7. Special Relativity
Although Newtonian mechanics gives an excellent description of Nature, it is not uni-
versally valid. When we reach extreme conditions — the very small, the very heavy or
the very fast — the Newtonian Universe that we’re used to needs replacing. You could
say that Newtonian mechanics encapsulates our common sense view of the world. One
of the major themes of twentieth century physics is that when you look away from our
everyday world, common sense is not much use.
One such extreme is when particles travel very fast. The theory that replaces New-
tonian mechanics is due to Einstein. It is called special relativity. The e↵ects of special
relativity become apparent only when the speeds of particles become comparable to
the speed of light in the vacuum. The speed of light is
1
c = 299792458 ms
This value of c is exact. It may seem strange that the speed of light is an integer
when measured in meters per second. The reason is simply that this is taken to be
the definition of what we mean by a meter: it is the distance travelled by light in
1/299792458 seconds. For the purposes of this course, we’ll be quite happy with the
approximation c ⇡ 3 ⇥ 108 ms 1 .
The first thing to say is that the speed of light is fast. Really fast. The speed of
sound is around 300 ms 1 ; escape velocity from the Earth is around 104 ms 1 ; the
orbital speed of our solar system in the Milky Way galaxy is around 105 ms 1 . As we
shall soon see, nothing travels faster than c.
The theory of special relativity rests on two experimental facts. (We will look at the
evidence for these shortly). In fact, we have already met the first of these: it is simply
the Galilean principle of relativity described in Section 1. The second postulate is more
surprising:
• Postulate 1: The principle of relativity: the laws of physics are the same in all
inertial frames
• Postulate 2: The speed of light in vacuum is the same in all inertial frames
On the face of it, the second postulate looks nonsensical. How can the speed of light
look the same in all inertial frames? If light travels towards me at speed c and I run
away from the light at speed v, surely I measure the speed of light as c v. Right?
Well, no.
– 107 –
, This common sense view is encapsulated in the Galilean transformations that we
met in Section 1.2.1. Mathematically, we derive this “obvious” result as follows: two
inertial frames, S and S 0 , which move relative to each with velocity v = (v, 0, 0), have
Cartesian coordinates related by
x0 = x vt , y0 = y , z0 = z , t0 = t (7.1)
If a ray of light travels in the x direction in frame S with speed c, then it traces out
the trajectory x/t = c. The transformations above then tell us that in frame S 0 the
trajectory if the light ray is x0 /t0 = c v. This is the result we claimed above: the
speed of light should clearly be c v. If this is wrong (and it is) something must be
wrong with the Galilean transformations (7.1). But what?
Our immediate goal is to find a transformation law that obeys both postulates above.
As we will see, the only way to achieve this goal is to allow for a radical departure in
our understanding of time. In particular, we will be forced to abandon the assumption
of absolute time, enshrined in the equation t0 = t above. We will see that time ticks at
di↵erent rates for observers sitting in di↵erent inertial frames.
7.1 Lorentz Transformations
We stick with the idea of two inertial frames, S and S 0 , moving with relative speed v.
For simplicity, we’ll start by ignoring the directions y and z which are perpendicular to
the direction of motion. Both inertial frames come with Cartesian coordinates: (x, t)
for S and (x0 , t0 ) for S 0 . We want to know how these are related. The most general
possible relationship takes the form
x0 = f (x, t) , t0 = g(x, t)
for some function f and g. However, there are a couple of facts that we can use to
immediately restrict the form of these functions. The first is that the law of inertia
holds; left alone in an inertial frame, a particle will travel at constant velocity. Drawn
in the (x, t) plane, the trajectory of such a particle is a straight line. Since both S and
S 0 are inertial frames, the map (x, t) 7! (x0 , t0 ) must map straight lines to straight lines;
such maps are, by definition, linear. The functions f and g must therefore be of the
form
x0 = ↵ 1 x + ↵ 2 t , t0 = ↵3 x + ↵4 t
where ↵i , i = 1, 2, 3, 4 can each be a function of v.
– 108 –
,Secondly, we use the fact that S 0 is travelling at speed v relative t
S’
to S. This means that an observer sitting at the origin, x0 = 0,
of S 0 moves along the trajectory x = vt in S shown in the figure.
Or, in other words, the points x = vt must map to x0 = 0. (There
is actually one further assumption implicit in this statement: that x
the origin x0 = 0 coincides with x = 0 when t = 0). Together with
the requirement that the transformation is linear, this restricts
the coefficients ↵1 and ↵2 above to be of the form, Figure 43:
x0 = (x vt) (7.2)
for some coefficient . Once again, the overall coefficient can be a function of the
velocity: = v . (We’ve used subscript notation v rather than the more standard (v)
to denote that depends on v. This avoids confusion with the factors of (x vt) which
aren’t arguments of but will frequently appear after like in the equation (7.2)).
