THE SPECIAL THEORY OF RELATIVITY

THE SPECIAL THEORY OF RELATIVITY
Lecture Notes prepared by
J D Cresser
Department of Physics
Macquarie University
July 31, 2003

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CONTENTS
1
Contents
1
Introduction: What is Relativity?
2
2
Frames of Reference
5
2.1
A Framework of Rulers and Clocks
. . . . . . . . . . . . . . . . . . . . . .
5
2.2
Inertial Frames of Reference and Newton’s First Law of Motion
. . . . . .
7
3
The Galilean Transformation
7
4
Newtonian Force and Momentum
9
4.1
Newton’s Second Law of Motion
. . . . . . . . . . . . . . . . . . . . . . . .
9
4.2
Newton’s Third Law of Motion . . . . . . . . . . . . . . . . . . . . . . . . .
10
5
Newtonian Relativity
10
6
Maxwell’s Equations and the Ether
11
7
Einstein’s Postulates
13
8
Clock Synchronization in an Inertial Frame
14
9
Lorentz Transformation
16
9.1
Length Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
9.2
Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
9.3
Simultaneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
9.4
Transformation of Velocities (Addition of Velocities) . . . . . . . . . . . . .
24
10 Relativistic Dynamics
27
10.1 Relativistic Momentum
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
10.2 Relativistic Force, Work, Kinetic Energy . . . . . . . . . . . . . . . . . . . .
29
10.3 Total Relativistic Energy
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
10.4 Equivalence of Mass and Energy . . . . . . . . . . . . . . . . . . . . . . . .
33
10.5 Zero Rest Mass Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
11 Geometry of Spacetime
35
11.1 Geometrical Properties of 3 Dimensional Space . . . . . . . . . . . . . . . .
35
11.2 Space Time Four Vectors
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
11.3 Spacetime Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
11.4 Properties of Spacetime Intervals . . . . . . . . . . . . . . . . . . . . . . . .
41

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1
INTRODUCTION: WHAT IS RELATIVITY?
2
1
Introduction: What is Relativity?
Until the end of the 19th century it was believed that Newton’s three Laws of Motion
and the associated ideas about the properties of space and time provided a basis on
which the motion of matter could be completely understood. However, the formulation
by Maxwell of a unified theory of electromagnetism disrupted this comfortable state of
affairs – the theory was extraordinarily successful, yet at a fundamental level it seemed to
be inconsistent with certain aspects of the Newtonian ideas of space and time. Ultimately,
a radical modification of these latter concepts, and consequently of Newton’s equations
themselves, was found to be necessary. It was Albert Einstein who, by combining the
experimental results and physical arguments of others with his own unique insights, first
formulated the new principles in terms of which space, time, matter and energy were to
be understood. These principles, and their consequences constitute the Special Theory
of Relativity. Later, Einstein was able to further develop this theory, leading to what
is known as the General Theory of Relativity. Amongst other things, this latter theory
is essentially a theory of gravitation. The General Theory will not be dealt with in this
course.
Relativity (both the Special and General) theories, quantum mechanics, and thermody-
namics are the three major theories on which modern physics is based. What is unique
about these three theories, as distinct from say the theory of electromagnetism, is their
generality. Embodied in these theories are general principles which all more specialized or
more specific theories are required to satisfy. Consequently these theories lead to general
conclusions which apply to all physical systems, and hence are of enormous power, as well
as of fundamental significance. The role of relativity appears to be that of specifying the
properties of space and time, the arena in which all physical processes take place.
It is perhaps a little unfortunate that the word ‘relativity’ immediately conjures up
thoughts about the work of Einstein. The idea that a principle of relativity applies to
the properties of the physical world is very old: it certainly predates Newton and Galileo,
but probably not as far back as Aristotle. What the principle of relativity essentially
states is the following:
The laws of physics take the same form in all frames of reference moving with
constant velocity with respect to one another.
This is a statement that can be given a precise mathematical meaning: the laws of physics
are expressed in terms of equations, and the form that these equations take in different
reference frames moving with constant velocity with respect to one another can be cal-
culated by use of transformation equations – the so-called Galilean transformation in the
case of Newtonian relativity. The principle of relativity then requires that the transformed
equations have exactly the same form in all frames of reference, in other words that the
physical laws are the same in all frames of reference.
This statement contains concepts which we have not developed, so perhaps it is best at
this stage to illustrate its content by a couple of examples. First consider an example
from ‘everyday experience’ – a train carriage moving smoothly at a constant speed on a
straight and level track – this is a ‘frame of reference’. Suppose that in this carriage is a
pool table. If you were a passenger on this carriage and you decided to play a game of
pool, one of the first things that you would notice is that in playing any shot, you would
have to make no allowance whatsoever for the motion of the train. Any judgement of
how to play a shot as learned by playing the game back home, or in the local pool hall,

