There’s a great little book on special relativity by the physicist N. David Mermin in which he gets to the heart of the astonishing consequences of Einstein’s special relativity in a particularly elegant fashion and with only very basic mathematics. In this and the following note we’ll closely follow Mermin’s treatment. The crucial fact of life which we have to come to terms with is that whether or not two events which are spatially separated happen at the same time is a matter of perspective. This flies in the face of our intuition ¹. We’re wired to think of time as a kind of universal clock and that we and the rest of the universe march forward with its tick-tock relentlessly and in unison.

Let us begin by reconsidering the relativity of velocities. Our intuition, and Galilean relativity, tells us that if you are riding a train and throw a ball in the direction of travel then to someone stationary with respect to the tracks the speed of the ball is simply the sum of the train’s speed and the speed with which the ball  leaves your hand. But thanks to special relativity we know that, at least for light, this isn’t the case. A photon (particle of light) emitted from a moving train moves at light speed \(c\) with respect to the train and with respect to the tracks. This surely has consequences for the relativity of motion in general.

Following Mermin we employ the neat device of measuring the velocity of an object by racing it against a photon. With a corrected velcocity addition rule as our goal we conduct this race on a train carriage.

The particle, black dot, whose velocity \(v\) we seek, sets off from the back of the carriage towards the front in a race with a photon which, as we know, travels at speed \(c\). We arrange that the front of the carriage is mirrored so that once the photon reaches the front it’s reflected back. The point at which the particle and photon meet is recorded (perhaps a mark is made on the floor of the carriage – this is a gedanken experiment!). At that point the particle has travelled a fraction \(1-f\) of the length of the carriage whilst the photon has travelled \(1+f\) times the length of the carriage. The ratio of those distances must be proportional to the ratio of the velocities, that is,
\begin{equation}
\frac{1-f}{1+f}=\frac{v}{c},
\end{equation}
which we can rewrite as an equation for \(f\),
\begin{equation}
f=\frac{c-v}{c+v}\label{eq:f1}.
\end{equation}
The velocity is thus established in an entirely unambiguous manner. This may strike you as a somewhat indirect approach to measuring speed but notice that we’ve avoided measuring either time or distance. As we’ll soon see, in special relativity such measurements are rather more subtle than we might imagine.

Now let’s consider the same race but from the perspective of the track frame relative to which the train carriage is travelling (left to right) with velocity \(u\).

We’re after the correct rule for adding the velocity \(v\), of the particle relative to the train, to the velocity \(u\), of the train relative to the track, to give the velocity \(w\), of the particle relative to the track. To facilitate the calculations we’ll allow ourselves to use some lengths and times. However their values aren’t important — as we’ll see they fall out of the final equation. We’re really just using their ‘existence’. As indicated in the diagram, after time \(T_0\) the photon is a distance \(D\) in front of the particle, that is,
\begin{equation}
D=cT_0-wT_0,
\end{equation}
but this distance is then also the sum of the distances covered respectively by the photon and particle in time \(T_1\),
\begin{equation}
D=cT_1+wT_1.
\end{equation}
So we can write the ratio of the times as
\begin{equation}
\frac{T_1}{T_0}=\frac{c-w}{c+w}\label{eq:time-ratio1}.
\end{equation}
If the length of the carriage in the track frame is \(L\) then we also have that the distance covered by the photon in time \(T_0\) is
\begin{equation}
cT_0=L+uT_0
\end{equation}
and in time \(T_1\) is
\begin{equation}
cT_1=fL-uT_1.
\end{equation}
Combining these we eliminate \(L\) to obtain another expression for the ratio of times,
\begin{equation}
\frac{T_1}{T_0}=f\frac{(c-u)}{(c+u)}\label{eq:time-ratio2}.
\end{equation}
The two equations, \eqref{eq:time-ratio1} and \eqref{eq:time-ratio2} provide us with a second equation for \(f\),
\begin{equation}
f=\frac{(c+u)}{(c-u)}\frac{c-w}{c+w},
\end{equation}
which in combination with the first, \eqref{eq:f1}, leads to
\begin{equation}
\frac{c-w}{c+w}=\frac{c-u}{c+u}\frac{c-v}{c+v}\label{eq:velocity-addition1},
\end{equation}
which expresses the velocity \(w\) of the particle in the track frame in terms of the velocity \(u\) of the train in the track frame and the velocity \(v\) of the particle in the train frame. With a bit more work this can be rewritten as
\begin{equation}
w=\frac{u+v}{1+uv/c^2}\label{eq:velocity-addition2},
\end{equation}
which should be compared to the Galilean addition rule, \(w=u+v\).