There is actually a small, but important, restriction on the form of v : it must be
an even function, so that v = v . There are a couple of ways to see this. The first
is by using rotational invariance, which states that can depend only on the direction
of the relative velocity v, but only on the magnitude v 2 = v · v. Alternatively, if this
is a little slick, we can reach the same conclusion by considering inertial frames S̃ and
S̃ 0 which are identical to S and S 0 except that we measure the x-coordinate in the
opposite direction, meaning x̃ = x and x̃0 = x0 . While S is moving with velocity
+v relative to S 0 , S̃ is moving with velocity v with respect to S̃ 0 simply because we
measure things in the opposite direction. That means that
x̃0 = v x̃ + v t̃
Comparing this to (7.2), we see that we must have v = v as claimed.
We can also look at things from the perspective of S 0 , relative to t’
S
which the frame S moves backwards with velocity v. The same
argument that led us to (7.2) now tells us that
x = (x0 + vt0 ) (7.3) x’
Now the function = v . But by the argument above, we know
that v = v . In other words, the coefficient appearing in (7.3) Figure 44:
is the same as that appearing in (7.2).
– 109 –
, At this point, things don’t look too di↵erent from what we’ve seen before. Indeed, if
we now insisted on absolute time, so t = t0 , we’re forced to have = 1 and we get back
to the Galilean transformations (7.1). However, as we’ve seen, this is not compatible
with the second postulate of special relativity. So let’s push forward and insist instead
that the speed of light is equal to c in both S and S 0 . In S, a light ray has trajectory
x = ct
While, in S 0 , we demand that the same light ray has trajectory
x0 = ct0
Substituting these trajectories into (7.2) and (7.3), we have two equations relating t
and t0 ,
ct0 = (c v)t and ct = (c + v)t0
A little algebra shows that these two equations are compatible only if is given by
s
1
= (7.4)
1 v 2 /c2
We’ll be seeing a lot of this coefficient in what follows. Notice that for v ⌧ c, we
have ⇡ 1 and the transformation law (7.2) is approximately the same as the Galilean
transformation (7.1). However, as v ! c we have ! 1. Furthermore, becomes
imaginary for v > c which means that we’re unable to make sense of inertial frames
with relative speed v > c.
Equations (7.2) and (7.4) give us the transformation law for the spatial coordinate.
But what about for time? In fact, the temporal transformation law is already lurking in
our analysis above. Substituting the expression for x0 in (7.2) into (7.3) and rearranging,
we get
⇣ v ⌘
t0 = t x (7.5)
c2
We shall soon see that this equation has dramatic consequences. For now, however, we
merely note that when v ⌧ c, we recover the trivial Galilean transformation law t0 ⇡ t.
Equations (7.2) and (7.5) are the Lorentz transformations.
– 110 –
Although Newtonian mechanics gives an excellent description of Nature, it is not uni-
versally valid. When we reach extreme conditions — the very small, the very heavy or
the very fast — the Newtonian Universe that we’re used to needs replacing. You could
say that Newtonian mechanics encapsulates our common sense view of the world. One
of the major themes of twentieth century physics is that when you look away from our
everyday world, common sense is not much use.
One such extreme is when particles travel very fast. The theory that replaces New-
tonian mechanics is due to Einstein. It is called special relativity. The e↵ects of special
relativity become apparent only when the speeds of particles become comparable to
the speed of light in the vacuum. The speed of light is
1
c = 299792458 ms
This value of c is exact. It may seem strange that the speed of light is an integer
when measured in meters per second. The reason is simply that this is taken to be
the definition of what we mean by a meter: it is the distance travelled by light in
1/299792458 seconds. For the purposes of this course, we’ll be quite happy with the
approximation c ⇡ 3 ⇥ 108 ms 1 .
The first thing to say is that the speed of light is fast. Really fast. The speed of
sound is around 300 ms 1 ; escape velocity from the Earth is around 104 ms 1 ; the
orbital speed of our solar system in the Milky Way galaxy is around 105 ms 1 . As we
shall soon see, nothing travels faster than c.
The theory of special relativity rests on two experimental facts. (We will look at the
evidence for these shortly). In fact, we have already met the first of these: it is simply
the Galilean principle of relativity described in Section 1. The second postulate is more
surprising:
• Postulate 1: The principle of relativity: the laws of physics are the same in all
inertial frames
• Postulate 2: The speed of light in vacuum is the same in all inertial frames
On the face of it, the second postulate looks nonsensical. How can the speed of light
look the same in all inertial frames? If light travels towards me at speed c and I run
away from the light at speed v, surely I measure the speed of light as c v. Right?
Well, no.
– 107 –
, This common sense view is encapsulated in the Galilean transformations that we
met in Section 1.2.1. Mathematically, we derive this “obvious” result as follows: two
inertial frames, S and S 0 , which move relative to each with velocity v = (v, 0, 0), have
Cartesian coordinates related by
x0 = x vt , y0 = y , z0 = z , t0 = t (7.1)
If a ray of light travels in the x direction in frame S with speed c, then it traces out
the trajectory x/t = c. The transformations above then tell us that in frame S 0 the
trajectory if the light ray is x0 /t0 = c v. This is the result we claimed above: the
speed of light should clearly be c v. If this is wrong (and it is) something must be
wrong with the Galilean transformations (7.1). But what?
