mcglinn w.d. introduction to relativity

Introduction to Relativity
Introduction to Relativity
William D. McGlinn
The Johns Hopkins University Press
Baltimore and London
© 2003 The Johns Hopkins University Press
All rights reserved. Published 2003
Printed in the United States of America on acid-free paper
9 8 7 6 5 4 3 2 1
The Johns Hopkins University Press
2715 North Charles Street
Baltimore, Maryland 21218-4363
www.press.jhu.edu
Library of Congress Cataloging-in-Publication Data
McGlinn, William D.
Introduction to relativity/ William D. McGlinn.
p. cm.
Includes bibliographical references and index.
ISBN 0-8018-7047-X (hc. : alk. paper) ISBN 0-8018-7053-4
(pbk. : alk. paper)
1. Relativity (Physics) I. Title.
QC173.55 .M38 2002
530.11 dc21
2002070073
A catalog record for this book is available from the British
Library.
To Louise
Contents
Preface xi
1 Foundations of Special Relativity 1

1.1 Introduction 1
1.2 Kinematics: The Description of Motion 1
1.3 Newtonian Mechanics and Galilean Relativity 3
1.4 Maxwell’s Equations and Light Propagation 7
1.5 Special Relativity: Einsteinian Relativity 10
1.5.1 Lorentz Transformation 10
1.5.2 Lorentz Transformation of Velocities 12
1.5.3 Lorentz Transformation with Arbitrary Relative
Velocity 13
1.6 Exercises 15
2 Geometry of Space-Time 17
2.1 Introduction 17
2.2 Invariant Length for Rotation and Euclidean
Transformations 17
2.3 Invariant Interval for Lorentz and Poincaré
Transformations 18
2.4 Space-Time Diagrams 19
2.4.1 Causality 21
2.4.2 Longest Elapsed Proper Time between
Two Events: The Twin Paradox 23
2.4.3 Length Contraction 26
2.4.4 Time Dilation 26
2.4.5 Doppler Shift 27
2.5 Vectors and Scalars 30
2.5.1 Euclidean Vectors and Scalars 30
2.5.2 Lorentzian Vectors and Scalars 31
2.5.3 The Doppler Shift Revisited 33
vii
2.6 Rotation and Lorentz Transformations as Groups 34
2.7 Exercises 35

3 Relativistic Dynamics 39
3.1 Introduction 39
3.2 Momentum in Galilean Relativity 40
3.3 Momentum-Energy in Einsteinian Relativity 41
3.4 The Geometry of the Energy-Momentum Four-Vector 44
3.4.1 “Elastic” Collisions 46
3.4.2 “Inelastic” Collisions 47
3.4.3 Particle Production 47
3.5 Relativistic Form of Newton’s Force Law 48
3.6 Dynamics of a Gyroscope 48
3.7 Exercises 50
4 Relativity of Tensor Fields 53
4.1 Introduction 53
4.2 Transformations of Tensors 53
4.2.1 Three-Tensors 53
4.2.2 Four-Tensors 55
4.3 Relativity of Maxwell’s Equations 58
4.4 Dynamics of a Charged Spinning Particle 61
4.5 Local Conservation and Gauss’s Theorem 63
4.6 Energy-Momentum Tensor 65
4.6.1 Energy-Momentum Tensor of Dust 67
4.6.2 Energy-Momentum Tensor of a Perfect Fluid 68
4.6.3 Energy-Momentum Tensor of the
Electromagnetic Field 72
4.6.4 Total Energy-Momentum Tensor of Charged
Dust and Electromagnetic Field 73
4.7 Exercises 74
5Gravitation and Space-Time 77
5.1 Introduction 77
5.2 Gravitation and Light 77

