Physics in Spacetime
Lecture notes
Linus Wulff
Spring 2020
2
Chapter 1
Space and Time
This course deals with the special theory of relativity laid out by Einstein more than 100
years ago. The traditional way of introducing special relativity is to derive it in much the
same way that Einstein did from the two postulates of
1. The principle of relativity
2. The constancy of the speed of light
From these assumptions the notion of a spacetime with (inertial) observers being connected
by Lorentz transformations follows. This is a natural way to proceed if one starts from a
knowledge of classical mechanics and of Maxwell’s equations of electrodynamics. However
it is not the best way to understand the geometrical aspects of spacetime which is part
of the reason it took Einstein another ten years to come up with the general theory of
relativity, where the geometry of spacetime is the key player.
In this course we will follow a different route1
which leads more directly to a geometric
picture. Rather than follow the traditional route starting with the principles described
above we will derive the same physics from something called
• The principle of maximum proper time
Before we explain what this principle says we must first decide what we mean by the
concepts of space and time in physics.
1.1 What is space and time?
The notions of Space and Time are central to physics. We typically want to answer
questions of the sort
Given some configuration, say of particles with given momenta in space, at
some initial time ti what will the configuration look like at some later time tf .
1
The approach to relativity taken here is inspired by lecture notes and a book by B. Laurent (Introduction
To Spacetime, World Scientific, 1994).
3
4 CHAPTER 1. SPACE AND TIME
For such questions to make sense we must have a precise way of defining what we mean
by time and also what we mean by a particle’s position in space. So what are space and
time? Rather than get into a philosophical discussion of the nature of space and time, a
more useful approach when faced with such deep questions questions in physics is to try
to replace it by a different, more down to earth, question. After all, in physics we can only
talk about things that can be measured and therefore a better question to ask is
How do we measure space and time?
In fact answering this question lets us define operationally, i.e. by measurements, what
space and time are. How do we measure distances in space? The most basic way is to
take a reference object, say a 1 meter ruler, and use it to measure the distance between
two points. Of course a good ruler should not bend or change its length with temperature
etc. so we will assume it is always possible to find a sufficiently good ruler (or equivalent)
that we can always measure lengths to the precision we need. How do we measure time?
To measure time we need a clock. It does not have to be what we normally think of as a
clock, it can be any physical system which has a known time dependence, e.g. a pendulum
or atoms in an excited state with known half-life. Again we will assume that there exist
such clocks with good enough precision for the time measurements we need to perform.
We have established that each person, or more generally, each observer measures time
with his clock and spatial distances with his reference ruler. We will assume that these
are small enough that the observer can carry them with him, i.e. they will be in the same
state of motion as the observer and experience the same forces he experiences. How are
the measurements of two different observers related? Newton assumed that there was an
absolute notion of time so that all observers clocks tick at the same rate. We now know
that this assumption was wrong. Taking two synchronized atomic clocks, putting one on
a plane circling the earth and leaving one on the ground, one finds when comparing them
at the end that they differ (by a few hundred nanoseconds). This observation is clearly
inconsistent with the Newtonian idea of an absolute time.
1.2 The principle of maximum proper time
Experiments show that time runs differently for different observers. We must assign each
observer her own time, her proper time, which is the time measured by her clock. We now
come to the principle that will allow us to compare the measurements of different observers
If two observers are separated and then meet the one that does not experience
any acceleration (inertial observer) always measures the longest time.
This is the principle of maximum proper time, that proper time is maximized for inertial,
or unaccelerated, observers. There is plenty of experimental evidence to support this
principle, such as the experiments with atomic clocks on planes, or the operation of GPS
satellites. We will take this principle as the starting point from which we will derive the
theory of special relativity.
1.3. SPACETIME 5
We are familiar with the fact that to specify the position of an object in our three
dimensions we need to give three numbers – the coordinates with respect to some coordinate
system. For positions on the earth we might for example give the longitude, the latitude
and the height above sea level. To specify an event – something happening at a certain
place at a certain time – we must give one more number namely the time on a clock
associated to our coordinate system. In our example this could be the time GMT.
1.3 Spacetime
We have seen that we must allow each observer to measure distances and times according
to her own coordinate system defined by her standard ruler and clock. Each observer will
therefore associate to a given event four numbers (t, x, y, z) – the spacetime coordinates
relative to her coordinate system. Note that we are defining an event here in an idealized
way as a single point in spacetime, i.e. something that happens at a point in space at a
single instant of time. The set of all events make up the four-dimensional spacetime. Note
that each observer will in general assign different coordinates to the same event because
they are using different coordinate systems, there is no special preferred coordinate system
in spacetime. One of our first tasks will be to understand how to relate the observations
of different observers.
The trajectory of an object traces out a path in spacetime – a worldline, or, if the
object is not point-like, a tube-like worldvolume. In ordinary Euclidean space we are
familiar with that fact the there is a shortest path between any two points. This path is
called a straight line. It is the path a particle follows if it is not acted upon by any external
forces. Similarly we will assume that there is precisely one straight worldline connecting
any two events in spacetime and that any object not acted on by external forces, i.e. not
experiencing any acceleration, follows such a straight line. To a worldline connecting two
events in spacetime we can associate a number – the proper time along that worldline.
Recall that this is the time an observer traveling along the worldline measures on her clock
between the two events. The principle of maximum proper time says that a straight line
in spacetime corresponds to the longest proper time. Therefore the analog of shortest
length in Euclidean space is longest proper time in spacetime and a clock can be thought
of as measuring distance in spacetime. When we draw spacetime diagrams we will draw
the worldlines of unaccelerated objects as straight lines. Curved lines will correspond to
worldlines of accelerated objects (Figure 1.1).
1.4 Parallel worldlines
An important notion in Euclidean geometry is the notion of two lines being parallel. In
spacetime we can similarly have the notion of two observers being on the same course.
How can two observers, e.g. two spaceships traveling in outer space, determine whether
they are on the same course? One way to do so uses a construction from Euclidean space
6 CHAPTER 1. SPACE AND TIME
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
t
t
event 1
event 2
Figure 1.1: Spacetime diagram showing and accelerated (curved line) and an unaccelerated
(straight line) observer meeting at the spacetime points marked event 1 and event 2.
adapted to spacetime. Imagine that the two observers each send out a probe fitted with
a clock, which travels freely until it is picked up by the other observer at some later time
(Figure 1.2). If the two probes happen to meet halfway, i.e. after half of the proper time
(from being emitted to being picked up) has elapsed on each clock, then we will say that
the observers are on the same course, or that their worldlines are parallel. From the figure
we see that this also implies that the lines AB and CD (not drawn) are parallel.
Note that to carry out the experiment we really need to send clocks that also have a
recording device that records the time they were sent and the time they met. We would
also need to do the experiment several times to get the clocks to meet halfway. It is also
important to note that we never refer to space or time separately, only to the full spacetime
or the proper time measured by a particular clock. This is in sharp distinction to how we
would describe such an experiment in Newtonian physics.
Just like in Euclidean space the line AC in Figure 1.2 defines a vector which we can
draw as an arrow starting at A and ending at C. The length of the vector is given by the
proper time elapsed from A to C. The construction in the figure gives us a way to parallel
transport vectors. The vector AC can be parallel transported to the vector BD. Taking
the two worldlines to approach each other we obtain the special case of parallel transport
of the vector along the worldline. A general parallel transport is obtained by a sequence
of such ”elementary” parallel transports.
We will now make a very important assumption. We will assume that the vector one
obtains by such a sequence of elementary parallel transports from a point A to a point A
in spacetime does not depend on how one chooses the sequence of parallel transports, i.e.
it does not depend on the path taken. This assumption is not true close to gravitating
bodies and in that case one must use general relativity. The assumption is true if gravity
1.4. PARALLEL WORLDLINES 7
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡¡












