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Given a global hyperbolic space-time $M$, Geroch's work implies existence of a three dimensional Cauchy surface $\Sigma$ (the 1+3 splitting of space-time). Then,

  1. Given a metric $g_{\mu\nu}$ in $M$, are there any known ways to construct $\Sigma$ using mathematical objects derived entirely from the metric?

  2. If yes, then are such constructions generally covariant (tensorial)?

John Doe
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  • Geroch splitting theorem: https://en.wikipedia.org/wiki/Geroch%27s_splitting_theorem . More on 3+1 splitting (or "foliation"): https://physics.stackexchange.com/q/233516/226902 https://physics.stackexchange.com/q/761289/226902 https://physics.stackexchange.com/q/328722/226902 https://physics.stackexchange.com/q/519905/226902 https://physics.stackexchange.com/q/411189/226902 https://physics.stackexchange.com/q/160118/226902 https://physics.stackexchange.com/q/573587/226902 – Quillo Apr 26 '23 at 12:54
  • References to the original work of Geroch, for the interested reader: "Topology in GR" J. Math. Phys. 8, 782–786 (1967) https://doi.org/10.1063/1.1705276 and "Domain of Dependence", J. Math. Phys. 11, 437–449 (1970) https://doi.org/10.1063/1.1665157 – Quillo Apr 26 '23 at 13:15
  • Here a good answer with some examples: https://physics.stackexchange.com/a/761360/226902 – Quillo Apr 26 '23 at 13:23

1 Answers1

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I think they are completely independent concepts. In particular, I think you can study foliations by talking about sets and maps between sets (think about how you would foliate $\mathbb{R}^2$, it is just a bunch of straight lines with a random slope).

This does not mean, however, it is not useful to use the metric to define a foliation. For example, one could compute the Killing vectors, and for each of them define a leave as the surface with the Killing vector as its normal vector.

I have not studied this in detail, but I would say that following this procedure your foliation may have nicer properties than an arbitrary foliation.

JGBM
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  • Thank you JGBM. Indeed, the use of a metric-derived object to demonstrate foliation is advantageous because it then becomes a universal method - an intrinsic property of Nature, rather than an arbitrary choice. This is my key motivation here. – John Doe Feb 24 '22 at 10:43
  • I feel that it will still be an arbitrary choice (and besides that, you may have several Killing vectors). You are just choosing to follow this procedure to help you make the choice. What could eventually not be an arbitrary choice is if you ask about foliations with given properties. – JGBM Feb 27 '22 at 16:29