝛾𝑚𝐯, we have
𝑚1 𝛾(𝑣1 ) + 𝑚2 𝛾(𝑣2 ) = 𝑚3 𝛾(𝑣3 ) + 𝑚4 𝛾(𝑣4 ) (7.2b)
𝐩1 + 𝐩2 = 𝐩3 + 𝐩4 . (7.2c)
Now recall that, as 𝑣 → 0, we have 𝛾(𝑣) → 1, so that the low-speed limit of
the spatial part of the vector 𝐏, Eq. (7.1), is just 𝑚𝐯, so that the spatial part
of the conservation equation, Eq. (7.2c), reduces to the statement that 𝑚𝐯
is conserved. Both of these prompt us to identify the spatial part of the
vector 𝐏 as the linear momentum, and to retrospectively justify both giving
the 4-vector 𝐏 the name 4-momentum and supposing that it is conserved.
What, then, of the time component of Eq. (7.1)? Let us (with, admittedly,
a little fore-knowledge) write this as 𝑃0 = 𝐸, so that
𝐸 = 𝛾𝑚. (7.3)
What is the low-speed limit of this? Taylor’s theorem tells us that
( )−1∕2 𝑣2
𝛾 = 1 − 𝑣2 =1+ + 𝑂(𝑣 4 ), (7.4)
2
so that, when 𝑣 is small, Eq. (7.3) becomes
1
𝐸 = 𝑚 + 𝑚𝑣2 + 𝑂(𝑣 4 ). (7.5)
2
At this point we can (a) spot that 𝑚𝑣2 ∕2 is the expression for the kinetic
energy in newtonian mechanics, and (b) recall that Eq. (7.2b) tells us that
this quantity 𝐸 is conserved in collisions, so that we have persuasive support
for identifying the quantity 𝐸 in Eq. (7.3) as the relativistic energy of a particle
with mass 𝑚 and velocity 𝑣.
If we rewrite Eq. (7.3) in physical units, we find
𝐸 = 𝛾𝑚𝑐2 , (7.6)
the low-speed limit of which (remember 𝛾(0) = 1) recovers what has been
called the most famous equation of the twentieth century.
Note that the units in Eq. (7.3) are kg on both sides, but after rewriting in
units where 𝑐 ≠ 1, Eq. (7.6), the units on both sides are kg m2 s−2 , or joules,
as expected.
The argument presented after Eq. (7.2a) has been concerned with giving
names to quantities, and, reassuringly for us, linking those newly named
128 7 Dynamics

things with quantities we already know about from newtonian mechanics.

This may seem artificial, and it is certainly not any sort of proof that the
4-momentum is conserved as Eq. (7.2a) says it might be. What we are doing
here, however, is that we are postulating that 4-momentum is conserved as
in Eq. (7.2a), which allows us to deduce that energy and 3-momentum are
separately conserved, as we expect. No proof is necessary, however: it turns
out from experiment that Eq. (7.2a) is indeed a law of nature, and nothing
more really needs to be said.

In case you are worried that we are pulling some sort of fast one, that
we never had to do in newtonian mechanics, note that we do have to
do a similar thing in newtonian mechanics. There, we postulate Newton’s third
law (action equals reaction), and from this we can deduce the conservation of
momentum; in this case, we work in the opposite direction, so that we postulate
the conservation of 4-momentum, and would then be able to deduce a relativistic
analogy of Newton’s third law. In each case, we are adding the same amount
of physics to the mathematics. I don’t discuss relativistic force here, but merely
mention it in passing in Section 7.6; this question is discussed in a little more
detail in Rindler (2006, §6.10).

We can see from Eq. (7.5) that, even when a particle is stationary and 𝑣 =
0, the energy 𝐸 is non-zero. In other words, a particle of mass 𝑚 has an
energy 𝛾𝑚 associated with it simply by virtue of its mass.
The low-speed limit of Eq. (7.2b) simply expresses the conservation of
mass, but we see from Eq. (7.3) that it is actually expressing the conservation
of energy. In SR there is no fundamental distinction between mass and
energy – mass is, like kinetic, thermal and strain energy, merely another
form into which energy can be transmuted – albeit a particularly dense store
of energy, as can be seen by calculating the energy equivalent, in joules,
of a mass of 1 kg. It turns out from GR that it is not mass that gravitates,
but energy-momentum (most typically, however, in the particularly dense
form of mass), so that thermal and electromagnetic energy, for example,
and even the energy in the gravitational field itself, all gravitate (it is the
non-linearity implicit in the last remark that is part of the explanation for
the mathematical difficulty of GR). In each of these cases, the amount of
(thermal, electromagnetic, strain) energy would be a quantity with the units
of kilogrammes.
Although the quantities 𝐩 = 𝛾𝑚𝐯 and 𝐸 are frame-dependent, and thus
not physically meaningful by themselves, the quantity 𝐏 defined by Eq. (7.1)
has a physical significance.
Let us now consider the magnitude of the 4-momentum vector. Like
any such magnitude, it will be frame-invariant, and so will express some-
 7.1 Energy and Momentum 129

thing fundamental about the vector, analogous to its length. Since this is
the momentum vector we are talking about, this magnitude will be some
important invariant of the motion, indicating something like the ‘quantity of
motion’. From the definition of the momentum, Eq. (7.1), and its magnitude,
Eq. (6.14), we have

𝐏 ⋅ 𝐏 = 𝑚2 𝐔 ⋅ 𝐔 = 𝑚2 , (7.7)

and we find that this important invariant is the mass of the moving particle.
Now using the definition of energy, Eq. (7.3), we can write 𝐏 = (𝐸, 𝐩),
and find

𝐏 ⋅ 𝐏 = 𝐸 2 − 𝐩 ⋅ 𝐩. (7.8)

Writing now 𝑝2 = 𝐩 ⋅ 𝐩, we can combine these to find

𝑚2 = 𝐸 2 − 𝑝2 . (7.9)

Equation (7.9) appears to be simply a handy way of relating 𝐸, 𝑝 and 𝑚,

but that conceals the more fundamental meaning of Eq. (7.7). The 4-
momentum 𝐏 encapsulates the important features of the motion, through
the energy and spatial momentum. The quantities 𝐸 and 𝑝 are frame-
dependent separately, but combine into a frame-independent quantity.

It seems odd to think of a particle’s momentum as being always non-

zero, irrespective of how rapidly it’s moving; this is the same oddness
that has the particle’s velocity always being of length 1. One way of thinking
about this is that it shows that the 4-momemtum vector (or energy-momentum
or momenergy vector) is a ‘better’ thing than the ordinary 3-momentum: it’s
frame-independent, and so has a better claim to being something intrinsic to
the particle. Another way of thinking about it is to conceive of the 4-velocity as
showing the particle’s movement through spacetime. In a frame in which the
particle is ‘at rest’, the particle is in fact moving at speed 𝑐 into the future. If you
are looking at this particle from another frame, you’ll see the particle move in
space (it has non-zero space components to its 4-velocity in your frame), and it
will consequently (in order that 𝐔 ⋅ 𝐔 = 1) have a larger time component in your
frame than it has in its rest frame. In other words, the particle moves through
more time in your frame than it does in its rest frame – yet another manifestation
of time dilation. Multiply this velocity by the particle’s mass, and you can imagine
the particle moving into the future with a certain momentum in its rest frame;
observe this particle from a moving frame and its spatial momentum becomes
non-zero, and the time component of its momentum (its energy) has to become
bigger – the particle packs more punch – as a consequence of the length of the
momentum vector being invariant.

[Exercises 7.1–7.3]
130 7 Dynamics

7.1.1 Another Route to the Momentum 4-vector

Above, we saw how we can define the momentum 4-vector in a rather
obvious way, by defining it as 𝑚𝐔, in direct analogy with the newtonian
momentum 𝑚𝐯. Another route to this, which brings out different points of
note, is to work by analogy with the way we developed the frequency vector
in Section 6.5, and ask ‘what would a momentum 4-vector have to look like?’
Start by thinking of a mass 𝑚 at rest. What is its 4-momentum 𝐐 =
(𝑄0 , 𝐪)? Based on our experience in Chapter 6, we can suppose that the spa-
tial part of a 4-momentum will be simply related to the particle’s newtonian
momentum, and in particular that it will reduce to the newtonian value 𝑚𝐯
at low speed, so that if the mass is at rest, 𝐪 = 0.
The obvious next step is to suppose that 𝑄0 = 0 for the particle at rest,
but if a 4-vector has all-zero components in one frame, then Eq. (6.3a) tells
us that it would have all-zero components in every frame, so this won’t work.
So we’ll presume that 𝑄0 ≠ 0, restrict ourselves to motion along the 𝑥-axis,
and write
𝐐 = (𝑄0 , 0).
What does this 4-vector look like in another frame? In particular, let’s
transform it into the frame 𝑆 ′ , which is moving with respect to the particle
at speed −𝑣 along the 𝑥-axis, so that the particle is moving at speed +𝑣 in 𝑆 ′ .
Looking again at Eq. (6.3a), we find
𝑄′0 = 𝛾𝑄0
𝑄′1 = 𝛾𝑄0 𝑣.
The low-speed limit of the second equation is (from Eq. (7.4)) 𝑄′1 = 𝑄0 𝑣 +
𝑂(𝑣3 ). If this is to reduce to the newtonian momentum 𝑄′1 = 𝑚𝐯 at low
speed, then we must identify 𝑄0 = 𝑚. Looking now at the first expression,
we have
( 𝑣2 ) 1
𝑄′0 = 𝑄0 1 + + 𝑂(𝑣 4 ) = 𝑚 + 𝑚𝑣2 + 𝑂(𝑣 4 ),
2 2
and spotting 𝑚𝑣2 ∕2 again prompts us to interpret the zeroth component
of the 4-momentum as an energy (or at least something closely related to
newtonian energy), and recovers what we now call the (relativistic) energy
of the moving particle as 𝑄′0 = 𝛾𝑚, as in Eq. (7.3).
This 4-momentum 𝐐 therefore has a plausible dynamical interpretation,
to which we can add the supposition that the 4-momentum is conserved in
collisions (a physical, rather than a mathematical statment), and pick up
the argument after Eq. (7.6).
 7.2 Photons 131

We have not added anything in this section that we didn’t know from
the previous one. But simply by demanding that a quantity of interest is a
4-vector, as we did here and in Section 6.5, we have imposed a significant
structure on that quantity, and discovered that two quantities that newtonian
physics sees as related but distinct – frequency and wavelength, or energy
and momentum – are in SR different components of a single geometrical
quantity. We did the same thing in Section 6.3 when we remarked that
we could define the 4-velocity of a particle by simply declaring it to be the
vector which had components (1, 𝟎) in the particle’s rest frame, and letting
Eq. (6.3a) do the rest of the work. This shows the fruitfulness of an approach
which focuses on the geometry of the spacetime we are examining; such
an approach may feel a little abstract, but it pays off when we look at the
description of gravity in general relativity.

7.2 Photons
For a photon, the interval represented by d𝐑 ⋅ d𝐑 is always zero (d𝐑 ⋅ d𝐑 =
d𝑡2 − d𝑥 2 − d𝑦 2 − d𝑧2 = 0 for photons). But this means that the proper
time d𝜏2 is also zero for photons. This means, in turn, that we cannot define
a 4-velocity vector for a photon by the same route that led us to Eq. (6.9),
and therefore cannot define a 4-momentum in the way we did in Eq. (7.1).
We can do so, however, by a different route. Recall that we defined (in
the paragraph below Eq. (6.14)) the 4-velocity as a vector pointing along the
worldline, which resulted in the 4-momentum being in the same direction.
From the discussion of the momentum of massive particles above, we see
that the 𝑃0 component is related to the energy, so we can use this to define
a 4-momentum for a massless particle, and again write

𝐏𝛾 = (𝐸, 𝐩𝛾 ).

Since the photon’s velocity 4-vector is null, the photon’s 4-momentum must
be also (since it is defined above to be pointing in the same direction). Thus
we must have 𝐏𝛾 ⋅ 𝐏𝛾 = 0, thus 𝐩𝛾 ⋅ 𝐩𝛾 = 𝐸 2 , recovering the 𝑚 = 0 version
of Eq. (7.9),

𝐸 2 = 𝑝2 (massless particle), (7.10)

so that even massless particles have a non-zero momentum.

In quantum mechanics, we learn that the energy associated with a
quantum of light – a photon – is 𝐸 = ℎ𝑓, where ℎ is Planck’s constant,
132 7 Dynamics

P1

P3

P2

Figure 7.2 The momenta involved in a collision.

ℎ = 6.626 × 10−34 J s (or 2.199 × 10−42 kg m in natural units), so that

𝐏 = (ℎ𝑓, ℎ𝑓, 0, 0) (photon). (7.11)

Thinking back to the approach of Section 7.1.1, we realise that Eq. (7.11)
is the conclusion we must come to if we speculate that the zeroth com-
ponent of a photon’s 4-momentum is its energy, 𝐸 = ℎ𝑓, and then demand that
that 4-vector is null.

7.3 Relativistic Collisions and the

Centre-of-Momentum Frame
Consider two particles, of momenta 𝐏1 and 𝐏2 , which collide and produce a
single particle of momentum 𝐏3 – you can think of this as describing either
two balls of relativistic putty, or an elementary particle collision which
produces a single new outgoing particle (this example is adapted from the
characteristically excellent discussion in Taylor & Wheeler (1992, §8.3));
note also that the subscripts denote the different particle momenta, but
the superscripts denote the different momentum components, so that (for
example) 𝑃21 is component 1 (the ‘𝑥-component’) of vector 𝐏2 .
Recalling Eq. (7.1), the particles have momenta

𝐏𝑖 = 𝛾𝑖 𝑚𝑖 (1, 𝐯𝑖 ), (7.12)

where the three particles have velocities 𝐯𝑖 , and 𝛾𝑖 ≡ 𝛾(𝑣𝑖 ). From momentum
conservation, we also know that

𝐏1 + 𝐏2 = 𝐏3 . (7.13)
 7.3 Relativistic Collisions and the Centre-of-Momentum Frame 133

Table 7.1 The momenta of the particles in Figure 7.2, in the ‘lab frame’

𝐏𝑖 = (𝐸𝑖 , 𝑝𝑖 ) 𝑣𝑖 𝛾𝑖 𝑚𝑖
√
1 (17, 15) 15∕17 17∕8 172 − 152 = 8
2 (8, 0) 0 1 8
√
3 (17 + 8, 15 + 0) 3∕5 5∕4 25 − 152 = 20
2

= (25, 15)

Of course, this also indicates that each component of the vectors is separately
conserved.
It’s useful to add some numbers to the discussion at this point.
Let’s simplify this, and imagine the collision taking place in one dimen-
sion, with an incoming particle moving along the 𝑥-axis to strike a stationary
second particle. The resulting particle will also move along the 𝑥-axis. In
this subsection we will, to avoid clutter, write only the 𝑥-component of the
spatial part of vectors, missing out the 𝑦 and 𝑧 components, which are zero
in this one-dimensional setup. Thus we’ll write (𝑡, 𝑥) rather than (𝑡, 𝑥, 𝑦, 𝑧)
or (𝑡, 𝐫), and (𝑃0 , 𝑃1 ) rather than (𝑃0 , 𝑃1 , 𝑃2 , 𝑃3 ).
Let the ‘incoming’ particles have masses 𝑚1 = 𝑚2 = 8 units; let the first
be travelling with speed 𝑣1 = 15∕17 along the 𝑥-axis in the ‘lab frame’ (so
that 𝛾1 = 𝛾(15∕17) = 17∕8); and let the second be stationary, 𝑣2 = 0 (so
𝛾2 = 1). As discussed above, the appropriate measure of the ‘length’ of
the 𝐏 vectors is the magnitude 𝑚2 of Eq. (7.7): 𝑚2 = 𝐏 ⋅ 𝐏. Recall that
𝐏𝑖 = (𝐸𝑖 , 𝑝𝑖 ) = (𝛾𝑖 𝑚𝑖 , 𝛾𝑖 𝑚𝑖 𝑣𝑖 ). Thus we can make a table of the various kine-
matical parameters in Table 7.1, where the first two rows are simply tran-
scribed from this paragraph.
To obtain the values in row 3, we add the 𝐏𝑖 of the first column to get
a 𝐏3 = (25, 15). We can get the speed by noticing that 𝑣𝑖 = 𝑃𝑖1 ∕𝑃𝑖0 , from
Eq. (7.12), and get the mass via the magnitude of the momentum 4-vector,
𝑚2 = (𝑃0 )2 − (𝑃1 )2 (that’s the most straightforward route; alternatively we
could note that if 𝑣3 = 3∕5, then 𝛾3 = 5∕4, so that if 𝑃30 = 𝛾3 𝑚3 = 25, then
𝑚3 = 20).
Notice the following points.

1. The two incoming particles have energy-momentum vectors with the

same magnitude, 𝑚1 = 𝑚2 = 8. This may seem surprising, since one is
moving much faster than the other, but recall that the scalar product 𝐏1 ⋅
134 7 Dynamics

𝐏1 is an invariant, so it has the same value in the frame in which particle 1

is moving at speed 𝑣1 = 15∕17 as it has in the frame in which particle 1
is stationary; in the latter frame, 𝐏′1 = (𝛾(0)𝑚1 , 𝑣1 ) = (8, 0), showing the
same values as 𝐏2 in the lab frame.
2. The mass in the ‘outgoing’ particle 3 is not merely the sum of the two
incoming masses: 𝑚1 + 𝑚2 ≠ 𝑚3 . It is the components of 𝐏, including
total energy 𝑃10 + 𝑃20 = 𝑃30 = 𝛾𝑚3 , that are the quantities conserved in
collisions.
3. The quantity 𝑚2 = 𝐏 ⋅ 𝐏 is frame-invariant, but not conserved; 𝑃𝜇 is
conserved, but not frame-invariant.

We must carefully distinguish the terms ‘conserved’, ‘covariant’, ‘invariant’

and ‘constant’ (cf. Taylor & Wheeler (1992, Box 7-3)):

Conserved refers to a quantity which is not changed by some process

(most typically unchanged in time) – momentum, for example, is
conserved in a collision. This refers to one frame at a time, and a
conserved quantity will typically have different numerical values in
different frames.
Covariant refers to an equation which does not change its form when
it is transformed from one frame to another.
Invariant refers to a quantity which is not changed by some transfor-
mation – the radius 𝑟 is an invariant of the rotation in Figure 4.9, and
we have just discovered that 𝑚2 is an invariant of the Lorentz trans-
formation. The term necessarily refers to more than one reference
frame; there is no reason to expect that an invariant quantity will be
conserved in a particular process. We will also sometimes refer to this
as ‘frame-independent’.
Constant refers to a quantity which does not change in time, such as
the mass of a test particle.

The speed of light is one of the very few things which has the full-house of
being conserved, invariant, and constant. [Exercises 7.4 & 7.5]

7.3.1 The Centre-of-Momentum Frame

From Eq. (7.12) and Eq. (7.13) we can write

𝐏3 = (𝛾1 𝑚1 + 𝛾2 𝑚2 , 𝛾1 𝑚1 𝑣1 + 𝛾2 𝑚2 𝑣2 ).
 7.3 Relativistic Collisions and the Centre-of-Momentum Frame 135

Table 7.2 The momenta of the particles in Figure 7.2, in

the ‘centre-of-momentum frame’

𝐏′𝑖 = (𝐸𝑖′ , 𝑝𝑖′ ) 𝑣𝑖 𝑚𝑖

√
1 (10, 6) 3∕5 102 − 62 = 8
2 (10, −6) −3∕5 8
3 (20, 0) 0 20

Now use the LT, Eq. (6.3), to transform this to a frame which is moving with
speed 𝑉 with respect to the lab frame. We therefore have

𝑃3′1 = 𝛾(𝑉)(𝑃31 − 𝑉𝑃30 )

= 𝛾(𝑉)(𝛾1 𝑚1 𝑣1 + 𝛾2 𝑚2 𝑣2 − 𝑉(𝛾1 𝑚1 + 𝛾2 𝑚2 )).