Q
d
d
d
d
d
ds
s
s
s
s
c
A
C
B
D
τ
2
τ
2
τ
2
τ
2
Figure 1.2: The worldlines of two observers are parallel if they can send out clocks, at A
and B, that meet halfway before encountering the other observer at D and C.
is very weak, which is the case we consider in this course, and we say that we work with
special relativity. In fact the change of a vector under parallel transport is related to the
curvature of a space so in special relativity we are working in a flat spacetime.
8 CHAPTER 1. SPACE AND TIME
Chapter 2
Spacetime vectors
In the last chapter we defined a vector on the straight worldline of an observer as an arrow
from one event to another on the worldline with length given by the proper time elapsed
between the two events. The notion of vectors is familiar from Euclidean space and we
will use the same notation ¯v for a vector in spacetime. Such vectors are often called fourvectors
since spacetime is four-dimensional. Just as any point in Euclidean space R3
can
be associated with a vector going from the origin to that point, any event in spacetime can
be associated to a spacetime vector from an origin (which we can choose as we please) to
the point in question. We have also seen that we can move vectors around using parallel
transport. Two vectors related by parallel transport will be considered the same vector.
Spacetime vectors obey the usual axioms familiar from Euclidean space
• Associativity of addition: ¯u + (¯v + ¯w) = (¯u + ¯v) + ¯w
• Commutativity of addition: ¯u + ¯v = ¯v + ¯u
• Identity element of addition: ¯v + ¯0 = ¯v
• Inverse elements of addition: Given ¯v the exists a vector −¯v such that ¯v + (−¯v) = 0
• Compatibility of scalar multiplication: a(b¯v) = (ab)¯v
• Identity element of scalar multiplication: 1¯v = ¯v
• Distributivity of scalar multiplication with respect to vector addition: a(¯u + ¯v) =
a¯u + a¯v
• Distributivity of scalar multiplication with respect to addition: (a + b)¯v = a¯v + b¯v
Addition of spacetime vectors can be done by the geometric construction familiar from
Euclidean space, Figure 2.1. Recall that a basis for a vector space is a set of linearly
independent vectors ¯vi with i = 1, . . . , n which span the space, so that any vector is
expressed uniquely as a linear combination
a1¯v1 + a2¯v2 + . . . + an¯vn , (2.1)
9
10 CHAPTER 2. SPACETIME VECTORS
d
d
d
d
d
d
d
d
ds
¯u







Q
¯v T
¯u + ¯v
Figure 2.1: Addition of vectors ¯u and ¯v to produce a third vector ¯u + ¯v.
for some numbers ai with i = 1, . . . , n. The vector can be denoted in this basis as
(a1, a2, . . . , an) and n is called the dimension of the vector space. A basis of spacetime
vectors therefore consists of four spacetime vectors.
2.1 Inner product
An important operation in linear algebra is the inner product (also often called scalar
product) between two vectors. Given two vectors their inner product is a number. We will
denote the inner product with a dot, e.g. ¯u · ¯v denotes the inner product between vectors
¯u and ¯v. The inner product satisfies the following axioms
• Symmetry: ¯u · ¯v = ¯v · ¯u
• Linearity: (a¯u) · ¯v = a(¯u · ¯v) and (¯u + ¯v) · ¯w = ¯u · ¯w + ¯v · ¯w
• Non-degenerate: If ¯u · ¯v = 0 for all vectors ¯v then ¯u = ¯0
Often the inner product is required to be positive definite, so that ¯u2
= ¯u · ¯u ≥ 0. In
Euclidean space we are used to identifying ¯u2
with the length-squared of a vector, which
is obviously positive. We will see below that this is not true for spacetime vectors, in fact,
for a vector that is the straight worldline of an object from point A to point B we will take
¯u2
= −τ2
, (2.2)
2.2. TIMELIKE, SPACELIKE AND NULL 11
where τ is the proper time along the worldline from A to B. The minus sign seems strange
at this point but we will see shortly that it is needed if we want lengths in space to have
positive square. All the differences between Euclidean space and spacetime are due to the
fact that the inner product of spacetime vectors are not positive definite. This is what
makes it possible to separate a time-direction from the spatial directions.
The assumption the ¯u2
= −τ2
for vectors corresponding to a piece of a straight worldline
determines also ¯u · ¯v for some other straight worldline vector ¯v. To see this consider three
such worldline vectors satisfying
a¯u = ¯v + ¯w , (2.3)
for some number a. Writing this as ¯w = a¯u − ¯v and squaring both sides we get
¯w2
= (a¯u)2
− 2(a¯u) · ¯v + ¯v2
= a2
¯u2
− 2a¯u · ¯v + ¯v2
. (2.4)
Rearranging this we have
¯u · ¯v =
1
2a
a2
¯u2
+ ¯v2
− ¯w2
(2.5)
and the right-hand-side is clearly expressed only in terms of the proper times corresponding
to the length-squared of ¯u, ¯v and ¯w.
It is important to understand that the assumption that there exists an inner product
for spacetime vectors satisfying the above axioms is not a trivial statement. The mere
existence of this inner product has physical consequences. To see this consider the identity
(¯u + ¯v)2
+ (¯u − ¯v)2
= 2¯u2
+ 2¯v2
. (2.6)
Consider the case that all these vectors are part of straight worldlines of observers. Since the
expression contains only squares it only involves proper times measured by these observers.
With four spaceships traveling along these straight worldlines it is then possible to arrange
an experiment (Figure 2.2) test whether the proper times they measure satisfy the above
identity which amounts to τ2
1 + τ2
2 = 2τ2
3 + 2τ2
4 .
2.2 Timelike, Spacelike and Null
Let ¯u, ¯v be two straight worldline vectors. Then
¯u2
= −τ2
u , ¯v2
= −τ2
v . (2.7)
We now define a new vector which is a linear combination of ¯u and ¯v
¯y = a¯u + b¯v . (2.8)
Its inner product with ¯u is
¯u · ¯y = a¯u2
+ b¯u · ¯v (2.9)
and picking a = −b¯u·¯v
¯u2 we find
¯u · ¯y = 0 . (2.10)
We say that ¯y is orthogonal to ¯u.
Now consider the following situation. Two spaceships part and then meet again:
12 CHAPTER 2. SPACETIME VECTORS
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡¡!
τ3
τ2 τ1
τ4
τ4
I
A












U
£
£
£
£
£
£
£
£
£
££#
Figure 2.2: Experiment involving four spaceships to test equation (2.6).
T
ship 1
ship 2
ship 2











