Relative proper times in a compactified universe

[spaghetti3451]

Homework Statement

Imagine that space (not spacetime) is actually a finite box, or in more sophisticated terms, a three-torus, of size $L$. By this we mean that there is a coordinate system $x^{\mu}=(t,x,y,z)$ such that every point with coordinates $(t,x,y,z)$ is identified with every point with coordinates $(t,x+L,y,z)$, $(t,x,y+L,z)$ and $(t,x,y,z+L)$. Note that the time coordinate is the same. Now consider two observers; observer $A$ is at rest in this coordinate system (constant spatial coordinates), while observer $B$ moves in the $x$-direction with constant velocity $v$. $A$ and $B$ begin at the same event, and while $A$ remains still, $B$ moves once around the universe and comes back to intersect the worldline of $A$ without ever having to accelerate (since the universe is periodic). What are the relative proper times experienced in this interval by $A$ and $B$? Is this consistent with your understanding of Lorentz invariance?

Homework Equations

The Attempt at a Solution

Observer $A$ is at rest, so the time elapsed for observer $A$ between observer $B$'s exit and subsequent return is the proper time for observer $A$, which is given by $\tau_{A}=\frac{L}{v}$

The proper time for observer $B$ is given by $\tau_{B}=\frac{L/\gamma}{v}$.

Are my answers correct? I am a little hesitant about my understanding of what the question actually requires (with regards to the relative proper times), since, as far as I know, there is only one proper time connecting two events, and that's the time elapsed in a frame moving at a velocity $v$ such that it intersects the two events. But, in this problem, I found two proper times.

 

[TSny]

Observer $A$ is at rest, so the time elapsed for observer $A$ between observer $B$'s exit and subsequent return is the proper time for observer $A$, which is given by $\tau_{A}=\frac{L}{v}$

The proper time for observer $B$ is given by $\tau_{B}=\frac{L/\gamma}{v}$.

Are my answers correct? I am a little hesitant about my understanding of what the question actually requires (with regards to the relative proper times), since, as far as I know, there is only one proper time connecting two events, and that's the time elapsed in a frame moving at a velocity $v$ such that it intersects the two events. But, in this problem, I found two proper times.

Yes, your answer for the times of A and B are correct. So, in this universe, are different inertial observers equivalent?

There is a lot of discussion that you can find on this topic. For example, here at the forum you can look at https://www.physicsforums.com/threads/the-cosmological-twin-paradox.51197/ which contains some links to papers on the web.

Also,
https://www.physicsforums.com/threads/twin-paradox-in-a-closed-universe.375432/

https://www.physicsforums.com/threads/closed-flat-space-twin-paradox.668559/