Geroch's splitting theorem
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In the theory of causal structure on Lorentzian manifolds, Geroch's theorem or Geroch's splitting theorem (first proved by Robert Geroch) gives a topological characterization of globally hyperbolic spacetimes.
The theorem
Let [math]\displaystyle{ (M, g_{ab}) }[/math] be a globally hyperbolic spacetime. Then [math]\displaystyle{ (M, g_{ab}) }[/math] is strongly causal and there exists a global "time function" on the manifold, i.e. a continuous, surjective map [math]\displaystyle{ f:M \rightarrow \mathbb{R} }[/math] such that:
- For all [math]\displaystyle{ t \in \mathbb{R} }[/math], [math]\displaystyle{ f^{-1}(t) }[/math] is a Cauchy surface, and
- [math]\displaystyle{ f }[/math] is strictly increasing on any causal curve.
Moreover, all Cauchy surfaces are homeomorphic, and [math]\displaystyle{ M }[/math] is homeomorphic to [math]\displaystyle{ S \times \mathbb{R} }[/math] where [math]\displaystyle{ S }[/math] is any Cauchy surface of [math]\displaystyle{ M }[/math].
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![]() | Original source: https://en.wikipedia.org/wiki/Geroch's splitting theorem.
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