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1
INTRODUCTION: WHAT IS RELATIVITY?
3
would apply equally well on the train, irrespective of how fast the train was moving. If
we consider that what is taking place here is the innate application of Newton’s Laws to
describe the motion and collision of the pool balls, we see that no adjustment has to be
made to these laws when playing the game on the moving train.
This argument can be turned around. Suppose the train windows are covered, and the
carriage is well insulated so that there is no immediate evidence to the senses as to whether
or not the train is in motion. It might nevertheless still be possible to determine if the
train is in motion by carrying out an experiment, such as playing a game of pool. But,
as described above, a game of pool proceeds in exactly the same way as if it were being
played back home – no change in shot-making is required. There is no indication from
this experiment as to whether or not the train is in motion. There is no way of knowing
whether, on pulling back the curtains, you are likely to see the countryside hurtling by,
or to find the train sitting at a station. In other words, what the principle of relativity
means is that it is not possible to determine whether or not the train carriage is moving.
This idea can be extended to encompass other laws of physics. To this end, imagine a
collection of spaceships with engines shut off all drifting through space. Each space ship
constitutes a ‘frame of reference’, an idea that will be better defined later. On each of
these ships a series of experiments is performed: a measurement of the half life of uranium,
a measurement of the outcome of the collision of two billiard balls, an experiment in
thermodynamics, e.g. a measurement of the specific heat of a substance, a measurement
of the speed of light radiated from a nearby star: any conceivable experiment. If the
results of these experiments are later compared, what is found is that in all cases (within
experimental error) the results are identical. In other words, the various laws of physics
being tested here yield exactly the same results for all the spaceships, in accordance with
the principle of relativity.
Thus, quite generally the principle of relativity means that it is not possible, by considering
any physical process whatsoever, to determine whether or not one or the other of the
spaceships is ‘in motion’. The results of all the experiments are the same on all the space
ships, so there is nothing that definitely singles out one space ship over any other as being
the one that is stationary. It is true that from the point of view of an observer on any one
of the space ships that it is the others that are in motion. But the same statement can be
made by an observer in any space ship. All that we can say for certain is that the space
ships are in relative motion, and not claim that one of them is ‘truly’ stationary, while the
others are all ‘truly’ moving.
This principle of relativity was accepted (in somewhat simpler form i.e. with respect to
the mechanical behaviour of bodies) by Newton and his successors, even though Newton
postulated that underlying it all was ‘absolute space’ which defined the state of absolute
rest. He introduced the notion in order to cope with the difficulty of specifying with
respect to what an accelerated object is being accelerated. To see what is being implied
here, imagine space completely empty of all matter except for two masses joined by a
spring. Now suppose that the arrangement is rotated, that is, they undergo acceleration.
Naively, in accordance with our experience, we would expect that the masses would pull
apart. But why should they? How do the masses ‘know’ that they are being rotated?
There are no ‘signposts’ in an otherwise empty universe that would indicate that rotation
is taking place. By proposing that there existed an absolute space, Newton was able to
claim that the masses are being accelerated with respect to this absolute space, and hence
that they would separate in the way expected for masses in circular motion. But this
was a supposition made more for the convenience it offered in putting together his Laws
of motion, than anything else. It was an assumption that could not be substantiated, as

