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I would like to know can any Lorentzian manifold be 3+1 foliated? I would like to know when a 3+1 foliation of a Lorentzian manifold exists and when it does not exist. If a 3+1 foliation does not exist, can it have other kinds of foliations?

Qmechanic
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Dr. user44690
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    Not all Lorentzian four-dimensional spacetimes can be foliated by 3-manifolds in a "space and time splitting" way. The result you seem to be looking for is that globally hyperbolic spacetimes admit such foliation. These are spacetimes containing a Cauchy surface, which is a hypersurface such that every inextendible causal curve pierces exactly once. If $\Sigma$ is such a surface, the result is that the spacetime is topologically $\mathbb{R}\times \Sigma$ and can be foliated by copies of said Cauchy surface. See Wald's GR book for details. – Gold Dec 16 '19 at 11:52
  • Possible duplicate: https://physics.stackexchange.com/q/233516/2451 – Qmechanic Dec 16 '19 at 12:13
  • See also links here: https://physics.stackexchange.com/q/761289/226902 and https://physics.stackexchange.com/q/696289/226902 – Quillo Apr 26 '23 at 12:51

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