Flat Donut Universes & Special Relativity

[Strilanc]

Suppose there was a universe that was small, and flat, but globally it was connected like a torus. Basically, pac-man's universe (except special relativity is in effect). The space $(\mathbb{R} \pmod{1})^3$.

Does this assumed universe have a distinguished inertial frame where the loop-around-distance is maximized (due to not being length-contracted)? For example, if you pushed a small ball away from yourself along the x axis, can the amount of time until it smacks into your back depend on your velocity w.r.t. some frame? Or are the differences canceled out by time dilation? What if there was a clock on the ball?

(Obviously this question is about how the Lorentz transform behaves in unusual situations, not about how reality actually is.)

 

[Dale]

Does this assumed universe have a distinguished inertial frame where the loop-around-distance is maximized (due to not being length-contracted)?

The short answer is yes.

We had a thread quite some time ago where we talked about this, but the thread is of rather poor quality due to some persistent off topic posts by one particular now-banned user. But here it is anyway: https://www.physicsforums.com/threads/twin-paradox-in-a-closed-universe.375432/

 

[Strilanc]

Thanks, that answers my questions. I particularly liked the "only one frame's present lines up with itself, instead of making candy-cane strips across spacetime" for making it seem kinda obvious in hindsight.

(I also like how that thread links to another thread that links to another thread and so on for a decade.)