Special Relativity in a closed universe

[kochanskij]

TL;DR

If a spacecrafts travels close enough to C in a closed finite unbounded universe, length contraction will make the circumference of the universe smaller than the length of the craft, in its own reference frame. Its nose will bump into its tail. But, in another frame, its length is far less than the universe. Its nose and tail don't bump. Is this a contradiction? What did I miss?

In this thought experiment, I am assuming a finite but unbounded flat space with a 3-torus topology. There is no concern about curvature. This problem is completely special relativity. No GR needed.
When the spacecraft travels very near C, the universe contracts, relative to the craft's frame. If its circumference becomes less than the length of the craft, its nose will crash into its tail. But in earth's frame, this never happens, since it is the craft length that contracts.
Is this a physical contradiction? Does the nose of the craft get crushed or not?

 

[robphy]

In this thought experiment, I am assuming a finite but unbounded flat space with a 3-torus topology. There is no concern about curvature. This problem is completely special relativity.

The underlying manifold of special relativity is $R^4$.
Flatness is not enough.

 

[Dale]

In such a universe the first postulate does not hold. So you have to be careful with any "these frames contradict" type of reasoning. They can not possibly contradict, but the rules may be different in the different frames.

When the spacecraft travels very near C, the universe contracts, relative to the craft's frame.

In particular, this is incorrect. In a closed universe the preferred (stationary) observers measure the smallest universe. The moving observers all measure a universe that is larger. So if the proper length is smaller than the length of the universe according to the preferred frame then it is impossible for the nose to crash into the tail.

https://arxiv.org/abs/gr-qc/0503070

 

[PeterDonis]

In this thought experiment, I am assuming a finite but unbounded flat space with a 3-torus topology. There is no concern about curvature.

But the change in topology does create something that is not present in SR on $R^4$: a preferred frame.

To see how this works, consider a simpler example: SR in two spacetime dimensions on a cylinder, $S \times R$, instead of a flat plane. (The paper @Dale referenced does this with two more spatial dimensions added, but those extra two dimensions don't really play any role in the analysis.) There will be one particular inertial frame in which the time axis is parallel to the cylinder; that's the preferred frame. In any other frame, the time axis winds around the cylinder in a helix, and things get more complicated. In particular, you can't assume that things like length contraction work the same as you're used to in SR on $R^4$.