Cauchy surface

Intuitively, a Cauchy surface is a plane in space-time which is like an instant of time; its significance is that giving the initial conditions on this plane determines the future (and the past) uniquely.

More precisely, a Cauchy surface is any subset of space-time which is intersected by every inextensible, non-spacelike (i.e. causal) curve exactly once.

A partial Cauchy surface is a hypersurface which is intersected by any causal curve at most once.

It is named for French mathematician Augustin Louis Cauchy.

Given a Lorentzian manifold, if ${\displaystyle {\mathcal {S}}}$ is a space-like surface (i.e., a collection of points such that every pair is space-like separated), then ${\displaystyle D^{+}({\mathcal {S}})}$ is the future of ${\displaystyle {\mathcal {S}}}$, i. e.:

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