Relative proper times in a compactified universe

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?

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
Homework Helper
Gold Member
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,

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