We can choose the speed 𝑉 to be such that this spatial momentum is zero,
𝑃3′1 = 0, giving

𝛾1 𝑣1 + 𝛾2 𝑣2 3
𝑉= =
𝛾1 + 𝛾2 5

(since here 𝑚1 = 𝑚2 ). We want to obtain 𝐏′𝑖 by starting with the energy

momentum vectors 𝐏𝑖 in the table above, and Lorentz-transforming them
into this new frame. Using 𝛾(𝑉 = 3∕5) = 5∕4, we have for example 𝑃𝑖′0 =
𝛾(𝑃𝑖0 − 𝑉𝑃𝑖1 )) in Table 7.2.
Notice:

1. In this frame also, 𝐏′1 + 𝐏′2 = 𝐏′3 : the conservation of a vector quantity in
one frame means that it is conserved in all frames.
2. Also 𝑚1′ = (𝐏′1 ⋅ 𝐏′1 )1∕2 = 8, 𝑚2′ = 8 and 𝑚3′ = 20: these are frame-
invariant quantities.
3. Both the 3-momenta and energy of the vectors in this frame are different
from the lab frame, even though their magnitudes are the same. We are
talking about the same 4-vector here, 𝐏𝑖 , but since we have decided to
change reference frame, we (unsurprisingly) change the vector’s frame-
dependent components.

This frame, in which the total spatial momentum is zero, so that the
incoming particles have equal and opposite spatial momenta, is known as
the centre-of-momentum (CM) frame, and the energy available for particle
136 7 Dynamics

0
production in this frame (𝑃cm ) is known as the centre-of-mass energy.
[Exercise 7.6]

7.4 But Where’s This Mass Coming From?

In the collision discussed in Section 7.3.1, we saw 𝑚1 + 𝑚2 ≠ 𝑚3 . We ex-
plained this by noting that although the rest masses of the before and after
particles are frame-invariant, they are not conserved in the collision.
What’s going on here? How much mass is actually present here? Put in
physical terms, is this telling us that there would be more gravity after the
collision than before? Excellent questions.
Imagine putting a box round the collision in Figure 7.2, and asking the
only apparently vague question ‘How much dynamics is there in this box?
– how much “punch”, how much “oomph” is here?’ Saying ‘there are two
particles of mass 𝑚’ doesn’t seem an adequate answer, because it doesn’t
account for the fact that the particles are moving towards each other. Differ-
ent observers will disagree about which particle is moving, and how quickly,
but there can be no doubt that the particles are going to hit each other, quite
hard. As well as the two particles, there’s a lot of kinetic energy in this box
(Table 7.2 tells us that, in the centre-of-momentum frame, the KE of the
incoming particles is 𝐸𝑖 − 𝑚𝑖 = 2, a significant fraction of the mass energy
𝑚 = 8).
If we imagine that these balls of relativistic putty are each a mere
eight grammes in mass, and reverting to physical units for the mo-
ment, the incoming particle in the lab frame, which is moving at a
healthy 15∕17 × 𝑐 (see Table 7.1), has an energy of 𝐸 = 𝛾𝑚𝑐2 = 17∕8 ×
(8 × 10−3 kg) × (3 × 108 m s−1 )2 = 1.53 PJ (about a third of a megaton of
TNT), only 720 TJ of which is accounted for by the energy equivalent of
the putty’s 8 g of mass, and the rest of which, amounting to 810 TJ, is kinetic
energy. That’s a lot of energy, and it may not then be surprising that that
energy of motion makes as important a ‘dent’ in spacetime as the energy
represented by the particles’ masses.
That is, there is more mass after this collision than before, but not more
‘gravitational charge’. A gravity detector (that is, another nearby mass)
would not notice a change from immediately before to immediately after
this collision.1

1 To be very precise, movement of masses like this would in principle generate gravitational
waves, but that’s a complication we should not indulge in at this particular point.
 7.5 More Unit Fun: an Aside on Electron-volts 137

Thus, as we have seen above, it is not mass that is conserved in collisions,

but 4-momentum (or energy-momentum), 𝐏. In General Relativity, it is not
mass that is the source of gravitation, but 4-momentum.
Taylor & Wheeler (1992, ch. 7) is an excellent discussion of the puzzles
here.

7.5 More Unit Fun: an Aside on Electron-volts

We’re going to start off using physical units in this section, partly in or-
der to show why they’re inconvenient for one of relativity’s most common
application domains.
If an electron (mass 9.11 × 10−31 kg) is moving close enough to the speed
of light that it has a 𝛾 = 150, then from Eq. (7.6) we discover that it has a total
energy of 1.23 × 10−11 J. Is that a large number in this context? Certainly,
it’s clear that the joule isn’t a convenient unit here.
If an electron is accelerated through a potential difference of one volt,
then it acquires a kinetic energy of 𝐸 = 𝑞𝑉 = 1.602 × 10−19 J. This is a
convenient unit of energy in particle physics, where it is known as the
electron-volt, written ‘eV’. The electron-volt is nothing more than a name for
a convenient quantity of energy, just as the jansky, at 10−26 W, is the name
for a small quantity of power, convenient for radio astronomers. Using this
unit, this electron has an energy of 76.8 × 106 eV, or 76.8 MeV. Does that
count as a large number?
If this electron, and a positron moving at the same speed, were to collide
with one another, the total energy in the CM frame would be 154 MeV.
Now, a neutral pion (𝜋0 ) is a subatomic particle which has a rest mass
of 2.40 × 10−28 kg, which you can therefore confirm has a rest energy of
135 MeV. Thus there is enough energy in this electron–positron collision
for the two incoming particles to annihilate and result (briefly) in a pion.
In such a context as this – where we are asking if there is enough energy
available to create new particles – we are much less interested in the mass of
a particle than in its rest energy, and so it is much more natural to quote the
particle’s mass, in a data table, in terms of that rest energy. Keeping the units
correct, we would therefore write that 𝑚(𝜋0 ) = 135 MeV∕𝑐2 , and similarly
that the mass of the electron is 𝑚e = 0.511 MeV∕𝑐2 : just as the convenient
unit of energy is the eV, the convenient unit of mass is the ‘eV∕𝑐2 ’, which
you can confirm has the dimensions of mass.
138 7 Dynamics

Similarly, we might look at Eq. (7.9), which in physical units is

𝐸 2 = 𝑝2 𝑐2 + 𝑚2 𝑐4 ,
and observe that 𝑝𝑐 has the dimensions of energy, and thus can be expressed
in eV, so that eV∕𝑐 is a useful unit for momentum.
As you will recall, in natural units the speed 𝑐 = 1, and this extra factor
of 𝑐2 becomes redundant. Thus in natural units the units of mass are the
same as the units of energy, and we habitually write 𝑚e = 0.511 MeV.

7.6 Relativistic Force

Another way of thinking about this is through the idea of relativistic force.
Possibly surprisingly, force is a less useful concept in relativistic dynamics,
than it is in newtonian dynamics – it’s more often useful to talk of momen-
tum and energy, and indeed energy-momentum. However, there are a few
useful things we can discover.
We can define force in the usual way, in terms of the rate of change of
momentum. Defining the 4-vector
d𝐏
𝐅= ,
d𝜏
we can promptly discover that 𝐅 = 𝑚𝐀. Referring back to Eq. (6.16), and
writing 𝐟 for the spatial components of the 4-vector, 𝐅, we discover that 𝐟 =
𝑚𝛾(𝛾𝐯 ̇ + 𝛾𝐚) (observing that in the non-relativistic limit 𝐟 = 𝑚𝐚 reassures
us that this quantity 𝐅 does have at least something to do with the thing
we are familiar with as ‘force’, and that we are justified in calling 𝐟 the
‘relativistic 3-force’). But this tells us that 𝐟 is not parallel to 𝐚, as we (and
Newton) might expect.
What else might we expect? In newtonian dynamics, 𝐟 ⋅ d𝐬 is the work
done on a particle by a force which displaces it by an amount d𝐬 – what is
the analogue in relativity? We can write 𝐅 ⋅ d𝐑 = 𝐅 ⋅ 𝐔 d𝜏, but
d𝐏 d𝐏 1 d
𝐅⋅𝐔=𝑚 ⋅𝐔=𝐏⋅ = (𝐏 ⋅ 𝐏) = 0,
d𝜏 d𝜏 2 d𝜏
since we know that 𝐏 ⋅ 𝐏 is conserved. So, although in newtonian dynamics
the effect of a force on a particle is to do ‘work’ on it, and so change its
energy, the effect of a relativistic force 𝐅 is not to change the particle’s energy-
momentum, but instead to simply ‘rotate’ the particle in Minkowski space
– it changes the direction of the particle’s velocity vector, 𝐔, but not its
(invariant) length.
 7.7 An Example: Compton Scattering 139

E, p

Q1 θ
φ
Q2

Figure 7.3 Compton scattering: a photon being scattered from a charged par-
ticle.

Of course this does increase the particle’s energy and momentum sep-
arately. If the particle starts off at rest in some frame, then after the
force has been applied it will have 𝑃0 = 𝑚𝛾, and 𝐩 = 𝛾𝑚𝐯. However, the
time and space components of the momentum 𝐏 are related by Eq. (7.8),
𝐸 2 − 𝐩 ⋅ 𝐩 = 𝐏 ⋅ 𝐏 = 𝑚2 , the mass of the particle, which is unchanged by
the application of the force. Because of the minus sign in this expression, the
time and space components of the momentum vector can increase without
the magnitude of the vector increasing, and this is closely analogous to the
role of the minus sign in our analysis of the twins paradox, in Section 5.7.1:
there, the (travelling) twin who took the indirect route into the future trav-
elled through less spacetime than the stay-at-home who took the direct route;
here, the accelerated vector has a momentum with larger (frame-dependent)
components, which nonetheless has the same length as it started with.
In both cases, the unexpected conclusions follow ultimately from the
observation that Minkowski space is not euclidean space, and that in
Minkowski space, instead of Pythagoras’s theorem, we have Eq. (6.6).

7.7 An Example: Compton Scattering

As a further example of a relativistic collision, we can examine the collision
between a photon – a quantum of light energy – and an electron. This is
Compton scattering, and is not the same as the classical Thomson scattering
of light by electrons: that is a purely electromagnetic effect whereas this
is an inherently relativistic and quantum-mechanical effect, where we are
treating both the electron and the incoming light as relativistic particles.
See also French (1968, pp. 194–196).
The collision is as shown in Figure 7.3. An incoming photon strikes a
stationary electron and both recoil. The incoming photon has energy 𝑄1 =
ℎ𝑓1 = ℎ∕𝜆1 and the outgoing one 𝑄2 = ℎ𝑓2 = ℎ∕𝜆2 ; the outgoing electron
140 7 Dynamics

has energy 𝐸, spatial momentum 𝐩, and mass 𝑚. The four momentum

4-vectors are therefore
𝐏1e = (𝑚, 0, 0, 0) 𝐏2e = (𝐸, 𝑝 cos 𝜃, 𝑝 sin 𝜃, 0)
𝐏1𝛾 = (𝑄1 , 𝑄1 , 0, 0) 𝐏2𝛾 = (𝑄2 , 𝑄2 cos 𝜙, 𝑄2 sin 𝜙, 0) ,
where subscripts 1 and 2 denote momenta before and after the collision,
respectively.
Momentum conservation implies 𝐏1e + 𝐏1𝛾 = 𝐏2e + 𝐏2𝛾 . Comparing
components, we find

𝑄1 + 𝑚 = 𝑄 2 + 𝐸 (7.14a)
𝑄1 = 𝑝 cos 𝜃 + 𝑄2 cos 𝜙 (7.14b)
0 = 𝑝 sin 𝜃 + 𝑄2 sin 𝜙. (7.14c)

Writing Eq. (7.14b) as 𝑄1 − 𝑄2 cos 𝜙 = 𝑝 cos 𝜃, and squaring and adding this
and Eq. (7.14c), we obtain

(𝑄1 − 𝑄2 cos 𝜙)2 + (−𝑄2 sin 𝜙)2 = 𝑝2 . (7.15)

Using the relation 𝑝2 = 𝐸 2 − 𝑚2 in this equation, then substituting for 𝐸

from Eq. (7.14a) and rearranging, we find

𝑄1 𝑄2 (1 − cos 𝜙) = (𝑄1 − 𝑄2 )𝑚, (7.16)

which, on dividing through by 𝑄1 𝑄2 𝑚, gives

1 1 1 − cos 𝜙
− = (7.17)
𝑄2 𝑄1 𝑚
or, in terms of wavelength,
ℎ
𝜆2 − 𝜆1 = (1 − cos 𝜙). (7.18)
𝑚
We can see that the scattered photon will have a longer wavelength (smaller
energy) than the incident one. Note also that this is not any type of Doppler
effect: the photon has a different energy not because we are observing it in
any sort of moving frame, but has a lower energy because it has had to give
some up to the electron.

7.8 Not The End

That is all we have time and space for. Over the course of this text, we
have examined the axioms on which SR is based, derived the immediate
 Exercises 141

consequences of these axioms in the form of the Lorentz transformation

equations, and finally examined how our understanding of dynamics has to
be transformed within the context of relativity.
It is also just about all of the Special Relativity that you will need in your
future study of physics. It is also, possibly surprisingly, just about all the SR
that there is – there is no ‘Advanced Special Relativity’ text to move on to.
More advanced courses that use SR depend on it as a foundation: they build
on it but do not extend it. Even fascinating texts such as Schwarz & Schwarz
(2004) use it as a starting point for excursions into neighbouring areas of
physics and maths, and Gourgoulhon (2013) covers much the same ground
with more sophisticated mathematical tools, but they do not discover more
of it, as such.
In Appendix A, I provide a compact introduction to General Relativity,
which builds directly on what we have learned about Special Relativity
here, but necessarily without the extra maths which is required for a full
understanding.
Has it been worth it? While you will only very rarely have to calculate
the speed of clocks on passing spaceships, and rather less often on passing
trains, you may well have to calculate the speed of a clock on a passing
pion. And stepping back from calculation, the physical world picture we
have looked at in this book is what underlies areas of physics as diverse
as Relativistic Quantum Mechanics, Quantum Field Theory (underlying
particle physics), and General Relativity, which are simply unintelligible
otherwise. Even if you do not go on to study those areas, you have come
to have a more fundamental view of the universe than you had from your
intuitive understanding of Newton’s physics, and gained access to one of
the twentieth century’s most radical intellectual adventures.
Enjoy!

Exercises
Exercise 7.1 (§7.1)
Recall the definition of the momentum 4-vector:
𝐏 = 𝑚𝐔 = 𝑚𝛾(1, 𝐯).
What are the time- and 𝑥-components of the momentum of a particle of
mass 2 kg moving with speed 𝑣 = 3∕5 (so 𝛾(𝑣) = 5∕4) along the 𝑥-axis?
1. 𝑃0 = 5∕2, 𝑃1 = 3∕2
142 7 Dynamics

2. 𝑃0 = 5∕2, 𝑃1 = 6∕5
3. 𝑃0 = 1, 𝑃1 = 3∕5
4. 𝑃0 = 5∕4, 𝑃1 = 3∕2 [ 𝑑− ]

Exercise 7.2 (§7.1)

We were a bit sloppy in Exercise 7.1. What are the units of 𝑃0 and 𝑃1 ?

1. kg m s−1 and kg m s−1

2. kg and kg m s−1
3. kg and kg [ 𝑑− ]

Exercise 7.3 (§7.1)

The luminosity of the Sun is 𝐿⊙ = 3.86 × 1026 W; this is powered by nuclear
fusion. How much mass does it consume, in fusion reactions, per second?

Exercise 7.4 (§7.3)

Conserved quantities: are the following statements true or false?

1. Momentum is conserved in collisions.

2. Energy is conserved in collisions.
3. Invariant mass (𝑚2 ) is conserved in collisions. [ 𝑑− ]

Exercise 7.5 (§7.3)

Invariant quantities: are the following statements true or false?

1. Momentum is an invariant quantity.

2. Energy is an invariant quantity.
3. Mass (𝑚2 ) is an invariant quantity. [ 𝑑− ]

Exercise 7.6 (§7.3.1)

A fast-moving particle of mass 𝑚1 = 1 μg is moving at speed 𝑣 = 0.5 when
it hits a stationary particle of mass 𝑚2 = 1000 μg. What is the speed of
the centre-of-momentum frame? What are the two particles’ energies and
momenta in the lab frame (that is, the one in which particle 2 is stationary),
and in the CM frame?
 Exercises 143

Give the answers in both natural units and physical units (SI) (remember
Exercise 4.1).
After you have read Section 7.5, additionally give the answers, including
the two masses, in physical units in terms of eV.
 Appendix A
An Overview of General Relativity

Mathematics, physics, chemistry, astronomy, march in one front. Whichever

lags behind is drawn after. Whichever hastens ahead helps on the others.
The closest solidarity exists between astronomy and the whole circle of exact
science.
Karl Schwarzschild, in 1913, quoted in Eddington’s obituary of him (1917)

Although, as I mentioned at the end of the last chapter, we have now covered
a substantial fraction of what there is to say about Special Relativity, this is
very far from all we can say about relativity in general. Special Relativity
discusses the special case of motion within, and transformation between,
inertial frames moving with constant velocity relative to each other. If we re-
lax this restriction, and ask about transformations between arbitrary frames
(i.e., between arbitrary coordinate systems), which may be accelerating with
respect to each other or moving under the influence of gravity, then we are
asking the questions of General Relativity – GR.
General Relativity is mathematically much more challenging than SR,
and so the account below does give in to ‘it can be shown that. . . ’ on more
than one occasion, and it makes reference to mathematical technology
such as differential geometry which we can talk about only schematically.
However, the way we have covered SR in this course, with its emphasis
on an axiomatic approach and on geometry, gives us a starting point from
which we are able to do more than merely sketch out the landscape here.
You can find alternative explanations of some of the fundamental ideas –
such as the Equivalence Principle, curvature, and cosmological expansion
– online and in popular books. You can also learn something from the
introductory chapters of GR textbooks, before they get properly started on
the tensor calculus. I mention one or two such books in Section 1.8. In
particular, I’ll draw attention to Schutz (2009) and Rindler (2006); and also

145
146 Appendix A An Overview of General Relativity

Misner, Thorne & Wheeler (1973), which is justly famous, though it has a
slightly idiosyncratic take on the subject. I emphasise, however, that these
are all graduate texts, so that the level of maths required will quickly outstrip
what I can prepare you for in this short chapter.

What is the problem that General Relativity is trying to solve?

We examine the problem of gravity in Section A.1, through a sequence
of thought experiments, which bring out some immediate consequences of
the ideas. The way we can think about the solution is by sticking with the
theme of geometry, which we have laid so much stress on up to this point.
In Section A.2 we learn about the ideas of the metric of a space, of the curva-
ture of a space, and of geodesics through that space. Then, in Section A.3,
we discover, in general terms, how we can put these ideas together to get
a relativistic theory of gravity, in the form of Einstein’s equations for the
structure of spacetime around a gravitating mass. The mathematical details
are beyond the scope of this course, but we can understand the shape of the
ideas. Finally, in Section A.4, we look briefly at some solutions to Einstein’s
equations, and discover the continuity with Newton’s theory of gravity.

A.1 Some Thought Experiments on Gravitation

In the text so far, we have dealt with gravity by ignoring it or, where we
couldn’t do that, for example because we were talking of throwing balls
or juggling, by quietly presuming it to be an uncomplicated ‘downward-
pointing’ force.
General Relativity – Einstein’s theory of gravitation – adds further sig-
nificance to the idea of the inertial frame. Indeed, at this point we slightly
redefine what we mean by an inertial frame. In the context of GR, an inertial
frame is a frame in which SR applies, and thus the frame in which the laws
of nature take their corresponding simple form. This definition, crucially,
applies even in the presence of large masses where (in newtonian terms) we
would expect to find a gravitational force. The frames thus picked out are
those which are in free fall, either because they are in deep space far from
any masses, or because they are (attached to something that is) moving
under the influence of ‘gravitation’ alone. I put ‘gravitation’ in scare-quotes
because it is part of the point of GR to demote gravitation from its newtonian
status as a distinct physical force to a status as a mathematical fiction – a
 A.1 Some Thought Experiments on Gravitation 147

conceptual convenience – which is real only in the same way that centrifugal
force is real, as a way of explaining the physical results of a change of frame.
In this section, we work through a number of thought experiments, which
one by one introduce some of the key ideas, and some of the key conclusions,
of GR.

