Recently I've had some discussions about time in GR. I've always read in different places that people usually want a spacetime to have a hypersurface-orthogonal timelike Killing vector field so that they can assign a time dimension to that spacetime. But Why is this needed?(adsbygoogle = window.adsbygoogle || []).push({});

I can understand it that a hypersurface-orthogonal timelike vector field allows you to define the worldlines of some observers and the hypersurfaces will be their "space" in each instant of "time". So this seems natural. But why do we need it to be a Killing vector field? What's wrong with associating a time to a spacetime that tells you the spacetime is changing with that time?

What condition should a spacetime satisfy to allow a globally hypersurface-orthogonal timelike vector field?

Also, in GR, we work with Lorentzian manifolds, so the signature is always (-+++)(or the other equivalent convention). But sometimes the metric can't be diagonalized. So is it still as easy as checking the sign of the diagonal terms to check the signature? Can we always put a metric in form such that one of the diagonal elements is always negative(or positive in the other convention)?

Another question I have, is that is it always possible to foliate a spacetime into spacelike hypersurfaces? If not, what condition should a spacetime satisfy to allow such a foliation?

Sorry if my questions are too many and too diverse. But I asked them in one thread because I thought there can be a discussion about time in GR which contains the answer of all of the questions above.

Thanks

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# Time, spacelike foliations and timelike vector fields in GR

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