Timelike homotopy

On a Lorentzian manifold, certain curves are distinguished as timelike. A timelike homotopy between two timelike curves is a homotopy such that each intermediate curve is timelike. No closed timelike curve (CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be multiply connected by timelike curves (or timelike multiply connected). A manifold such as the 3-sphere can be simply connected (by any type of curve), and at the same time be timelike multiply connected. Equivalence classes of timelike homotopic curves define their own fundamental group, as noted by Smith (1967). A smooth topological feature which prevents a CTC from being deformed to a point may be called a timelike topological feature.

References

*cite journal
author = J. W. Smith
year = 1967
title = Fundamental groups on a Lorentz manifold
journal = Amer. J. Math.
volume = 82
pages =873–890
doi = 10.2307/2372946

*cite journal
author = André Avez
year = 1963
title = Essais de géométrie riemannienne hyperbolique globale. Applications à la relativité général
journal = Annales de l’institut Fourier
volume = 13
pages =105–190