A.1.1 Standing in a Rocket

The first step of that demotion is to observe that the force of gravitation (I’ll
omit the scare-quotes from now on) is strangely independent of the nature
of the things that it acts upon. Imagine a frame sitting on the surface of the
Earth, and in it a person, a bowl of petunias, and a radio, at some height
above the ground: we discover that, when they are released, each of them
will accelerate at the same rate towards the floor (Galileo is supposed to
have demonstrated this same thing using the Tower of Pisa, careless of the
health and safety of passers-by). Newton explains this by saying that the
force of gravitation on each object is proportional to its gravitational mass
(the gravitational ‘charge’, if you like); and the acceleration of each object,
in response to that force, is proportional to its inertia, which is proportional
to its inertial mass. Newton doesn’t put it in those terms, of course, but he
also fails to explain why the gravitational and inertial masses, which a priori
have nothing to do with each other, turn out experimentally to be exactly
proportional to each other, even though the person, the plant, the plant pot,
and the radio broadcasting electromagnetic waves all exhibit very different
physical properties.1
Einstein supposed that this was not a coincidence, and that there was
a deep equivalence between acceleration and gravity (we will see later, in
Section A.3, that the force of gravity we feel standing in one place is the
result of us being accelerated away from the path we would have if we were

1 Newton does mention that ‘a bit of fine down and a piece of solid gold descend with equal
velocity’ in the General Scholium to the second edition (1713) of his Principia (see for
example https://isaac-newton.org/general-scholium/), but that text is better known for
Newton’s acknowledgement that, although he has described gravity, he cannot explain its
source: ‘Hitherto we have explaind the phaenomena of the heavens and of our sea, by the
power of Gravity, but have not yet assignd the cause of this power. [. . . But] I frame no
hypotheses.’ Part of the problem was that the expectations of the time presumed that all
forces were transmitted by contact, so that the idea of a pervasive gravitational field was
contrary to all intuition, and a much more radical conceptual innovation than we might
expect. Indeed it was possibly only Newton’s occult and alchemical interests – research into
which he pursued in parallel with his scientific work – that allowed him to think in such
terms.
148 Appendix A An Overview of General Relativity

Figure A.1 Objects in a spaceship.

in free fall). He raised this to the status of a postulate: the Equivalence

Principle.
Putting Galileo to one side for a moment, imagine this same frame – or, for
the sake of concreteness and the containment of a breathable atmosphere,
a spacecraft – isolated in space (Figure A.1). Since spacecraft, observer,
petunias and radio are all equally floating in space, none will move with
respect to another (or, if they are initially moving, they will continue to
move with constant relative velocity). That is, Newton’s laws of motion
work in their simple form in this frame, which we can therefore identify as
an inertial frame.
If, now, we turn on the spacecraft’s engines (Figure A.2), then the space-
craft will accelerate, but the objects within it will not, until the spacecraft
collides with them, and starts to accelerate them by pushing them with
what we will at that point decide to call the cabin floor. Before they hit
the floor, they will move towards the floor at an increasing speed, from the
point of view of someone in the box.2 Crucially – and, from this point of
view, obviously – the sequence of events here is independent of the details
of the structure of the ceramic plant pot, the biology of the observer and
the petunias, and the electronic intricacies of the radio. If the spacecraft
continues to accelerate at, say, 9.81 m s−2 , then the objects now firmly on
the cabin floor will experience a continuous force of one standard Earth
gravity, and observers within the cabin will find it difficult to tell whether
they are in an accelerating spacecraft or in a uniform gravitational field.
In fact we can make the stronger statement – and this is another physical
statement which has been verified to considerable precision in, for example,
Eötvös-type experiments3 – that the observers will find it impossible to tell

2 By ‘point of view’ I mean ‘as measured with respect to a reference frame fixed to the box’, but
such circumlocution can distract from the point that this is an observation we’re talking
about – we can see this happening.
3 These experiments use a torsion balance to directly confirm that masses of different
materials experience the same gravitational force; for a review, see Adelberger et al. (2009).
 A.1 Some Thought Experiments on Gravitation 149

Figure A.2 Objects on the floor of an accelerating spaceship.

the difference between acceleration and uniform gravitation, and we can

elevate this remark to a physical principle.

The Equivalence Principle (EP): Uniform gravitational fields are equiv-

alent to frames that accelerate uniformly relative to inertial frames.
(A.1)

The Equivalence Principle is a physical statement rather than a mathematical

one. By a ‘physical statement’ I mean a statement that picks out one of
multiple mathematically consistent possibilities, and says that this one is the
one which matches our universe. Mathematically, we could have a universe
in which the EP did not hold, or in which the galilean transformation works
for all speeds and the speed of light is infinite; but we don’t.
That is, the EP (or at least this version of it) has now explained why all
objects fall at the same rate in a gravitational field.
Note that I am at this point carefully not using the word ‘accelerate’ for
the change in speed of the objects in the box with respect to that frame. We
reserve that word for the physical phenomenon measured by an accelerome-
ter, and the result of a real force, and try to avoid using it (not, I fear, always
successfully) to refer to the second derivative of a position. Depending on
the coordinate system, the one does not always imply the other, as we will
see shortly.

A.1.2 The Falling Lift

Recall from Special Relativity that we may define an inertial frame to be
one in which Newton’s laws hold, so that particles that are not acted on by
an external force move in straight lines at a constant velocity. In Misner,
Thorne and Wheeler’s words, inertial frames and their time coordinates
are defined so that motion looks simple. This is so if we are in a box far
150 Appendix A An Overview of General Relativity

Figure A.3 An observer in a floating box.

Figure A.4 Falling down a lift-shaft.

away from any gravitational forces, and so we may identify that as a local
inertial frame (LIF; we will see the significance of the word ‘local’ below).
Another way of removing gravitational forces, less extreme than going into
deep space, is to put ourselves in free fall. Einstein asserted that these two
situations are indeed fully equivalent, and defined an inertial frame, firstly,
as one in free fall, and declared, secondly, that just as all inertial frames in
SR are equivalent, all (redefined) inertial frames in GR are equivalent, too.

Equivalence Principle, version 2: All local, freely falling, non-rotating

laboratories (i.e., LIFs) are fully equivalent for the performance of all
physical experiments. (A.2)

Imagine being in a box floating freely in space, and shining a torch hori-
zontally across it from one wall to the other (Figure A.3). Where will the
beam end up? Obviously, the beam will end up at a point on the wall directly
opposite the torch. There’s nothing exotic about this. The EP, however, tells
us that the same must happen for a box in free fall, Figure A.4, left. That is,
a person inside a falling lift would observe the torch beam to end up level
with the point at which it was emitted, in the (inertial) frame of the lift. This
is a straightforward and unsurprising use of the EP.
How would this appear to someone watching the lift fall? Since the light
takes a finite time to cross the lift cabin, the spot on the wall where it strikes
will have dropped some finite (though small) distance, and so will be lower
 A.1 Some Thought Experiments on Gravitation 151

m E

m + mgz = E

Figure A.5 A falling mass and rising photon.

than the point of emission, in the frame of someone watching this from a
position of safety (Figure A.4, right). That is, this non-free-fall observer will
measure the light’s path as being curved in the gravitational field. The EP
forces us to conclude that even massless light is affected by gravity.

Note that it is useful to be clear exactly where the EP enters this argu-
ment. It lets us go from a scenario we are confident we can fully analyse
– namely the box far away from gravitating matter – to a scenario we might be
more hesitant about – namely motion under the influence of gravity. We will use
a version of the EP again in Section A.3, to move from a context we understand –
physics in SR – to one we initially don’t – physics in GR.

A.1.3 Gravitational Redshift

Next, imagine dropping a particle of mass 𝑚 through a distance 𝑧. The
particle starts off with energy 𝑚 (𝐸 = 𝑚𝑐2 , with 𝑐 = 1), and ends up with
energy 𝐸 = 𝑚 + 𝑚𝑔𝑧 (see Figure A.5). Now convert all of this energy into a
single photon of energy 𝐸, and send it up towards the original position. It
reaches there with energy 𝐸 ′ , which we convert back into a particle.4 Now,
either we have invented a perpetual motion machine, or else 𝐸 ′ = 𝑚:
𝐸
𝐸′ = 𝑚 = , (A.3)
1 + 𝑔𝑧
and we discover that a photon loses energy as a necessary consequence of
climbing a distance 𝑧 through a gravitational field, and as a consequence of
our demand that energy be conserved.
Let’s look more closely at this photon, and take it to be climbing through
a positive distance Δ𝑟. As Planck and Einstein learned, the photon’s energy

4 As described, this is kinematically impossible, since we cannot do this and conserve

momentum, but we can imagine sending distinct particles back and forth, conserving just
energy; this would have an equivalent effect, but be more intricate to describe precisely.
152 Appendix A An Overview of General Relativity

is 𝐸 = ℎ𝑓, where ℎ is Planck’s constant and 𝑓 the photon’s frequency, in

hertz. From Eq. (A.3), we can write 𝐸 = 𝐸 ′ (1 + 𝑔Δ𝑟), and writing 𝐸 ′ =
ℎ𝑓 ′ = ℎ(𝑓 + Δ𝑓), we discover that ℎ𝑓 = ℎ(𝑓 + Δ𝑓)(1 + 𝑔Δ𝑟). Taking the
differences to be infinitesimal, and discarding terms in 𝑂(d𝑟 d𝑓), we then
find
d𝑓
= −𝑔 d𝑟.
𝑓
Now we know that this acceleration 𝑔 is the result of a gravitational force
𝐹 = 𝑚𝑔, or 𝐹 = 𝑚∇𝜙, where 𝜙 is Newton’s gravitational potential. If we
suppose that this potential is that of a body such as the Earth, and 𝑟 is a
radial coordinate, then 𝜙 = 𝜙(𝑟), and 𝑔 = ∇𝜙 = d𝜙∕d𝑟. Thus 𝑔d𝑟 = d𝜙, and
d𝑓
= −d𝜙.
𝑓
We can integrate this between, say, radii 𝑟0 and 𝑟1 to find
𝑓1 e−𝜙1
= , (A.4)
𝑓0 e−𝜙0
where we can if we wish choose the constant of integration to set either one
of the potentials to zero.
The frequency of radiation is of course inversely related to its period,
so we can repurpose Eq. (A.4) to obtain an expression for the change in
periods Δ𝑡 of radiation at the two potentials:
Δ𝑡0 e−𝜙1
= . (A.5)
Δ𝑡1 e−𝜙0
This frequency shift, corresponding to an energy loss, is termed gravita-
tional redshift,5 and it has been confirmed experimentally, in the ‘Pound–
Rebka experiments’ of 1959–64,6 which were able to detect the frequency

5 This is also sometimes referred to as ‘gravitational Doppler shift’, but inaccurately, since it is
not a consequence of relative motion, and so has nothing to do with the Doppler shift you
are familiar with. Also, if the photon is descending, it is blue-shifted, so that ‘gravitational
frequency shift’ would be a better term. But ‘redshift’ is conventional.
6 Multiple research groups, including Pound’s in Harvard, USA, and John Paul Schiffer’s in
Harwell, UK, realised in 1959 that they could use the Mössbauer effect (discovered in 1958)
to make this measurement, and published data-less announcements of their plans to do so.
Schiffer’s group were the first to publish experimental results, in February 1960, followed
only a few weeks later by the Harvard group’s first publication; both groups announced a
detection of the redshift, within experimental error. The Harwell group’s results were,
however, arguably the result of a systematic chemical effect which supplied or amplified the
redshift result, and only Pound’s group seems to have gone on to refine the measurement in
a sequence of publications in the succeeding years. It is not unreasonable, therefore, that
 A.2 Geometry 153

changes of gamma rays falling a mere 22.5 m, in which the factor 𝑔𝑧∕𝑐2 in
Eq. (A.3)7 is only 2.5 × 10−15 .
Light, it seems, can tell us about the gravitational field it moves through.

A.2 Geometry
To get from here to GR we need to answer three questions: (i) what do
we mean by ‘the geometry of a space’? (ii) how do we describe geometry
mathematically? and (iii) how do we make the link from geometry to gravity?
Geometry is the mathematical discussion of shapes. We tend to think of
this as a fixed thing, which we learned about in school, and haven’t had to
think much more about since. Euclid famously established an axiomatic
basis for geometry, setting out a small number of axioms, or postulates, on
the basis of which a large number of geometrical theorems can be proved.
The space he described, with the fairly straightforward extension from the
plane to higher dimensions, is referred to as euclidean space. In this space,
the internal angles of a triangle add up to 180°, parallel lines never meet, the
circumference of a circle is 𝜋 times its diameter, and Pythagoras’s theorem
gives the distance between points.
That was thought to be the end of the matter for a very long time.
One of Euclid’s postulates is that there exist parallel lines – lines which
never cross each other. This postulate can be expressed in a variety of ways,
but it also seems either to be self-evident, or to be such a trivial addition to
the other postulates that it must surely be provable from them; and for more
than 2000 years after Euclid, mathematicians tried to do exactly that. It was
only in the nineteenth century that the mathematical community discovered
that one can consistently discuss geometry without the parallel postulate, or
with alternative postulates such as the demand that an arbitrary line in the
space has zero parallel lines, or that it has more than one. These alternative
postulates lead to different conclusions about, for example, the sum of the

these are memorialised as the ‘Pound–Rebka’ measurements, but the episode usefully
illustrates the contingency of experimental races. For a history of the episode, including
references to the various original papers, see Hentschel (1996). The race, the
pre-announcements, and the science-by-press-release, are also discussed rather
unflatteringly in Reif (1961). Further, see Vessot et al. (1980) for a description of the 1976
Gravity Probe A experiment, which measured the effect between the ground and a
space-based instrument at an altitude of 10 000 km.
7 Where did the 𝑐2 come from? Equation (A.3) is written in units where 𝑐 = 1, as usual; in
order to get a numerical value from ‘22.5 m’, we must examine the dimensions of the
expression to discover that the denominator, in physical units, must be 1 + 𝑔𝑧∕𝑐2 .
154 Appendix A An Overview of General Relativity

internal angles of a triangle, or the ratio of a circumference to a diameter.

So when we talk about ‘the geometry of a space’, we mean by ‘space’ a set
of points such as a plane or a volume, or Minkowski space; and by that
space’s ‘geometry’ we mean the set of ‘rules’ for distances, angles, and so
on. If we use Euclid’s rules, we get euclidean geometry; if we want to talk
about Minkowski space, we have very similar rules, but with a definition of
‘distance’ which we have seen is different from the pythagorean distance.
But before we can talk about distance, we have to step back and think a
little more clearly about coordinates.

A.2.1 Coordinates
Where are you? What time is it?
At this stage in your physics education, you are comfortable with the
idea of describing the positions of things, and their movement, by giving
them coordinates, and talking about how those coordinates change. Those
coordinates are a mathematical fiction, of course, but it’s useful to stress
how completely arbitrary those coordinates are.
In addressing one or other physics problem, we are taught to make things
easy for ourselves by choosing cartesian coordinates, or polar coordinates, or
one of the various other more exotic alternatives. Our choice here will have
an effect on, for example, how we write down velocities, and the form of the
gradient operator that we must use. Two key points are (i) the coordinates
we use are a choice, and (ii) they are not physically significant.
The way we describe geometry mathematically (or at least the way which
is of most relevance to GR) is using differential geometry. Unfortunately, this
subject is too mathematically challenging to go into at the level of this text.
What I can do instead is to give you a sense of the key ideas, in just enough
depth that we can understand how they are used to make the link to gravity.
A coordinate system is a way of systematically drawing grid-lines through-
out a region of spacetime, which is curved or flat, and thus attaching coordi-
nates to each event. These coordinates don’t have any intrinsic meaning –
they’re just labels attached to points in space and time, and the difference in
coordinates between two events doesn’t tell you anything immediate about
how far apart they are. Also, we discover that, in our universe, we will al-
ways need three space and one time dimension to fully and non-redundantly
locate a point.
This isn’t too hard to think about for space dimensions: we can of course
imagine grid-lines drawn on the ground, and perhaps imagine some ar-
rangement with balloons or helicopter drones, with numeric coordinates
 A.2 Geometry 155

painted on the sides of them, and perhaps observers sitting in some of them.
It’s a little harder to think about for time. For Newton, this was easy: the
newtonian picture is of a sequence of three-dimensional ‘snapshots’, each of
which has a time-value attached to it. There’s some clear arbitrariness to
the timescale, in terms of how fast the clock ticks, or where the zero of time
is, but there’s equally well a clear association between each snapshot and
‘the time’. A brief reflection on Chapter 4 will tell us that things aren’t going
to be so simple in a spacetime built on Minkowski space.
Probably the most straightforward way to define a time coordinate is to
decide that the time coordinate of an event is the number showing on the
watch of an observer located there. Or, more generally, that each one of our
balloons or drones has a clock attached to it, which ticks in some systematic
way, and ‘the time’ is the number on that clock face. It might be that these
clocks run differently on different drones, so that it’s not a given that they
all tick ‘together’.

A.2.2 The Metric of Euclidean Space

In Chapter 5 we derived the Lorentz transformation by demanding that the
interval 𝑠2 is invariant under that transformation. Our eventual definition
of the interval is in Eq. (4.4), and as such we can now recognise this as the
first appearance of the metric. Quite generally, we regard the metric of a
space as the definition of length in that space. What does that mean?
The metric in euclidean space is
d𝑠2 = d𝑥 2 + d𝑦 2 , (A.6)
which you will recognise as a differential form of Pythagoras’s theorem. This
definition doesn’t pin down everything – it doesn’t say what length units we
use, nor where the origin is – but it does give a systematic recipe for turning
coordinate differences d𝑥 and d𝑦 into a length d𝑠.
How would we use this to calculate the distance between two points
𝑃0 = (𝑥1 , 𝑦) and 𝑃1 = (𝑥2 , 𝑦)? That’s not hard: we simply calculate
𝑃1 𝑃1 √ 𝑃1 √
𝑙=∫ d𝑠 = ∫ d𝑠2 = ∫ d𝑥 2 + d𝑦 2
𝑃0 𝑃0 𝑃0
√𝑥1
=∫ d𝑥 2 , since d𝑦 = 0 along this path
𝑥0
𝑥1
= ∫ d𝑥 = 𝑥1 − 𝑥0 .
𝑥0
156 Appendix A An Overview of General Relativity

That’s not unexpected.

However, we can also describe points in the plane using polar coordinates,
(𝑟, 𝜃). The distance element – that is, the metric for polar coordinates – is
d𝑠2 = d𝑟2 + 𝑟2 d𝜃 2 , (A.7)
which you have also probably seen before, in one guise or another. We
could also use this metric to obtain the distance between points 𝑃0 and 𝑃1 .
This is a slightly fiddly calculation, because (𝑟, 𝜃) would be a poor choice of
coordinates to use, but the end result would be the same distance: the two
metrics are defined in relation to different coordinates, and so they provide
different recipes for going from coordinate differences to distances in the
space; but the space they are describing – flat euclidean space in this case –
is the same, so the distances they produce must be identical.
That is, the metric is describing a measurable property of a space – the
length of a path through that space – but each version of that metric works
with a particular set of coordinates. The set of coordinates is otherwise
(largely) arbitrary. Dealing with that arbitrariness, and learning how to talk
about distance, and differentiation, in the presence of that arbitrariness, is
what differential geometry is all about.

A.2.3 The Metric of a Non-euclidean Space: Curvature

As an example of a space which is not euclidean, let us look at coordinates
on the surface of a sphere. Here, the line element on the surface of the
sphere (in terms of co-latitude 𝜙) is
2
d𝑠2 = 𝑅2 (d𝜃 2 + sin 𝜃d𝜙2 ). (A.8)
Here, we are talking about the two-dimensional surface (2-d) of a sphere of
radius 𝑅, and are not referring to the volume element of three-dimensional
(3-d) space in spherical polar coordinates.
It is not obvious, just from looking at this, that it describes a non-euclidean
space. To learn more, we must at least mention ‘spherical trigonometry’.
This is the study of triangles on the surface of a sphere, which navigators or
astronomers learn about, since they want to be able to calculate distances or
angles on the surface of the ocean, or on the sphere of the sky.
In Figure A.6 we see a triangle on a sphere of radius 𝑅, with sides of
length 𝑎, 𝑏 and 𝑐, and internal angles 𝐴, 𝐵 and 𝐶 (note that most treatments
of spherical trigonometry will quote expressions with a nominal sphere
radius of 𝑅 = 1).
On the surface of a sphere, these internal angles of the triangle add up
 A.2 Geometry 157

C
a
b
R B
c
A

Figure A.6 A spherical triangle.

to a value of 𝑠 = 𝐴 + 𝐵 + 𝐶 which is greater than 𝜋 radians, and if we draw

a circle on the surface, consisting of all of the points a distance 𝐿 from the
pole, we would find its circumference to be less than 2𝜋𝐿 (for example, the
equator is fairly obviously shorter than 2𝜋 times the distance from the pole
to the equator). That is, these basic geometrical rules are not the same as
they are in euclidean space.
Further, it may be clear that this excess 𝑠 − 𝜋 depends on 𝑅; and specif-
ically, as 𝑅 gets larger, this excess gets smaller: that is, the geometrical
features of the space depend on a parameter, 𝑅, which is related to a prop-
erty called the curvature of the space. Specifically, because of how it is
defined, the curvature of this space is 1∕𝑅, and as 𝑅 gets larger, and the
surface becomes locally more like flat euclidean space, the curvature tends
to zero. In this example, the ‘curvature’ is simply related to the radius of the
sphere, but the notion is more general.
A key thing, here, is that if we lived on this surface, we could draw a
triangle on the 2-d surface, make careful measurements of it, and discover
this excess, and thus measure the parameter 𝑅 which appears in the metric,
without knowing anything about the 3-d volume that the surface is embed-
ded within. That is, the curvature is a property of the (2-d) space which
is detectable by local measurements within the surface, and it is not, for
example, an artefact of our perception of the surface in 3-d space.
In the case of the surface of a sphere, the curvature is the same over the
entire space. If you think of the surface of an ellipsoid, in contrast, you can
see that the curvature can vary as you move around the space. The curvature
in these two cases is positive.
In contrast, a hyperbolic paraboloid, as in Figure A.7, is a surface with
constant negative curvature (which therefore doesn’t have such a straight-
forward relationship to anything like a radius). On this surface, the internal
angles of triangles add up to less than 𝜋, and the circumferences of circles
158 Appendix A An Overview of General Relativity

Figure A.7 A hyperbolic paraboloid.

are greater than 𝜋 times the diameter.

