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.

Discussion

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

$D^+(\mathcal{S}):=\{p\in\mathcal{M} \ \ \text{such that every inextensible, past-directed, non-spacelike curve through }p\text{ intersects }\mathcal{S})$

Similarly $D^{-}(\mathcal{S})$, the past of $\mathcal{S}$, is the same thing going forward in time.

When there are no closed timelike curves, $D^{+}$ and $D^{-}$ are two different regions. When the time dimension closes up on itself everywhere so that it makes a circle, the future and the past of $\mathcal{S}$ are the same and both include $\mathcal{S}$. The Cauchy surface is defined rigorously in terms of intersections with inextensible curves in order to deal with this case of circular time. An inextensible curve is a curve with no ends: either it goes on forever, remaining timelike or null, or it closes in on itself to make a circle, a closed non-spacelike curve.

When there are closed timelike curves, or even when there are closed non-spacelike curves, a Cauchy surface still determines the future, but the future includes the surface itself. This means that the initial conditions obey a constraint, and the Cauchy surface is not of the same character as when the future and the past are disjoint.

If there are no closed timelike curves, then 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.

Cauchy Horizon

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 $\mathcal{S}$. A clear physical example of a Cauchy horizon is the second horizon inside a charged or rotating black hole. The outermost horizon is an event horizon, beyond which information cannot escape, but where the future is still determined from the conditions outside. Inside the inner horizon, the Cauchy horizon, the singularity is visible and to predict the future requires additional data about what comes out of the singularity.

Since a black hole Cauchy horizon only forms in a region where the geodesics are outgoing, in radial coordinates, in a region where the central singularity is repulsive, it is hard to imagine exactly how it forms. For this reason, Kerr and others suggest that a Cauchy horizon never forms, instead that the inner horizon is in fact a spacelike or timelike singularity.

A homogeneous space-time with a Cauchy horizon is anti-de Sitter space.

See also

• causal structure

References

• P.K. Townsend, Black Holes, lecture notes, Section 3.3, arXiv:gr-qc/9707012, 1997.