# Foliations of Lorentzian manifolds by Spacelike Hypersurfaces

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Suppose that $M$ is a Lorentzian manifold (not necessarily satisfying Einstein's equations). What conditions do we need in order to guarantee that $M$ admits a foliation by codimension-$1$ spacelike submanifolds?

Geroch showed that global hyperbolicity is equivalent to admitting a foliation by Cauchy hypersurfaces, and Bernal and Sánchez showed this foliation can be taken to be smooth (arXiv:gr-qc/0306108). But a Cauchy hypersurface can contain a null geodesic segment (unless I'm quite confused about that). My question then is under what conditions we can take this foliation to consist of spacelike Cauchy hypersurfaces?

This post imported from StackExchange MathOverflow at 2015-03-11 13:20 (UTC), posted by SE-user Blake
retagged Mar 11, 2015

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In the case of a globally hyperbolic spacetime, what you want is a smooth Cauchy temporal function (the gradient is everywhere timelike, not just causal, and each level set is a Cauchy surface that is necessarily spacelike). That global hyperbolicity is also sufficient the the existence of a smooth temporal function was also shown by Bernal and Sanchez, in the followup [arXiv:gr-qc/0512095] to their original paper that you cite.

I'm not sure if you are also interested in foliations of non globally hyperbolic spacetimes. I'm not completely sure what the right conditions would be then. See comment about stable casuality below.

This post imported from StackExchange MathOverflow at 2015-03-11 13:20 (UTC), posted by SE-user Igor Khavkine
answered Feb 26, 2015 by (420 points)
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).

This post imported from StackExchange MathOverflow at 2015-03-11 13:20 (UTC), posted by SE-user 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.

This post imported from StackExchange MathOverflow at 2015-03-11 13:20 (UTC), posted by SE-user Pedro Lauridsen Ribeiro
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).

This post imported from StackExchange MathOverflow at 2015-03-11 13:20 (UTC), posted by SE-user Blake
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.

This post imported from StackExchange MathOverflow at 2015-03-11 13:20 (UTC), posted by SE-user Igor Khavkine
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.

This post imported from StackExchange MathOverflow at 2015-03-11 13:20 (UTC), posted by SE-user 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).

This post imported from StackExchange MathOverflow at 2015-03-11 13:20 (UTC), posted by SE-user Ben Whale

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