5.3 Geometry Change in the Presence of Gravity 81
5.4 Deflection of Light in a Gravitational Field 83
6 General Relativity 87
6.1 Introduction 87
6.2 Tensors of General Coordinate Transformations 90
viii Contents
6.3 Path of Freely Falling Particles: Timelike Geodesics 92
6.4 Covariant Differentiation 94
6.5 Parallel Transport: Curvature Tensor 96
6.6 Bianchi Identity and Ricci and Einstein Tensors 102
6.7 The Einstein Field Equations 104
6.8 The Cosmological Constant 107
6.9 Energy-Momentum Tensor of a Perfect Fluid in
General Relativity 108
6.10 Exercises 110
7Static Spherical Metrics and Their Applications 113
7.1 Introduction 113
7.2 The Static Spherical Metric 114
7.3 The Schwarzschild Solution 116
7.4 Gravitational Redshift 117
7.5 Conserved Quantities 118
7.6 Geodesic Motion for a Schwarzschild Metric 120
7.6.1 Gravitational Deflection of Light 123
7.6.2 Precession of the Perihelia of Orbits 125
7.7 Orbiting Gyroscopes in General Relativity 128
7.8 Stellar Interiors 131
7.8.1 Constant Density Newtonian Star 134
7.8.2 Constant Density Relativistic Star 135
7.9 Black Holes 136
7.10 Exercises 139

8 Metrics with Symmetry 143
8.1 Introduction 143
8.2 Metric Automorphisms 143
8.3 Killing Vectors 145
8.3.1 Conserved Momentum 146
8.4 Maximally Symmetric Spaces 146
8.5 Maximally Symmetric Two-Dimensional
Riemannian Spaces 153
8.5.1 Two-Dimensional Space Metric of Positive
Curvature 154
8.5.2 Two-Dimensional Space Metric with Zero
Curvature (Flat) 155
8.5.3 Two-Dimensional Space Metric with Negative
Curvature 155
Contents ix
8.6 Maximally Symmetric Three-Dimensional
Riemannian Spaces 156
8.6.1
k 0=
159
8.6.2
k 1=+
159
8.6.3
k 1=-
159
8.7 Maximally Symmetric Four-Dimensional
Lorentzian Spaces 160
8.8 Exercises 161
9 Cosmology 163

9.1 Introduction 163
9.2 The Robertson-Walker Metric 165
9.3 Kinematics of the Robertson-Walker Metric 167
9.3.1 Proper Distance 167
9.3.2 Particle Horizons 168
9.3.3 Event Horizons 170
9.3.4 Cosmological Redshift: Hubble’s
Constant 171
9.3.5 Luminosity Distance 173
9.3.6 Cosmological Redshift: Deceleration
Parameter 174
9.4 Dynamics of the Robertson-Walker Metric 175
9.4.1 Critical Density 179
9.4.2 Cosmological Redshift: Distant Objects 180
9.4.3 Cosmological Dynamics with ␭ϭ0 182
9.4.4 Cosmological Dynamics with ␭ ≠ 0 186
9.5 The Early Universe 187
9.5.1 The Cosmic Microwave Background
Radiation 188
9.5.2 Inflation 189
9.5.3 Cosmic Microwave Background and
Cosmological Parameters 193
9.6 Exercises 196
Suggested Additional Reading 197
References 199
Index 201
xContents
Preface
This book provides an introduction to the theory of relativity, both
special and general, to be used in a one-term course for undergradu-

ate students, mainly physics and math majors, early in their studies. It
is important that students have a good understanding of relativity to
appreciate the unifying principles and constraints it brings to both
classical and quantum theories. The understanding of such a beauti-
ful subject can bring a great deal of satisfaction and excitement to the
student.
The history of both special and general relativity is given short
shrift in the book, however. I believe that an initial study of relativity
is best done by a straightforward, linear development of the theory,
without the twists and turns that are inevitably a part of a theory’s
history. With such an approach, students can understand and appre-
ciate the structure of the resulting theory more quickly than if the
historical path is followed. The historical path has an interest and
value of its own, but the illumination of this path is best done by histo-
rians of science. Therefore, for example, I have not discussed Mach’s
principle—-the principle that inertial frames are determined by the
distribution of mass in the universe. Undoubtedly, Einstein was influ-
enced by Mach, but in the end the answer to the question “Does
Einstein’s general relativity obey Mach’s principle?” is elusive.
The target audience imposes constraints on what material is
included and on the level of sophistication, especially mathematical
sophistication, employed. I assume the student has had an introduc-
tory course in physics, a knowledge of basic calculus, including simple
differential equations and partial derivatives, and linear algebra,
including vectors and matrices. In addition, it is useful, though not
necessary, that students be able to use a symbolic mathematical
program such as Mathematica or Maple. Most physics majors at
American universities have the required background at the begin-
ning of their third year, many as much as a year earlier. With such
students in mind, I use the traditional tensor index notation and not