Q











k
Spaceship 1 is unaccelerated throughout the whole duration of its journey, while spaceship
two flies unaccelerated for awhile then accelerates hard for a short time to reverse its
direction of motion and then again floats freely until it meets spaceship 1 again.
We take the spacetimes vectors corresponding to this situation as in Figure 2.3 with
¯v1 =
1
2
¯u + ¯y , ¯v2 =
1
2
¯u − ¯y , (2.11)
where ¯y is the vector introduced above which is orthogonal to ¯u and is a small number.
Note that ¯v1 + ¯v2 = ¯u so that the spaceships indeed meet at the end. The proper time for
2.2. TIMELIKE, SPACELIKE AND NULL 13
T
¯u
¯v1
¯v2
¢
¢
¢
¢
¢¢
f
f
f
f
ffw
Figure 2.3: Spacetime vectors corresponding to two spaceships parting and meeting again.
the journey of ship 1 is
τ1 =
√
−¯u2 . (2.12)
The proper time for the journey of ship 2 is the sum of the proper time for the two segments
of the journey
τ2 = −¯v2
1 + −¯v2
2 . (2.13)
Since ¯u · ¯y = 0 we have ¯v2
1 = 1
4
¯u2
+ 2
¯y2
= ¯v2
2 so that finally
τ2 = −¯u2 − 4 2 ¯y2 =
√
−¯u2 1 +
4 2 ¯y2
¯u2
. (2.14)
The principle of maximum proper time says that the unaccelerated observer measures the
longest proper time, i.e. τ1 > τ2 (provided that ¯y = 0). This in turn implies that
¯y2
> 0 . (2.15)
This result was obtained assuming ¯u2
< 0. With the opposite convention that ¯u2
> 0 we
would instead find ¯y2
< 0. We see that, in contrast to what we are used to from Euclidean
geometry, it is not possible for all spacetime vectors to have positive square.
Clearly no observer can travel along ¯y because then his clock would need to show an
imaginary time, which is absurd.
Consider the vector
¯w = a¯u + b¯y . (2.16)
Squaring we find
¯w2
= a2
¯u2
+ b2
¯y2
. (2.17)
If we take a2
= −b2 ¯y2
¯u2 (note that the RHS is positive which is consistent with a, b being
real numbers) we get ¯w2
= 0! We therefore conclude that there exist spacetime vectors
¯w = ¯0 such that ¯w2
= 0.
14 CHAPTER 2. SPACETIME VECTORS
To summarize we have learned that there are 3 classes of spacetime vectors ¯v which we
give names as follows
• ¯v2
negative: Timelike
• ¯v2
positive: Spacelike
• ¯v2
= 0: Null (light-like)
Vectors that are part of a straight worldline of an observer are timelike. We have also seen
above that if ¯u is timelike and ¯u · ¯v = 0 then either ¯v = ¯0 or ¯v is spacelike. This is a very
useful result to remember when working with spacetime vectors.
Chapter 3
Simultaneity and spatial distance
An observer traveling along in a spaceship has only direct access to the interior of the
spaceship. Nevertheless she must be able to make statements and inferences about what
happens in the outside world. To be able to do this she needs in particular to be able
to say when an event, which is not on her worldline, occurred. To do this she needs to
have a way to determine whether an event far away is simultaneous with an event on her
worldline, e.g. a supernova explosion far away happens when her clock shows 10:00.
The natural way for her to define simultaneity is via the construction in figure 3.1.
She sends out a probe which travels on a straight worldline to the event and on a straight
worldline back. She arranges it so that the probe reaches the event precisely when half the
proper time of its journey has elapsed. Then she will say that the event on her worldline
halfway between sending out and receiving the probe is simultaneous with the distant
event.
From the figure we have
τ2
= −(¯v + ¯r)2
= −(¯v − ¯r)2
⇒ ¯v · ¯r = 0 , (3.1)
so that ¯r is spacelike.
Let us now consider a family of straight parallel worldlines L0, L1, . . . defined by the
equation
¯Rn = λn ¯u + n¯ρ , n = 0, 1, 2, . . . (3.2)
where ¯u, ¯ρ are timelike vectors and λn parameterizes a point on Ln. This is illustrated in
figure 3.2. This could be a fleet of identical spaceships traveling unaccelerated arranged
head to tail. Consider an observer traveling from the front of the fleet to the back counting
how many ships he passes. This number is a measure of how far it is from the head of the
fleet to the tail. The distance is expressed in units of ‘standard spaceship’.
There is an alternative way to measure this distance. There is only one vector going
15
16 CHAPTER 3. SIMULTANEITY AND SPATIAL DISTANCE
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡¡
r
rrrrj
$$$$$$$$
v
v
v
v
v
v
v
¡
¡
¡
¡¡!
¡
¡
¡
¡¡!
s
s
s
s P
P1
P2
P3
t
t
τ
τ¯v
¯r
¯v
Figure 3.1: Via this construction the observer decides that P2, halfway between P1 and P3,
is simultaneous with P.
from L0 to Ln with the property that it is orthogonal to ¯u.1
It is given by
¯rn = n¯ρ −
n¯ρ · ¯u
¯u2
¯u . (3.3)
Note that ¯rn, and therefore also its length, is proportional to n, the number of spaceships.
We can use ¯r2
n as a measure of the distance. We just need to work out the conversion
factor to go between ¯r2
n and the number of spaceships.
Looking at figure 3.1 (replacing ¯v with ¯u and dropping the subscript on ¯rn in the
following) we read off
τ2
= −(¯u + ¯r)2
= t2
− ¯r2
, or l2
= ¯r2
= t2
− τ2
. (3.4)
The advantage of this method is that we don’t need the fleet of spaceships (other than
to fix the unit of distance). Later we will find an even more practical way to measure
distance.
How we pick the unit of distance is up to us. Nothing prevents us from choosing units
such that
√
¯r2 itself is the distance. This is in fact the most natural choice to make.
From (3.4) we see that now space and time acquire the same dimensions. In the theory
of relativity this is as natural as height and width having the same dimensions and being
measured in the same units.
1
Proof: Assume there are two such vectors ¯r1, ¯r2. We may assume their foot-point is the same point
on L0. The fact the ¯u · ¯r1 = ¯u · ¯r2 = 0 implies ¯u · (¯r1 − ¯r2) = 0. But ¯r1 − ¯r2 = λ¯u and the previous equation
implies λ = 0 so that ¯r1 = ¯r2.
3.1. ORTHOGONAL SPACE 17
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢¢
¢
¢
¢
¢
¢
¢¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢¢



Q








Q












Q

















Q
¯u ¯ρ
L0 L1 L2 L3 · · ·
Figure 3.2: A family of parallel worldlines.
3.1 Orthogonal space
To every unaccelerated observer there corresponds a straight worldline. Such a worldline
is characterized by a timelike vector which we can normalize to a unit vector ˆu. We will
always us a ‘hat’ to denote a unit vectors. A timelike unit vector satisfies ˆu2
= −1 and a
spacelike unit vector ˆr2
= 1. Given an observer with unit vector ˆu there there exist vectors
¯r such that
ˆu · ¯r = 0 . (3.5)
They form the orthogonal space to the observers worldline. This is a vector space since
linear combinations of such vectors clearly belong to the space. In fact, since all such ¯r are
spacelike (or zero), it is a Euclidean vector space. Since we are imposing one condition on
the four components of ¯r this orthogonal space is three-dimensional. Recall that
√
¯r2 is the
(spatial) distance from the observer. The orthogonal space is the space used in Newtonian
physics. The difference is that in the theory of relativity each observer constructs her own
orthogonal space.
18 CHAPTER 3. SIMULTANEITY AND SPATIAL DISTANCE
Given the worldline of an observer with direction ˆu we can split any spacetime vector
¯R into a component along ˆu and a component orthogonal to it as
¯R = tˆu + ¯r with ˆu · ¯r = 0 . (3.6)
This is illustrated in figure 3.3. According to the figure an observer following the worldline
¢
¢¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢













Q
€€€€€€€€€€q
tˆu
¯R
¯r
L
Figure 3.3: Split of a spacetime ¯R vector with respect to a timelike direction ˆu.
L measures the event corresponding to ¯R to happen at a time t and spatial position ¯r. The
distance to the event is
√
¯r2. From the equation we find t = −ˆu · ¯R and ¯r = ¯R + (ˆu · ¯R)ˆu
from which we see that t and ¯r are uniquely fixed in terms of ˆu and ¯R.
3.2 Linearly independent vectors
Consider four spacetime vectors
¯A, ¯B, ¯C, ¯D . (3.7)
They are linearly independent of none of them can be expressed as a linear combination
of the others, or equivalently if the equation
a ¯A + b ¯B + c ¯C + d ¯D = 0 (3.8)
3.2. LINEARLY INDEPENDENT VECTORS 19
has only the trivial solution a = b = c = d = 0. Note that since spacetime is fourdimensional
we cannot have more than four linearly independent vectors.
In Euclidean space we are used to two orthogonal vectors being linearly independent.
This is not true in spacetime, e.g. a null vector is orthogonal to itself but clearly not
linearly independent of itself. What is true is that if ¯v · ¯u = 0 for all ¯u then ¯v = 0. To see
this take ¯u timelike. We conclude that ¯v must be spacelike but if it is orthogonal to all
other spacelike vectors it must vanish since they form a Euclidean vector space.
To test if ¯A, ¯B, ¯C, ¯D are lineraly independent we consider the determinant of the
matrix of inner products
¯A · ¯A ¯A · ¯B ¯A · ¯C ¯A · ¯D
¯B · ¯A ¯B · ¯B ¯B · ¯C ¯B · ¯D
¯C · ¯A ¯C · ¯B ¯C · ¯C ¯C · ¯D
¯D · ¯A ¯D · ¯B ¯D · ¯C ¯D · ¯D
(3.9)
The vectors are linearly dependent if and only if this determinant vanishes. To see this assume
they are linearly dependent. Then (3.8) holds for some non-zero coefficients. Taking
this linear combination of rows in the matrix we obtain a row of zeros so that the determinant
vanishes. Conversely, if the determinant vanishes there exists a linear combination
of the rows that gives zero. This means that there exists a vector a ¯A + b ¯B + c ¯C + d ¯D,
with a, b, c, d not all zero, which is orthogonal to ¯A, ¯B, ¯C, ¯D. Assuming they are linearly
independent gives a contradiction, therefore they are linearly dependent.
Note that this test works only for four vectors. It does not work for lower-dimensional
subspaces of spacetime.
20 CHAPTER 3. SIMULTANEITY AND SPATIAL DISTANCE
Chapter 4
Velocity and light signals
Consider an observer and an object following straight worldlines L and L respectively,
figure 4.1. The observer describes the objects position at a time t to be ¯r, with ¯r = 0 at














