Someone here once said to me, via post, that "any compact spacetime must have closed timelike curves". Are there any good references out there on why that is / how that is derived? As an after thought... Isn't it true that a particle traveling in one direction in time is equivalent to its anti-particle traveling in the opposite direction in time? So, for a closed geodesic in a compact space-time, a particle traveling in this "loop" would appear the same to us as a particle and anti-particle popping out of the void and then coming back together again and disappearing no? As the particle comes around and travels back in time, we would interpret that (or at least the equations make it equivalent to) as it's anti particle moving forward in time. I would think that a closed universe, in space and time, would allow for regions of time where energy would be isolated. If we could look at this universe from the outside, as we do say looking at the surface of the earth (a closed 2d example), we would see that on the whole energy is conserved. But if time to those on the surface is like the motions from the south pole to the north (big bang, big crunch), they would see energy being "created" and then "destroyed".