xi
the coordinate-free notation of modern differential geometry.
Though the modern notation gives deeper and quicker insight into
the structure of the geometry of space-time, for a first introduction I
think the traditional tensor index notation is preferable—-and this
notation is almost always used in calculations. Also, by the use of
symbolic programs students can do calculations that would be some-
what prohibitive without them and, thus, develop understanding by
way of calculation.
An author of such a textbook faces a choice as to how and when to
introduce the required new mathematics, the most important “new,”
in this case, being tensor analysis (or differential geometry). Should
one introduce the mathematics separately so that it stands alone, or
should one introduce it within the context of the physics to be
described, whereby it can be physically motivated? I do both to some
degree. In the study of special relativity, I introduce the concept of
the space-time metric and associated tensors within the context of the
physics and in analogy with the comparable entities in three-dimen-
sional Euclidean space of which students would be expected to have
some understanding. The concepts of a metric and associated tensors
under the general coordinate transformations, required in the theory
of general relativity, are introduced somewhat abstractly but are
quickly tied to the metric of space-time and physical tensors. This
path entails a loss of rigor in the discussion. At times, an appeal is
made to the reasonableness of a result rather than giving rigorous
arguments: it is clear in the context when this is done. One mathe-
matical subject is presented in a “stand alone” manner: in Chapter 8,
symmetries of a metric, called isometries, are characterized by the exis-
tence of Killing vectors. This is a subject one might think is unneces-
sary in such a book. However, symmetries of the metric are so

important in studying dynamically conserved quantities and in the
discussion of cosmological models that I feel this introduction to
Killing vectors is useful.
The book is roughly divided into two parts. The first part, Chapters
1 through 4, concerns special relativity. General relativity, including a
chapter on cosmology, is covered in Chapters 6 through 9. Given the
exciting theoretical and observational work being done at the present
time, the student should find the study of the cosmology chapter
particularly satisfying. Chapter 5 is a reading of Einstein’s first
attempt at incorporating the equivalence principle into physics,
thereby predicting the bending of light rays by the gravitational field
of massive bodies, a reading in which the result is viewed as a change
in the Minkowskian geometry of space-time. It serves as a physical
motivation to changes introduced in his general theory of relativity
xii Preface
and is used to relate the coupling constant introduced in Einstein’s
field equations to Newton’s gravitational constant.
Except for Chapter 5, there are exercises associated with each
chapter. The number of exercises are few enough so that it is not
unreasonable to expect the student to do all of them. The impor-
tance of students doing these exercises cannot be emphasized
enough. Exercises are particularly important in the study of relativity.
The student is susceptible to thinking the material is understood
because he or she can “read” the equations. The understanding
comes only through serious thought and by working out problems.
Furthermore, some important results are contained only in the exer-
cises.
A word about units. A unit of time is introduced so that the speed
of light has value one. (In such a set of units, velocities are dimen-
sionless.) This is the only change from standard units, such as MKS

units. I do not go “all the way” by defining the Planck units wherein
the velocity of light, Newton’s gravitational constant G, and Planck’s
constant h are all chosen to have value one. Such units are particu-
larly useful when studying quantum gravity, a subject we are, most
definitely, not considering here.
I am grateful to my late friend, Jim Cushing, and to Steven Shore,
who both took the time to read an early version of the notes upon
which this book is based and to suggest improvements. Jim’s encour-
agement was instrumental in my decision to put the notes in book
form suitable for publication.
Preface xiii
Introduction to Relativity
Chapter 1
Foundations of Special
Relativity
1.1 Introduction
Before studying Einstein’s modification of the scientist’s view of space
and time as held at the beginning of the last century, we will take a
quick look at the theories of physical phenomena that were operative
at that time, for the space-time concepts of scientists were shaped by
these theories.
At that time, an overall aim of physicists was to give a mechanical
explanation of physical phenomena based on Newton’s laws of
motion. Theories existed for two basic forces: Newton’s theory for the
gravitational force, and Maxwell’s theory for the electromagnetic
force.
A relativity principle, recognized by Newton and, before him, by
Galileo, applies in Newton’s mechanics. With the advent of electro-
magnetic theory, as incorporated in Maxwell’s equations, it seemed
that the relativity of Newtonian mechanics could not be the