E
t τ
¯r
L
L
Figure 4.1: Observer L describes the object L as having position ¯r at time t.
t = 0. He assigns the object the velocity
¯v =
d¯r
dt
=
¯r
t
, (4.1)
where the last equality follows from the fact that they are following straight worldlines so
¯r is proportional to t. This velocity clearly depends on the observer, since ¯r and t refer
21
22 CHAPTER 4. VELOCITY AND LIGHT SIGNALS
to the observer. For this reason it is called the relative velocity of the observer and the
object.
The unit vector along the objects worldline ˆv tells us how the relative velocity ¯v is
directed relative to ˆu. We call ˆv the four-velocity of the object. Any timelike unit vector
(pointing forwards in time) can be a four-velocity since it could be the direction of an
objects worldline.
4.1 Standard velocity split
Let ˆv be the four-velocity of the object and ˆu that of the observer. The objects position is
given by ¯R = τˆv and from figure 4.1 we see that
τˆv = tˆu + ¯r = t(ˆu + ¯v) , (4.2)
where we used the definition of the relative velocity in (4.1). We can write this as
ˆv = γ(ˆu + ¯v) where γ =
t
τ
and ˆu · ¯v = 0 . (4.3)
This formula for the split of the four velocity of the object into the four-velocity of the
observer and the relative velocity is very useful and has many applications. For example,
squaring this equation gives
1 = γ2
(1 − v2
) or
t
τ
= γ =
1
√
1 − v2
, (4.4)
where v2
= ¯v2
, the relative velocity squared. Note that v2
= 1− τ2
t2 ≤ 1. This is the famous
formula for time dilation. Since t = γτ and γ ≥ 1 the observer sees the objects clocks
running slower that his by a factor of γ.
Taking the inner product of (4.3) with ˆu we find
−ˆu · ˆv = γ =
1
√
1 − v2
. (4.5)
Notice that this implies that the definition of the relative velocity v is symmetric between
the observer and object. The object judges the observer to have the same velocity as the
observer assigns to the object.
4.2 Light signals
So far we have not discussed null lines, which are on the border between timelike and
spacelike lines. Now let us assume that L in figure 4.1 is such a null line. Then
τ2
= − ¯R2
= 0 , (4.6)
4.2. LIGHT SIGNALS 23
but it is still true that
τˆv = tˆu + ¯r . (4.7)
Squaring this we get
−t2
+ ¯r2
= 0 or v2
= 1 . (4.8)
Note that this result is independent of the observer. All observers would measure the
relative velocity of such a signal to have magnitude v = 1.
It does not follow from the theory of relativity itself that particles following such null
lines exist. It would be meaningless to assign them a clock since it would not tick (τ = 0
along such a line).
Nevertheless experience tells us that there exist signals in nature that can travel along
null lines. Light being the most important example and v = 1 is called the ”velocity of
light”. More generally, as we will see later, a particle can follow a null worldline if and
only if it is massless. The quantum of light is the massless particle called the photon.
The existence of such signals is of great practical importance. Recall that to determine
simultaneity we used the setup in figure 3.1. The observer has to arrange the situation
so that the proper time of the probe to and from the event are the same. To achieve
this in practice presents great difficulties. The existence of light, or more generally electromagnetic,
signals solves this problem since using such a signal instead of the probe
τ = 0 always. Equivalently the observer knows that v = 1 and can therefore calculate the
distance directly from the time it takes there and back.
Imagine an observer who sends out a flash of light at t = 0. The light pulse travels out
in all directions with unit velocity forming a sphere of light. To draw the corresponding
spacetime diagram we must go down to two dimensions of space where the light forms a
circle traveling outwards from the observer, figure 4.2. In this three-dimensional spacetime
picture the light forms a cone. For this reason it is referred to as the light-cone. It is the
surface given by the equation
r = t , where r =
√
¯r2 . (4.9)
What we have drawn is really only half of the light-cone, called the future light-cone. There
is also the past light-cone given by
r = −t . (4.10)
It describes a sphere of light contracting towards the origin. The full light cone is given by
the equation
r2
= t2
(4.11)
and is illustrated in figure 4.3. Note that this equation is true for any observer. This means
that the light-cone looks the same to any observer. Don’t be mislead by the picture which
might seem to suggest otherwise!
24 CHAPTER 4. VELOCITY AND LIGHT SIGNALS
T
E 
 
 
 
 
 
 
 
  
d
d
d
d
d
d
d
d
dd
ˆˆˆˆz
¢
¢
¢
¢
¢
¢
¢
¢
¢
¯n
¯r
t
Figure 4.2: An observer’s future light-cone. The circle of light is at distance r at time t
and traveling out in all null directions, e.g. ¯n.
4.3 Split of null vectors
As we have seen any spacetime vector can be split with respect to some four-velocity ˆu
into a component along ˆu and a component orthogonal to it. For a null vector ¯n we get
¯n = a(ˆu + ¯m) (4.12)
but ¯n2
= 0 implies ¯m2
= 1 so that
¯n = a(ˆu + ˆm) , (4.13)
proportional to the sum of a timelike and a spacelike unit vector. If a vector ¯k is orthogonal
to ¯n it is either spacelike or proportional to ¯n.1
In particular we have that two orthogonal
null vectors must be parallel.
1
Proof: Writing ¯k = b(ˆu + ¯r) we find the condition (we may assume a, b = 0)
ˆm · ¯r = 1 .
since ˆm and ¯r belong to the orthogonal subspace they are Euclidean. Therefore we must have either√
¯r2 > 1, in which case ¯k is spacelike, or
√
¯r2 = 1 and ¯r is parallel to ˆm, in which case ¯k = b
a ¯n.
4.4. FUTURE AND PAST 25
 
 
 
 
 
 
  
d
d
d
d
d
d
dd
 
 
 
 
 
 
  
