In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. 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.)
They are named for French mathematician Augustin-Louis Cauchy (1789-1857) due to their relevance for the Cauchy problem of general relativity.
Although it is usually phrased in terms of general relativity, the formal notion of a Cauchy surface can be understood in familiar terms. Suppose that humans can travel at a maximum speed of 20 miles per hour. This places constraints, for any given person, upon where they can reach by a certain time. For instance, it is impossible for a person who is in Mexico at 3 o'clock to arrive in Libya by 4 o'clock; however it is possible for a person who is in Manhattan at 1 o'clock to reach Brooklyn by 2 o'clock, since the locations are ten miles apart. So as to speak semi-formally, ignore time zones and travel difficulties, and suppose that travelers are immortal beings who have lived forever.
The system of all possible ways to fill in the four blanks indefines the notion of a causal structure. A Cauchy surface for this causal structure is a collection of pairs of locations and times such that, for any hypothetical traveler whatsoever, there is exactly one location and time pair in the collection for which the traveler was at the indicated location at the indicated time.
There are a number of uninteresting Cauchy surfaces. For instance, one Cauchy surface for this causal structure is given by considering the pairing of every location with the time of 1 o'clock (on a certain specified day), since any hypothetical traveler must have been at one specific location at this time; furthermore, no traveler can be at multiple locations at this time. By contrast, there cannot be any Cauchy surface for this causal structure that contains both the pair (Manhattan, 1 o'clock) and (Brooklyn, 2 o'clock) since there are hypothetical travelers that could have been in Manhattan at 1 o'clock and Brooklyn at 2 o'clock.
There are, also, some more interesting Cauchy surfaces which are harder to describe verbally. One could define a function τ from the collection of all locations into the collection of all times, such that the gradient of τ is everywhere less than 1/20 hours per mile. Then another example of a Cauchy surface is given by the collection of pairs
\{(p,\tau(p)):palocationonEarth\}.
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. Hence one can speak of Cauchy surfaces for these causal structures as well.
Let be a Lorentzian manifold. One says that a map is an inextensible differentiable timelike curve in if:
A subset of is called a Cauchy surface if every inextensible differentiable timelike curve in has exactly one point of intersection with ; if there exists such a subset, then is called globally hyperbolic.
The following is automatically true of a Cauchy surface :It is hard to say more about the nature of Cauchy surfaces in general. The example of
| ||||
| \{(t,x,y,z):t |
\}
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):
Let be a time-oriented Lorentzian manifold. One says that a map is a past-inextensible differentiable causal curve in if:
One defines a future-inextensible differentiable causal curve by the same criteria, with the phrase "as decreases to " replaced by "as increases to ". Given a subset of, the future Cauchy development of is defined to consist of all points of such that if is any past-inextensible differentiable causal curve such that for some in, then there exists some in with . One defines the past Cauchy development by the same criteria, replacing "past-inextensible" with "future-inextensible".
Informally:
The Cauchy development is the union of the future Cauchy development and the past Cauchy development.
When there are no closed timelike curves,
D+
D-
l{S}
l{S}
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
l{S}
D+(l{S})\cupl{S}\cupD-(l{S})=l{M}
l{S}
t
If
D+(l{S})\cupl{S}\cupD-(l{S})\not=l{M}
D\pm(l{S})
l{S}
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. The inner horizon corresponds to the instability due to mass inflation.[2] [3]
A homogeneous space-time with a Cauchy horizon is anti-de Sitter space.
Research articles
Textbooks