Geroch's splitting theorem

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

A Cauchy surface can possess corners, and thereby need not be a differentiable submanifold of the spacetime; it is however always continuous (and even Lipschitz continuous). By using the flow of a vector field chosen to be complete, smooth, and timelike, it is elementary to prove that if a Cauchy surface S is C^k-smooth then the spacetime is C^k-diffeomorphic to the product S × R, and that any two such Cauchy surfaces are C^k-diffeomorphic.^[1]

Robert Geroch proved in 1970 that every globally hyperbolic spacetime has a Cauchy surface S, and that the homeomorphism (as a C⁰-diffeomorphism) to S × R can be selected so that every surface of the form S × {a} is a Cauchy surface and each curve of the form {s} × R is a continuous timelike curve.^[2]

Various foundational textbooks, such as George Ellis and Stephen Hawking's The Large Scale Structure of Space-Time and Robert Wald's General Relativity,^[3] asserted that smoothing techniques allow Geroch's result to be strengthened from a topological to a smooth context. However, this was not satisfactorily proved until work of Antonio Bernal and Miguel Sánchez in 2003. As a result of their work, it is known that every globally hyperbolic spacetime has a Cauchy surface which is smoothly embedded and spacelike.^[4] As they proved in 2005, the diffeomorphism to S × R can be selected so that each surface of the form S × {a} is a spacelike smooth Cauchy surface and that each curve of the form {s} × R is a smooth timelike curve orthogonal to each surface S × {a}.^[5]