d
d
d
d
d
d
dd
Figure 4.3: The full light-cone.
4.4 Future and past
Consider a timelike vector ¯v at the origin. We can split it with respect to the four-velocity
of an observer as
¯v = tˆu + ¯r , ˆu · ¯r = 0 . (4.14)
Squaring we find
¯v2
= r2
− t2
< 0 , (4.15)
which implies that every timelike vector is pointing insider the light-cone (see figure 4.2).
Replacing ¯v with a spacelike vector we similarly find that every spaclike vector points
outside the light-cone. Null vectors point along the light-cone.
Let ¯u, ¯v be timelike vectors with negative inner product
¯u · ¯v < 0 . (4.16)
Construct the linear combination
¯w = a¯u + b¯v , with a, b ≥ 0 , a + b = 0 . (4.17)
Squaring we find
¯w2
= a¯u2
+ 2ab¯u · ¯v + ¯v2
< 0 , (4.18)
since all terms are negative. By varying a, b we can continuously go from the vector ¯u to
the vector ¯v via only timelike vectors. This would be impossible if one was pointing into
the future light-cone and the other into the past light-cone. We therefore conclude that
two timelike vectors with negative inner product must be pointing in the same direction,
26 CHAPTER 4. VELOCITY AND LIGHT SIGNALS
i.e. either both into the future light-cone or both into the past light-cone. It is easy to see
that the same argument goes through if we take one timelike and one null vector.
If instead ¯u · ¯v > 0 then ¯u and −¯v are point in the same direction so that ¯u and ¯v are
pointing in opposite directions. Conversely two timelike vectors pointing inside the same
part (future/past) of the light-cone have ¯u · ¯v < 0 (note that ¯u · ¯v cannot vanish). Again
this goes through if one of them is null.
Let ˆu, ˆv be timelike and future directed. Then
(ˆu + ˆv)2
= −2 + 2ˆu · ˆv < 0 and (ˆu + ˆv) · ˆv = ˆu · ˆv − 1 < 0 , (4.19)
from which we conclude that ˆu + ˆv is also timelike and future directed. This must hold
for any sum of timelike future directed vectors. An important consequence of this is that
no spaceship (or other object) can reverse its four-velocity and travel to the past to arrive
before it departed. Time travel is therefore impossible within the theory of relativity.
Related to this, if two spaceships part and then meet again they will always agree they
part before they meet.
We have seen that timelike (and null) vectors can be divided into two classes: future
directed and past directed. Note that no such division is possible for spacelike vectors.
Chapter 5
Lorentz transformation
In this chapter we will restrict to situations where all the interesting spacetime vectors lie
in a two-dimensional plane. This means that they can all be expressed in terms of two
linearly independent vectors. Many problems one encounters are in fact of this type.
In the cases of interest here this plane contains a timelike future directed unit vector
which could be the four-velocity of an observer. Every vector can be split with respect to
this four-velocity, figure 3.3. Note that this corresponds to a one-dimensional problem in
Newtonian physics.
Consider the situation in Euclidean space with two sets of orthogonal vectors rotated
with respect to each other. Thinking of this as two coordinate systems we are familiar
with how two relate them via the rotation. Consider not the corresponding situation in
spacetime, illustrated in figure 5.1, which occurs often. We have two sets of orthogonal
unit vectors ˆu, ˆr with ˆu · ˆr = 0 and ˆv, ˆs with ˆv · ˆs = 0 and ˆu2
= ˆv2
= −1, ˆr2
= ˆs2
= 1.
Obviously ˆu and ˆr are linearly independent since one is timelike and one is spacelike. Since
we are restricting to a two-dimensional plane can express ˆv, ˆs in terms of ˆu, ˆr. Up to a
proportionality factor we have
ˆv ∝ ˆu + αˆr , ˆs ∝ ˆr + αˆu , (5.1)
where we have used the fact that ˆv · ˆs = 0. Let us compute the length of the vectors on
the RHS to fix the normalization. We have
(ˆr + αˆu)2
= −(ˆu + αˆr)2
= 1 − α2
(5.2)
and therefore
ˆv = ±
1
√
1 − α2
(ˆu + αˆr) , ˆs = ±
1
√
1 − α2
(ˆr + αˆu) , (5.3)
so that ˆs2
= −ˆv2
= 1. Demanding that ˆv → ˆu and ˆs → ˆr as α → 0 we see that we need to
pick the plus signs.
Recalling the standard velocity split, eq. (4.3)
ˆv =
1
√
1 − v2
(ˆu + ¯v) , ˆu · ¯v = 0 , (5.4)
27
28 CHAPTER 5. LORENTZ TRANSFORMATION
T
Ee
e
e
e
e
e
e
e
e
eeu
¨
¨¨¨
¨¨¨
¨¨¨B
ˆu
ˆv
ˆr
ˆs
Figure 5.1: Two sets of orthogonal unit vectors in spacetime.
we read off α = ±v with v =
√
¯v2, the magnitude of the relative velocity. We therefore
have
ˆv = γ(ˆu + vˆr) , ˆs = γ(ˆr + vˆu) , γ =
1
√
1 − v2
, (5.5)
where we have absorbed the sign into v so that v is positive if the relative velocity is along
ˆr and negative if it is along −ˆr. This is the famous Lorentz transformation. It tells us
how the measurements of two inertial observers are related. To see this consider an event
specified by the spacetime vector ¯R. We can write it in two ways as
¯R = tˆu + xˆr or ¯R = t ¯v + x ¯s . (5.6)
Observer u assigns coordinates (t, x) to the event while observer v assigns it coordinates
(t , x ). Equating the two expressions for ¯R and using the formula for the Lorentz transformation
we find
tˆu + xˆr = t γ(ˆu + vˆr) + x γ(ˆr + vˆu) (5.7)
or
t = γ(t + vx ) , x = γ(x + vt ) . (5.8)
This is the Lorentz transformation that relates the time and space measurements of the
two observers.
5.1 Addition of velocities
In Euclidean space we can perform first one rotation and then another. The result is a
third rotation. Similarly we can perform first one Lorentz transformation with parameter
5.2. LORENTZ CONTRACTION 29
(relative velocity) v1 and then another with parameter v2. This clearly gives a third
Lorentz transformation. The only question is how the parameter of the third Lorentz
transformation is related to v1, v2.
To answer this question we note that the final timelike unit vector is proportional to
(ˆu + v1ˆr) + v2(ˆr + v1 ˆu) = (1 + v1v2)ˆu + (v1 + v2)ˆr . (5.9)
From the equation ˆv = γ(ˆu + vˆr) we see that we can read off v as ˆr-component/ˆucomponent.
In this way we find
v =
v1 + v2
1 + v1v2
. (5.10)
This is the relativistic formula for addition of velocities. It has been tested to high accuracy
in experiments with light propagating through flowing liquids.
In the special case v2 = −v1 we get v = 0. This means that the inverse Lorentz
transformation is obtained by changing the sign of v. This is easily verified directly by
performing first a Lorentz transformation with parameter v and then one with parameter
−v.
5.2 Lorentz contraction
Consider a spaceship of length . The observer on the spaceship has four-velocity ˆu.
Another observer has four-velocity ˆv. Both measure the length of the spaceship from their
perspective getting the answer and respectively. The setup is illustrated in figure 5.2.
From the figure we see that
ˆs − ˆr ∝ ˆu (5.11)
and taking the inner product with ˆr gives
ˆr · ˆs − = 0 . (5.12)
Using the Lorentz transformation ˆs = γ(ˆr + vˆu) we find ˆr · ˆs = γ so that
/ = 1/γ =
√
1 − v2 , (5.13)
where we is the relative velocity of the two observers.
This is the equation for the Lorentz contraction. A moving object appears shortened,
or contracted, in its direction of motion relative to the observer. This effect is similar to
slicing a sausage in that the length of the slice depends on the cutting angle. In that case
the perpendicular slice has the shortest length, whereas in the spaceship case the observer
at rest measures the longest length. Note that in this respect the Euclidean figure 5.2 is
very misleading.
30 CHAPTER 5. LORENTZ TRANSFORMATION
T
¡
¡
¡
¡
¡
¡
¡
¡
¡¡
¡
¡
¡¡!
¡
¡
¡¡
EE
rrr
rrrj
rrrrj
¡
¡
¡
¡
¡
¡
¡
¡
¡¡
¡
¡
¡¡
L
L1 L2
ˆv ˆu
ˆs
ˆr
Figure 5.2: A spaceship stretching between L1 and L2, with length , is observed by the
observer following the worldline L who measures its length to be .
Chapter 6
Waves
Sound waves are familiar from Newtonian physics. The pressure P(¯r, t) varies in space and
with time around the mean pressure P0. For a plane sound wave p = P − P0 has the form
p = A sin(−ωt + ¯k · ¯r + φ0) . (6.1)
The constants A and φ0 are the amplitude and phase shift of the wave, while ω is the
angular frequency and the three-vector ¯k is the wave vector. We may assume that ω > 0.
The surfaces where p takes a constant value, e.g. its maximum, are 2-dimensional planes
in our 3-dimensional space, see figure 6.1, which is the reason for the name plane waves.
Sound waves are just one example. Many other types of waves exist in nature and they
can all be described in the same way.
Consider now a plane wave in spacetime given by
ψ = A sin( ¯K · ¯R + φ0) , (6.2)
where now ¯K is the wave four-vector and ¯R is the four-vector of a spacetime point (event).
Introduce an observer with four-velocity ˆu. We can then write
¯R = tˆu + ¯r , ¯r · ˆu = 0 , (6.3)
¯K = ωˆu + ¯k , ¯k · ˆu = 0 . (6.4)
Using these expressions the wave takes the form
ψ = A sin(−ωt + ¯k · ¯r + φ0) , (6.5)
which is the same as before. Notice that ω and ¯k depend on the observer since they are
defined using his four-velocity. We have for example
ωu = −ˆu · ¯K , (6.6)
the angular frequency of the wave as measured by the observer u.
The superposition principle says that we can add together waves of the form (6.2) to
form new waves. For physical waves the angular frequency is determined by the wave vector
31
32 CHAPTER 6. WAVES
T
E 
 