The examples above have illustrated curvature in 2-d surfaces. The same
general ideas carry over to surfaces of higher dimensions, even if they then
become harder to visualise: a 3-d space can be similarly curved or flat. The
volume of a sphere in three euclidean dimensions is, as you’ll recall, 4𝜋𝑅3 ∕3,
but a sphere in a curved space would have a volume greater or smaller than
this, depending on the sign of the curvature.
The case we are most interested in, of course, is that of spacetime, which
is a space (in the mathematical sense) with one time and three spatial
dimensions. Minkowski space, as we have seen it so far, is the special case
of a spacetime of constant zero curvature, but, just as with the ellipsoid, the
curvature can vary over the spacetime.

A.2.4 Geodesics
When we draw a triangle in euclidean space, we obviously connect the three
vertices with straight lines. The lines joining the vertices in the triangles
of Section A.2.3 are not straight lines in 3-d space, but they are the nearest
thing to a ‘straight line’ on the 2-d surface of the sphere (they are ‘great
circles’). This is telling us that we might have to generalise our notion of
‘straight line’ when talking about non-euclidean geometries. The result of
this generalisation is the geodesic.
There are a couple of ways we can think of geodesics. One is to see them
as the shortest distance between two points in the space. This works for
euclidean space, and for the great circles on the surface of a sphere, but if
you think back to Section 5.7.1, you might recall that I mentioned in passing
that the straight line followed by the non-travelling twin was in fact the
 A.2 Geometry 159

Figure A.8 Walking forwards on a sphere.

Figure A.9 A trivial geodesic in Minkowski space.

longest distance between two points in Minkowski space. We can think of

geodesics instead as the path of extremal length between two points (and see
also Exercise 5.8). A great circle between two points which goes ‘the long
way round’ is the maximal distance between the points on the surface of the
sphere, and is also a geodesic.
An alternative way of thinking about geodesics is to conceive of them as
the path you trace out in a space if you ‘walk forwards’. If, in flat euclidean
space you walk, cycle, or drive straight ahead then you will trace out a
straight line; if you do the same on the surface of a sphere, you will trace out
a great circle. This definition can be made more precise, but the key thing is
that it is a local definition – you don’t have to know the overall structure of
the space in order to work out, or pace out, the geodesic.
What are the geodesics in Minkowski space?
The instruction ‘walk forwards’ in Minkowski space means ‘move in a
straight line at a constant speed’, so the sequence of events associated with,
say, a flickering light carried by an observer, traces out a geodesic in that
space. We can also note – this will become important later – that for any
observer moving in this way, there is an inertial frame in which they are at
rest, and thus have a geodesic worldline lying along the 𝑡-axis (Figure A.9).
160 Appendix A An Overview of General Relativity

A.3 Gravity
We now have the set of ideas, and the language, in which we can describe a
relativistic theory of gravity – Einstein’s General Relativity.

A.3.1 How Matter Moves: Geodesics

Our first question is to ask: if you are moving under the influence of gravity,
what direction are you moving in?
For that, we can return to the EP, quoted above as Statement A.2. This is
similar to our statement about the universal equivalence of inertial frames,
back in Section 2.1, but it is more general in that, by expanding the set of
equivalent frames to ‘all freely falling non-rotating’ frames – which in the
context of GR we refer to as ‘inertial frames’ – it has made it possible to start
to talk about gravity as well.
We can make a still more restrictive statement and say

Equivalence Principle, version 3: Any physical law that can be ex-

pressed geometrically in SR has exactly the same form in a locally
inertial frame of a curved spacetime. (A.9)

This is why we have been so concerned with geometry all the way through
this text.
Imagine doing some physics experiments in a free-falling frame, within
a spacetime which is not necessarily the simple one we have come to un-
derstand as Minkowski space. This free-falling frame can be regarded as an
inertial frame, within which SR tells us the rules. From Chapter 7 we know
that 𝐅 = d𝐏∕d𝜏 in this inertial frame. What this final version of the EP is
telling us is that, even if the spacetime we are falling within is significantly
more complicated (i.e., curved) than Minkowski space, this curvature does
not change the physical laws; there is no ‘curvature coupling’. We could
imagine the law 𝐅 = d𝐏∕d𝜏 acquiring some extra terms due to the curvature
– there’s no mathematical reason why it shouldn’t – but it doesn’t. That’s a
profound physical statement.
This tells us that we already understand the physics of how things move
in the context of GR – particles move along the 𝑡-axis of their local inertial
frame, and according to the force laws of Chapter 7. But at this point we
cannot translate that understanding into the other frames that we care about,
such as ones that are accelerating, or which are otherwise not free-falling
(for example because they are stationary on the surface of the gravitating
Earth). Since we saw above that our normal motion within an SR inertial
 A.3 Gravity 161

frame is along a geodesic in Minkowski space, this EP tells us that this is

true in GR as well, so that our motion in the spacetime of GR is to follow
geodesics in that space. That is, the effect of the curvature of spacetime is to
define the geodesics along which free-falling particles move. This is usually
summed up by the slogan:
Spacetime tells matter how to move. . .

This discussion is about free-falling under gravity. How does this tell us
something new about our familiar gravitational experience of standing on
the ground? Whether we are standing on the ground or sitting in a chair, or
holding desperately onto a trapeze, what we are not doing is falling towards
the centre of the Earth. But if we were to compare ourselves to a ghostly
alternative self which is indeed falling, we would find the gap between us
increasing at a constant 9.81 m s−2 , and see ourselves being accelerated away
from that falling figure by the force transmitted through our feet or arms,
ultimately from the ground. That is, the ‘force of gravity’ that we feel is
merely the acceleration away from our otherwise free-fall trajectory.

The conceptual material in this section is discussed in many GR text-

books, without that discussion requiring much mathematical apparatus,
so that it can be generally intelligible. For example, see Schutz (2009, §7,1) and
Rindler (2006, ch. 1 and to some extent §8.4) for details. The various ‘versions’
of the EP quoted here have subtly but significantly different physical content
(Rindler is good on the distinctions), but I have treated them as mere rephras-
ings for our present purpose, and referred to them all as just ‘the Equivalence
Principle’.

A.3.2 How Space Curves: Einstein’s Equations

We can at this point draw an analogy with newtonian gravitation, in which
the gravitational force is the gradient of the gravitational potential, 𝜙(𝐫), and
that potential is given by Poisson’s equation with mass density as the source
term:
∇2 𝜙 = 4𝜋𝐺𝜌(𝐫).
The mass on the right-hand side constrains the second derivative of the
potential. We can solve this equation to find the potential 𝜙, and its gradient
𝐟 = ∇𝜙, which gives the gravitational force at a point. Einstein’s equations
for the gravitational field are broadly analogous to this.
You learned in Chapter 7 that (rest) mass is a frame-invariant quantity,
but you also learned in Section 7.4 that, when things are moving at high
162 Appendix A An Overview of General Relativity

speed, there is more energy-momentum in a system than is accounted for

by the obvious mass, and that while the amount of mass is not conserved in
a collision, the amount of energy-momentum is. It therefore makes sense
that it is the energy-momentum that appears on the right-hand side of our
relativistic gravity equation, but in what form?
This appears in the form of the ‘energy-momentum tensor’, where a ‘ten-
sor’ is a geometrical object with a family relationship to vectors. We can
think of a vector as thing which will provide an answer to a ‘one-vector
question’ – meaning a question which is phrased in terms of a single vec-
tor. For example a force vector is something which, when combined in
an inner product with a displacement vector, gives the work done in that
displacement. The strain tensor (for example) can answer a ‘two-vector
question’: ‘what is the component of force in this direction across a surface
oriented in that direction?’ For our present purposes, you don’t need to
know anything more about tensors than this: that they are as geometrical,
and as frame-independent, as the 4-vectors we have already learned about.
What, then, appears on the left-hand side of our analogue of Poisson’s
equation? Just as that equation has the mass density constrain the second
derivative of the potential, we can define a tensor, called the Einstein tensor,
which is a combination of second derivatives of the metric.
Through a sequence of physically motivated but nonetheless heuristic
arguments, Einstein asserted in his 1915 paper that this tensor 𝖦 was simply
proportional to the energy-momentum tensor:
𝖦 = −𝜅𝖳. (A.10)
This is Einstein’s equation for gravity, in which the constant 𝜅 plays the same
broad role as the gravitational constant 𝐺 in Newton’s theory. This is a
physical statement, rather than a mathematical deduction from anything,
and it is a statement which is corroborated by experiment.
We have omitted a lot of mathematics here. This equation, simple though
it might look here, represents a set of ten coupled second-order partial
differential equations, and is fiendishly difficult to solve. It has been solved
in a few cases, however, and we will look briefly at a couple of these in the
next section.
The key thing to note, however, is that, just as with Poisson’s equation,
Einstein’s equation represents a sequence of constraints. The distribution
of energy and matter is represented by the energy-momentum tensor 𝖳.
This constrains 𝖦, which in turn constrains the metric of the space. In less
mathematical terms, the presence of a lump of matter changes the metric
of the spacetime around it. Because the spacetime is curved, the path of
 A.4 Solutions of Einstein’s Equation 163

the ‘straight line’ geodesic traced out by a particle in free fall is different
from the geodesic worldline that the particle would trace out if the mass
weren’t there. We interpret that deflection of the worldline as the result of
the ‘gravity’ of the central mass, so that we see the straight line in spacetime
as the movement in space and time of a ball thrown through the air.
Newton explains gravity by saying that a mass creates a field around it,
which results in a force on test particles, which is proportional to their mass
and directed towards the centre. Einstein explains the same behaviour by
saying that the effect of a mass is to change (the metric of) the spacetime
around it. Nearby free-falling test particles ‘know’ nothing of the central
mass, but simply move straight ahead along geodesics in their local inertial
frame (Figure A.9). The shape of the spacetime, however, has the effect of
causing the particle to move along a path which we describe as an ‘orbit’.
An orbit is simply a straight line in a curved space.
It is astonishing that Newton’s and Einstein’s models, which start from
such different places, with such different motivations, nonetheless produce
predictions for the behaviour of free-falling particles which so very precisely
match each other, and match reality.
This relationship allows us to complete the other half of the famous
slogan, originally framed by John Wheeler,
Spacetime tells matter how to move;
matter tells spacetime how to curve.

A.4 Solutions of Einstein’s Equation

At this point, we have Einstein’s equations, which relate the metric for a
spacetime to the energy-momentum within it. The process of solving these
equations is, as I have mentioned, famously hard, and there is no general
solution, but only a variety of solutions to various special cases. These
special cases – for example, the case of an isolated mass, or a small mass –
are typically approximations or idealisations of the more general case.
As well as exact solutions, it is possible to solve the equations numerically,
and this is the locus of most of the currently mainstream astronomical
research in GR.
Confining ourselves to the analytic solutions, however, the various spe-
cial cases are important, and cover a large fraction of the astronomically
important cases. We will look at a few of these in the remaining sections of
this appendix.
164 Appendix A An Overview of General Relativity

I can go into very little mathematical detail here, but I hope that I can
give you at least a flavour of how we would use the mathematical structures
of SR, in the generalisation to gravity.

A.4.1 The Weak-Field Solution, and Newton

The first special case to look at is the case where the gravitating mass is
small, such as something around the mass of a planet. This creates what is
known as the ‘weak-field’ approximation to the solution. We are also going
to restrict ourselves to ‘slow’ movement, at speeds much less than that of
light. Our goal is to describe the spacetime around this mass, and in doing
so (as we have learned in the previous section) describe the effects of gravity
there.
Our first step is to choose a coordinate system. With that choice made, our
goal translates into finding a metric for our ‘weak-field’ spacetime, which is
consistent with what we have learned about relativity and gravitation so far,
and expressed using those coordinates.
In Section A.2.1, we saw that we can imagine a grid drawn on the ground,
and extended upwards with drones or balloons. We can use our weak-field
approximation immediately by deciding to draw this grid naively: we’ll
presume that the space is approximately flat, and draw a euclidean cartesian
grid on it, or a spherical polar one, or whatever one we prefer. We’ll write
the metric of this flat space as d𝜎2 , and not care, for the present, whether
2
this is d𝜎2 = d𝑥 2 + d𝑦 2 + d𝑧2 , or d𝜎2 = d𝑟2 + 𝑟2 (d𝜃2 + sin 𝜃d𝜙2 ), or even
just d𝜎2 = d𝑥 2 if we’re caring only about movement along the 𝑥-axis.
For our time coordinate, we’ll tell each one of our observers to station
themself at a particular spatial coordinate, and issue each of them with an
identical clock. This is a clock which will reliably tick out proper time (that
is, the amount of time which has passed for it; think back, again, to the
taffrail clock of Figure 1.4), but which is adjustable by the observer so that
the time shown on the clock face advances systematically faster or slower
than proper time. We will decide that the number shown on a particular
clock face is the coordinate time at that location. At this point, any event in
spacetime can be given one time and three space coordinates, and we can
write down a metric, in these coordinates, as
d𝑠2 = (1 + 𝐴)d𝑡 2 − d𝜎2 , (A.11)
where 𝐴 is some position- and time-dependent correction, yet to be deter-
mined, which we will shortly discover is small, in the sense of 𝐴 ≪ 1. As
you can see, at 𝐴 = 0 this reduces to the familiar Minkowski metric.
 A.4 Solutions of Einstein’s Equation 165

In writing this down, we have also assumed that we are dealing with
‘slow’ movement, in the sense that d𝜎2 ≪ d𝑡 2 . That means that we have not
included a factor (1 + 𝐵)d𝜎2 , where the ‘small × small’ term 𝐵d𝜎2 would
be promptly discarded.
We have decided that the clocks can have their rates adjusted, but to
what value? In Section 2.2.1 we saw how we can synchronise clocks in
flat Minkowski space. Imagine doing a similar thing, but in the context
of a spacetime with a non-flat metric. How do I, stationed at radius 𝑟me ,
instruct an observer at radius 𝑟 how to set their clock? Let’s decide that my
clock is the reference standard, and ask ‘what would we have to do to make
the coordinate time difference between events at radius 𝑟 the same as that
at 𝑟me ?’ We have to slow down the clock’s displayed time with respect to
its proper time, by a factor which depends on the gravitational potential, 𝜙,
at that point, as we saw in Eq. (A.5). Specifically, and arbitrarily taking
𝜙(𝐫me ) = 0, we can write
d𝜏2 = e2𝜙(𝑟) d𝑡 2 , (A.12)
to define the interval in coordinate time between two events, co-located at
radius 𝑟, which are separated by a proper time d𝜏. At this point we can
rewrite Eq. (A.11) to get
d𝑠2 = e2𝜙 d𝑡2 − d𝜎2 . (A.13)
The newtonian potential 𝜙 = −𝐺𝑀∕𝑟 has dimensions L2 T−2 , so that in phys-
ical units we would expect instead 𝜙∕𝑐2 here: putting in standard physical-
units values8 for 𝐺𝑀⊙ and 𝑟⊕ gives 𝜙∕𝑐2 ≈ 10−8 , so that 𝜙, in Eq. (A.13),
is indeed small, this leading term is very close to 1, and we can write this
alternatively as
d𝑠2 = (1 + 2𝜙)d𝑡2 − d𝜎2 , (A.14)
omitting terms in 𝜙2 . This is a weak-field metric: a metric for the spacetime
around a ‘small’ central mass. For small 𝜙, this is close to the metric of
Eq. (4.4) or Eq. (6.7), but it is not quite the same. Our argument here has
told us that the presence of the mass has changed the definition of distance
in the spacetime around it.
How do particles move in this spacetime? We learned in Section A.3.1
that particles move along geodesics, and in Section A.2.4 that those geodesics
are the lines of extremal length.

8 Here, as is conventional in astronomical contexts, I use subscripts ⊙ and ⊕, to refer to the

Sun and Earth respectively.
166 Appendix A An Overview of General Relativity

If you have done some advanced classical mechanics (Hamill 2013) you
will have learned that it is possible to obtain the equations of motion of an
object in (for example) a gravitational field by extremising a quantity known
as the ‘lagrangian’, formed from the difference between the kinetic and
gravitational potential energy, or 𝐿 = 𝑚𝑣 2 ∕2 − 𝑚𝜙. It is possible to show
that the demand that our geodesic extremises Eq. (A.14) produces the same
constraint as this lagrangian formalism. In other words, the solution to our
geometrical problem produces the same path 𝐱(𝑡) as we would otherwise
classically predict on the basis of motion under gravity.9
In other other words, we have rediscovered Newton’s theory of gravity,
as a ‘weak-field’ approximation to the behaviour of objects in free-falling
inertial frames, as constrained by the Equivalence Principle. This is the first
basic test of GR.
Notice that, in this argument, we have not used Einstein’s equation, or
any part of the argument of Section A.3.2 (though we briefly touched on the
physics of Section A.3.1); that is, to be precise, this isn’t really a weak-field
‘solution’. Instead, we brought gravity into the argument by using what is
essentially Newton’s gravity: the argument was built directly on the idea
of gravitational redshift (Section A.1.3), which in turn depends on the idea
of a mass losing energy 𝑚𝑔ℎ when it drops through a given distance. It is
because we have imported gravity by this route that this argument is limited
to low-speed (i.e., newtonian) motion within the field.

A.4.2 The Weak-field Solution, and Einstein

An alternative route to a theory of weak-field gravity is to solve Einstein’s
equations in the same framework as the previous section – namely an iso-
lated single mass – and the same low-mass approximation, but this time
starting with Einstein’s equation, rather than heuristically including New-
ton’s gravitation. Unlike the previous section, this route necessarily involves
the mathematics of differential geometry, so all we can do in this section is
to indicate how the problem is set up, and then jump to the solution.
As before, we imagine a spacetime containing only a single gravitating
mass. We assume that the mass is ‘small’ (which in this context means just
about everything less dense than a neutron star). In this case, the spacetime
must be approximately the same as that of empty space: it is approximately

9 The argument here is elegant, but unfortunately very compressed. It is ultimately derived
from Rindler (2006, §§9.1–9.4), where you can find the steps I have skipped, and a number
of subtleties which I have elided.
 A.4 Solutions of Einstein’s Equation 167

Minkowski space. Put mathematically, that means that the metric of this
spacetime will be approximately the metric of Eq. (4.4), plus some small
perturbation.
Skipping the details, we discover that our invariant interval in this space
– our metric – becomes

d𝑠2 = (1 + 2𝜙)d𝑡2 − (1 − 2𝜙)d𝜎2 , (A.15)

where d𝜎2 = d𝑥 2 + d𝑦 2 + d𝑧2 , and where the function 𝜙(𝑟) is a solution of

𝜅
∇2 𝜙 = − 𝜌(𝑟),
2
and 𝜅 is a constant, the value of which we can fix by demanding compatibility
with Newton’s gravity. You can see that this reduces to Eq. (4.4) when 𝜙 is
small. This is obviously also similar to Eq. (A.14) with the addition of the
factor in front of d𝜎2 . It is an approximation for the case of small masses,
but because of the way it has been obtained, this version is relativistic, and
doesn’t have the restriction to d𝜎∕d𝑡 ≪ 1.
We learned in Section A.3.1 that objects move along geodesics, so our next
question is: what are the geodesics in this spacetime? We can discover that
these geodesics are the result of the momentum 4-vector being constrained
by the potential 𝜙, with the result that we rediscover

𝐟 = −𝑚∇𝜙.

