Friday 15 July 2016

general relativity - Does any spacetime admit a global foliation in spacelike hypersurfaces?


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.


foliation of spacetime


I would be interested to know:



  1. Does there always exist a global foliation of spacetime into spacelike hypersurfaces?

  2. In such a foliation, are all leaves homeomorphic?





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