d
d
d
d
d
d
d
d
d
d
dd
d
d
d
d
d
d
d
d
d
d
dd
d
d
d
d
d
d
d
d
d
d
dd
¯k
x
y
Figure 6.1: Plane wave in two space dimensions. The wave fronts, surfaces of constant
phase at say t = 0, given by ¯k · ¯r = const, are straight parallel lines orthogonal to ¯k. In 3
dimensions they are planes.
ω = ω(¯k). The relation between ω and ¯k is called the dispersion relation and depends on
the type of wave. Let us consider the sum of two one-dimensional plane waves with similar
wave number k1 = (1 − )k and k2 = (1 + )k and φ0 = 0,
ψ = sin(−ω1t + k1x) + sin(−ω2t + k2x) . (6.7)
We have
−ω1t + k1x = −ω(k1)t + k1x = −ω(k)t + kx + k(ω t − x) , (6.8)
where ω = dω/dk, and for k2 we find the same expression with the sign of the last term
changed. Therefore, using the formula for sin of a sum of angles, we have
ψ = sin(−ωt + kx + k(ω t − x)) + sin(−ωt + kx − k(ω t − x))
= sin(−ωt + kx) cos( k(ω t − x)) + cos(−ωt + kx) sin( k(ω t − x))
+ sin(−ωt + kx) cos(− k(ω t − x)) + cos(−ωt + kx) sin(− k(ω t − x))
=2 cos( k(ω t − x)) sin(−ωt + kx) . (6.9)
This is a plane wave with wave number k and frequency ω = ω(k) but with an amplitude
which is modulated by the cos factor. The cosine factor describes the envelope of the wave
packet. The envelope of the wave packet depends on the position through x − ω t and
therefore travels with the group velocity
v = ω =
dω
dk
. (6.10)
6.1. DOPPLER SHIFT AND ABERRATION 33
This is the velocity of signals sent with such waves. For light we should clearly have v = 1
and indeed light (electromagnetic radiation) has the dispersion relation
ω = k = ¯k2 (6.11)
or
¯K2
= (ωˆu + ¯k)2
= −ω2
+ ¯k2
= 0 , (6.12)
i.e. light rays have a wave four-vector which is null. This means that for light (electromagnetic
radiation) we can write the wave four-vector as
¯K = ω(ˆu + ˆk) ˆk · ˆu = 0 , (6.13)
where ˆk is a unit vector which guarantees that the wave four-vector ¯K is null.
6.1 Doppler shift and aberration
Consider two observers with four-velocities ˆu and ˆv observing a light wave ( ¯K2
= 0).
Splitting vectors with respect to the first observer we have
ˆv = γ(ˆu + ¯v) ¯v · ˆu = 0 , (6.14)
the standard velocity split, and
¯K = ωu(ˆu + ˆku) ˆku · ˆu = 0 . (6.15)
We have put a subscript u to emphasize that ωu and ˆku are the quantities measure by
observer u. Since ¯v and ˆku are in the orthogonal space to ˆu they can be treated as ordinary
Euclidean vectors and we can write
¯v · ˆku = v cos θu , (6.16)
where θu is the angle between the relative velocity ¯v of observers u and v and the direction
the light is traveling, given by ˆku, as measured by observer u.
The angular frequency measured by observer v is given by
ωv = −ˆv · ¯K = −ωuγ(−1 + ¯v · ˆku) = ωuγ(1 − v cos θu) , (6.17)
or
ωv
ωu
=
1 − v cos θu
√
1 − v2
. (6.18)
This is the formula for the Doppler shift. It says how the frequency seen by two observers
depends on their relative velocity and the angle of the velocity to that of the light. Let
the observer u be at rest with respect to the source emitting the light (radiation). Let us
consider the two special cases when v is moving directly towards the source and directly
away from it. We have
34 CHAPTER 6. WAVES
Towards source cos θu = −1 ω
ω
= 1+v
1−v
> 1 ”Blue shift”
Away from source cos θu = +1 ω
ω
= 1−v
1+v
< 1 ”Red shift”
An observer moving towards the source sees a higher frequency, the light is blue shifted,
while an observer moving away sees a lower frequency, the light is red shifted. The Doppler
effect is very important in astronomy where it is used for example to measure the velocity
of stars and galaxies relative to us by looking at the shifts of their spectral lines.
We could have used the observer v instead of u and we would have found the formula
for the Doppler shift
ωu
ωv
=
1 + v cos θv
√
1 − v2
. (6.19)
Note the plus sign in the numerator which is due to the fact that the relative velocity
according to v has opposite sign compared to what u measures. Multiplying this with the
previous formula for the Doppler shift we find
(1 + v cos θv)(1 − v cos θu) = 1 − v2
. (6.20)
This is the formula for aberration derived by Einstein in his 1905 paper. It can also be
written as
cos θv =
v − cos θu
1 − v cos θu
. (6.21)
For θu << 1 we have θv ≈ π since ¯v is directed from u to v. Changing the sign of cos θv so
that θu and θv are measured in the same way we find, using cos θu ≈ 1 − θ2
u/2, that
θv ≈
1 + v
1 − v
θu , (6.22)
which shows that an observer traveling away from the object (since ¯v and ˆk almost point
in the same direction in this case) measures a larger angle. Conversely, switching the sign
of v the observer is traveling toward the light source and measures a smaller angle by a
factor 1−v
1+v
< 1. Therefore light is concentrated in the direction of motion and an observer
traveling towards a star sees it as smaller and brighter while one traveling away sees it as
larger and fainter. This effect again has important consequences in astronomy where it
leads to seasonal variations in the angular separation of objects in the sky.
Chapter 7
Particle kinematics
An important application of special relativity is to processes involving elementary particles,
e.g. in particle accelerators, where they can often reach velocities close to the speed of light.
In order to discuss what happens in particle collisions we first need to introduce the notion
of four-momentum.
7.1 Four-momentum
Recall the split of a four-velocity ˆv with respect to another one ˆu
ˆv = γ(ˆu + ¯v) ˆu · ¯v = 0 , γ =
1
√
1 − v2
. (7.1)
If v is an object and u an observer ¯v is the velocity of the object relative to the observer.
In the Newtonian limit v << 1, i.e. for velocities much smaller than the speed of light, we
have, dropping terms of order v3
and higher,
ˆv ≈ (1 +
v2
2
)ˆu + ¯v . (7.2)
To relate to something familiar let us multiply this by the mass of the object m giving
mˆv ≈ (m +
mv2
2
)ˆu + m¯v . (7.3)
We recognize the kinetic energy of the particle 1
2
mv2
and its momentum m¯v. Note that
every object has a well-defined mass (just consider an observer traveling along with the
object, i.e. whose ˆu is (nearly) parallel to ˆv, who can define the mass in the usual Newtonian
way). We define the four-momentum of an object to be its mass times its four-velocity
¯P = mˆv . (7.4)
It satisfies
¯P2
= −m2
. (7.5)
35
36 CHAPTER 7. PARTICLE KINEMATICS
The split of ˆv gives a corresponding split of the four-momentum
¯P = Eˆu + ¯p ˆu · ¯p = 0 , (7.6)
with
E = mγ =
m
√
1 − v2
≈ m +
mv2
2
, ¯p = m¯vγ =
m¯v
√
1 − v2
≈ m¯v . (7.7)
In analogy with the Newtonian case we call E the energy and ¯p the momentum (or threemomentum)
of the object. Note that they are not intrinsic properties of the object but
depend on the observer.
The four-momentum ¯P = mˆv is time-like since ˆv is time-like. Furthermore, since m > 0,
it is also future directed. The energy measure by observer u can be written
E = −ˆu · ¯P (7.8)
and we see that it is always positive.
The usefulness of the notion of momentum in Newtonian mechanics comes from the
fact that it is conserved (in the absence of external forces): The sum of all momenta before
a collision is equal to the sum of momenta after the collision. This is actually a special
case of the relativistic conservation of four-momentum
before
¯Pi =
after
¯Pi . (7.9)
It says that there are four conserved quantities: The total energy and the three components
of the momentum.
The fact that four-momentum is conserved is intimately tied to symmetries in nature.
In fact you will learn later in your studies that the conservation of momentum is equivalent
to the statement that physical laws look the same regardless of position in space. The laws
of physics are the same here as on the moon or in the Andromeda galaxy. Similarly the
conservation of energy is equivalent to the fact that the laws of physics look the same at
all times. Just like the theory of relativity combines space and time into a spacetime it
combines the notions of momentum and energy into the single concept of four-momentum.
Note that in the Newtonian limit the total energy is
E ≈
i
mi +
mi¯v2
i
2
. (7.10)
But we know that the kinetic energy is not conserved in general, it is only conserved in
elastic collisions. This means that the first term must change to keep the total energy
conserved, i.e. the sum of the masses before and after the collision will in general differ
contrary to what is assumed in Newtonian mechanics. In inelastic collisions the kinetic
energy decreases so the total mass must increase. Of course in everyday situations v << 1
so the kinetic energy is much smaller than the total mass, 1
2
m¯v2
<< m.
7.2. MASSLESS PARTICLES 37
This fact, that mass is also a form of energy, is the content of one of the most famous
equations in physics
E = mc2
. (7.11)
Remember that we are using units where the speed of light is unity, c = 1. Note also that
E = m is true only for an observer at rest with respect to the object in question. It is
often referred to as the rest energy.
7.2 Massless particles
Knowing all but one four-momentum of the particles in a given reaction one can use the
conservation of four-momentum to determine the remaining one. In this way one can
experimentally establish the existence of particles with null four-momentum
¯P2
= 0 . (7.12)
Since we may define the mass through ¯P2
= −m2
we call such particles massless. The most
important example of such a particle is the photon, the quantum of light (more generally
electromagnetic radiation). The four-momentum and energy of such a particle does not
vanish, if it did we would not be able to detect them, and therefore we conclude from the
expression
E =
m
√
1 − v2
(7.13)
that such a particle must have v = 1, i.e. travel at the speed of light. It therefore follows
a null line with four-velocity proportional to the null vector ¯P. Note that for a massless
particle we have
¯P = Eˆu + ¯p = E(ˆu + ˆp) , (7.14)
since it is null.
7.3 Tachyons
Sometimes on hears about particles with space-like four-momentum
¯P2
> 0 . (7.15)
Such particles are referred to as tachyons. Such a particle would have an imaginary mass
from ¯P2
= −m2
and always travel along space-like lines. This turns out not to be consistent
with the laws of quantum mechanics and such particles therefore cannot exist in nature.
Instead in our current best theories particles emerge as ”ripples” of a field with a
quantum of energy. In such quantum field theories it is not unusual to have m2
< 0.
However, the resulting particles do not travel faster than light. Instead such an imaginary
mass signals an instability of the vacuum. In fact this is part of the mechanism by which
particles can acquire mass through the so-called Higgs mechanism, related to the famous
Higgs particle discovered at CERN.
38 CHAPTER 7. PARTICLE KINEMATICS
7.4 Particle reactions and kinematics
The two most important types of particle reactions are
• Decay: One particle goes into two or more
• Collision: Two particles go into one or more
The allowed configurations of four-momenta are constrained by
1. ¯P2
i = −m2
i for all particles involved
2. Conservation of four-momentum
i
¯Pi,in =
j
¯Pj,out
These equations form the basis of particle kinematics. It is often convenient to refer all
quantities to some particular observer, say with four-velocity ˆu. Then we have
¯Pi = Ei ˆu + ¯pi ˆu · ¯pi = 0 . (7.16)
The inner product the four-momenta of two particles becomes
¯Pi · ¯Pj = −EiEj + ¯pi · ¯pj , (7.17)
and in the special case i = j we find
m2
i = − ¯P2
i = E2
i − ¯p2
i . (7.18)
A typical kinematical calculation involves two steps
1. Take the inner product of the conservation of for momentum
i
¯Pi,in =
j
¯Pj,out
with some ¯Pi, or rearrange it and take the square.
2. Replace ¯P2
i by −m2
i everywhere and inner products by (7.17).
7.4. PARTICLE REACTIONS AND KINEMATICS 39
E 
 
