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Length Contraction

The invariance of the spacetime interval enables us to determine by just how much lengths of moving objects seem contracted in comparison to their proper lengths.

Consider the following thought experiment wherein we measure the length of a train, using a timing method, in two different inertial reference frames:

The train is moving to the right at velocity V relative to the ground. We send a flash of light from the left end of the train toward a mirror mounted on the right end. This reflects the light back to a detector which is mounted at the same position on the train as the light source. To determine the train's length, we measure the length of time between when the pulse is emitted and the return pulse is detected. We want to consider the result of this experiment, as measured on the train and on the ground.

First consider the train's frame S', for which we use primed spatial and temporal coordinates. In this case, the relation between the length of the train, delta x', and the measured time interval, delta t', is trivial, as shown in the following figure:

Thus, in S' we have that

That is, the time interval is simply the round-trip path length divided by the speed of light. Note that the time interval is a proper time interval, since the light is emitted and detected at the same place in S'. Also, the length is called a proper length for reasons that will become apparent shortly.

Now consider the measurement in the ground's frame S, for which we use unprimed coordinates. How would an observer in this frame measure the length of the train? The problem is analogous to the boat-buoy problem used to illustrate the Michelson-Morley experiment :

In the present case, the observer on the ground in frame S is analogous to the water frame because the swimmer swims at a constant and well-defined speed relative to the water, just as the light travels at c relative to the frame S. The only real difference is trivial: the swimmer starts on the boat and swims left while the boat and buoy move to the right, while the pulse of light starts at the source and travels right while the source and mirror move in the same direction. These relative directions are just reversed on the return trip, so the round trip times are the same in the two cases. Using similar diagrams:

and slightly modified equations:

we come to the conclusion that the time interval measured in S is

Note that this is an improper time interval, since the beginning and ending times are measured at different locations in S since the train is moving in this frame. Also, we have introduced the length delta x, which the length of the train measured by an observer in S. This is an improper or non-proper length since it is measured in a frame in which the object is not at rest. Using the time dilation result, we can rewrite this last equation as:

This last equation is the result we want since is constitutes length contraction. Recall that gamma is always greater than one, so the improper length is always shorter than the proper length.

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