relativity of Maxwell’s equations. We study this (Galilean) relativity
of Newtonian mechanics to understand its conflict with the
(Einsteinian) relativity of Maxwell’s equations.
1.2 Kinematics: The Description of Motion
The basic question answered (differently) by Galilean and
Einsteinian relativity is, “How do different observers see the same
motion?” To address this question, one has to be able to describe the
1
motion of particles quantitatively. A notion prior to motion is that of
an event, that is, a position and a time, for motion is a sequence of
events. To characterize the position of an event, an observer must
choose a coordinate system. One can imagine constructing a lattice
frame of orthogonal (micro) meter sticks so that the position of an
event that occurs at the intersection of three sticks has coordinates
(
,,)(,,)ml nl ol x y z=
, where
l
is the length of the sticks, and
(,,)mno
are
integers. The position
(, , )000
is referred to as the origin of the coor-
dinate system and is not a special point, if the space is homogeneous.
We assume one can place the origin of the coordinates at any point
in space without prejudice and can pick the mutually orthogonal axis
pointing in any direction. This is clearly an assumption that space is
homogeneous and isotropic. Of course, an event may not occur
precisely at an intersection, but one can choose the length

l
as small
as one wishes for the accuracy required. Further, we assume that the
space is Euclidean (flat) and of infinite extent, that there is a distance
between points in the space
(, ,)xyz
111
and
(, ,)xyz
222
given by
()()()dxx yy zz
2
12
2
12
2
12
2
=- +- +-
, and that this distance does not
depend on the choice of origin or the direction of the chosen axis.
This Euclidean nature of space is clearly an assumption, one that is
relaxed in Einstein’s theory of general relativity. (Imagine trying to
set up a two-dimensional Euclidean lattice on a two-dimensional
sphere.)
The lattice permits one to record the position of an event. How
might the time of an event be characterized? One would like to have
a set of “synchronized” standard clocks at each lattice intersection so
that an event’s time measurement can be recorded locally. But how

does one synchronize the clocks? Imagine firing “free” particles from
the origin with a known velocity, in a direction of a lattice junction.
Note that the velocity can be measured locally with nonsynchronized
clocks. Knowledge of the space coordinates of the junction, and thus
the distance from the origin, permits the clock at this junction to be
synchronized with the origin clock, if the velocity of the free particle
is constant and independent of direction. Frames relative to which
free particles, particles that have no forces acting on them, move with
constant velocity are a special class called inertial frames. These are
frames in which free particles obey Newton’s first law. In discussing
special relativity, we restrict our description of motion to such inertial
frames.
An event is then characterized by a set of space coordi-
nates and a time coordinate. That is, we have four numbers
(
,,,) ( , , , )txyz x x x x
0123
=
, which we also write as
(, )t r
. The motion of
a particle is described by the position
r
of the particle as a func-
2 Chapter 1. Foundations of Special Relativity
tion of the time
t
—-a sequence of events
(, ())ttr
with the

t
ranging
continuously over some interval. From this parameterization of the
motion, we obtain the instantaneous velocity
/ddtvr=
or, in compo-
nent form,
(, ,)(/,/,/)vvv dx dt dy dt dz dt
xyz
=
, and the acceleration
//ddtd dtav r
22
==
.
1.3 Newtonian Mechanics and Galilean Relativity
Newton’s force law relates the acceleration
a
of a particle with the
force
F
through the equation
mFa=
. (1.1)
Here
m
is a fixed property of the particle, called the (inertial) mass.
The value of the force
F
must be given by some force law if this equa-

tion is not to be merely a definition of
F
in terms of
m
and
a
. This law
implies that a particle under the influence of no force (a free parti-
cle) has
a 0=
, and thus its velocity
v
is constant. The law is valid in an
inertial frame.
How are these inertial frames related? What is the relation between
the descriptions of the motion of a particle referred to in one inertial
frame and another? What is the relation between the force law as
viewed in one inertial frame and another? These questions can be
answered by a knowledge of the transformation (translation) of
events between inertial frames.
Consider two frames: an unprimed inertial frame
S
and a primed
frame
S
l
moving with a constant velocity
v
0
relative to the unprimed

