A 20 foot pole is moving towards a 10 foot barn fast enough that the pole appears to be only 10 feet long. As soon as both ends of the pole are in the barn, slam the doors. How can a 20 foot pole fit into a 10 foot barn?

This is the beginning of the Pole and Barn Paradox. It's bad enough trying to imagine what happens to the pole when it suddenly stops and finds itself in a barn which is too small. But what does the pole see?

Since length contraction is symmetric, if the barn sees the pole shortened by a factor of 2, then the pole sees the barn shortened by the same factor of 2. This factor is just $\cosh\beta$, where $|\beta|$ is the same in both cases. So the 20 foot pole sees this 5 foot barn approaching. No way is the pole going to fit in the barn!

So what does the pole see?

The spacetime diagrams for this situation are shown in Figure 8.1, as seen first from the barn's reference frame and then from the pole's reference frame. The two dots represent the events of closing the barn doors when the ends of the pole are even with the corresponding door.

 
Figure 8.1: In each diagram, the heavy straight lines represent the ends of the pole and the lighter straight lines represent the front and back of the barn. The hyperbolas of “radius” 10 and 20 are also shown. The dot at the origin labels the event where the back of the pole enters the barn, and the other dot labels the event where the front of the pole leaves the barn.

In the barn's frame, these events happen simultaneously, so it is possible in principle to trap the pole in the barn by shutting the doors “at the same time”. (We omit speculation about what happens when the pole hits the closed door!)

In the pole's frame, the exit door is closed long before the rear of the pole enters the barn. Assuming the pole keeps going, for instance by virtue of the door opening again, then the entrance door is closed much later, when the rear of the pole finally gets there. The pole thinks it silly to try to catch it by waiting to close the entrance door until most of the pole has already escaped through the exit door!