Notes on Relativity and Cosmology - Physics Department, UCSB

76 CHAPTER 3. EINSTEIN AND INERTIAL FRAMES
that signals can only be sent at speed c or slower, the earliest possible time that
the back of the train could discover the exit explosion is at the event marked
D on the diagram. Note that, at event D, the back of the train does find itself
inside the tunnel and also finds that event 1 has already occurred. The entrance
is closed and the train cannot escape.
There are two things that deserve more explanation. The first is the above
comment about the shock wave. Normally we think of objects like trains as
being perfectly stiff. However, this is not really so. Let’s think about what
happens when I push on one end of a meter stick. I press on the atoms on the
end, which press on the atoms next to them, which press on the atoms next
to them ..... Also, it takes a (small but finite) amount of time for each atom
to respond to the push it has been given on one side and to move over and
begin to push the atom on the other side. The result is known as a “shock
wave” that travels at finite speed down the object. Note that an important
part of the shock wave are the electric forces that two atoms use to push each
other around. Thus, the shock wave can certainly not propagate faster than
an electromagnetic disturbance can. As a result, it must move at less than the
speed of light.
For the other point, let’s suppose that the people at the front of the train step
on the brakes and stop immediately. Stopping the atoms at the front of the
train will make them push on the atoms behind them, stopping them, etc. The
shock wave results from the fact that atoms just behind the front slam into
atoms right at the front; the whole system compresses a bit and then may try
to reexpand, pushing some of the atoms farther back.
⋆⋆ What we saw above is that the shock wave cannot reach the back of the train
until event D. Suppose that it does indeed stop the back of the train there. The
train has now come to rest in the tunnel’s frame of reference. Thus, after event
D, the proper length of the train is less than 100m!!!!
In fact, suppose that we use the lines of simultaneity in the train’s original frame
of reference (before it tries to stop) to measure the proper length of the train.
Then, immediately after event 2 the front of the train changes its motion, but
the back of the train keeps going. As a result, in this sense the proper length
of the train starts to shrink immediately after event 2. This is how it manages
to fit itself into a tunnel that, in this frame, is less than 100m long.
What has happened? The answer is in the compression that generates the shock
wave. The train really has been physically compressed by the wall of debris at
the exit slamming into it at half the speed of light 6 ! This compression is of
course accompanied by tearing of metal, shattering of glass, death screams of
passengers, and the like, just as you would expect in a crash. The train is
completely and utterly destroyed. The robbers will be lucky if the gold they
wish to steal has not been completely vaporized in the carnage.
Now, you might want to get one more perspective on this (trying to show some
hidden inconsistency?) by analyzing the problem again in a frame of reference
6 Or the equivalent damage inflicted through the use of the train’s brakes.