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]

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

• Bernal, Antonio N.; Sánchez, Miguel (2003). "On smooth Cauchy hypersurfaces and Geroch's splitting theorem". Communications in Mathematical Physics . 243 (3): 461–470. arXiv: gr-qc/0306108 . doi:10.1007/s00220-003-0982-6. MR   2029362. Zbl   1085.53060.
• Bernal, Antonio N.; Sánchez, Miguel (2005). "Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes". Communications in Mathematical Physics . 257 (1): 43–50. arXiv: gr-qc/0401112 . doi:10.1007/s00220-005-1346-1. MR   2163568. Zbl   1081.53059.
• Geroch, Robert (1970). "Domain of dependence". Journal of Mathematical Physics . 11 (2): 437–449. doi:10.1063/1.1665157. MR   0270697. Zbl   0189.27602.
• Hawking, S. W.; Ellis, G. F. R. (1973). The large scale structure of space-time. Cambridge Monographs on Mathematical Physics. Vol. 1. London−New York: Cambridge University Press. doi: 10.1017/CBO9780511524646 . ISBN   9780521099066. MR   0424186. Zbl   0265.53054.
• Minguzzi, Ettore; Sánchez, Miguel (2008). "The causal hierarchy of spacetimes". In Alekseevsky, Dmitri V.; Baum, Helga (eds.). Recent developments in pseudo-Riemannian geometry. ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society. pp. 299–358. arXiv: gr-qc/0609119 . doi:10.4171/051-1/9. ISBN   978-3-03719-051-7. MR   2436235. Zbl   1148.83002.
• Wald, Robert M. (1984). General relativity. Chicago, IL: University of Chicago Press. ISBN   0-226-87032-4. MR   0757180. Zbl   0549.53001.