frame. For simplicity we set up the coordinates of the frames in a par-
ticular way. We choose the relative velocity of
S
l
in the
x
direction and
have the origin of the
S
l
frame move along the
x
axis, the origin of the
S
frame move along the
x
l
axis, and set the time of the event when the
1.3 Newtonian Mechanics and Galilean Relativity 3
Figure 1.1 Two inertial frames.
Image not available.
origins agree to be zero for both sets of clocks. This is possible for ho-
mogeneous and isotropic space-times. Furthermore, we will assume
that we can choose the
y
l
and
z
l
axes such that the the locus of all

events with
y 0=
(
z 0=
) agree with that corresponding to
y 0=
l
(
z 0=
l
),
with the
>z 0
(
>y 0
) events agreeing with
>z 0
l
(
>y 0
l
). Figure 1.1 de-
picts a single event “O” with space-time coordinates
(, , , )txyz
and
(
,,,)txyz
llll
as measured in the unprimed and primed frames, respec-
tively. From this figure we see that it is reasonable that events charac-

terized by
x 0=
l
are the set of events characterized by
vxt0
0
-=
in
terms of unprimed coordinates.
We might expect that the translation of event “O” between the two
frames is given by
.
v
tt
xx t
yy
zz
0
=
=-
=
=
l
l
l
l
(1.2)
The first equation assumes that time is universal, that is, that all stan-
dard clocks run synchronized. This translation of events is the
Galilean transformation for the special coordinates chosen.

This derivation of the Galilean transformation is at best ad hoc. We
merely wrote down the transformation equations as one might expect
them to be from viewing Figure 1.1. We now tighten up the derivation
a bit in a manner that can be generalized to the Einsteinian case and
to understand the assumptions hidden in this derivation.
1
For
instance, no mention was made about transforming between inertial
frames—-ones for which a constant velocity observed in one frame
implies a constant velocity observed in the transformed frame. It
should be reasonable to the reader, and it can be shown, that for this
to be so the transformation between the frames must be linear, that
is, of the form
n
xxAT
v 0
3
= +
=
vvnn
!
l
, (1.3)
where
A
no
and
T
n
are constants and

xt
0
/
. We assume this to be true.
We can choose the space coordinates and the time coordinates so
that the event
(, , , )0000
, referred to the primed coordinates, is the
4 Chapter 1. Foundations of Special Relativity
1
The derivations of the Galilean and Lorentz transformations that follow are much
influenced by those in Rindler (1991).
event
(, , , )0000
, referred to the unprimed coordinates, so that
T 0=
n
.
Eq. (1.3) then becomes
n
xxxAA.
v 0
3
= =
=
vv vvnn
!
l
(1.4)
In the last expression here, we have used the Einstein convention that

repeated indexes indicate summation (a convention we will follow
hereafter).
What are these constants
A
no
for the transformation between the
specially chosen coordinates described above? Again, for our choice
of
,xy
ll
and
z
l
axes, the events characterized by
x 0=
l
are characterized
by
vxt0
0
-=
. Thus, we have
(),vxAtAxAyAzAtAxAx t
10 11 12 13 10 11 0
=+ + + =+ =-
l
(1.5)
where
A
is some constant. If one considered the transformation from

the primed coordinates to the unprimed, one would obtain
(),vxAx t
0
=+
ll l
(1.6)
where
A
l
is some constant.
How are
A
l
and
A
related? If one assumes an equality of the two
frames in the sense that no observation can be made that distin-
guishes one frame as being different or special, then
A
l
must equal
A
. For instance, consider an event
O
, say
(, , , )xL000=
o
, simultaneous
with the origin event in the unprimed frame. From Eq. (1.5), for this
event

xAL=
l
. Similarly, a different event
O
l
, say
(, , , ),xL000=
o
l
simul-
taneous with the origin event in the primed frame, from Eq. (1.6),
has
xAL=
l
. The assumed equality of the two frames requires
AA=
l
and thus Eq. (1.6) becomes
()vxAx t
0
=+
ll
. (1.7)
Similarly, for our choice of axes, since events characterized by
y
0
=
l
are characterized by
y 0=

, the relation
yAtAxAyAzAy
20 21 22 23 22
=+ + + =
l
(1.8)
follows. But, if we considered the transformation from the primed to
the unprimed frame, we would obtain
22
,yAy=
ll
(1.9)
1.3 Newtonian Mechanics and Galilean Relativity 5
and, again with the assumed equality of the frames,
AA
22 22
=
l
. With
this, Eqs. (1.8) and (1.9) imply
A 1
22
2
=
. Since we chose our axis such
that
>y 0
corresponds to
22
>,yAA01