That is, as we would hope, the small mass limit of Einstein’s theory of gravity
is Newton’s law of gravity – the law of gravity that we are familiar with. This
demonstrates explicitly that Newton’s theory of gravity is the non-relativistic
limit of Einstein’s.

A.4.3 The Schwarzschild Solution

Now imagine the same configuration – a spacetime with a single point mass
in it – but relax the constraint that the central mass be ‘small’. We now
have a harder problem to solve, but one which was nevertheless solved
fairly rapidly. In 1916, only a few months after Einstein published his 1915
paper describing GR, Karl Schwarzschild published the first exact solution
to Einstein’s equation.
The so-called Schwarzschild solution is a metric in terms of the coordi-
nates (𝑡, 𝑟, 𝜃, 𝜒), where 𝜃 and 𝜒 are the usual polar coordinates. In these
168 Appendix A An Overview of General Relativity

coordinates, the metric is

−1
𝑅 𝑅
d𝑠2 = (1 − ) d𝑡2 − (1 − ) d𝑟2 − 𝑟2 dΩ2 . (A.16)
𝑟 𝑟
2
Here, dΩ2 = d𝜃2 + sin 𝜃d𝜒 2 is the usual differential surface element, and
the parameter
𝑅 = 2𝐺𝑀
is written in terms of the mass of the central object, 𝑀, and Newton’s gravita-
tional constant 𝐺; the parameter 𝑅 is known as the Schwarzschild radius (or
sometimes the ‘gravitational radius’). Notice that, unlike Eq. (A.15) and the
metrics before it, the spatial sector here is not flat. Also, and again unlike
the previous expressions, this is an exact solution to an idealised problem
and not a low-energy, low-mass or low-speed approximation.
The first thing to notice about this metric is that it is compatible with the
weak-field solution Eq. (A.15) when 𝑅 ≪ 𝑟; that is, when the mass is small,
or when an observer is far away from it. The above value of the constant 𝑅
is determined by demanding that the solution matches Newton’s law of
gravitation in this limit.
That also implies that the metric is ‘asymptotically flat’, in the sense
that, at large 𝑟, it reduces to the Minkowski metric. That means that, as
long as 𝑅∕𝑟 is reasonably small, it is natural to interpret the coordinates 𝑟
and 𝑡 as being approximately the same as the corresponding coordinates in
Minkowski space.
In physical units, the parameter 𝑅 is 𝑅 = 2𝐺𝑀∕𝑐2 . For the Sun, the so-
called ‘gravitational parameter’10 𝐺𝑀⊙ = 1.327 × 1020 m3 s2 , giving a value
for the Schwarzschild radius of the Sun as
𝑅⊙ = 2.95 km.
As you can see, the ratio 𝑅⊙ ∕𝑟 is therefore an extremely small number
everywhere in the solar system, so that the Schwarzschild solution is an
excellent approximation to the spacetime in our vicinity. It is this solution
that is used for all practical relativistic corrections to newtonian physics.

10 The combination 𝐺𝑀⊙ is what governs the motions of objects in the solar system, and thus
planetary observations can determine the value of this to great accuracy, to one part in 1010 .
The mass of the Sun in kilogrammes is obtained by dividing this number by 𝐺, but the
gravitational constant is known only to around one part in 105 , from terrestrial
measurements; so the mass of the Sun in kilogrammes has the same uncertainty. This
means that the mass of the Sun is more precisely known in metres than it is in kilogrammes.
See the IAU’s list of ‘current best estimates’ at https://iau-a3.gitlab.io/NSFA/NSFA_cbe.html.
For reference, the gravitational radius of the Earth is 𝑅⊕ = 8.86 × 10−3 m.
 A.4 Solutions of Einstein’s Equation 169

t

4


2 
3


1
r
r1 r2

Figure A.10 A pair of outward-going light-rays.

A.4.3.1 Gravitational Redshift

Consider the Minkowski diagram in Figure A.10, which represents a pair
of events ○1 and ○2 happening at a single location, at radius 𝑟 = 𝑟1 . Each of

these events produces a flash of light which travels outwards along a null
geodesic to the events ○
3 and ○ 4 , which are also co-located, at radius 𝑟 = 𝑟2 .

What are the path-lengths between these various events?

The two events ○ 1 and ○ 2 happen at the same location, as do ○ 3 and ○4 ,

so d𝑟 = 0, and the interval between them in the figure is the proper time
separating them (that is, the time showing on a clock which is present at
both events by virtue of staying still at the radius 𝑟1 ), and this is obtainable
directly from the metric Eq. (A.16):
2 2
𝜏12 = (1 − 𝑅∕𝑟1 )Δ𝑡12
2 2
(A.17)
𝜏34 = (1 − 𝑅∕𝑟2 )Δ𝑡34 .

Since the metric is not changing in time, the pair of events ○ 2 and ○ 4 must

stand in the same relation to each other as the pair of events ○ 1 and ○3 ,

and thus the time coordinate at ○ 4 must bear the same relationship to that

at ○2 as the coordinate at ○ 3 does to ○

1 ; that is, we must have Δ𝑡34 = Δ𝑡12 .

But this means that 𝜏12 is slightly less than 𝜏34 . That is, more time elapses
between events ○ 3 and ○4 , on the watch of an observer at 𝑟 = 𝑟2 , than elapses

between ○ 1 and ○2 on the watch of an observer at 𝑟 = 𝑟1 ; if the time interval

𝜏12 is the period of an EM wave, then this implies that the wave will change
its frequency between 𝑟1 and 𝑟2 . Rewriting Eq. (A.17), we find
−1∕2
𝜏12 1 − 𝑅∕𝑟1
=( ) . (A.18)
𝜏34 1 − 𝑅∕𝑟2

This is gravitational redshift or, if you prefer, gravitational time dilation,

and is the quantitative equivalent of the argument of Section A.1.3. It must
be taken into account when making precise timing observations of events
170 Appendix A An Overview of General Relativity

both on Earth and in space. When national standards bodies compare their
atomic clocks, in order to establish the consensus International Atomic
Time (TAI), they must take into account the altitude, on Earth, of their
observatories. And when systems such as GPS or Galileo broadcast their
time signal for navigation purposes, they must include both a correction for
the altitude of the spacecraft (orbital radius of 26 600 km, for GPS), which
causes spacecraft time to advance with respect to the Earth at 45.7 μs per
day, and a correction from SR due to the orbital velocity, which causes the
spacecraft to lose 7.2 μs per day, for a total GPS advance of 38.5 μs per day.

A.4.3.2 Geodesics in Schwarzschild Spacetime: Orbits

The Schwarzschild solution includes some geodesics which spiral around
the 𝑡-axis in Figure A.12 – that is, they are ‘orbits’. It is possible to calculate
what these orbits are (another calculation we must skip here), and we can
discover that, in terms of their path in the (𝑟, 𝜃, 𝜒) part of spacetime, they
are nearly ellipses. They deviate from elliptical orbits to the extent that
they precess, with the principal axis of the ellipse slowly moving around the
central mass, at a speed of Δ𝜒 per orbit, where
6𝜋𝑀
Δ𝜒 = , (A.19)
𝑎(1 − 𝑒2 )
where 𝑎 is the semi-major axis and 𝑒 the eccentricity, as usual.
The nearest planet to the Sun is Mercury. The actual orbit of Mer-
cury is not quite a kelperian ellipse, but instead is observed to precess at
574′′ /century. This is almost all explicable by newtonian perturbations aris-
ing from the presence of the other planets in the solar system, and over the
course of the nineteenth century most of this anomalous precession had
been accounted for in detail. The process of identifying the corresponding
perturbations had also been carried out for Uranus, with the anomalies in
that case resolved by predicting the existence of, and then finding, Nep-
tune. At one point it was suspected that there was an additional planet near
Mercury which could explain the anomaly, but this was ruled out on other
grounds, and a residual precession of 43′′ /century remained stubbornly
unexplained.
Mercury has semi-major axis 𝑎 = 5.79 × 1010 m = 193 s, and eccentricity
𝑒 = 0.2056. Taking the Sun’s mass to be 𝑀⊙ = 1.48 km = 4.93 × 10−5 s, we
find Δ𝜒 = 5.03 × 10−7 rad∕orbit. The orbital period of Mercury is 88 days,
so that converting Δ𝜒 for Mercury to units of arcseconds per century, we
find
Δ𝜒 = 43′′. 0/century.
 A.4 Solutions of Einstein’s Equation 171

B
α
d A

Figure A.11 Deflection of light near a mass.

Einstein published this calculation in 1916 (Einstein 1916). It is the first of

the classic tests of GR, which also include the phenomena we will look at in
the next section.

A.4.3.3 Geodesics in Schwarzschild Spacetime: Photons

We have looked at the orbits of massive particles in the Schwarzschild space-
time, but massless photons also have their paths changed by the change
in the spacetime (as we might expect by recalling the thought experiment
of Section A.1.2). The calculations described in this section can be done
exactly in the Schwarzschild spacetime or approximately in the weak-field
limit of Section A.4.1.
In particular, by the same calculation process that led to Eq. (A.19), a
photon which passes nearby a gravitating mass, 𝑀, with impact parameter 𝑑
(see Figure A.11) will be deflected by an angle 𝛼, where
4𝐺𝑀
𝛼= . (A.20)
𝑑
This deflection is observable: a star which is known to be at position 𝐴 in
Figure A.11 will be observed to appear in a telescope at position 𝐵. Such
an observation, however, is far from easy. Taking the mass and impact
parameter in Eq. (A.20) to be the mass and radius of the Sun, we find a
deflection angle, for a photon which grazes the limb of the Sun, of 1′′. 75
(if you calculate this for yourself, remember that we are using units where
𝑐 = 1). One way of observing this is to measure the position of a star which
appears near the Sun during a solar eclipse (allowing us to see the star, with
the Sun’s light cut off), and compare it to the same star’s position when the
Sun is elsewhere in the sky.
Such an experiment was performed in 1919, only a few years after the
1915 publication of GR (and incidentally after Einstein’s early paper (1911),
where he first predicted that there would be a deflection of light due to gravity,
deriving a (mistaken) figure of half of that in Eq. (A.20)). Arthur Eddington,
working with Frank Dyson (respectively directors of the Cambridge and
Greenwich observatories), organised expeditions to Principe, off Western
Africa, and to northern Brazil, to observe the 1919 eclipse and measure
172 Appendix A An Overview of General Relativity

r = 2GM

Figure A.12 Light cones in a Schwarzschild spacetime. We also see a world-

line of a particle escaping outwards.

the deflection. They found a deflection which matched the predicted one,
which was widely taken as a robust confirmation of the validity of Einstein’s
theory. There is a little more to say about this in Section B.3.

A.4.3.4 Black Holes

A final thing to notice about this metric is that, at the radius 𝑟 = 2𝐺𝑀,
the coefficient of d𝑟2 is divergent – there appears to be a singularity here.
Initially, it was thought that something extraordinary happened to spacetime
at that point, but further study revealed that it is merely the coordinate
system that goes wrong there, in much the same way that the coordinates on
the Mercator map ‘go wrong’ at the poles. In fact the spacetime, as observed
locally, is perfectly regular at this radius, and an observer falling past this
point would not notice anything amiss (that the spacetime is well behaved at
𝑟 = 𝑅 is far from obvious, and it took several years, after the Schwarzschild
solution was published, before this good behaviour was confirmed).
Is there anything special about this radius? In Figure A.12 we can see a
representation of the light cones, or null cones, of an observer at various radii
(see Section 4.7). These represent the range of possible future worldlines for
the observer, and we can see that the effect of the Schwarzschild spacetime
is to tilt these light cones by an increasing amount as we move in towards
the mass. For 𝑟 > 2𝐺𝑀, you can see it is possible to have a worldline which
heads towards larger 𝑟, if it moves quickly enough. At 𝑟 = 2𝐺𝑀, however,
this becomes impossible: even a particle travelling at the speed of light,
and thus along the diagonal of the light cone, can only remain at 𝑟 = 2𝐺𝑀;
and further inwards of this, at 𝑟 < 2𝐺𝑀, the only possible worldlines are
inward-pointing.
The Schwarzschild radius 𝑅 = 2𝐺𝑀 is the radius of the event horizon.
This close in to the centre, even a photon cannot travel quickly enough to
get onto an outward-directed trajectory: even light cannot escape at this
point. The Schwarzschild solution – or more precisely the spacetime when
 A.4 Solutions of Einstein’s Equation 173

the central mass is physically compact enough that it is entirely contained

within the radius 2𝐺𝑀 – is the spacetime of a black hole.

A.4.4 Gravitational Waves

Up to this point, we have discussed metrics which are static, meaning
(slightly loosely) that they are constant in time, and which arise from the
presence of a central mass. The next important solution to look at is not
static, and has no source mass: it is an oscillatory solution, that of gravita-
tional waves. Yet again, we cannot usefully go into any mathematical details,
and can only describe the main results.
If we consider Einstein’s equation, Eq. (A.10), in the absence of a source
term (thus 𝖳 = 0), and once again in the weak-field limit of Eq. (A.15), then
it is possible to cast the equation in the form of a wave equation, ∇2 𝑋 = 0,
where the 𝑋 is a tensor rather than a scalar, and the ∇2 is an appropriate
laplacian operator (this is where the mathematical complications come in11 ).
This equation admits of solutions which describe curvature in spacetime in
the absence of mass, where the curvature of one part of spacetime causes cur-
vature in a neighbouring part, in a way broadly similar to the way an electric
or magnetic field will induce a magnetic or electric field in a neighbouring
part of space.
Thus these gravitational waves propagate through spacetime at the speed
of light, and carry energy. In particular, they carry energy away from a source
consisting of one or more accelerating masses; the masses in question are
most typically a pair of compact masses, such as neutron stars or black holes,
orbiting each other. It is possible to predict the amount of energy being
carried away from such a system by gravitational waves, as a function of the
masses and orbital parameters of the objects.
The binary pulsar PSR 1913+16 (also known as the ‘Hulse–Taylor binary’)
is a pair of neutron stars orbiting each other. The arrival times of the signal
from the pulsar have been measured to high accuracy over the decades
since the binary was discovered, and the orbital period has been found to
be increasing – the binary is slowing down. The behaviour of the binary
pulsar, and in particular the rate of spindown, is consistent with GR to high
accuracy, and this, in the absence of any other mechanism for energy loss,
is taken as an early indirect test of the existence of gravitational waves.
What do gravitational waves look like?

11 To be honest, this is where they come in and kick over the furniture and scare the cat.
174 Appendix A An Overview of General Relativity

y
x

x
y
y
x

x
y

Figure A.13 A sketch of a gravitational wave.

M1

M2
M4 M3

Figure A.14 An interferometer.

In Figure A.13 we can see a sketch of the analogue of a gravitational wave

in a two-dimensional space, in two different phases of its oscillation. If you
imagine measuring the diameters of the top ‘circle’ in the 𝑥- and 𝑦-directions
by pacing from one end to the other, you will see that they will be different
from the 𝑥- and 𝑦-diameters of the bottom ‘circle’. If we placed a set of test
particles around the edge of this circle, then the distances between them
would change over the period of a gravitational wave, even though they
did not move, and specifically did not accelerate. If the test particles were
mirrors, at either end of the cross drawn at the bottom of the figure, and
we shone a laser between them, we would expect different times of flight
between the mirrors at different phases of the passing gravitational wave.
We have just described an interferometer, of the type which is used in the
LIGO and Virgo collaborations to attempt to directly detect gravitational
waves. In the interferometer sketched in Figure A.14, we see a laser shining
into a beam-splitter, which sends the beam down both the arms of the
apparatus, where it is reflected back and forth between the four mirrors
M1 to M4, before being recombined at the detector, at the bottom. All four
mirrors are carefully suspended, so that they do not vibrate from any local,
 A.4 Solutions of Einstein’s Equation 175

terrestrial, cause. If a gravitational wave passes, then one or both of the

M1–M2 or M3–M4 distances will change, in the way we have seen above,
without any of the mirrors accelerating. Thus the path length of one or other
arm will change, causing a detectable change in the interference pattern at
the detector. The two LIGO detectors, in Louisiana and Washington State
in the USA, each have arms 4 km long and send the light back and forth
hundreds of times. It is a fine irony that this apparatus is broadly the same
design as the Michelson–Morley experiment which made its appearance at
the beginning of the history of SR.
The measurement is a little tricky, however, since the anticipated change
in the 4 km arm length, for a plausibly detectable gravitational wave source,
is of the order of 10−18 m. After decades of effort, on improving lasers, mir-
rors, mirror suspensions, and data analysis techniques, this measurement
was finally made, and jointly announced by the LIGO and Virgo collabora-
tions in 2015. In doing so, the collaborations not only directly confirmed the
existence of gravitational waves, and thus further corroborated General Rel-
ativity, but also opened up a new non-electromagnetic messenger (alongside
neutrinos and cosmic rays) for the observation of the universe.

A.4.5 Cosmology
In the last few sections, we have looked at the spacetime around a small mass,
around a large mass, and waves in a spacetime produced by an accelerating
mass. The next step: the universe!
To find a metric for the universe as a whole, we start with the observa-
tion that, on the largest scales, the universe appears to be homogeneous
and isotropic – that is, it is the same at all spatial points, and in all spatial
directions. This is known as the copernican principle. Apart from encour-
aging very proper modesty about our place in the universe, this permits a
considerable mathematical simplification. It is possible to write down, from
purely mathematical considerations, the most general metric which has
these two properties, namely the Friedman–Lemaître–Robertson–Walker
metric.12 In the most convenient choice of coordinates, it looks like
d𝑟2
d𝑠2 = d𝑡 2 − 𝑎2 (𝑡) [ + 𝑟2 dΩ2 ] , (A.21)
1 − 𝜅𝑟2

where dΩ2 is the angular area element, as before, 𝑟 is a radial coordinate,

12 Usually referred to as ‘FLRW’, or by whatever subset and ordering of the four independent
discoverers’ names is demanded by your national prejudices.
176 Appendix A An Overview of General Relativity

scaled so that the parameter 𝜅 has one of the values {−1, 0, +1}, and 𝑎(𝑡) is
a dimensionless scaling factor which depends only on the time coordinate.
The effect of the scaling factor 𝑎(𝑡) is, in effect, to change the size of the
universe. It turns out that, when we apply Einstein’s equation, Eq. (A.10),
to this metric and a reasonable energy-momentum tensor, we deduce 𝑎̈ ≠ 0,
and thus a universe which is changing in size.
That Einstein’s equations admit of a non-stationary solution – an expand-
ing universe – was initially thought to be a physically implausible defect
in the theory. To amend this, Einstein observed that the argument leading
up to Eq. (A.10) can be slightly extended to allow an additional term on the
left-hand side, Λ𝗀, including a scaling factor now known as the cosmolog-
ical constant; this modified version of Einstein’s equation does admit of a
stationary solution, albeit an unstable one. The discovery of the Hubble–
Lemaître expansion of the universe meant that the need for this fell away,
but the last couple of decades have seen the term reintroduced, responding
to otherwise-inexplicable observational evidence, and corresponding to a
further exotic source of energy-momentum in the universe, not identifiable
as either matter or radiation.
Moving from maths to physics, the next step is to consider various dispo-
sitions of energy-momentum, and then use Eq. (A.10) to constrain the form
of 𝑎(𝑡), and thus its first and second time derivatives 𝑎̇ and 𝑎,̈ with different
solutions depending on the value of 𝜅. At that point, we cross over into the
study of cosmology.
One extreme case of this metric is where 𝑎(𝑡) = 0. The solutions to the
FLRW metric generally include an in-principle initial state where this is so,
and they have a singularity at this point. This is the big bang. At this point,
the curvature of the metric becomes infinite, which implies an infinite energy
density, which is enough to disrupt some of the mathematical foundations of
GR. At this point, quantum effects become important, obliging us to attempt
to develop a theory of quantum gravity. That is the next challenge.
 Appendix B
Relativity’s Contact with Experimental Fact

For, if I had believed that we could ignore these eight minutes, I would have
patched up my hypothesis accordingly. But since it was not permissible to
ignore them, those eight minutes point the road to a complete reformation
of astronomy: they have become the building material for a large part of this
work. . . .
Johannes Kepler, on the discrepancy of 8′ of arc between the predicted and
actual positions of Mars, in Astronomia Nova, II, Cap.19, quoted in
Arthur Koestler, The Sleepwalkers (1959)

Special Relativity is a physical theory – a theory of how our universe works.

