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 $\mathcal{S}$ .