appot89 said:
Assume that space is a three-dimensional torus ( a 3D donut) .
For special relativity to apply, it has to be a
flat 3-torus. SR only applies if the spacetime geometry is flat. It is possible to have a flat geometry with a 3-torus spatial topology, but it won't be like what you are imagining. See below.
appot89 said:
Which clock ticks faster and why?
A simpler case to analyze is the 1+1 spacetime in which the spatial topology is a circle. In other words, the spacetime as a whole has the topology of a cylinder. Everything about this case carries over to the flat 3-torus case.
The nice thing about the 1+1 cylinder case, though, is that you can simply "flatten out" the cylinder without changing its geometry (since an ordinary cylinder already has a flat intrinsic geometry--it only looks curved because of how it is embedded in 3-D Euclidean space, but nothing about that embedding affects how we analyze the 1+1 cylinder spacetime). When you "flatten out" the cylinder, you see that, unlike the ordinary 1+1 Minkowski spacetime (which has an ordinary infinite plane topology), the 1+1 cylinder spacetime has a "preferred frame": the inertial frame whose spatial axis is a closed circle going around the cylinder (and which thus is exactly "horizontal" when the cylinder is flattened out). An observer at rest in this frame will have the fastest ticking clock (more precisely, will age more between meetings with any other observer in relative motion), and this observer's worldline will go "straight up" the cylinder (and will be exactly "vertical" when the cylinder is flattened out).
Any other observer in relative motion will have a worldline that winds around the cylinder, and the "spatial axis" of such an observer's rest frame will not be a closed circle, but a helix (and in fact this poses some technical issues when defining coordinates in such a frame). This should be evident from looking at how such a frame's axes look in the "flattened out" version, and then "rolling up" that flattened out picture into a cylinder again.