It is a hugely successful theory, which has been corroborated both implicitly
and explicitly by numerous high-precision measurements in the years since
the theory was first proposed.
That said, SR’s contact with experiment is a little odd, compared with
other parts of the physical sciences. It is foundational to physics, and its
validity essentially unquestioned, to the extent that there is paradoxically
rather little to test: any deviation between SR and experiment would require
very large areas of physics to be rethought. This means that what current
efforts there are, are not asking routine questions of ‘is this theory correct?’.
Instead, they are looking for any discrepancies at all since, however small or
subtle such differences might turn out to be, they would be ground-shifting.
This means that when we describe ‘experimental support for SR’, we are
almost always doing so with pedagogical or historical intent, rather than
as part of an open claim about validity. That pedagogical intent can be to
deepen our understanding of the physics, but in this appendix I want instead
to discuss the way in which the link with experiment throws up questions
of what it means to test a theory. For me, the interesting thing about the
Michelson–Morley test (mentioned in Section 2.1.2), and to a greater extent
about the Dyson–Eddington eclipse test (Section A.4.3.3 and below), is that

177
178 Appendix B Relativity’s Contact with Experimental Fact

they show the extent to which science is a social as well as a logical process:
to the extent there is a ‘scientific method’ which demarcates science from
non-science, it’s a social one of asking certain types of questions which are
answerable in a particular way, with these questions being asked within a
community which aims and is able to arrive at provisional consensuses by a
more-or-less systematic route.
‘What is science?’ is an interesting question, but it is not a scientific
one. However, this question, along with ‘what does science know?’, ‘how
does science justify its statements?’, and ‘how do scientists choose between
theories?’ are all importantly different questions, however similar they may
sound at first. The last of these questions is the most interesting one, because
it is perhaps the one least often asked. I don’t plan to offer any confident
opinions about these questions – they are too large for that – but I hope to
illustrate that they are distinct questions, and to persuade you that the last
one is one that should be asked more often.
A very conventional starting point, when talking about science, is to ob-
serve (this is Karl Popper’s formulation) that science proceeds by ‘conjecture
and refutation’, meaning that the ‘scientific method’, such as it is, consists
of theories being proposed, and then those theories being either refuted by
experiment, or not refuted yet. Though this badly oversimplifies the practice
of science – there are epistemological, methodological and even sociologi-
cal complications that have to be taken into account – it does capture the
essential asymmetry of scientific logic: experiments can only prove theories
wrong, but never really prove them right, except from exhaustion. So rela-
tivity’s contact with experiment consists of searching for an experimental
failure.

B.1 Special Relativity

As we have seen earlier in this text, relativity rests on only two physical
statements, which we can rephrase here as:
1. all inertial frames are equivalent, and there is no preferred frame;
2. – there is an upper limit to speeds in nature;
– or, the speed of light is the same for all observers;
– or, the elapsed time measured in a moving frame is 𝑡 ′ = 𝑡∕𝛾;
– or, the interval 𝑠2 is invariant between inertial frames
(see the discussion in Section 5.8.1). Tests of SR consist of attempts to devise
an experiment which lets us observe one or other of these statements, or
 B.1 Special Relativity 179

their necessary consequences, to be false. We only have to find one failure,

for physics to be turned on its head.
The first link is electromagnetism (EM). As I described in Section 5.8.2,
Maxwell’s equations are not invariant under the galilean transformation;
the Lorentz transformation was developed, before Einstein’s relativity, in
order to establish the transformation that did leave Maxwell’s equations
invariant; and it is possible to imagine an alternative history in which SR
was developed by extrapolating from this invariance of electromagnetism to
the rest of physics. From a point of view which embraces SR as foundational,
however, the experimental success of Maxwell’s equations, and the fact that
they are Lorentz invariant, is not merely a trigger for SR, but a powerful
corroboration of it. Einstein demands that any plausible physical theory
must be Lorentz invariant, and – look! – we already have a successful theory
which illustrates that.
Indeed, it is possible to start with the non-relativistic Lorentz force law
𝐟 = 𝑞(𝐄 + 𝐯 × 𝐁), and Coulomb’s law for a stationary charge, ∇ ⋅ 𝐄 = 4𝜋𝜌,
add the demand of Lorentz invariance, and deduce Maxwell’s equations
as a consequence.1 In a way, it’s Maxwell’s equations that are the first
experimental test of SR.
There is a difference of interpretation, of course. Starting from SR, it
is (fairly) natural to develop a view where space and time are facets of a
single structure, rather than rigidly demarcating time from space, in the
way that (what is now called) the newtonian picture does. This is quite
a change of perspective, and a disorienting one, and it was natural to be
sceptical. Lorentz and his colleagues did not think of Eq. (5.16) in terms of
spacetime rotations, of course, but it meant they were obliged to come up
with mechanisms which would contract measuring rods in the direction
of motion, and slow down clocks. FitzGerald’s and Lorentz’s suggestion,2
that the aether somehow compressed measuring rods in the direction of
travel, can be regarded as an adequate temporary hypothesis, but one which
started running out of steam when experiments failed to demonstrate the
aether behaving as consequently required.
The 1887 Michelson–Morley experiment used an optical arrangement
very much like the LIGO and Virgo experiments of Figure A.14, but with
arms a few metres long. Michelson and Morley expected to demonstrate

1 This is illustrated in Rindler (2006, §7.3). It is illuminating, but the maths is a little beyond
the level of this text.
2 See FitzGerald (1889) and Lorentz (1895); Lorentz reports that he had first suggested this
idea in 1892, and also notes that Oliver Lodge referred to the idea in 1893.
180 Appendix B Relativity’s Contact with Experimental Fact

that the effective light path changed length when the arms were moving or
not moving with respect to the aether, resulting in observable interference
effects. If they had observed this, they would have refuted the statement
that all inertial frames are equivalent – they would have shown that the
aether frame had a special status, and that Maxwell’s equations only work
properly in that frame. Famously, they did not see this, and nor did the
various people who attempted to re-do the experiment in one or other vari-
ant, encouraged at various times by both Lorentz and Einstein. Attempts
were made with instruments of different materials, at altitude, housed in
buildings with thick or thin walls, at different seasons of the year, and using
sunlight or starlight instead of artificial light, in case any of these made a
difference. In no case, as far as I am aware, was there any principled reason
why such a difference would be expected; for all that they were experiment-
ing with light, these observers were exploring in darkness. As late as 1933,
Dayton Miller claimed to have detected an absolute motion with respect
to the aether, which was smaller than expected but still inconsistent with
zero, requiring (he asserted) adjustments to either the aether theory or the
FitzGerald–Lorentz contraction, but still in contradiction to the precisely-
null result required by relativity theory (Miller 1933). Miller’s one-time
research assistant reanalysed the data in 1955, however, and concluded that,
despite all efforts, the measurements were still consistent with a null result
(Shankland et al. 1955), and thus that there was still no refutation.3
I’ve described this history at a little length, partly because it’s a famous and
important experiment, but also because it is illuminating about the way that
the status of a measurement changes within the scientific community. As
usefully discussed in Collins & Pinch (2012, ch. 2), Michelson and Morley’s
null result was initially understood as an anomaly for aether theory; later,
it was re-perceived as consistent with relativity, and a confirmation of it;
finally, Miller’s claimed non-null result changed, and from being seen as an
anomaly in relativity theory, it itself became an experimental anomaly, to
be explained away by Shankland et al. as a systematic observational error.

3 Both Miller’s and Shankland et al.’s papers provide fascinating and detailed accounts of the
history of this measurement. Reading these papers one is repeatedly reminded of the
technical challenges of highly sensitive equipment, challenges which were and are also
faced by the gravitational wave communities, who also experienced frustrations and
disruptions from distant traffic, microseismic events, weather, and gruelling observational
campaigns. Rather desperately in retrospect, Miller ends his paper with a list of other
observations which seem to hint at an absolute Earth motion, the last one of which,
ironically, is Karl G. Jansky’s report of ‘a peculiar hissing sound in short wave radio
reception, which comes from a definite cosmic direction’.
 B.1 Special Relativity 181

The Michelson–Morley experiment is at heart an EM experiment and,

although this is not the way it is usually framed, it shows that EM really is
fully consistent with the Lorentz transformation.
After electromagnetism, the next and possibly most powerful implicit test
of relativity is its tight integration with other fundamental areas of physics.
If Special Relativity were not correct, then General Relativity, which builds
precisely on top of it, would be incorrect, too. However, SR is also the
foundation on which relativistic quantum mechanics (RQM) is built, which
is in turn the foundation, alongside classical field theory, for the gauge
field theories of the Standard Model of particle physics. RQM is quantum
mechanics rebooted on the assumption that it is taking place in Minkowski
space. If this assumption were not correct, then none of contemporary
particle physics could be observed, particle accelerators would not work as
they do, and – put most crudely – the engineers who built CERN would
have different mechanical and electrical problems to solve.
There is a further well-known implicit test in the satellite navigation
systems such as GPS, GLONASS, and Galileo. These work by each of the net-
work of satellites transmitting its precise position and time at regular inter-
vals; receivers pick up these signals from four or more satellites, and can de-
duce their current position by comparing the multiple reports (Ashby 2003).
As we saw near Eq. (A.18), the clocks on board such a satellite advance,
relative to clocks on the ground, at about 38 μs per day. That immediately
translates to a position error, if uncorrected, of 38 light-microseconds (in
units where 𝑐 = 1), or 11 km per day in physical units. That the receiver in
your phone is more accurate than this is not because your phone is doing
any relativity calculations, but because the satellites are designed to report
their current time with this correction factored in. In addition, the satellite
clocks are running in an inertial frame, but transmitting their reports to, and
receiving calibration updates from, a rotating Earth. For this to work, the
system as a whole has to take account of the Sagnac effect, which changes
the phase of a light signal which propagates in a rotating reference frame.4
These satellites, like the design of CERN, are not designed as ‘tests’ of SR; but
as a fact of engineering none of these systems would work if the spacecraft
clocks, and their relationships with clocks on the Earth’s surface, did not
behave as SR and GR predict.

4 The Sagnac effect is also used for practical navigation in a class of high-precision gyroscopes
called ‘ring-laser gyroscopes’. The Sagnac experiment was devised by Georges Sagnac in
order to demonstrate the presence of the aether; it appeared to, at the time, but turns out to
corroborate relativity on closer examination. The Sagnac effect is discussed in Ashby (2003),
and there is a history in Pascoli (2017).
182 Appendix B Relativity’s Contact with Experimental Fact

In the presence of these powerful implicit tests of SR, I don’t feel there’s
really a great deal extra that’s useful to say about the various explicit ones.
There is an extensive annotated bibliography of experimental tests by Roberts
& Schleif (2007) and a summary in Will (2014, §2.1.2): it seems sufficient
to say that SR has been tested with great ingenuity, at length, and to great
precision, and it has not failed any test so far. At the end of Roberts &
Schleif’s bibliography is a section on papers which claim but do not sustain
results inconsistent with SR, which is worth reading as a series of (unwitting)
demonstrations of how hard experimental work is, in this area.
Although the underlying theory is not in doubt, there nonetheless are
significant current efforts to make measurements which look for deviations
from SR’s predictions, using observations which stretch from sub-nuclear
physics to astrophysics. Various exotic physical theories – and there is no
current shortage of these, mostly to do with quantum gravity, theories of
the quantum mechanical structure of spacetime at the microscopic level,
or exotic cosmologies, but also including theories of particle physics at very
high energies – include predictions of some violation of Lorentz invariance.
That is, a proposed theory includes some preferred frame (in the sense
that the aether was a ‘preferred frame’, of absolute rest) or, what is much
the same thing, that equations of motion or measured physical properties
would have different values in different inertial frames. The goal here is not
to ‘prove Einstein wrong’, but instead to discover some way in which the
axioms of SR are inadequate or incomplete, and so to uncover new physics
beyond the Standard Model. Finding a preferred frame would contradict
the first postulate, above, turning it from a fundamental organisational
principle of our universe into a special case or low-energy approximation.
No matter how special the special case, or how precise the approximation,
if any deviation from perfect Lorentz invariance were found, that would
be of major significance, and provoke sudden huge interest in any theory
which predicted it; if no deviation is found, then that usefully rules out those
theories which propose it or depend upon it; for a review, see Mattingly
(2005). All of these experiments are technically challenging. None so far
has found anything.
 B.2 General Relativity: Classical and Post-classical Tests 183

B.2 General Relativity: Classical and Post-classical

Tests
Unlike Special Relativity, General Relativity is not so embedded into the
fabric of physics, that doubting it verges on the irrational. Although the
two subjects are fundamentally linked in their concerns and approach, and
although GR builds directly on top of SR – remember that GR is the attempt
to repeat the success of SR outside of the special case of no gravity, and that
Statement A.9 has SR acting as the ‘bridge’ into GR – SR sits at the centre of
physics while GR stands at least a little way closer to the border. There is
very little about SR that is in any sense optional: the first axiom is a deeply
fundamental principle which it seems reckless to reject, and the second is a
binary choice between there being an upper speed limit in nature, or not. If
SR isn’t true, very little makes sense.
In contrast, there is significantly more that is contingent in GR. The ver-
sion of the EP in Statement A.9 is a physical statement, which appears to
be extensively confirmed by experience, but which might be wrong – per-
haps there really are curvature-coupling terms in some exotic circumstances.
Einstein added a ‘cosmological constant’ to the field equations, as we saw
in Section A.4.5, so there is already some choice in the matter of the field
equations to be solved, and it is possible in principle that our Eq. (A.10) is
incomplete. The argument leading up to Eq. (A.10) stresses the extent to
which Einstein’s equation is a (well-motivated) guess: experiments corrobo-
rate the physical validity of this guess, but while a failure to confirm some
detailed prediction of GR would be an event of extraordinary importance,
it would not be the fundamental earthquake represented by any failure to
confirm SR.
The three so-called ‘classical tests’ of GR are:
r the advance in the perihelion of Mercury, beyond that predicted by new-
tonian mechanics, as we saw in Section A.4.3.2;
r the deflection of light by the Sun, as we saw in Section A.4.3.3, and as
observed by Eddington and Dyson in 1919; and
r the red-shifting of light, which was finally directly confirmed in the
Pound–Rebka experiments of 1959–64, as we described in Section A.1.3.

These tests’ ‘classic’ status derives in part from Einstein’s description of them
in the final section of his 1916 paper, but also because the successful tests
are famous, and between them seem generally regarded as demonstrating
GR to be correct.
It is remarkable how quickly and completely the scientific community
184 Appendix B Relativity’s Contact with Experimental Fact

arrived at the consensus that Eddington and Dyson’s eclipse measurements

had confirmed the correctness of GR. Though some scientists retained a
principled scepticism, there is little sign of serious scientific disagreement
after the compatible results from the (much less famous) 1922 eclipse; and
the similar Yerkes Observatory observations of the 1952 eclipse, and the
Texas observations of the 1973 one, were reported as routine confirmations
of a well-established result. This is all the more remarkable because the
eclipse measurement is very challenging, and remained at the limit of what
was measurable whenever it was repeated – the 1952 and 1973 observations
reported similar errors (around 10%) to the 1919 ones. The light-bending
measurement did not become routine until the emergence of radio and
pulsar astronomy, which quickly achieved sub-arcsecond position resolution:
since the Sun isn’t dazzlingly bright in radio, there is no need to wait for an
eclipse, so that measurement of the solar deflection becomes straightforward.
The deflection of electromagnetic radiation by the Sun, and indeed the
general effects of the line of sight passing through the Sun’s Schwarzschild
spacetime, are now merely standard observational corrections for radio
astronomers.
Confirmation of gravitational redshift took longer. In comparison with
the fame of the 1919 eclipse measurements, it is less often mentioned that
gravitational redshift should produce a measurable shift in the absorption
lines in the solar spectrum, and that attempts to observe these, in the 1910s,
produced contradictory results. The theoretical prediction, on the part of
various authors including Eddington, was unclear (surprisingly so, given
how straightforward the argument now seems in retrospect, in Section A.1.3),
but did agree that there should be a redshift (and this was the consensus after
1921); and an open question at the time was whether quantum phenomena
(i.e., photons) would experience relativistic effects in the same way that
classical electromagnetic fields do (again, this seems fairly obvious from
our point of view in the twenty-first century). By 1919 multiple observers
had attempted to detect the redshift in the Fraunhofer lines, without clear
success, and until the mid-1920s it was possible to enlist these observations
on both sides of the argument. Although the consensus from then appears
to have moved towards corroboration (more so after the 1919 observations),
it was only in the 1960s that the redshift was measured convincingly. See
Earman & Glymour (1980a) and Hentschel (1996) for the history, and Will
(2014, §2.1.3) for discussion and further references.
It was also in the 1960s that it became technically feasible for Pound,
Rebka and co-workers to directly measure the gravitational redshift in a
terrestrial laboratory, as mentioned above. Other GR tests, from a further
 B.2 General Relativity: Classical and Post-classical Tests 185

detection of redshift by Gravity Probe A in 1976 to the detection of gravita-

tional waves in 2015, are demonstrations of huge experimental skill, at the
edge of what is measurable, but in no case has the outcome been seriously
in doubt. All scientific statements are in principle permanently provisional,
but a consensus emerges, in a particular area, when it becomes fruitlessly
difficult to generate or sustain plausible alternatives to the core theories.
That GR is ‘correct’, in this sense, has been a settled fact since about 1919.
So is that the end of GR’s confrontation with experiment? No, it’s not,
but the nature of the confrontation has changed.
The ‘classical’ tests represented the first phase of the theory’s confronta-
tion with reality, addressing the question ‘is this theory correct?’. As we have
seen, this phase effectively finished in 1919 when, broadly speaking, the
statement ‘GR is correct’ became conventional knowledge, and the subject
became something one could consider teaching to undergraduates. Then
there was a ‘mopping up’ phase, during which the remainder of GR’s pre-
dictions were elaborated and then tested. There was little expectation that
these tests would show a failure in the theory, but they were interesting
and valuable as technical challenges which would tax and advance other
sub-disciplines within physics, always with the off-chance of uncovering a
thrillingly unexpected anomaly. This phase might be said to have concluded
with the detection of gravitational waves, and that sub-discipline’s transition
from a test of GR, to another messenger for astronomy. This broadly fits
in with the historically motivated account of the social dynamic of science
famously described by Thomas Kuhn (1996), which sees it work round a
cycle of ‘revolution’, where a new idea changes the ground rules, to ‘normal
science’, a technical puzzle-solving phase where details are worked out and
anomalies resolved, then to a phase where a residue of unresolved anomalies
builds up to the point where it creates a ‘crisis’, which is finally resolved by
a new conceptual revolution, and the beginning of another cycle.
So where are the ‘anomalies’ with GR? General Relativity is now so firmly
established as a foundational theory of our universe that any deviation be-
tween theory and observation would be a major scientific event (if not quite
the earthquake that would result from a negative result in SR). This search
for anomalies is the new ‘normal science’ for GR, and is represented by a sys-
tematic parameterisation of the various ways that a theory of gravity might
plausibly differ from Einstein’s.5 Thus the ‘Parameterised Post-Newtonian’

5 A similar process is currently taking place in experimental particle physics, where ‘normal
science’ is the increasingly desperate attempt to find some measurements which are
demonstrably at odds with the Standard Model of particle physics.
186 Appendix B Relativity’s Contact with Experimental Fact

formulation of GR is a theoretically motivated distillation of the question ‘in

what ways could GR be incorrect?’, which is concrete enough to allow critical
measurement. So far, no deviations have been found, but the attempts to
find them have been summarised at exhaustive length in Will (2014).

