Cauchy surface
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A Cauchy surface, named after Augustin Louis Cauchy, is a subset of a region in space-time, which is intersected by every non-spacelike, inextensible curve exactly once. A partial Cauchy surface is a hypersurface which is intersected by any causal curve no more than once.
If is a space-like set in the space-time then is the future Cauchy development of , the set of points through which every past-directed non-spacelike curve intersects . Similarly for every future-directed non-spacelike curve. Given appropriate information on the regions have their states completely determined.
Given a partial Cauchy surface and if , the entire manifold, then is a Cauchy surface. Any surface of constant t in Minkowski space-time is a Cauchy surface.
If then there exists a Cauchy horizon between and regions of the manifold not completely determined by information on .
An example of such a space-time is anti de Sitter space since is space-like and so no Cauchy surface exists.
[edit] References
- P.K. Townsend, Black Holes, lecture notes, Section 3.3, arXiv:gr-qc/9707012, 1997.