Timeline for Foliations of Lorentzian manifolds by Spacelike Hypersurfaces
Current License: CC BY-SA 3.0
9 events
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| Mar 2, 2015 at 3:19 | comment | added | Ben Whale | Sorry for second comment - ran out of characters. Note that we haven't said much about the level sets of these functions. The construction guarantees that at least one such surface is entirely composed of null surfaces. It is my expectation that, because of the relationship to the distance, the set of all points in the manifold that lie on a null level surface of one of these functions should have measure zero (or some statement like this will hold). | |
| Mar 2, 2015 at 3:18 | comment | added | Ben Whale | Just in case you're interested in weakening your conditions to conditions that hold almost everywhere; then the condition that you want is that the Lorentzian distance is finite between any two points (which is strictly weaker than stable causality). Under these conditions my co-author and I have shown that the manifold carries functions that are strictly increasing along any timeline curve, that are continuous and differentiable a.e., whose gradient is uniformly bounded away from light cones and have a very nice relationship to the distance. See arxiv.org/abs/1412.5652. | |
| Feb 27, 2015 at 8:34 | history | edited | Igor Khavkine | CC BY-SA 3.0 |
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| Feb 27, 2015 at 8:32 | comment | added | Igor Khavkine | Pedro, (facepalm) you're right of course! Stable causality is the right condition, which had slipped my mind. Hawking originally proved that stably causal spacetimes admit time functions (causal gradient) and Sanchez improved that to temporal functions in Theorem 4.15 of arXiv:gr-qc/0411143. | |
| Feb 27, 2015 at 6:34 | comment | added | Blake | Thanks! I'd also add that the existence of the foliation by spacelike Cauchy surfaces is addressed in arxiv.org/abs/gr-qc/0401112, theorem 1.1 (especially 1.1.2). | |
| Feb 27, 2015 at 6:27 | vote | accept | Blake | ||
| Feb 27, 2015 at 1:22 | comment | added | Pedro Lauridsen Ribeiro | By the way, my former comment addresses the OP's first question. As for OP's second question, global hyperbolicity is also necessary, since the existence of a foliation by spacelike Cauchy hypersurfaces implies that such a hypersurface exists and hence the space-time is globally hyperbolic. In this case, Bernal and Sánchez have shown that one can even choose the function to have one of its level sets match the given hypersurface. | |
| Feb 27, 2015 at 1:11 | comment | added | Pedro Lauridsen Ribeiro | A weaker sufficient condition would be to require that the space-time is stably causal. In this case, you also have a real-valued smooth function with everywhere (say, past-directed) timelike gradient, hence all its level sets are regular and hence are spacelike hypersurfaces which foliate the space-time manifold. In this more general case, they are even allowed to change topology (in this case, the manifold must be disconnected, of course). | |
| Feb 26, 2015 at 23:35 | history | answered | Igor Khavkine | CC BY-SA 3.0 |