 
  
e
e
e
e
e
e
e…
θ
¯q
¯p = 0
¯p
¯q
Figure 7.1: Scattering of electron off a proton at rest (¯p = 0) in the observers orthogonal
space. The recoil angle of the proton is θ.
Example
Consider an electron (e−
) scattering off a proton (p+
). We set me− = m and mp+ = M
and let ¯P, ¯Q be the initial four-momenta and ¯P , ¯Q the final ones. We have
¯P2
= ¯P 2
= −M2
, ¯Q2
= ¯Q 2
= −m2
(7.19)
while the conservation of four-momentum says that
¯P + ¯Q = ¯P + ¯Q . (7.20)
Let’s say we are interested in the situation where the proton is at rest before the collision
and we want to find the recoil angle of the proton. The situation is illustrated in the
observers orthogonal space in figure 7.1. One there is one particle which we do not know
anything about, in this case the outgoing electron with four-momentum ¯Q , it is often a
useful strategy to rearrange the conservation of four momentum so that its four momentum
appears alone on one side and then square. We therefore write
¯P + ¯Q − ¯P = ¯Q . (7.21)
Squaring this equation we get
¯P2
+ ¯Q2
+ ¯P 2
+ 2 ¯P · ¯Q − 2 ¯P · ¯P − 2 ¯Q · ¯P = ¯Q 2
. (7.22)
40 CHAPTER 7. PARTICLE KINEMATICS
Using fact that ¯P2
= ¯P 2
= −M2
and ¯Q2
= ¯Q 2
= −m2
this can be written
0 = − M2
+ ¯P · ¯Q − ( ¯P + ¯Q) · ¯P
=( ¯P + ¯Q) · ( ¯P − ¯P ) . (7.23)
This is a useful equation for two-particle elastic collisions. Since we are assuming the
proton is initially at rest with respect to the observer we have
¯P = M ˆu , ¯Q = Eˆu + ¯q , ¯P = E ˆu + ¯p . (7.24)
Using this we find
0 = ( ¯P + ¯Q)·( ¯P − ¯P ) = ((M+E)ˆu+¯q)·((M−E )ˆu−¯p ) = −(M+E)(M−E )−¯q·¯p . (7.25)
Since ¯q and ¯p are in the orthogonal space to ˆu they can be treated as ordinary Euclidean
vector and we can write
¯q · ¯p = qp cos θ , (7.26)
where θ is the recoil angle, see figure 7.1. We have
−m2
= ¯Q2
= −E2
+ q2
⇒ q =
√
E2 − m2 (7.27)
and similarly we have p =
√
E 2 − M2 giving
cos θ =
¯q · ¯p
qp
=
(E + M)(E − M)
√
E2 − m2
√
E 2 − M2
=
E + M
√
E2 − m2
E − M
E + M
. (7.28)
This expresses the recoil angle of the proton in terms of the initial energy of the electron
E and the final energy of the proton E .
7.5 Center-of-mass observer
For more complicated processes it is often useful to regard all (or part) of the out-going
particles together. Their total four-momentum is
¯P =
i
¯Pi . (7.29)
Then the ”mass” defined by ¯P2
= −M2
does not have a fixed value. It depends on the
relative motion of the particles. It does have a lower bound though. Since a sum of
time-like future directed vectors is again time-like and future directed we can consider an
observer with four-velocity ˆu = ¯P/M. Then we find
¯P2
= −E2
= −
i
Ei
2
= −
i
m2
i + p2
i
2
. (7.30)
7.5. CENTER-OF-MASS OBSERVER 41
Therefore
¯P2
=
i
¯Pi
2
≥
i
mi
2
(7.31)
with equality occurring only if ¯pi = 0 for all i, i.e. if all particles are at rest with respect to
each other. This inequality leads to so-called threshold conditions for when certain particles
can be produced, for example a Higgs boson at the LHC. Note that there is no reference
to any observer in the last equation – the result is observer independent. Nevertheless it
was convenient to introduce an observer in the derivation of the result and this turns out
to often be the case.
An observer, such as the one considered above, whose four-momentum is parallel to
the total four-momentum i
¯Pi is called a center-of-mass observer (often center-of-mass
frame or center-of-mass system), since such an observer sees the particles with zero total
three-momentum, i ¯pi = 0. The introduction of such an observer can often simplify the
calculations. Note that using Lorentz transformations one can of course go freely from one
observer to another.
42 CHAPTER 7. PARTICLE KINEMATICS
Chapter 8
Curved worldlines
So far we have dealt almost exclusively with straight worldlines. Unaccelerated spaceships
and particles not influenced by external forces travel along such worldlines. Conversely
spaceships that run their engines or particles influenced by external forces follow curved
worldlines. While it is possible to treat accelerated observers in special relativity it is usually
avoided for practical reasons. We will therefore continue to assume that all observers
are unaccelerated, unless otherwise stated.
The natural parameter along a (massive) particle’s worldline is the proper time τ. The
spacetime position of the particle is given as a function of τ
¯R = ¯R(τ) . (8.1)
We can also calculate the rate of change of the position vector with τ as
d ¯R
dτ
= lim
∆τ→0
¯R(τ + ∆τ) − ¯R(τ)
∆τ
. (8.2)
Since τ is the proper time along the worldline, which measures its length, we have
dτ2
= −(d ¯R)2
. (8.3)
Therefore
ˆv =
d ¯R
dτ
(8.4)
is a time-like unit vector, ˆv2
= −1. It is tangent to the worldline at the event ¯R(τ). For a
straight worldline d¯r/dτ =constant and ˆv is the four-velocity. We will continue to define
it as the four-velocity also for curved worldlines.
The four-acceleration is defined as
¯A =
dˆv
dτ
=
d2 ¯R
dτ2
. (8.5)
Differentiating the condition ˆv2
= −1 we find
ˆv · ¯A = 0 . (8.6)
43
44 CHAPTER 8. CURVED WORLDLINES
This implies that ¯A is space-like. Note that it is not, in general, a unit vector.
Let us now analyze how an unaccelerated observer views an accelerated worldline.
Consider an observer whose worldline is tangent to the curved worldline at a certain point
¯R(τ0), as depicted in figure 8.1. Splitting with respect to the four-velocity ˆu of the observer
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡¡
¡
¡
¡
¡
¡
¡
¡
¡
¡!
r
rrj
 
