Exercise 1.02 Three torus space

The question was

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^μ = (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 world line of A without 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?

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