Timelike simply connected

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Suppose a Lorentzian manifold contains a closed timelike curve (CTC). No CTC can be continuously deformed as a CTC (is timelike homotopic) to a point, as that point would not be causally well behaved. [1] Therefore, any Lorentzian manifold containing a CTC is said to be timelike multiply connected. A Lorentzian manifold that does not contain a CTC is said to be timelike simply connected.

Any Lorentzian manifold which is timelike multiply connected has a diffeomorphic universal covering space which is timelike simply connected. For instance, a three-sphere with a Lorentzian metric is timelike multiply connected, (because any compact Lorentzian manifold contains a CTC), but has a diffeomorphic universal covering space which contains no CTC (and is therefore not compact). By contrast, a three-sphere with the standard metric is simply connected, and is therefore its own universal cover.

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  1. Monroe, Hunter (2008-10-29). "Are Causality Violations Undesirable?". Foundations of Physics. 38 (11): 1065–1069. arXiv: gr-qc/0609054 . Bibcode:2008FoPh...38.1065M. doi:10.1007/s10701-008-9254-9. S2CID   119707350.