B.3 A Closer Look at the 1919 Eclipse Observations

It is worthwhile spending a little time looking again at the swiftness and
completeness of the community’s acceptance of Eddington and Dyson’s
observations and conclusion. This is because it leads us to interesting ques-
tions about how science works in practice, rather than in widely accepted
myth.
Eddington’s and Dyson’s observations of the solar eclipse of 1919 May 29
were technically challenging, and although the conclusions are solid, the ob-
servational analysis is not beyond comment. Indeed, this set of observations
has become something of a case-study in the subtleties of ‘theory choice’ in
science – how is it that a particular, potentially disputable, observation is
or is not taken to be conclusive by the scientific community? That the case-
study has gone on to achieve that most unwelcome type of fame, notoriety,
is separately interesting.
The joint expedition was planned starting in 1917, as a collaboration
between Frank Dyson, who was both the Astronomer Royal and the director
of the Royal Greenwich Observatory (RGO), and Arthur Eddington, the di-
rector of the Cambridge University observatory. The expedition was planned
during, and undertaken just after, the First World War, and wartime logistics
meant that the equipment was not everything the observers could hope for.
It is also at least interesting that Eddington, who was of conscriptable age,
but who, as a devout Quaker, had made it clear he was prepared to object
to military service on conscientious grounds (an unpopular position which
could have resulted in him being detained), led the Principe expedition as
an explicit part of a negotiated scientific deferment of his conscription (see
Earman & Glymour (1980b), quoting Chandrasekhar).
The observational goal was to measure a deflection in stellar positions of
around one arcsecond, in the field (literally) rather than at an observatory,
and in the few minutes of totality available during an eclipse. The obser-
vations were framed to distinguish between three possibilities, namely no
deflection, the 1′′. 75 ‘einsteinian’ prediction of GR, and a so-called ‘newto-
nian’ deflection of half of that, 0′′. 87. The full deflection is the prediction of
Eq. (A.20), based on Einstein’s field equations and a non-euclidean spatial
 B.3 A Closer Look at the 1919 Eclipse Observations 187

sector; whereas the ‘half-einstein’ prediction arises from an argument related

to that of Section A.4.1, in the sense that it depends only on the Equivalence
Principle (this half value was the one that Einstein predicted when he first
discussed the deflection in 1911). In his 1918 ‘Report’ on relativity, Edding-
ton discusses (1920, §32) the newtonian deflection only as a pedagogical
contrast to the einsteinian one, so it seems likely that the latter was the value
he expected to observe on the expedition; but the derivation of the result was
a lot less clear in Eddington’s 1918 work than it is now, and both non-zero
possibilities were clearly scientifically realistic outcomes. There is a detailed
description of the history of the theoretical work in Earman & Glymour
(1980b, §1).
Measurements were made by Dyson’s RGO colleagues at Sobral in Brazil,
and by Eddington and assistants at Principe, off the coast of West Africa;
Dyson was responsible for the data reduction for the Brazil observations,
and Eddington did the data reduction for Principe. There were three tele-
scopes taken to the two sites: an astrographic (survey) instrument in both
places, each equipped with a clockwork coelostat, plus, in Sobral, a 4-inch
instrument with a narrower field of view.
The results were reported briefly to a joint Royal Society and Royal As-
tronomical Society meeting on 1919 November 6.6 Despite being originally
intended as a backup, the 4-inch instrument produced the cleanest results,
giving a deflection of 1′′. 98 ± 0′′. 18.7 Both of the astrographic observations
had problems, partly to do with cloud cover and focus problems, and partly
because of the technical need for a more elaborate data reduction process;
and Eddington and Dyson reported (1920), deflections of 1′′. 61 ± 0′′. 30 for
Principe and 0′′. 93 (no error quoted) for Sobral. Dyson concluded that, al-
though the Sobral astrographic result was closest to the ‘newtonian’ figure,
the result was too unreliable, after analysis, to give much information, and
it was effectively discarded.8 Eddington’s analysis led him to the large error

6 The joint meeting is briefly summarised in Thomson (1919), which includes a record of the
discussion afterwards. This was accompanied by a much more detailed account (Dyson
et al. 1920), which was ‘read’ to the Royal Society on the same date in 1919. You can find
fuller accounts of the expedition and experimental analysis in Earman & Glymour (1980b),
and in a brief but lucid account in Kennefick (2009), which is related to a more detailed
version, which also talks about the ‘myths’ surrounding the observation, in Kennefick
(2012).
7 It was usual at the time to quote errors in terms of ‘probable error’ rather than standard
deviations: the probable error is half the interquartile range, and for a normal distribution is
a little more than 2/3 of the standard deviation. Here and below I have quoted the errors
rescaled to standard deviations.
8 The Sobral plates were remeasured at the RGO in 1979 (Harvey 1979) with the result that
the error on the 4-inch observations was tightened, and the result from the (Sobral)
188 Appendix B Relativity’s Contact with Experimental Fact

quoted here, but he argued that there were no systematic errors which would
otherwise undermine this result (indeed that the cloud cover at Principe
had accidentally but usefully avoided the overexposures which had been
part of the problem at Sobral). Specifically, he stated that ‘the accuracy
seems sufficient to give a fairly trustworthy confirmation of Einstein’s the-
ory, and to render the half-deflection at least very improbable’, but that this
Principe result ‘has much less weight than [the 4-inch one].’ The logic of
the report, in its conclusion, is that the 4-inch observation is the main result,
inconsistent with the ‘newtonian’ prediction, that this is weakly corrobrated
by the Principe data, and that the Sobral astrographic observation was too
unreliable to rule out either possibility.
I have described this measurement in much more detail than you will
usually find in a text of this type – relativity textbooks tend to deal with
the 1919 measurements in a sentence or a short paragraph – but there is
little that is really unusual about it. It was a challenging measurement at
the limit of what was possible; the observers constructed and published an
argument that some of the data was unreliable and should be discarded, and
that some other data, though difficult to analyse, should be retained; they
came to a qualified but still confident conclusion about its consistency with
the ‘einsteinian’ result; and the community promptly and overwhelmingly
accepted this conclusion, and seems to have regarded the matter as largely
settled. It is still regarded as a famously conclusive measurement.
After discussing the history of the measurement, Earman & Glymour
(1980b) drew attention to a number of features of this measurement which
are not uncommon in scientific experimentation, but which were particu-
larly clearly exhibited here; their account has been much discussed since.
(i) The deflection measurements were motivated by a particular theoret-
ical prediction, and so were designed to rule in or rule out three distinct
possibilities, rather than making an unconstrained estimate of the value. In
other words, this was a hypothesis-testing experiment, and not the measure-
ment of a physical parameter.
(ii) Although they were in the service of a particular prediction, there
was no widespread understanding of the details of the theory. Eddington
was one of the few people able to do the calculations confidently, but even
there, it is clear that Eddington’s, Einstein’s and others’ understanding of the
theory, and of how to do calculations, was not as profound as it would later
become. That is, it would have been reasonable to criticise the theoretical

astrographic plates improved to 1′′. 55 ± 0′′. 34, both ruling out the ‘newtonian’ prediction and
suggesting that Dyson had been pessimistic in his analysis.
 B.3 A Closer Look at the 1919 Eclipse Observations 189

work, though no-one seems to have done so in fact. It appears likely, in

context, that the consistency of the measurement with the theory would
have increased confidence in the validity of the derivation of the predictions
from the theory, even though this would be circular in strict logical terms.
(iii) Eddington was already convinced of the correctness of GR, though he
was in something of a minority here. Dyson was sceptical of it, as were others
who took part in prior attempts to make this observation. Further, there was
open antipathy to German science in general, in the UK in the aftermath
of the war, and it seems very likely that a significant part of Eddington’s
personal motivation, and possibly part of Dyson’s as well, was to ‘put an
end to wild talk of boycotting German science’, as Eddington put it in his
obituary (1940) for Dyson (the obituary, written in the early years of the
Second World War, goes on to remark that the confirmation of a German
theory ‘kept alive the finest traditions of science; and the lesson is perhaps
still needed in the world to-day’).
(iv) Dyson and Eddington offered others access to their data, for reanaly-
sis, but no-one seems to have thought it useful to do so: their analyses of their
observations were taken at face value, on the grounds of their expertise (in
the obituary, Eddington notes that Dyson ‘had a reputation for soundness’,
and draws attention to his advances in plate reduction techniques).
(v) The difficult measurements of Fraunhofer redshift produced incon-
clusive results before 1919, but soon afterwards they were consistently found
to corroborate it. The goal of such measurements had, it seems, changed:
they had been unable to test the truth or falsity of the theory by themselves,
but by taking GR to be more-or-less confirmed, observers became able to
use these observations to achieve other scientific goals. If a particular mea-
surement appeared to be inconsistent with the redshift, then that would
be a prompt to re-examine it in search of a practical defect; but if it were
consistent, there would be no such prompt (this is not an empty process:
scientific progress sometimes comes when anomalies stubbornly remain
even after such re-examination).
None of these features – prior convictions, theoretical uncertainty, per-
sonal motivations, data selection, the appeal to authority, and apparently
sudden consensus – is unusual in science as it is actually practised; the
history of science is full of other examples, and we have already seem some
similar features in the fuller history of the Pound–Rebka measurements
(note 6 on p. 152). What is unusual in this story is that so many of these
features are present at once, and that the history of the case is so well known.
In addition, these various features tend not to be ones mentioned in popular
accounts of science (written by scientists as well as non-scientists), which
190 Appendix B Relativity’s Contact with Experimental Fact

usually tend to retail an account of science as an activity carried on with

dry rigour – this observation leads to that conclusion – and with a linear
inevitability from theory, to experiment, to certainty.
This is not a claim that science is somehow illogical (it is logical, and
an assertion that observation A is inconsistent with, or ‘falsifies’, theory B
will prompt critical examination of both9 ), or that it is unreliable (over time,
science does seem to ratchet towards a better and better understanding of a
certain category of knowledge). Instead, it is to assert that, when we look
at a case such as the 1919 eclipse, the deductive logic of falsification is far
from being the only thing going on. How is it that a community of scientists
will choose one conclusion over another, at a particular point in time? The
observational evidence is clearly a part of it – and science frames questions
so that this evidence is a much bigger influence on its conclusions than
is possible in other areas of human enquiry – but the other features are
not absent, even if they play less and less of a role in our acceptance of a
theory, as time moves on and we are able to see an observation in hindsight
and in an ever-enlarging context. But if we want to look back, and try to
understand how a particular confrontation between theory and experiment
was actually resolved, at a particular historical moment, then we cannot
use that hindsight to make sense of contemporary theory choice, which can
only be examined, and puzzled over, in its own, contemporary, terms.
This case became famous partly because, after drawing attention to some
of these features in their history of the case, Earman and Glymour remarked
in passing, in their final paragraph, that ‘[t]his curious sequence of reasons
might be cause enough for despair on the part of those who see in science a
model of objectivity and rationality’. Some people have chosen to see this
remark as an attack on the integrity of Science, or even on the personal
integrity of Eddington or Dyson; this interpretation makes little sense, not
least because the authors go on to remark bluntly, as the very last words in
their paper, that GR ‘still holds the truth about space, time and gravity’.
The case has also been of interest to those who are not primarily interested
in the history, or the science as such, but instead interested in what it and
similar cases tell us about the specifically social aspects of theory choice in
the past and in the present – the Sociology of Scientific Knowledge.10 This

9 I use the word ‘falsify’ because that fits in with the popperian logic of science, but
‘falsification’ is a complicated business, and the more evasive phrasing of ‘inconsistent with’
is probably more descriptive.
10 ‘In the present’ is important, because current controversies, perhaps scientific controversies
which interact with crucial public policy decisions, necessarily happen without the benefit
of any hindsight, even though ‘how do we know?’ and ‘what do we do now?’ may be
 B.3 A Closer Look at the 1919 Eclipse Observations 191

approach originated in the work of David Bloor in 1976 (i.e., Bloor (1991),
and see Laudan (1981) and Bloor (1981)), and is well summarised in Collins
& Pinch (2012). It represented, I suppose, one ‘side’ in the so-called ‘Science
Wars’ of the 1990s, an unedifying episode which has a useful synthesis in
Labinger & Collins (2001).
It’s also worth pointing out that Earman and Glymour were not the first
to talk of this case, nor the first to raise the suggestion (to the extent that they
did) that there was something amiss in Dyson and Eddington’s measurement.
Indeed, it seems to have become something of an urban myth in gravitation
physics: it was mentioned in a rather garbled aside by Everitt et al. (1979),
in an article proposing an alternative test of GR, and it reached possibly its
more extreme published form in Hawking’s A Brief History of Time (1988),
where he suggests, without pointing to a source, that ‘the errors were as great
as the effect they were trying to measure.’ There are at least two interesting
things about these two (historically inaccurate) accounts: one is that neither
author thinks it important, from a scholarly point of view, to give a precise
account of the history; secondly, neither author seems scandalised by the
incident, and indeed Hawking goes on to remark that this was ‘a case of
knowing the result they wanted to get, not an uncommon occurrence in
science’. When I have heard this story retold, in one variant or another, or
retold it myself, it it has never been received with either surprise or outrage,
but at most wry amusement or a facetious tut-tut-tut.
From one point of view, this doesn’t matter. Science tends to be cheerfully
ahistorical in practice, and most scientists would probably agree that, as a
question of how we know what we know, it doesn’t much matter just why
the community arrived at one conclusion or the other, as long as whatever
conclusion we have on the matter, now, can reasonably be agreed to be
correct (this touches on the distinctions between the various questions on
p. 178). Science gains its justifications from current consistency between
theory and experiment, and when we hear ‘Newton said. . . ’ or ‘Galileo
observed. . . ’ it’s generally a pedagogical point being made, rather than a
historical one, or any attempt to persuade. The history of science is much
more intricate (and indeed more interesting) than you would generally
learn in a science lecture, but that history is not part of the professed logic

scientific questions which demand immediate answers. I am writing this during the
COVID-19 pandemic, when demands that politicians ‘follow the science!’, though
reasonable, were met with some unease by the scientists in question, and when the contrasts
between scientific, medical, and political theory-choice – ‘how do we know?’, ‘what are the
chances?’, and ‘what do we do now?’ – were discussed in newspaper leading articles rather
than journals of the sociology of science.
192 Appendix B Relativity’s Contact with Experimental Fact

of science.
The reason I am describing this at such length is that I think that the
episode illustrates some important features of the relationship between the-
ory and experiment, in actual scientific practice. In the swift acceptance of
the 1919 result, we (broadly speaking) see the point at which a theory moves
from prediction to (broadly speaking) accepted fact, and in the Fraunhofer
redshift observations we see the effect this has on a related field, turning
relativistic effects into a background detail for solar spectroscopic measure-
ments. The fact that Dyson and Eddington are so scrupulously exhaustive
in describing their data analysis illustrates the continuing force of the Royal
Society’s motto ‘Nullius in verba’, conventionally glossed as ‘take nobody’s
word for it’ – the idea that scientific results should be described in a way
that lets the reader repeat the observation, or re-do the analysis, themself,
and which has logical, methodological and even ethical dimensions; the
fact that no reader seems to have thought this re-observation worth doing
illustrates the continuing force of scientists’ communal trust in each other’s
craft skills, and in their professional honesty, in reporting and analysing
data as fairly as possible.
So if, as I assert, this incident is more intricate than most people would
expect, but not more intricate than usual practice, we must ask why it has
become so famous, and so frequently and combatively discussed. Why, for
that matter, is this usually described as Eddington’s observation, though it
seems to have been a joint effort, with Dyson, in every important respect?
One answer is that it is a good story, in the practical sense that it incorporates
the multiple features I mentioned above, in a way which illustrates the
strands of motivation and persuasion which are present in actual scientific
practice. Another answer is that it incorporates human involvement with
the process – the result was personally important to Eddington, for more
than merely professional reasons. I think it is those human aspects of the
story – aspects which are essential to the story – which make some people
uncomfortable, as if they suggest that scientific conclusions are arbitrary, or
personal, or are somehow mere consensus (in a deprecatory sense, which is
distinct from the pragmatic sense in which I have used the word above), and
as if they undermine the thoughtful confidence we should have in scientific
conclusions here and elsewhere. But that confidence is emphatically not
undermined by the observation that scientists are people, that the logic of
science is not only the simplistic one described in popular science books (or
press releases), and that uncertainty and doubt are integral to the process.
So after ‘a closer look’ at the 1919 observations, what conclusions can
we come to? The ‘closer look’ appears to have simply made things more
 B.3 A Closer Look at the 1919 Eclipse Observations 193

complicated, and added specifically human complications to what seemed

like a reassuringly rational, or logical, observation. A key thing to note
is that we are asking at least three questions here, namely (i) is general
relativity correct? (ii) how do astronomers conclude that general relativity
is correct? and (iii) how did astronomers conclude that in the past? The
last question is a historical and social question: what happened, as a matter
of history, in the year-long run-up to the 1919 November 6 meeting where
Dyson and Eddington reported their results, and what were the mechanisms
through which the attendees at that meeting decided, as a matter of ‘theory
choice’, that Dyson and Eddington had proved Einstein correct? Phrased
like that, it is surely obvious that subsequent observational work – from
later eclipse observations to the LIGO result – is simply irrelevant to the
question. A related question is to ask how can astronomers now address
that same ‘theory choice’? They can use the hindsight of a century of further
observation, and it is through that hindsight that this question shades into
the first one: how do we know that relativity is correct? To answer that
question would take us into the philosophy and pragmatics of science, and
thus even further adrift from our main topic.
Although the history is complicated, we remain supremely confident
(while respecting the detailed qualifications of Section B.2) that GR is a
reliably correct description of nature.
Our appreciation of the history of the observations makes it all the more
impressive that the evolution of scientific culture has produced a process
which is able to accommodate the apparent methodological problems de-
scribed on p. 188, working within a world of uncertainty, confidence, ambi-
tion, ambivalence, patriotism, bloody-mindedness, and every other human
passion, which still ratchets towards a certain type of truth. Science does
not remove these passions, but the process of accommodation is worthy of
admiration, and worthy of examination and critical study, and possibly even
of emulation. If we are clearer about science’s boundaries, when we look at
its work in current controversies, then we can be a little clearer about what
is outside those boundaries – in particular topics of social or human policy
where science or scientific approaches may have little to contribute – and a
little clearer about the great confidence we can have in science’s conclusions
and assistance when it is inside those boundaries, asking the questions it
can answer best.
 Appendix C
Maths Revision

In a couple of places in the text I rely on a little familiarity with particular

mathematical structures – namely complex numbers, hyperbolic ‘trigonom-
etry’, and matrix algebra. I presume you have at least encountered these
before, but appreciate that this text may be the first time you’ve used them
outside of a maths class. I’ve therefore included, below, some very compact
introductions to these bits of maths, which are self-contained, but probably
more useful as a reminder than as a fully standalone introduction.

C.1 Complex Numbers

We are most familiar with the real numbers, ℝ, which corresponds directly
to the number line, and using which we can define addition, subtraction,
multiplication, division, and so on. We might, however, ask ourselves, ‘what
about the number 𝑥, which is such that 𝑥 2 = −1? Where is that number on
the number line?’ One answer to this is that this simply isn’t a number, or
that the equation 𝑥 2 = −1 is meaningless.
Another answer is to say: what if there is a ‘number’ which is a solution
to this equation – call it i – what are its properties? Can we define operations
of addition, subtraction, multiplication and division with such numbers?
You will not be surprised to discover that we can; you may be surprised to
discover that this extra ‘number’ i is the only new thing we need, to uncover
a completely new set of numbers. We refer to this set as the complex
numbers, ℂ.
The rules for complex numbers follow fairly naturally:

The number i is a ‘complex number’ The number i is defined such

that i2 = −1, and is the prototype complex number.

194
 C.1 Complex Numbers 195

Any real multiple of i is a complex number Thus 2i and −999i and

𝜋 × i are complex numbers. Multiples of i are known as imaginary
numbers (there’s nothing unreal about them, but this is the usual name,
in contrast to the real numbers).
We can add real numbers and complex numbers So 1 + i is a com-
plex number, as is 2 + i − 3 + i4 = −1 + i5. Here, we have gathered
the real numbers in this sum, and gathered the multiples of i, but we
cannot simplify this expression further. That means that. . .
All complex numbers can be expressed in the form 𝑧 = 𝑎 + i𝑏 for
𝑎, 𝑏 ∈ ℝ. That is, we can gather (and simplify) the terms involving i,
and the terms not involving i, but we can’t simplify further. We refer
to 𝑎 as the real part of the number 𝑧, and 𝑏 as the imaginary part.
All real numbers are complex numbers That is to say, ℝ ⊂ ℂ, since
the real numbers are just those complex numbers for which 𝑏, above,
is zero.

The above points give us enough information to describe the full range
of arithmetic operations.
Addition: Given two complex numbers 𝑧1 = 𝑎 + i𝑏 and 𝑧2 = 𝑐 + i𝑑, the
sum of the two numbers is obtained by collecting and simplifying as much
as possible:

𝑧1 + 𝑧2 = (𝑎 + i𝑏) + (𝑐 + i𝑑) = (𝑎 + 𝑐) + i(𝑏 + 𝑑)

𝑧1 − 𝑧2 = (𝑎 + i𝑏) − (𝑐 + i𝑑) = (𝑎 − 𝑐) + i(𝑏 − 𝑑).

