Q & A: Lorentz contraction paradox?

Q:
I have a question about special relativity. Let's suppose a ball is thrown on a straight road at a near light speed, and the road has some large, circular holes that can make the ball fall into them. From the point of view of the ball, the distances in the same direction of the motion shorten, so the holes should become elliptical, and at last the ball will no longer fall into them. From the point of view of an external watcher, the holes are still circular and so the ball should indeed fall. There is a clear contradiction. How can it be solved?
- Andrea F. (age 20)
Italy
A:

This is a very well known apparent paradox in Special Relativity, with the story usually being told about a car driving into a garage. In the garage frame, the car fits in, but in the car frame it doesn't. (See this nice article for various versions: .)

What will really happen in your story, which is slightly more complicated than the garage story because the ball has to go down to get in the hole? Let's simplify slightly by replacing the ball with a hockey puck. As soon as the center of the puck slides over the hole region, the puck starts to fall. In the road frame, there's plenty of room for the puck to fall so it will indeed get stuck in the hole. As the front edge collides with the other side of the hole, the puck and the road will distort a lot.  How does it look in the puck frame? It must still fall into the hole, because the occurrence of that sort of record-leaving event must be invariant under choice of frames. Anybody using any frame can check later whether the event happened. In the puck frame, the collision with the hole happens to one side of the puck before the hole moves under the other side. The puck squashes, and becomes skinny enough to fit in the hole.

The two frames agree except one says the squashing happened when both sides of the puck were in the hole and the other says it started when only one side was in the hole. They both get the same consequences.

What if the puck were rigid? Wouldn't we have a contradiction? Yes, but that's just a reminder that rigid objects can't exist. () When you push on one end of a rod of length L, the other end can't respond until after a delay of L/c. So in the meantime the rod has to deform.

Mike W.

(published on 07/01/2015)