TITLE: Passage of time on Moebius strip/Klein bottle/higher dimensional equivalents
QUESTION [0 upvotes]: Space time are related, is there any relationship regarding the type of space? for example would time space relationship from relativity theories be any different within a 3 dimensional equivalent of Klein bottle and Moebius strip?

REPLY [2 votes]: There is nothing specific that happens with respect to time in a manifold that has spatial slices homeomorphic to a non-orientable surface like the Klein bottle, Moebius strip or projective plane. There will be some other issues (relating to the spin structure of the spacetime), but none are directly time related.
If the entirity of the spacetime is either the Klein bottle or the projective plane, or some other compact non-orientable manifold, then this spacetime will have closed causal curves everywhere. This isn't related to the non-orientability, just to the compact nature.
There is a more specific notion called time-orientability, which means the spacetime admits a global notion of a past and a future. Some spacetimes are not time-orientable, but this is not related to the orientability of the manifold itself. Any non-simply connected manifold admits some non-time orientable metric, including orientable ones, such as the cylinder, and non-orientable manifolds such as the Klein bottle can be time-orientable.
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