Multiplication: The product of 𝑧1 and 𝑧2 is obtained similarly, by multi-

plying the numbers out, and simplifying as much as possible:

𝑧1 𝑧2 = (𝑎 + i𝑏)(𝑐 + i𝑑)
= 𝑎𝑐 + i𝑏𝑐 + i𝑎𝑑 + i2 𝑏𝑑
= (𝑎𝑐 − 𝑏𝑑) + i(𝑏𝑐 + 𝑎𝑑),

using the fact that i2 = −1.

Division: We can define division slightly indirectly, relying on our defini-
tion of multiplication:
𝑧1 𝑎 + i𝑏
=
𝑧2 𝑐 + i𝑑
𝑎 + i𝑏 𝑐 − i𝑑
= × (multiplication by 1)
𝑐 + i𝑑 𝑐 − i𝑑
(𝑎𝑐 + 𝑏𝑑) + i(𝑏𝑐 − 𝑎𝑑)
= .
𝑐2 + 𝑑2
196 Appendix C Maths Revision

Either using this, or by inspection, you can see that 1∕i = −i.
The complex conjugate: There is a new arithmetic operation we can
perform on complex numbers, the complex conjugate: this is the operation of
negating the imaginary part of the number. Thus, given a complex number
𝑧 = 𝑎 + i𝑏, the conjugate, 𝑧∗ , is

𝑧∗ = 𝑎 − i𝑏.

̄ Using this, we can define the modulus of a

This is sometimes written 𝑧.
complex number as |𝑧|, where

|𝑧|2 = 𝑧𝑧∗ = (𝑎 + i𝑏)(𝑎 − i𝑏) = 𝑎2 + 𝑏2 .

This corresponds to the pythagorean distance in the complex plane.

With a limited set of exceptions, all of the arithmetic operations we
can use with real numbers, we can also use with complex numbers. The
only reason I’m mentioning complex numbers in this context is because in
Chapter 5 we use the hyperbolic functions, which are most naturally related
to the similar trigonometric functions, but using complex numbers.

C.2 The Hyperbolic Functions

You are, I presume, familiar with the trigonometric functions sin 𝜃, cos 𝜃,
and so on. And so you will have seen the range of identities associated
2
with those: sin 𝜃 + cos2 𝜃 = 1, sec2 𝜃 = 1 + tan2 𝜃, and the various addition
formulae.
There is a related set of functions, called the hyperbolic functions (some-
times called the ‘hyperbolic’ or ‘complex trigonometric functions’, plausibly
but inaccurately). These are defined by

1 𝑧
sinh 𝑧 = (e − e−𝑧 ) (C.1a)
2
1
cosh 𝑧 = (e𝑧 + e−𝑧 ), (C.1b)
2
and illustrated in Figure C.1. There are corresponding definitions

sinh 𝑧
tanh 𝑧 =
cosh 𝑧
1
sech 𝑧 = ,
cosh 𝑧
 C.2 The Hyperbolic Functions 197

c( 
s ( 


- -  
-

-

Figure C.1 The sinh 𝑧 and cosh 𝑧 functions.

and identities such as

2 2
cosh 𝑥 − sinh 𝑧 = 1
2 2
1 − tanh 𝑧 = sech 𝑧
sinh(𝑢 ± 𝑣) = sinh 𝑢 cosh 𝑣 ± cosh 𝑢 sinh 𝑣
cosh(𝑢 ± 𝑣) = cosh 𝑢 cosh 𝑣 ± sinh 𝑢 sinh 𝑣
tanh 𝑢 ± tanh 𝑣
tanh(𝑢 ± 𝑣) =
1 ± tanh 𝑢 tanh 𝑣

See your favourite source of mathematical tables for the rest, and for expres-
sions for the derivatives, and so on. You can see that these are quite similar
to the corresponding trigonometric identities.
We can find other expressions for these functions by looking at the series
expansion of the exponential function:

𝑧2 𝑧3 𝑧4
e𝑧 = 1 + 𝑧 + + + + ⋯. (C.2)
2! 3! 4!

By looking at Eq. (C.1), we can calculate that

𝑧3 𝑧5
sinh 𝑧 = 𝑧 + + +⋯
3! 5!
𝑧2 𝑧4
cosh 𝑧 = 1 + + + ⋯.
2! 4!

We may also now ask what is the exponential of a pure imaginary number, i𝑧?
This seems an odd thing to do, if you think of e𝑧 only as the number e raised
to a given power; but if you think of Eq. (C.2) as defining the exponential
198 Appendix C Maths Revision

function, then it is obvious what to do:

(i𝑧)2 (i𝑧)3 (i𝑧)4
ei𝑧 = 1 + i𝑧 + + + +⋯
2! 3! 4!
𝑧2 𝑧3 𝑧4
= 1 + i𝑧 − −i + +⋯
2! 3! 4!
𝑧2 𝑧4 𝑧3 𝑧5
= (1 − + + ⋯) + i (𝑧 − + + ⋯)
2! 4! 3! 5!
= cos 𝑧 + 𝑖 sin 𝑧. (C.3)
The final line comes when we spot the well-known series for cos 𝑧 and sin 𝑧,
𝑧2 𝑧4 𝑧3 𝑧5
cos 𝑧 = 1 − + + ⋯, sin 𝑧 = 𝑧 − + + ⋯.
2! 4! 3! 5!
From Eq. (C.3), or directly from Eq. (C.2), we can now see that
1 i𝜃
cos 𝜃 = (e + e−i𝜃 ) ⇒ cos i𝜃 = cosh 𝜃
2
1
sin 𝜃 = (ei𝜃 − e−i𝜃 ) ⇒ sin i𝜃 = i sinh 𝜃.
2i
The argument in Section 5.1 used complex arguments to the trigonometric
functions, in a way that may be surprising at first, but which turns out to be
a straightforward application of the identities we see here.
Equation (C.3), incidentally, is known as Euler’s equation, and rewritten
in the special case of 𝑧 = 𝜋, it is
ei𝜋 + 1 = 0,
which Richard Feynmann called ‘the most remarkable equation in mathe-
matics,’ involving as it does five of the key numbers in mathematics, in a
single equation.

C.3 Linear Algebra

The term ‘linear algebra’ refers to a quite general area of mathematics, but
the way we most often, or first, encounter it as physicists is through the
arithmetic of matrices. We encounter it first in this text in Section 6.1.
The key results are as follows.
If 𝐚 = (𝑎𝑥 , 𝑎𝑦 , 𝑎𝑧 ) and 𝐛 = (𝑏𝑥 , 𝑏𝑦 , 𝑏𝑧 ) are vectors, then 𝐚 + 𝐛 is also a
vector, with components (𝑎𝑥 + 𝑏𝑥 , 𝑎𝑦 + 𝑏𝑦 , 𝑎𝑧 + 𝑏𝑧 ). We can define a scalar
product 𝐚 ⋅ 𝐛 = 𝑎𝑥 𝑏𝑥 + 𝑎𝑦 𝑏𝑦 + 𝑎𝑧 𝑏𝑧 . We can define the length, |𝐚|, of a 3-
vector in terms of the scalar product of a vector with itself: |𝐚|2 = 𝐚 ⋅ 𝐚 =
 C.3 Linear Algebra 199

𝑎2 = 𝑎𝑥2 + 𝑎𝑦2 + 𝑎𝑧2 (we can also see this from Pythagoras’s theorem, or in-
deed from Eq. (6.1)), and we know that this is an invariant of a rotation –
that is, that it takes the same value irrespective of the coordinate system. It
is easy to see that |𝐚 + 𝐛|2 = 𝑎2 + 𝑏2 + 2𝐚 ⋅ 𝐛 and so, since both 𝑎2 and 𝑏2
are frame-invariant, the scalar product 𝐚 ⋅ 𝐛 must be frame-invariant also,
even though the individual coordinates 𝑎𝑖 and 𝑏𝑖 are not. Finally, if the
scalar product of two vectors vanishes, 𝐚 ⋅ 𝐛 = 0, we say that the two vectors
are orthogonal. If a euclidean vector is orthogonal to itself (𝐚 ⋅ 𝐚 = 0) then
we can deduce that 𝑎𝑖 = 0. In linear algebra, the scalar product is more
generally termed the inner product, and the length of a vector is termed its
norm.
In Minkowski space, in contrast, the scalar product of Section 6.2 can
be negative (and a mathematician would therefore insist that it cannot,
strictly, be called an ‘inner product’), which means that, if the scalar product
of two Minkowski vectors is zero, 𝐀 ⋅ 𝐀 = 0, we therefore cannot deduce
that 𝐀 = 0. Also, in this context, we can continue to talk of the magnitude
of a vector, 𝐀 ⋅ 𝐀, but can no longer (and indeed need no longer) talk of the
norm as the square root of this.
 Appendix D
How to Do Calculations: a Recipe

D.1 Key Things to Remember

Two frames are in standard configuration (see Section 1.7 and Figure 1.5)
when:

1. they are aligned so that the (𝑥, 𝑦, 𝑧) and (𝑥 ′ , 𝑦 ′ , 𝑧′ ) axes are parallel;
2. the frame 𝑆 ′ 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 =
𝑡′ = 0 (which means that the origin of the 𝑆 ′ frame is always at position
𝑥 = 𝑣𝑡).

All three conditions must be satisfied in order for the Lorentz transformation
equations to be valid.
The values 𝑥, 𝑡, and so on, are coordinates of events in the unprimed
frame, 𝑆 (they’re not displacements, or lengths, or the intervals between
events); the values 𝑥 ′ , 𝑡′ , and so on, are coordinates of the same events, as
measured in 𝑆 ′ .
Remember: an event is not ‘in’ one frame or another. An event is a frame-
independent thing which is ‘in’ every frame. The difference between frames
is that an event will have different coordinates in different frames.

D.2 A Checklist for Relativity Problems

The process of solving a relativity problem is, in many cases, pretty mechan-
ical. That’s not the same as easy, but there are pretty clear steps you can go
through to help you to the answers.
You won’t necessarily go through each of these items in precisely this
order – so this is perhaps more of a checklist than a recipe – but you’ll almost

200
 D.2 A Checklist for Relativity Problems 201

certainly need to complete all of the ‘identify’ steps before it makes sense to
do either of the ‘write down’ steps.

Draw the Minkowski diagram Drawing the diagram is generally a

useful step, since it forces you to work through the textual information
in the question, turning it into events. It may make sense to draw the
Minkowski diagram in both the ‘stationary’ and the ‘moving’ frame.
But that requires. . .
Identify the frames Usually these will be pretty obvious – something
is moving past something else – but always make it explicit.
Identify the origins of the frames Very often, one of the events will
be at the origin, but sometimes it will be more natural to have some-
thing else there (for example, the centre of the Earth, or the centre of
a rocket). Make sure you write down what’s at the origins of the two
frames, when, and make sure that your choice is compatible with stan-
dard configuration (𝑥 = 𝑥 ′ = 0 and 𝑡 = 𝑡 ′ = 0). If you have nothing at
the origin, think again: you’re very probably making things hard for
yourself.
Identify the events Relativity is all about the coordinates of events,
as measured by observers in different frames. Remember events are
things like explosions, or hand-claps, or light-flashes, that happen at
a specific place, at an instant of time. If no event is described in the
question, make one up (for example ‘the train snaps a ribbon at the
exit of the tunnel’). Very often, an event will be at the intersection of
two worldlines.
Identify the worldlines Something will be moving! Identify what its
worldline is in the ‘stationary frame’. Identify the worldlines of any-
thing not moving in this frame. Look at the description of each event:
what worldline(s) is it on? Sometimes worldlines will join the dots
between events; sometimes an event is, in effect, defined as happening
at the place where two worldlines cross.
Write down what you know There will be numeric values in the ques-
tion. What events do they correspond to? If it’s a distance or a time
for an event, it’s a distance or time in which frame? Whenever you
write down a space or time coordinate, you must make clear which
event it is the coordinate of: if it is (for example) the time coordinate
′
2 in the primed frame, write 𝑡 (and not just 𝑡 ′ ).
of event ○ 2
Write down what you have to find out The question will probably
(implicitly) require you to find the position or time coordinate of an
event as viewed in the ‘other’ frame. What coordinate (𝑥1 , say, or 𝑡2′ ?)
202 Appendix D How to Do Calculations: a Recipe

are you being asked to find?

Redraw your Minkowski diagram? At this point you might have
changed your mind about your Minkowski diagram. That’s OK: the
diagram is there to help you sort these steps out.
Head for home At this point, the rest is pretty mechanical. Use one of
the Lorentz transformation equations to find the coordinate you need
from the coordinates you have. You may have to do some rearrange-
ment. There are a few sanity-checks you can do at this point. If two
events are connected by a light flash, do their coordinates show that
the light is moving at speed 𝑐 in all frames? Have lengths contracted,
and times dilated, as you might expect? These checks are the new
intuitions you acquire through doing SR problems.

The first steps are the hard ones. After about half-way, you’re just turning
the handle.
The big wrinkle is that you might not need to use the Lorentz transfor-
mation, since there are a few other bits of dynamics you’re supposed to
know.

D.3 Which Equation When?

Lorentz transformation equations These relate the coordinates of
one event in two frames. Remember that, for the Lorentz trans-
formation equations to work, the two frames must be in standard
configuration. Given, say, 𝑥1 and 𝑡1 , you might be asked to find 𝑥1′
and/or 𝑡1′ (all coordinates of the same event ○ 1 ). Whenever you use

one of the Lorentz transformation equations, you must identify

which event you’re talking about. If you write down a Lorentz
transformation equation as part of a calculation, and the 𝑥s and 𝑡s
don’t have subscripts identifying an event (the same event), then you’re
doing it wrong.
Length contraction and time dilation These are for calculating in-
tervals – the length of a stick, or the interval between clock ticks – in
one frame, as measured in another. You can deduce them from the
Lorentz transformation, but also (as we saw in Chapter 3) directly from
the axioms. You won’t necessarily have to set up frames in standard
configuration in order to use these relations.
Distance is speed times time This is for movement within a frame
(possibly including the movement of one frame with respect to an-
 D.4 Rest, Moving and Stationary Frames: Be Careful! 203

other). If something is moving at speed 𝑣 in some frame, for a time Δ𝑡,

then it will have moved a distance 𝑣Δ𝑡 in that frame.

D.4 Rest, Moving and Stationary Frames: Be Careful!

Be careful when using the terms ‘rest frame’ or ‘moving frame’.
Frame 𝑆 ′ will often be referred to as the rest frame; however, it should
always be the rest frame of something. Yes, it does seem a little counter-
intuitive 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 fussy, 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 un-
primed frame’ to refer to ‘the moving frame’ unless you’ve previously made
clear which frame is which, through a diagram or through a textual expla-
nation. 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 be-
cause 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; and when people tie themselves in knots (for
example in exam answers), it is often because they have failed to set things
out in a systematic way (such as Section D.2).
204 Appendix D How to Do Calculations: a Recipe

D.5 Length Contraction, Time Dilation, and ‘Rest

Frames’
𝐿0 and 𝑇0 are the length and proper time in the rest frame of a particle/ob-
ject/train. That may or may not be the ‘primed frame’.
For length contraction and time dilation, I don’t try to memorise an
equation such as Eq. (3.3) or Eq. (3.5): instead, I simply remember that
‘length contraction’ means that ‘moving rods get shorter’, and ‘time dilation’
means that ‘moving clocks run slow’. Make sure you understand what these
slogans mean, and multiply or divide by 𝛾 as appropriate.
Finally, be careful about terminology. The rest frame of a particle (or train,
or person) is the frame in which that particle is not moving (everyone agrees
about this frame). The ‘moving frame’ and ‘stationary frame’ are informal
and relative terms – the station is the ‘moving frame’ to an observer on the
train. Similarly, the ‘primed frame’ is usually chosen as the ‘moving frame’,
but this is entirely arbitrary.
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 Index

acceleration, 87, 90, 149 electromagnetism, 20, 22–27, 95–98, 179, see
constant, equations of, 121 Maxwell’s equations
proper, see proper acceleration electron-volt, 137
vector, 111, 113 energy conservation, 128
aether, 24, 97, 179 energy equivalent of mass, 127
energy-momentum, 126, 133, 137, 162
big bang, 176 Equivalence Principle, 19, 148–150, 160
black hole, 173 euclidean space, 153
boost, 74 event horizon, 172
events, 3, 201
causality, 66–68
centre-of-mass energy, 136 falling lift, 149
centre-of-momentum frame, 132–136 FitzGerald, George Francis, 25, 96
clock force, 138
hypothesis, 10, 39 frequency 4-vector, 115–118
synchronisation, 10, 29
co-moving reference frame, 112 galilean transformation, 20
instantaneously, see ICRF Galileo, 19
Compton scattering, 139 gamma, 39, 74
conserved quantities, 134 general relativity, 5, 19, 110, 137, 160,
constant quantities, 134 163–176
coordinate system, 3, 154–155 geodesic, 158–159, 167
copernican principle, 175 geometry, 46, 153
cosmological constant, 176 good clock, 11
cosmology, 175–176 GPS, 170, 181
covariance, 22, 134 gravitational redshift, 152, 169, 183
curvature, 156–158, 160 gravitational waves, 173–175
gravity, 136, 160–163
differential geometry, 154, 166 group theory, 75–76
Doppler effect, 115–118
ICRF, 91, 113, 121
Eötvös experiment, 148 inertial frame, 5, 146, 149
Eddington–Dyson eclipse measurements, local, 150
172, 183, 186–193 inner product, 199
Einstein tensor, 162 invariant
Einstein’s equation, 162 interval, 56–61, 77–78, 86

211
212 Index

quantities, 134 and gravity, 163

conception of spacetime, 77
Kuhn, Thomas, 185 first law, 4
second law, 5, 21, 138
Langevin’s traveller, 87 notation
length components, 105
contraction, 32–34, 63, 79 sign of metric, 60
perpendicular, absence of, 37 null cone, 67, 172
measuring, 9–10
proper, see proper length orthogonal, 103, 199
light
clock, 37–42 paradoxes
deflection, 171, 183 pole in the barn, 88
speed of, 27 spaceships, 90
light-metre, 47 twins, 84
lightlike separation, 67 physical units, 47
linear algebra, 107 Poincaré transformation, 76
linearised gravity, see weak field solution Poisson’s equation, 161
local observers, 7 Popper, Karl, 178
Lorentz postulates
group, 76, 97 first, 18–27
transformation, 12, 25, 72–83, 105, 109, 116 second, 27–29
Lorentz, Hendrik Antoon, 25, 96 Pound–Rebka experiment, 152, 183
Principle of Relativity, 18–27
magnitude (of a 4-vector), 108 proper acceleration, 113, 121
mass, 125, 128–129 proper length, 41, 203
and gravity, 136 proper time, 60, 64, 78–79, 86, 111, 169
gravitational, 6 Pythagoras, 56, 139
relativistic (as a bad idea), 126
Maxwell’s equations, 22, 95–98, 179 quantum mechanics, 131, 139
measurement, 7, 9 relativistic, 50, 141, 181
Mercury, 170, 183
metric, 58, 108, 110, 155–158 rapidity, 74
euclidean, 155 reference frames, 3
FLRW, 175 rest frame, 8, 203
Minkowski, 58, 164, 168 rest length, 203
Schwarzschild, 168, 169, 172 rotation, 55, 76, 77
signature of, 61
weak-field, 165, 167 Sagnac effect, 181
Michelson–Morley experiments, 25, 175, 179 scalar, 111, 125
Minkowski scalar product, 108
diagram, 50–55, 63, 66, 101 Schwarzschild
space, 51 metric, see metric, Schwarzschild
Minkowski, Hermann, 51, 77 radius, 168
momentum solution, 167–173, 184
4-vector, 125 sign convention, 60
of photons, 131 simultaneity, 7–9, 32–34, 88
sociology of scientific knowledge, 190
natural units, 46–50, 83, 138 spacelike separation, 67
Newton spacetime, 45, 77–78
and gravitation, 6 standard configuration, 12–13, 45, 60, 72, 74,
 Index 213

200, 202

TAI, see time, atomic

tensors, 110
time
atomic, 11, 48, 170
coordinate, 164
dilation, 32–34, 63
gravitational, 169
timelike separation, 67

upper limit to speed, 94, 178

vectors
displacement, 104, 105
transformation of, 106
velocity
addition of, 75
vector, 111–113

weak-field solution, 164–167

worldlines, 50, 66