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Under what conditions splitting (e.g. $m + p$ foliation of a $m + p$ dimensional manifold) of a general pseudo-Riemannian manifold (with any arbitrary signature) possible? If it is too general then I am happy knowing the answer for manifolds with Euclidean and Lorentzian signatures.

If a manifold doesn't admit this type of foliation, hence we cannot specify field data on "spacelike" surfaces, can we still do field theory on that manifold?

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  • $\begingroup$ More on 3+1 splitting in GR. $\endgroup$
    – Qmechanic
    Jun 8, 2022 at 15:59
  • $\begingroup$ well, I shouldn't have used that 3+1 example; I meant any foliation of general manifolds. $\endgroup$
    – spacetime
    Jun 8, 2022 at 16:02
  • $\begingroup$ @spacetime relaxing from 3+1 to another strategy won't change the result, it will just change whether the evolution equation is hyperbolic, parabolic, or elliptical. If the space time is smooth, you can always foliate, and then (presumably) change coordinates after a caustic develops, and then foliate over the new patch $\endgroup$ Jun 8, 2022 at 16:41
  • $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$
    – Qmechanic
    Jun 8, 2022 at 17:46
  • $\begingroup$ @Qmechanic it definitely gets discussed in GR books and research, but I can see the argument. $\endgroup$ Jun 8, 2022 at 18:43

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There is a proof that we can always 3+1 foliate on a finite patch of a (smooth) spacetime. How large that patch is depends on the spacetime, of course, but if a caustic shows up, the expectation (I don't know if this expectation is just conjecture or if there is a formal proof), is that one could take the finite-volume slice from the initial evolution, re-foliate it in a way that deals with the caustic in the foliated coordinates, evolve the spacetime further, and eventually get N patches over the whole spacetime for some finite (or at least countable) N.

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