In the comments of this question the following new questions came up: in general relativity, local coordinates can be found around any point, that single out a time coordinate and a three dimensional spatial submanifold, but in general it won't be possible to write spacetime globally as a product of a spacelike and a timelike submanifold.
If local coordinates around all points such that the locally spatial hyperplanes map into hyperplanes under a change of coordinates, then we got a foliation of spacetime into spacelike hypersurfaces.
I would be interested to know:
- Does there always exist a global foliation of spacetime into spacelike hypersurfaces?
- In such a foliation, are all leaves homeomorphic?
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