Here’s a plot, with velocities in units of \(c\), comparing the Galilean with the special relativity velocity addition for an object fired at a speed \(v\) from a train carriage moving at half the speed of light:

Equation \eqref{eq:velocity-addition2} ensures that no matter how fast the particle travels with respect to the train (assuming it’s less than light speed), its velocity with respect to the track is always less than light speed. In the extreme case of a particle traveling at light speed with respect to a train which is also travelling at light speed, \(1+1=1\)!

Events, observers and measurements

In special relativity we often read that such and such an inertial observer measures the time between two events or such and such an inertial observer measures the distance between two events. On the face of it such assertions seem reasonably clear and straightforward and indeed very often their perspicuity is simply taken for granted. But as we’ll see their meanings in relativity are not what we’d expect and therefore its important to establish early exactly what is meant by an ‘observer’ an ‘event’ and what constitutes a measurement.

The adjective ‘inertial’ in ‘inertial observer’ has been dealt with already — whatever or whoever constitutes an observer should be in free-fall. Let’s also be clear that by an ‘event’ we mean a happening, somewhere, sometime, corresponding to a point in spacetime — perhaps a photon of light leaving an emitter or being absorbed by a detector, perhaps a particle passing through a particular point in space, perhaps a time being recorded by a clock at a particular point in space. Events, points in spacetime, are real, they care nothing for coordinate systems, frames of reference etc.

When we introduced the idea of a frame of reference we vaguely mentioned a laboratory in which lengths and times could be measured. Let’s be more concrete now and imagine an inertial frame of reference as a freely floating 3-dimensional latticework of rods and clocks with one node designated as the origin.

All the rods have the same length but the clocks at each node are rather special. Like all good clocks they can of course keep time. In addition though they are programmed with their respective locations with respect to the origin, so in particular they ‘know’ their distance from the origin. Furthermore they are sophisticated recording devices ready to detect any event and record its location and time for future inspection. In particular, this allows them all to be synchronized with the clock at the origin in the following way. A flash of light is sent out from the origin just as the clock there is set to 0. The spherical light front spreads out at the same speed \(c\) in all directions. As each clock in the lattice detects this light it sets its time equal to its distance from the origin divided by \(c\) and is then ‘in sync’ with the clock at the origin. We should imagine this latticework to be ‘fine-grained’ enough to ensure that to any required accuracy a clock is located ‘at’ the spatial location of any event. This is a crucial point. The time assigned to an event, with respect to an inertial frame of reference, is always that of one of the inertial frame’s clocks at the event. The spacetime location of the event is then given by the spatial coordinates of the clock there together with the clock’s time at the moment the event happens and is recorded along with a description of what took place. This would then constitute a ‘measurement’ and the inertial ‘observer’ carrying out the measurement should be thought of as the whole latticework. An observer is better thought of as the all-seeing eye of the entire inertial frame than as somebody located at some specific point in space with a pair of binoculars and a notepad! If we do speak of an observer as a person, and it is convenient and usual to do so, then we really mean such an intelligent latticework of rods and detecting clocks with respect to which that person is at rest.

Shortly we’ll see that when two or more events at different points in space occur simultaneously with respect to one inertial observer, with respect to another they generally occur at different times. Let’s be clear though that if two or more things happen at the same place at the same time then that’s an event and as such its reality is independent of any frame of reference. All observers must agree that it took place even if they assign to it different spacetime coordinates. Sometimes this is obvious. Consider two particles colliding somewhere. Then obviously the collision either took place or it didn’t and the question is merely what spacetime coordinates should be assigned to the location in spacetime of the collision. But other times it might seem a little more confusing. We might say that an observer, let’s call ‘her’ Alice, records that two spatially separated events, for example photons arriving at two different places, occur at the same time. Recall that this really means that at each location a clock records a time corresponding to the event there and these times turn out to be the same, let’s say 2pm. Now the clock striking 2 at a location just as the event takes place there is itself an event and so will be confirmed by any other inertial observer. Let’s call Bob our other observer. He will assign his own times to the two events, and, as we’ll see, he’ll find that his clocks record different times. However, recall that clock’s don’t just tell time — they also record the event — so Bob will certainly confirm that Alice’s clocks both struck 2pm as the photons arrived at those points in spacetime but Bob will conclude that Alice’s clocks aren’t synchronised since from his perspective these two events did NOT occur simultaneously!