Special Relativity in an (in)finite universe

Alkatran

We've all heard of the twin paradox, and the idea of an infinite but finite universe, on the "surface" of a hyper-sphere and what not..

Anyways, my question is, what does relativity predict about someone who would travel all the way around this sphere? Starting at earth and ending at earth without ever accelerating? (Unless the rotation around the hyper-sphere counts as acceleration)

pervect

For the standard Friedmann cosmologies, the prediction is that in the best case, a photon emitted at the big bang could just make it around the universe by the time of the big crunch. This happens onlyfor a universe that's totally mater dominated. If the universe were radiation dominated, the photon could only make it halfway around by the time of the big crunch.

So in the standard cosmologies, an observer won't be able to circumavigate a closed universe between the big bang and the big crunch (except for the case of a photon in a purely matter dominated universe, the best case, where it just barely makes it).

If the universe isn't closed, the observer (or photon) just keeps going and never arrives back at his starting point.

Note - this is discussed a little bit in Gravitation pg 734, and applies for a zero cosmological constant. Unfortunately the case for a non-zero cosmological constant is left as an exercise. If the universe expands indefinitely, though, the photon isn't going to ever reach it's starting point, and neither is any form of matter.