22
==
ll
. Similarly, we conclude
that
A 1
33
=
.
We see that a transformation between the inertial frames (linearity
of transformation) with the axes as chosen and with the assumption
that no inertial frame is special assumes the form
()vxAx t
0
=-
l
()vxAx t
0
=+
ll
yy=
l
zz=
l
.tAtAxAyAz
00 01 02 03
=+ + +
l
(1.10)
Without further assumptions—-such as, perhaps, that space is

isotropic—-these relations seem to be as far as we can go. The
assumption of a Newtonian universal time implies that
A 1
00
=
and
,,
A
i0123
i0
==
, and these, with the first two relations of Eq. (1.10),
imply
A 1=
. The relations Eq. (1.10) reduce to
vxx t
0
=-
l
yy=
l
zz=
l
,tt=
l
(1.11)
which are the Galilean transformation, Eq. (1.2), deduced before.
The transformation of velocity components, defined by
/, ,,
v

dx dt i 123
ii
==
for an arbitrarily moving particle, immediately
obtains
x
vvv
x 0
=-
l
y
vv
y
=
l
z
.vv
z
=
l
(1.12)
For the acceleration components,
/, ,,vad dti123
ii
==
, we have
x
aa
x
=

l
y
aa
y
=
l
aa
zz
=
l
. (1.13)
The two different frames record the same acceleration. If
mF
a
=
6 Chapter 1. Foundations of Special Relativity
is valid in the unprimed frame, then
mFa=
ll
is valid in the primed
frame if
.FF=
l
(1.14)
Thus, if Newton’s law of motion is valid in one frame, the law
is valid in all frames that move with constant velocity with respect
to it if (1) the transformation of events is given by a Galilean trans-
formation and if (2) all forces are the same in all such frames. The
forces cannot depend on the relative velocity. If the form of the
law is not changed by certain coordinate transformations, the law is

said to be invariant with respect to the transformations considered.
Newton’s law
mFa=
is invariant with respect to Galilean transforma-
tions.
Newton believed in an absolute space or reference frame and that
inertial frames were those at rest or in uniform motion with respect
to it. However, he recognized the difficulty in discovering which of all
inertial frames is the absolute frame because his laws of motion are
form invariant under change of frames if the transformation law of
events is given by a Galilean transformation and if, under these trans-
formations, the force doesn’t change. Such invariance of all physical
laws between all inertial frames is the principle of relativity. Note that if
this principle is valid, then no experiment could be performed that
would distinguish one particular frame as being special in any way.
(Recall that we used this in the derivation of the Galilean transfor-
mation.) The principle of relativity and the assumption of the exis-
tence of a universal time give rise to Galilean relativity. Force laws, to
be consistent with this relativity, must be such that they give an iden-
tical force in all inertial frames. Since the vector distance between two
simultaneous events is the same in all inertial frames, any law of force
between two bodies that depends only on the vector distance between
the bodies will satisfy this principle. Newton’s gravitational force is
such a law. However, a force law that depends upon both velocities of
the two bodies (and not just on their relative velocity) would violate
this principle.
1.4 Maxwell’s Equations and Light Propagation
In 1861 James Clerk Maxwell, born in Edinburgh in 1831, formulated
mathematical equations that describe electromagnetic phenomena.
These equations predict the existence of electromagnetic waves that

travel with a velocity that could be calculated from the theory—-calcu-
lated in terms of a parameter of the theory that was experimentally
1.4 Maxwell’s Equations and Light Propagation 7
determined by measuring the electric force between charges (or the
magnetic force between currents). The calculated velocity agreed
with that of light as determined by the Danish astronomer Olaus
Roemer in 1675 by observing the time lag of the eclipses of Jupiter’s
moons—-a remarkable agreement that could only imply that light is
an electromagnetic wave.
In spite of the great successes of Maxwell’s equations, most physi-
cists (including Maxwell himself ) demanded a mechanical explana-
tion of these waves and, thus, the existence of a medium, called
“ether,” whose mechanical vibration constituted electromagnetic
waves. The speed of light predicted by Maxwell’s equations would be
the speed in this all-encompassing ether. The principle of Galilean
relativity is clearly in trouble. If the ether were a valid concept, it
would provide the “absolute” reference frame, thus violating the prin-
ciple; if Maxwell’s equations are valid in all inertial frames, and thus
the velocity of light is the same in all inertial frames, the principle of
Galilean relativity, which predicts the simple addition of velocities, is
invalid.
Irrespective of the existence of the ether, it was crucial to detect the
motion of the earth with respect to the frame in which the speed of
light has a fixed standard value, independent of the direction of
propagation. The most famous attempt to detect this motion was an
experiment performed by two Americans, Albert A. Michelson and
Edward W. Morley. Michelson first performed the experiment in
1881 and, in collaboration with Morley, again in 1887, with more
sensitivity.
Michelson’s technique for measuring the motion of the earth

