Special Relativity question reworded?

Question

The following is an argument to prove that the direction perpendicular to motion doesnt contract. Is something wrong with the argument? (I know the results are correct) When two rings are at rest with respect to each other (when they are in the same frame of reference), they are exactly the same size. They are in a position similar to the tires on a car axle. Imagine that the rings travel toward each other. If the direction perpindicular to motion contracted then the postulate that all frames of reference are equally valid would be broken because ring 1 would see ring 2 to contract and go inside of ring 1 and ring 2 (because it is at rest in its own frame of reference) would see ring 1 contract go inside of ring 2.

Answer 1

The argument is actually OK.

An analogous argument might be used to show that lengths parallel to the line of motion don't contract (this is the famous "ladder fitting into the barn" paradox), but that fails to cause a contradiction, because there are two events in that case, relating to the opposite ends of the latter; so it becomes a question of simultaneity as well as length contraction.

However, in the case you are proposing, all the comparisons are made at the same time and same x coordinate. It is also a frame-invariant statement that one right fit around outside another. So the principle of relativity plus the symmetry of the situation DO imply that there can be no contraction in the perpendicular direction.

Congratulations!

Answer 2

Don't follow your assertion at all. Why would the rings see each other going "inside"? If the contraction were perpendicular to the direction of travel, seems to me an outside observer would see each ring as shrinking sideways, but not going inside each other. And, when the respective rings become the reference frame, the rings are still separated in the direction of travel wrt each other.

And, to a point, would one see the other shrinking or expanding sideways? If a ring sees its own width as being fixed at its rest frame width, while it is contracting wrt to that frame, wouldn't that ring see the other ring's width expanding if that other ring is going slower wrt to the rest frame of both? That is, that slower ring would also be going slower wrt to the faster ring; so, relativistically speaking, the observer on the faster ring should see the slower ring width expand.

Nope, don't follow your "inside" story.