Notes on Relativity and Cosmology - Physics Department, UCSB

74 CHAPTER 3. EINSTEIN AND INERTIAL FRAMES
So, we see that distance measurements also depend on the observer’s frame of
reference. Note however, that given any inertial object, there is a special inertial
frame in which the object is at rest. The length of an object in its own rest
frame is known as its proper length. The length of the object in any other inertial
frame will be shorter than the object’s proper length. We can summarize what
we have learned by stating:
An object of proper length L moving through an inertial frame at speed v
has length L √ 1 − v 2 /c 2 as measured in that inertial frame.
There is an important subtlety that we should explore. Note that the above
statement refers to an object. However, we can also talk about proper distance
between two events. When two events are spacelike related, there is a special
frame of reference in which the events are simultaneous and the separation is
“pure space” (with no separation in time). The distance between them in this
frame is called the proper distance between the events. It turns out that this
distance is in fact longer in any other frame of reference.....
Why longer? To understand this, look back at the above diagram and compare
the two events at either end of the students’ rod that are simultaneous in the
professor’s frame of reference. Note that the proper distance is the distance
measured in the professor’s reference frame, which we just concluded is shorter
than the distance measured by the student. The difference here is that we are
now talking about events (points on the diagram) where as before we were
talking about objects (whose ends appear as worldlines on the spacetime diagram).
The point is that, when we talk about measuring the length of an object,
different observers are actually measuring the distance between different pairs
of events.
3.7 The Train Paradox
Let us now test our new skills and work through some subtleties by considering
an age-old parable known as the train paradox. It goes like this:
Once upon a time there was a really fast Japanese bullet train that ran at 80%
of the speed of light. The train was 100m long in its own rest frame. The
train carried as cargo the profits of SONY corporation from Tokyo out to their
headquarters in the countryside. The profits were, of course, carried in pure
gold.
Now, some less than reputable characters found out about this and devised an
elaborate scheme to rob the train. They knew that the train would pass through
a 100m long tunnel on its route. Watching the train go by, they measured the
train to be quite a bit less than 100m long and so figured that they could easily
trap it in the tunnel.
Of course, the people on the train found that, when the train was in motion,
it was the train that was 100m long while the tunnel was significantly shorter.
As a result, they had no fear of being trapped in the tunnel by train robbers.
Now, do you think the robbers managed to catch the train?