through the ether was to split a beam of light in two, send the
split beams in two mutually perpendicular directions for approxi-
mately equal distances (
L
in Fig. 1.2), reflect them back to a
common point, and measure the difference in travel times of the two.
In the figure, the ether is assumed to have a velocity relative to the
earth of magnitude
v
e
, which is assumed to be perpendicular to the
direction traveled by the light in one path. One can show (see
Exercise 1) that the expected difference in travel time
td
of the two
beams is
.
v
t
c
L
c
e
2
.d
cm
(1.15)
The difference in arrival times for a reasonable length
L
is very small

—-it is proportional to the square of
/v c
e
. If
v
e
is taken to be the orbit
8 Chapter 1. Foundations of Special Relativity
speed of the earth around the sun, about 30 km/s,
/v c 10
e
4
.
-
, which
when squared is a very small number indeed. Michelson did not
measure the difference in arrival times directly. Rather, he measured
the shift in the interference pattern as the apparatus was slowly
rotated. That is, as the apparatus was rotated the arms would inter-
change the roles of lying parallel and perpendicular to the supposed
flow of the ether, thus changing the difference in arrival times of the
beams traversing the two arms, resulting in a shift in the fringes of the
interference pattern. The total change in the lag time is twice that of
Eq. (1.15). A change of one period
x
corresponds to a shift of one
fringe! The number of fringes
n
shifted when the apparatus is rotated
by

90
o
is thus
n
tct22
==
x
d
m
d
. .
v
L
c
2
e
2
m
cm
Here
m
is the wavelength of the light used. The 1887 experiment of
Michelson and Morley had
L 11.
m, which with
510
7
#=m
-
m, a typi-

cal wavelength of visible light, yields
.n 4.
. This is quite detectable—
Michelson and Morley had estimated they could detect
.n 01=
.
However, the experiment gave a null result. No fringe shift was
detected.
1.4 Maxwell’s Equations and Light Propagation 9
Figure 1.2 Michelson-Morley experimental setup.
Image not available.
1.5 Special Relativity: Einsteinian Relativity
It seems that Einstein was not influenced greatly by the Michelson-
Morley experiment although he probably knew of the null result.
Rather, he doubted the existence of an absolute frame of reference
that the Michelson-Morely experiment was attempting to detect. In
his later years, Einstein recalled that as a boy of sixteen he wondered
how a light wave would appear if one were moving along with it.
2
Galilean relativity would predict a static, nonoscillatory wave. But
such a static wave does not satisfy Maxwell’s equations if the parameters
that define the theory do not change from one frame to another. After all,
these parameters fix the speed of the wave. Thus, if there is a relativ-
ity principle and Maxwell’s equations are correct, then Galilean rela-
tivity and Newton’s mechanics are wrong.
In his remarkable paper of 1905 titled “On the Electrodynamics of
Moving Bodies,”
3
Einstein broke with Galilean relativity and its
concomitant view of space and time. In this paper he enunciated two