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1
INTRODUCTION: WHAT IS RELATIVITY?
4
Newton was well aware – he certainly felt misgivings about the concept! Other scientists
were more accepting of the idea, however, with Maxwell’s theory of electromagnetism for
a time seeming to provide some sort of confirmation of the concept.
One of the predictions of Maxwell’s theory was that light was an electromagnetic wave
that travelled with a speed c ≈ 3 × 108 ms−1. But relative to what? Maxwell’s theory
did not specify any particular frame of reference for which light would have this speed.
A convenient resolution to this problem was provided by an already existing assumption
concerning the way light propagated through space. That light was a form of wave motion
was well known – Young’s interference experiments had shown this – but the Newtonian
world view required that a wave could not propagate through empty space: there must
be present a medium of some sort that vibrated as the waves passed, much as a string
vibrates as a wave travels along it. The proposal was therefore made that space was filled
with a substance known as the ether whose purpose was to be the medium that vibrated
as the light waves propagated through it. It was but a small step to then propose that
this ether was stationary with respect to Newton’s absolute space, thereby solving the
problem of what the frame of reference was in which light had the speed c. Furthermore,
in keeping with the usual ideas of relative motion, the thinking then was then that if you
were to travel relative to the ether towards a beam of light, you would measure its speed
to be greater than c, and less than c if you travelled away from the beam. It then came
as an enormous surprise when it was found experimentally that this was not, in fact, the
case.
This discovery was made by Michelson and Morley, who fully accepted the ether theory,
and who, quite reasonably, thought it would be a nice idea to try to measure how fast
the earth was moving through the ether. They found to their enormous surprise that the
result was always zero irrespective of the position of the earth in its orbit around the sun
or, to put it another way, they measured the speed of light always to be the same value
c whether the light beam was moving in the same direction or the opposite direction to
the motion of the earth in its orbit. In our spaceship picture, this is equivalent to all
the spaceships obtaining the same value for the speed of light radiated by the nearby
star irrespective of their motion relative to the star. This result is completely in conflict
with the rule for relative velocities, which in turn is based on the principle of relativity
as enunciated by Newton and Galileo. Thus the independence of the speed of light on
the motion of the observer seems to take on the form of an immutable law of nature, and
yet it is apparently inconsistent with the principle of relativity. Something was seriously
amiss, and it was Einstein who showed how to get around the problem, and in doing so he
was forced to conclude that space and time had properties undreamt of in the Newtonian
world picture.
All these ideas, and a lot more besides, have to be presented in a much more rigorous
form. The independence of results of the hypothetical experiments described above on the
state of motion of the experimenters can be understood at a fundamental level in terms
of the mathematical forms taken by the laws of nature. All laws of nature appear to have
expression in mathematical form, so what the principle of relativity can be understood as
saying is that the equations describing a law of nature take the same mathematical form in
all inertial frames of reference. It is this latter perspective on relativity that is developed
here, and an important starting point is the notion of a frame of reference.

