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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?
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.
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