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?

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?