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2
FRAMES OF REFERENCE
5
2
Frames of Reference
Newton’s laws are, of course, the laws which determine how matter moves through space
as a function of time. So, in order to give these laws a precise meaning we have to specify
how we measure the position of some material object, a particle say, and the time at which
it is at that position. We do this by introducing the notion of a frame of reference.
2.1
A Framework of Rulers and Clocks
First of all we can specify the positions of the particle in space by determining its coor-
dinates relative to a set of mutually perpendicular axes X, Y , Z. In practice this could
be done by choosing our origin of coordinates to be some convenient point and imagining
that rigid rulers – which we can also imagine to be as long as necessary – are laid out from
this origin along these three mutually perpendicular directions. The position of the par-
ticle can then be read off from these rulers, thereby giving the three position coordinates
(x, y, z) of the particle1.
We are free to set up our collection of rulers anywhere we like e.g.
the origin could be
some fixed point on the surface of the earth and the rulers could be arranged to measure
x and y positions horizontally, and z position vertically. Alternatively we could imagine
that the origin is a point on a rocket travelling through space, or that it coincides with
the position of a subatomic particle, with the associated rulers being carried along with
the moving rocket or particle2.
By this means we can specify where the particle is. In order to specify when it is at
a particular point in space we can stretch our imagination further and imagine that in
addition to having rulers to measure position, we also have at each point in space a clock,
and that these clocks have all been synchronized in some way. The idea is that with these
clocks we can tell when a particle is at a particular position in space simply by reading off
the time indicated by the clock at that position.
According to our ‘common sense’ notion of time, it would appear sufficient to have only
one set of clocks filling all of space. Thus, no matter which set of moving rulers we use
to specify the position of a particle, we always use the clocks belonging to this single vast
set to tell us when a particle is at a particular position. In other words, there is only one
‘time’ for all the position measuring set of rulers. This time is the same time independent
of how the rulers are moving through space. This is the idea of universal or absolute time
due to Newton. However, as Einstein was first to point out, this idea of absolute time
is untenable, and that the measurement of time intervals (e.g. the time interval between
two events such as two supernovae occurring at different positions in space) will in fact
differ for observers in motion relative to each other. In order to prepare ourselves for this
possibility, we shall suppose that for each possible set of rulers – including those fixed
relative to the ground, or those moving with a subatomic particle and so on, there are a
different set of clocks. Thus the position measuring rulers carry their own set of clocks
around with them. The clocks belonging to each set of rulers are of course synchronized
with respect to each other. Later on we shall see how this synchronization can be achieved.
1Probably a better construction is to suppose that space is filled with a scaffolding of rods arranged in
a three dimensional grid.
2If, for instance, the rocket has its engines turned on, we would be dealing with an accelerated frame
of reference in which case more care is required in defining how position (and time) can be measured in
such a frame. Since we will ultimately be concerning ourselves with non-accelerated observers, we will not
concern ourselves with these problems. A proper analysis belongs to General Relativity.

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2
FRAMES OF REFERENCE
6
The idea now is that relative to a particular set of rulers we are able to specify where a
particle is, and by looking at the clock (belonging to that set of rulers) at the position of
the particle, we can specify when the particle is at that position. Each possible collection
of rulers and associated clocks constitutes what is known as a frame of reference or a
reference frame.
Z
Y
X
Figure 1: Path of a particle as measured in a frame of reference. The clocks indicate the times at
which the particle passed the various points along the way.
In many texts reference is often made to an observer in a frame of reference whose job
apparently is to make various time and space measurements within this frame of reference.
Unfortunately, this conjures up images of a person armed with a stopwatch and a pair
of binoculars sitting at the origin of coordinates and peering out into space watching
particles (or planets) collide, stars explode and so on. This is not the sense in which the
term observer is to be interpreted. It is important to realise that measurements of time are
made using clocks which are positioned at the spatial point at which an event occurs. Any
centrally positioned observer would have to take account of the time of flight of a signal
to his or her observation point in order to calculate the actual time of occurrence of the
event. One of the reasons for introducing this imaginary ocean of clocks is to avoid such
a complication. Whenever the term observer arises it should be interpreted as meaning
the reference frame itself, except in instances in which it is explicitly the case that the
observations of an isolated individual are under consideration.
If, as measured by one particular set of rulers and clocks (i.e. frame of reference) a particle
is observed to be at a position at a time t (as indicated by the clock at (x, y, z)), we can
summarize this information by saying that the particle was observed to be at the point
(x, y, z, t) in space-time. The motion of the particle relative to this frame of reference
would be reflected in the particle being at different positions (x, y, z) at different times
t. For instance in the simplest non-trivial case we may find that the particle is moving
at constant speed v in the direction of the positive X axis, i.e. x = vt. However, if
the motion of the same particle is measured relative to a frame of reference attached to
say a butterfly fluttering erratically through the air, the positions (x , y , z ) at different
times t (given by a series of space time points ) would indicate the particle moving on an
erratic path relative to this new frame of reference. Finally, we could consider the frame
of reference whose spatial origin coincides with the particle itself. In this last case, the
position of the particle does not change since it remains at the spatial origin of its frame
of reference. However, the clock associated with this origin keeps on ticking so that the
particle’s coordinates in space-time are (0, 0, 0, t) with t the time indicated on the clock at
the origin, being the only quantity that changes. If a particle remains stationary relative

