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 \mathcal{S} is a space-like set in the space-time then D^{+}(\mathcal{S}) is the future Cauchy development of \mathcal{S}, the set of points through which every past-directed non-spacelike curve intersects \mathcal{S}. Similarly D^{-}(\mathcal{S}) for every future-directed non-spacelike curve. Given appropriate information on \mathcal{S} the regions D^{\pm}(\mathcal{S}) have their states completely determined.

Given \mathcal{S} a partial Cauchy surface and if D^{+}(\mathcal{S})\cup \mathcal{S}\cup D^{-}(\mathcal{S}) = \mathcal{M}, the entire manifold, then \mathcal{S} is a Cauchy surface. Any surface of constant t in Minkowski space-time is a Cauchy surface.

If D^{+}(\mathcal{S})\cup \mathcal{S}\cup D^{-}(\mathcal{S}) \not= \mathcal{M} then there exists a Cauchy horizon between D^{\pm}(\mathcal{S}) and regions of the manifold not completely determined by information on \mathcal{S}.

An example of such a space-time is anti de Sitter space since \mathcal{J} is space-like and so no Cauchy surface exists.

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