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

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 Ck-smooth then the spacetime is Ck-diffeomorphic to the product S × R, and that any two such Cauchy surfaces are Ck-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]

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References

  1. ↑ Geroch 1970, Property 7; Bernal & Sánchez 2005, Section 2.
  2. ↑ Geroch 1970, Section 5; Bernal & Sánchez 2005, Section 2; Hawking & Ellis 1973, Proposition 6.6.8; Minguzzi & Sánchez 2008, Section 3.11.2.
  3. ↑ Hawking & Ellis 1973, p. 212; Wald 1984, p. 209.
  4. ↑ Bernal & Sánchez 2003, Theorem 1.1; Minguzzi & Sánchez 2008, Section 3.11.3.
  5. ↑ Bernal & Sánchez 2005, Theorem 1.1; Minguzzi & Sánchez 2008, Section 3.11.3.

Sources

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