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3
THE GALILEAN TRANSFORMATION
7
to a particular frame of reference, then that frame of reference is known as the rest frame
for the particle.
Of course we can use frames of reference to specify the where and when of things other
than the position of a particle at a certain time. For instance, the point in space-time
at which an explosion occurs, or where and when two particles collide etc., can also be
specified by the four numbers (x, y, z, t) relative to a particular frame of reference. In fact
any event occurring in space and time can be specified by four such numbers whether it
is an explosion, a collision or the passage of a particle through the position (x, y, z) at the
time t. For this reason, the four numbers (x, y, z, t) together are often referred to as an
event.
2.2
Inertial Frames of Reference and Newton’s First Law of Motion
Having established how we are going to measure the coordinates of a particle in space and
time, we can now turn to considering how we can use these ideas to make a statement
about the physical properties of space and time. To this end let us suppose that we have
somehow placed a particle in the depths of space far removed from all other matter. It
is reasonable to suppose that a particle so placed is acted on by no forces whatsoever 3.
The question then arises: ‘What kind of motion is this particle undergoing?’ In order to
determine this we have to measure its position as a function of time, and to do this we have
to provide a reference frame. We could imagine all sorts of reference frames, for instance
one attached to a rocket travelling in some complicated path. Under such circumstances,
the path of the particle as measured relative to such a reference frame would be very
complex. However, it is at this point that an assertion can be made, namely that for
certain frames of reference, the particle will be travelling in a particularly simple fashion
– a straight line at constant speed. This is something that has not and possibly could
not be confirmed experimentally, but it is nevertheless accepted as a true statement about
the properties of the motion of particles in the absences of forces. In other words we can
adopt as a law of nature, the following statement:
There exist frames of reference relative to which a particle acted on by no forces
moves in a straight line at constant speed.
This essentially a claim that we are making about the properties of spacetime. It is also
simply a statement of Newton’s First Law of Motion. A frame of reference which has this
property is called an inertial frame of reference, or just an inertial frame.
Gravity is a peculiar force in that if a reference frame is freely falling under the effects of
gravity, then any particle also freely falling will be observed to be moving in a straight line
at constant speed relative to this freely falling frame. Thus freely falling frames constitute
inertial frames of reference, at least locally.
3
The Galilean Transformation
The above argument does not tell us whether there is one or many inertial frames of
reference, nor, if there is more than one, does it tell us how we are to relate the coordinates
3It is not necessary to define what we mean by force at this point. It is sufficient to presume that if
the particle is far removed from all other matter, then its behaviour will in no way be influenced by other
matter, and will instead be in response to any inherent properties of space (and time) in its vicinity.

