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 be a globally hyperbolic spacetime. Then is strongly causal and there exists a global "time function" on the manifold, i.e. a continuous, surjective map such that:
- For all , is a Cauchy surface, and
- is strictly increasing on any causal curve.
Moreover, all Cauchy surfaces are homeomorphic, and is homeomorphic to where is any Cauchy surface of .
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![]() | Original source: https://en.wikipedia.org/wiki/Geroch's splitting theorem.
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