A Cauchy surface in a time-oriented Lorentzian manifold is a hypersurface that every inextendible causal curve meets exactly once. Its role in the initial-value problem is to specify data determining a globally hyperbolic development. A partial exterior region need not contain the full original Cauchy surface, so extending that region is distinct from extending the full development.
Cauchy Surface
See also
Globally Hyperbolic Spacetime, Initial Data Set, Lorentzian Manifold, Maximal Globally Hyperbolic DevelopmentExplore with Wolfram|Alpha
References
Chruściel, P. T.; Galloway, G. J.; and Pollack, D. "Mathematical General Relativity: A Sampler." Bull. Amer. Math. Soc. 47, 567-638, 2010. https://doi.org/10.1090/S0273-0979-2010-01304-5.Cite this as:
Weisstein, Eric W. "Cauchy Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CauchySurface.html