 
  
rrj
¡
¡
¡
¡!
¯R(τ0)
¯a0
¯R(τ)
¯r
tˆu
Figure 8.1: An observers worldline is tangent to the curved worldline of an accelerated
object at the point ¯R(τ0).
we have
¯R(τ) = tˆu + ¯r , ˆu · ¯r = 0 . (8.7)
Considering a small interval along the curve this gives
d ¯R = dtˆu + d¯r (8.8)
and squaring this we find
−dτ2
= −dt2
+ d¯r2
(8.9)
or
dτ = dt
√
1 − v2 , ¯v =
d¯r
dt
. (8.10)
8.1. CONSTANT ACCELERATION 45
Note that ¯v is the relative velocity and we can also write dτ = dt/γ. This gives the
four-velocity of the object
ˆv =
d ¯R
dτ
=
d ¯R
dτ
= γ
d ¯R
dt
= γ(ˆu + ¯v) . (8.11)
In fact this is the same equation we have before when expressing a constant four-velocity
ˆv in terms of another one ˆu, here however ˆv and ¯v are not constant. At the point ¯R(τ0) we
have ˆv = ˆu so the relative velocity ¯v = 0 at this point. For the four-acceleration we find
¯A0 =
dˆv
dτ 0
=
dˆv
dt 0
= (¯a)0 , (8.12)
where ¯a is the ordinary Newtonian acceleration measured by the observer momentarily at
rest with respect to the accelerated object. If this object happens to be a spaceship the
vector ¯A on a certain point of its worldline is the acceleration experience by the crew at
that instant.
8.1 Constant acceleration
Consider a two-dimensional spacetime, which could be the history of a spaceship traveling
in a straight line as seen by an observer. In two-dimensions we need only one condition to
determine a curve and we take
¯R2
= R2
, (8.13)
with ¯R the spacetime position and R a constant. Taking the derivative we find
¯R · ˆv = 0 (8.14)
and another derivative gives
¯R · ¯A = 1 . (8.15)
The vectors ¯R, ˆv and ¯A lie in a 2-plane and since both ¯R and ¯A are orthogonal to ˆv they
must be parallel, so that using the equation above we have
¯A =
¯R
R2
⇒ ¯A2
=
1
R2
, (8.16)
i.e. the four-acceleration is constant. This could be the worldline of a spaceship running
its engines so that the crew experiences a constant acceleration.
Let ¯R2
= R2
1 and ¯R2
= R2
2 be two such curves in the same 2-plane and with the same
origin. From (8.14) we see that any line through the origin cuts the curves orthogonally.
Therefore R1 − R2 is the orthogonal distance between the curves.
In fact one of the lines could be the worldline of the front of a spaceship and the other
the worldline of its tail. The ship’s length is constant but the acceleration of the front and
the tail are unequal, being 1/R2
1 and 1/R2
2 respectively.
46 CHAPTER 8. CURVED WORLDLINES
Introducing an observer with four-velocity ˆu we can split
¯R = tˆu + ¯r , ˆu · ¯r = 0 (8.17)
and
¯R2
= R2
⇒ −t2
+ ¯r2
= R2
. (8.18)
This means that in the (¯r, t)-plane of the observer the worldlines of constant acceleration
are hyperbolas, as illustrated in figure 8.2. Important properties are hidden in this figure,
T
E
t
r
R1 R2
¯R2
= R2
1
¯R2
= R2
2
Figure 8.2: Worldlines of constant acceleration as viewed by an observer.
e.g. that all points on the curves are equivalent and that all observers see the same
picture. It is also hard to see that lines through the origin are orthogonal to the curves
and that they have constant separation. These problems arise because we are representing
a spacetime situation in Euclidean space. The corresponding situation in Euclidean space
would be two concentric circles of radius R1 and R2 respectively, and here they properties
mentioned above become obvious. Note however that this Euclidean picture is also not
a reliable representation of the spacetime situation, e.g. the curves are closed which is
impossible in spacetime.
8.2 Fitting a car into a garage
A famous ’paradox’ in special relativity is this:
8.2. FITTING A CAR INTO A GARAGE 47
Imagine you just bought a very fast car. Unfortunately the car turns out to
be slightly too long to fit in your garage. Is it possible to exploit the Lorentz
contraction and drive the car very fast into the garage and slam the door?
From the point of view of the garage the car appears shorter due to the Lorentz contraction
which suggests that it should work. On the other hand, from the point of view of the car,
the garage appears shorter making the situation worse, not better. This is the ’paradox’.
Of course, when you analyze things carefully there is no contradiction.
Let ˆu be the four-velocity of the garage and ˆv that of the car. Lets also take units
where the length of the garage is 1 and let ˆg be orthogonal to ˆu and stretching from the
doors to the back of the garage. Similarly we let ¯c be orthogonal to ˆv and stretch from the
front of the car to the tail. We assume the front (tail) of the car continues unaccelerated
until it reaches the back wall (doors) of the garage at which point it suddenly stops. (This
is of course an idealized situation involving infinite acceleration.)
The spacetime diagram is given in figure 8.3. The following important events are
T T
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
E
d
d
d
dd‚
s s
s
E0
E1
E2
doors back wall
front
tail
ˆu
ˆv
ˆg
¯c
Figure 8.3: Spacetime diagram for the problem of fitting the car into the garage.
indicated in the figure
E0 : The garage doors close
E1 : The simultaneous event at the front of the car from the cars point of view
E2 : The front of the car collides with the back wall of the garage. Simultaneous with
E0 from the point of view of the garage.
48 CHAPTER 8. CURVED WORLDLINES
If ¯v is the relative velocity we can apply the Lorentz transformation relating the observations
from the garage to those from the car
ˆv = γ(ˆu + vˆg) , ˆc = γ(ˆg + vˆu) , (8.19)
or the inverse transformations
ˆu = γ(ˆv − vˆc) , ˆg = γ(ˆc − vˆv) . (8.20)
The last of these equations will interest us here. Let ¯V be the vector from E1 to E2. From
figure 8.3 we have
¯c + ¯V = ˆg = γ(ˆc − vˆv) (8.21)
and we find
¯c = γˆc , ¯V = −γvˆv . (8.22)
The length of the car is =
√
¯c2 = γ > 1, so the car is longer than the garage, consistent
with our assumptions.
Since ¯V is negative relative to ˆv, from the point of view of the car the front collides
with the back wall before the doors close, although these events are simultaneous from the
point of view of the garage. However, since both ˆg and ¯c are space-like no signal from
the collision can reach the tail of the car before it passes the doors. Therefore no material
strength can stop the car from being compressed (from its point of view) and you can shut
the door behind it. You have to be quick though if the car is elastic and tends to regain
its original length. This example illustrates the fact that no perfectly rigid body exists.
We have seen that it is indeed possible (though not recommended) to fit the car into
the garage by exploiting the Lorentz contraction. We have also seen that what appears at
first sight to be a paradox is just due to not analyzing the situation carefully enough, in
particular neglecting the issues due to the relativity of simultaneity. When the problem is
carefully analyzed there are of course no contradictions.
8.3 Rotating wheel
In a wheel at rest all constituent particles follows straight worldlines. When the wheel
starts spinning the worldlines become tilted and form helices in spacetime. Only the
center continues on a straight worldline.
Using the center as an observer and thinking of a ring of matter a distance ρ around it
we realize that such a ring must have circumference 2πρ whether the wheel is rotation or
not. In a sense the Lorentz contraction is prevented from taking place. Instead the tilting
of the worldlines has the effect that the orthogonal distance between them increases, i.e. a
deformation takes place.
This is an inevitable deformation which is always connected with rotation. How much
energy is needed to produce the deformation depends on the stresses in the wheel. Therefore
the moment of inertia of a wheel must also depend on the stresses within it. This is an
important consequence of the theory of relativity though it plays little role in everyday
circumstances.