Our immediate goal is to find a transformation law that obeys both postulates above.
As we will see, the only way to achieve this goal is to allow for a radical departure in
our understanding of time. In particular, we will be forced to abandon the assumption
of absolute time, enshrined in the equation t0 = t above. We will see that time ticks at
di↵erent rates for observers sitting in di↵erent inertial frames.
7.1 Lorentz Transformations
We stick with the idea of two inertial frames, S and S 0 , moving with relative speed v.
For simplicity, we’ll start by ignoring the directions y and z which are perpendicular to
the direction of motion. Both inertial frames come with Cartesian coordinates: (x, t)
for S and (x0 , t0 ) for S 0 . We want to know how these are related. The most general
possible relationship takes the form
x0 = f (x, t) , t0 = g(x, t)
for some function f and g. However, there are a couple of facts that we can use to
immediately restrict the form of these functions. The first is that the law of inertia
holds; left alone in an inertial frame, a particle will travel at constant velocity. Drawn
in the (x, t) plane, the trajectory of such a particle is a straight line. Since both S and
S 0 are inertial frames, the map (x, t) 7! (x0 , t0 ) must map straight lines to straight lines;
such maps are, by definition, linear. The functions f and g must therefore be of the
form
x0 = ↵ 1 x + ↵ 2 t , t0 = ↵3 x + ↵4 t
where ↵i , i = 1, 2, 3, 4 can each be a function of v.
– 108 –
,Secondly, we use the fact that S 0 is travelling at speed v relative t
S’
to S. This means that an observer sitting at the origin, x0 = 0,
of S 0 moves along the trajectory x = vt in S shown in the figure.
Or, in other words, the points x = vt must map to x0 = 0. (There
is actually one further assumption implicit in this statement: that x
the origin x0 = 0 coincides with x = 0 when t = 0). Together with
the requirement that the transformation is linear, this restricts
the coefficients ↵1 and ↵2 above to be of the form, Figure 43:
x0 = (x vt) (7.2)
for some coefficient . Once again, the overall coefficient can be a function of the
velocity: = v . (We’ve used subscript notation v rather than the more standard (v)
to denote that depends on v. This avoids confusion with the factors of (x vt) which
aren’t arguments of but will frequently appear after like in the equation (7.2)).
There is actually a small, but important, restriction on the form of v : it must be
an even function, so that v = v . There are a couple of ways to see this. The first
is by using rotational invariance, which states that can depend only on the direction
of the relative velocity v, but only on the magnitude v 2 = v · v. Alternatively, if this
is a little slick, we can reach the same conclusion by considering inertial frames S̃ and
S̃ 0 which are identical to S and S 0 except that we measure the x-coordinate in the
opposite direction, meaning x̃ = x and x̃0 = x0 . While S is moving with velocity
+v relative to S 0 , S̃ is moving with velocity v with respect to S̃ 0 simply because we
measure things in the opposite direction. That means that
x̃0 = v x̃ + v t̃
Comparing this to (7.2), we see that we must have v = v as claimed.
We can also look at things from the perspective of S 0 , relative to t’
S
which the frame S moves backwards with velocity v. The same
argument that led us to (7.2) now tells us that
x = (x0 + vt0 ) (7.3) x’
Now the function = v . But by the argument above, we know
that v = v . In other words, the coefficient appearing in (7.3) Figure 44:
is the same as that appearing in (7.2).
– 109 –
, At this point, things don’t look too di↵erent from what we’ve seen before. Indeed, if
we now insisted on absolute time, so t = t0 , we’re forced to have = 1 and we get back
to the Galilean transformations (7.1). However, as we’ve seen, this is not compatible
with the second postulate of special relativity. So let’s push forward and insist instead
that the speed of light is equal to c in both S and S 0 . In S, a light ray has trajectory
x = ct
While, in S 0 , we demand that the same light ray has trajectory
x0 = ct0
Substituting these trajectories into (7.2) and (7.3), we have two equations relating t
and t0 ,
ct0 = (c v)t and ct = (c + v)t0
A little algebra shows that these two equations are compatible only if is given by
s
1
= (7.4)
1 v 2 /c2
We’ll be seeing a lot of this coefficient in what follows. Notice that for v ⌧ c, we
have ⇡ 1 and the transformation law (7.2) is approximately the same as the Galilean
transformation (7.1). However, as v ! c we have ! 1. Furthermore, becomes
imaginary for v > c which means that we’re unable to make sense of inertial frames
with relative speed v > c.
Equations (7.2) and (7.4) give us the transformation law for the spatial coordinate.
But what about for time? In fact, the temporal transformation law is already lurking in
our analysis above. Substituting the expression for x0 in (7.2) into (7.3) and rearranging,
we get
⇣ v ⌘
t0 = t x (7.5)
c2
We shall soon see that this equation has dramatic consequences. For now, however, we
merely note that when v ⌧ c, we recover the trivial Galilean transformation law t0 ⇡ t.
Equations (7.2) and (7.5) are the Lorentz transformations.
– 110 –