postulates:
1. The principle of relativity: All physical laws have the same form
(i.e., they are invariant) in all inertial reference frames.
2. The speed of light is the same in all inertial reference frames.
Again, inertial frames are those in which isolated particles, those
with no “force” acting on them, move with constant velocity. The first
postulate implies that no measurement can be made that distin-
guishes one frame as being special or different from any other frame.
The second postulate of Einstein, the invariance of the speed of
light to all inertial observers, immediately contradicts Galilean trans-
formations of events, since such transformations imply that no speed
is invariant. Special relativity is a study of transformations of events,
called Lorentz transformations, that are consistent with the invariance of
the speed of light.
1.5.1 Lorentz Transformation
We now derive the form of the transformation between the two iner-
tial frames depicted in Figure 1.1. Because the speed of light is
10 Chapter 1. Foundations of Special Relativity
2
See Autobiographical Notes in Schilpp (1949), p.53. For a view questioning the accuracy
of Einstein’s recollection, see Bernstein (1973), p.38.
3
Annalen der Physik 17, 891 (1905), reprinted (in English translation) in Einstein et al.
(1923).
special—-we will see it is the only speed that has the same value for all
inertial observers—-it is useful to introduce a time unit that reflects
this special role. The time unit used is the amount of time it takes
light to travel a distance unit. Thus, if we measure distance in meters,
the time unit will be “one meter of light travel time.” In such a system
of units, a velocity is dimensionless since time and distance have the

same units. When a velocity is expressed in terms of dimensionless
units we will use the symbol
b
for velocity.
||1=b
for light. Note also
that we usually indicate the time coordinate of an event by “
x
0
” rather
then “
t
.”
Again, the transformation of events between the two frames is
linear, and we choose the space coordinates and the time coordinates
so that the “origin” events
( ,,,)0000
of the two coordinates are the
same event:
n
.xAx=
no o
l
(1.16)
As before, the events characterized by
x 0
1
=
l
are characterized by

x
x 0
r
10
-=b
, where
r
b
is the velocity of the primed coordinate with
respect to the unprimed. If one applies the principle of relativity as
before, we again obtain Eqs. (1.10):
2
()
()
,
xAx x
xAx x
xx
xx
xAxAxAxAx
r
r
110
110
2
33
0000011022 033
=-
=+
=

=
=+++
b
b
l
ll
l
l
l
(1.17)
where
A
is some constant dependent on
r
b
.
Now, we do not assume a Newtonian universal time
xx
00
=
l
as we
did before, and do not obtain
A 1=
. However, consider a sequence of
events of a light pulse traveling in the
x
1
+
(and thus

x
1
+
l
) direction
whose emission was the origin event
(, ,,)0000
. These events are char-
acterized (for light
1=b
) by
xx
10
=
and
xx
10
=
ll
. Using these relations
in the first two equations of Eqs. (1.17), we obtain
()()xAx x A xx1
rr
110 00
=- =- =bb
ll
0
()().xAx x A x x1
rr
11 0 0

=+ =+ =bb
ll l
(1.18)
From this it follows that
().A 1
/
r
2
12
=-b
-
(1.19)
1.5 Special Relativity: Einsteinian Relativity 11
The positive square root is chosen since we assumed the axes were
chosen so that a light pulse traveling in the
x
1
+
direction travels in the
x
1
+
l
direction. Furthermore, since the first two equations of Eqs.
(1.17) are valid for the transformation of any event , we can solve for
x
0
l
in terms of
x

0
and
x
1
, and the last of Eqs. (1.17) becomes
()( ).xxx1
/
rr
0
2
12
10
=- - +bb
-
l
(1.20)
The Lorentz transformation of events between these two frames is
given by
()xxx
r
010
=- +cb
l
()xxx
r
110
=-cb
l
2
xx

2
=
l
3
.xx
3
=
l
(1.21)
Here
()1
/
r
2
12
=-cb
-
, a standard notation. We refer to such a Lorentz
transformation, that is Eq. (1.21), as canonical. Solving these equa-
tions for the
x
n
in terms of
n
x
l
, thus obtaining the inverse transfor-
mation, one finds
0
()xxx

r
01
=+cb
ll
()xx x
r
11 0
=+cb
ll
xx
22
=
l
.
3
xx
3
=
l
(1.22)
These equations can also be obtained from Eqs. (1.21) by substituting
n
x
x)
n
l
and
rr
" -bb
.

1.5.2 Lorentz Transformation of Velocities
First note, because the Lorentz transformations are linear, the differ-
ence in the space-time coordinates of two events transforms in the
same way as the coordinates of a single event, that is,
()xxx
r
010
=- +cb⌬⌬⌬
l
()xxx
r
110
=-cb⌬⌬⌬
l
2
xx
2
=⌬⌬
l
.xx
33
=⌬⌬
l
(1.23)
12 Chapter 1. Foundations of Special Relativity