In the application of Lorentzian geometry to the physics of general relativity, a Cauchy surface is usually interpreted as defining an "instant of time".
In the mathematics of general relativity, Cauchy surfaces provide boundary conditions for the causal structure in which the Einstein equations can be solved (using, for example, the ADM formalism.)
Suppose that humans can travel at a maximum speed of 20 miles per hour.
There are, also, some more interesting Cauchy surfaces which are harder to describe verbally.
The physical theories of special relativity and general relativity define causal structures which are schematically of the above type ("a traveler either can or cannot reach a certain spacetime point from a certain other spacetime point"), with the exception that locations and times are not cleanly separable from one another.
One says that a map c : (a,b) → M is an inextensible differentiable timelike curve in (M, g) if: A subset S of M is called a Cauchy surface if every inextensible differentiable timelike curve in (M, g) has exactly one point of intersection with S; if there exists such a subset, then (M, g) is called globally hyperbolic.
The example of as a Cauchy surface for Minkowski space ℝ3,1 makes clear that, even for the "simplest" Lorentzian manifolds, Cauchy surfaces may fail to be differentiable everywhere (in this case, at the origin), and that the homeomorphism S × ℝ → M may fail to be even a C1-diffeomorphism.
Furthermore, at the cost of not being able to consider arbitrary Cauchy surface, it is always possible to find smooth Cauchy surfaces (Bernal & Sánchez 2003): Given any smooth Lorentzian manifold (M, g) which has a Cauchy surface, there exists a Cauchy surface S which is an embedded and spacelike smooth submanifold of M and such that S × ℝ is smoothly diffeomorphic to M.Let (M, g) be a time-oriented Lorentzian manifold.
One says that a map c : (a,b) → M is a past-inextensible differentiable causal curve in (M, g) if: One defines a future-inextensible differentiable causal curve by the same criteria, with the phrase "as t decreases to a" replaced by "as t increases to b".
Given a subset S of M, the future Cauchy development D+(S) of S is defined to consist of all points p of M such that if c : (a,b) → M is any past-inextensible differentiable causal curve such that c(t) = p for some t in (a,b), then there exists some s in (a,b) with c(s) ∈ S. One defines the past Cauchy development D−(S) by the same criteria, replacing "past-inextensible" with "future-inextensible".
When the time dimension closes up on itself everywhere so that it makes a circle, the future and the past of
The Cauchy surface is defined rigorously in terms of intersections with inextensible curves in order to deal with this case of circular time.
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.
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.
The inner horizon corresponds to the instability due to mass inflation.
[2][3][4] A homogeneous space-time with a Cauchy horizon is anti-de Sitter space.