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[edit]

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 .