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3
THE GALILEAN TRANSFORMATION
8
of an event as observed from the point-of-view of one inertial reference frame to the
coordinates of the same event as observed in some other. In establishing the latter, we
can show that there is in fact an infinite number of inertial reference frames. Moreover,
the transformation equations that we derive are then the mathematical basis on which it
can be shown that Newton’s Laws are consistent with the principle of relativity. To derive
these transformation equations, consider an inertial frame of reference S and a second
reference frame S moving with a velocity vx relative to S.
Z
Z
vx
S
S
‘event’
Y
Y
X
X
4'-5"
1'-5"
vxt
x
Figure 2: A frame of reference S is moving with a velocity vx relative to the inertial frame S.
An event occurs with spatial coordinates (x, y, z) at time t in S and at (x , y , z ) at time t in S .
Let us suppose that the clocks in S and S are set such that when the origins of the two
reference frames O and O coincide, all the clocks in both frames of reference read zero
i.e. t = t = 0. According to ‘common sense’, if the clocks in S and S are synchronized
at t = t = 0, then they will always read the same, i.e. t = t always. This, once again, is
the absolute time concept introduced in Section 2.1. Suppose now that an event of some
kind, e.g. an explosion, occurs at a point (x , y , z , t ) according to S . Then, by examining
Fig. (2), according to S, it occurs at the point
x = x + vxt ,
y = y ,
z = z
(1)
and at the time
t = t
These equations together are known as the Galilean Transformation, and they tell us how
the coordinates of an event in one inertial frame S are related to the coordinates of the
same event as measured in another frame S moving with a constant velocity relative to
S.
Now suppose that in inertial frame S, a particle is acted on by no forces and hence is
moving along the straight line path given by:
r = r0 + ut
(2)
where u is the velocity of the particle as measured in S. Then in S , a frame of reference
moving with a velocity v = vxi relative to S, the particle will be following a path
r = r0 + (u − v)t
(3)
where we have simply substituted for the components of r using Eq. (1) above. This last
result also obviously represents the particle moving in a straight line path at constant
speed. And since the particle is being acted on by no forces, S is also an inertial frame,
and since v is arbitrary, there is in general an infinite number of such frames.

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4
NEWTONIAN FORCE AND MOMENTUM
9
Incidentally, if we take the derivative of Eq. (3) with respect to t, and use the fact that
t = t , we obtain
u = u − v
(4)
which is the familiar addition law for relative velocities.
It is a good exercise to see how the inverse transformation can be obtained from the above
equations. We can do this in two ways. One way is simply to solve these equations so as to
express the primed variables in terms of the unprimed variables. An alternate method, one
that is mpre revealing of the underlying symmetry of space, is to note that if S is moving
with a velocity vx with respect to S, then S will be moving with a velocity −vx with
respect to S so the inverse transformation should be obtainable by simply exchanging the
primed and unprimed variables, and replacing vx by −vx. Either way, the result obtained
is
x = x − vxt





y = y




(5)
z = z







t = t.

4
Newtonian Force and Momentum
Having proposed the existence of a special class of reference frames, the inertial frames of
reference, and the Galilean transformation that relates the coordinates of events in such
frames, we can now proceed further and study whether or not Newton’s remaining laws
of motion are indeed consistent with the principle of relativity. FIrst we need a statement
of these two further laws of motion.
4.1
Newton’s Second Law of Motion
It is clearly the case that particles do not always move in straight lines at constant speeds
relative to an inertial frame. In other words, a particle can undergo acceleration. This
deviation from uniform motion by the particle is attributed to the action of a force. If
the particle is measured in the inertial frame to undergo an acceleration a, then this
acceleration is a consequence of the action of a force F where
F = ma
(6)
and where the mass m is a constant characteristic of the particle and is assumed, in
Newtonian dynamics, to be the same in all inertial frames of reference. This is, of course, a
statement of Newton’s Second Law. This equation relates the force, mass and acceleration
of a body as measured relative to a particular inertial frame of reference.
As we indicated in the previous section, there are in fact an infinite number of inertial
frames of reference and it is of considerable importance to understand what happens to
Newton’s Second Law if we measure the force, mass and acceleration of a particle from
different inertial frames of reference. In order to do this, we must make use of the Galilean
transformation to relate the coordinates (x, y, z, t) of a particle in one inertial frame S say
to its coordinates (x , y , z , t ) in some other inertial frame S . But before we do this, we
also need to look at Newton’s Third Law of Motion.

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