Cauchy horizon
Encyclopedia
In physics
, a Cauchy horizon is a light-like boundary of the domain of validity of a Cauchy problem
(a particular boundary value problem
of the theory of partial differential equation
s). One side of the horizon contains closed space-like geodesic
s and the other side contains closed time-like geodesics.
Waves traveling in Misner space approaching a Cauchy horizon would receive a frequency boost proportional to
each time they pass through the identification world line
. As this happens infinitely many times while approaching the horizon, the stress-energy tensor
diverges at the horizon. Presumably, this prevents spacetime
from developing closed time-like curves that would otherwise be feasible.
Under the averaged weak energy condition
(AWEC), Cauchy horizons are inherently unstable. However, cases of AWEC violation, such as the Casimir effect
, do exist. If we can assume that the spacetime inside the Cauchy horizon violates AWEC, then the horizon becomes stable and frequency boosting effects would be canceled out by the tendency of the spacetime to act as a divergent lens. Were this conjecture to be shown empirically true, it would provide a counter-example to the strong cosmic censorship
conjecture.
The simplest example is the internal horizon of a Reissner-Nordström black hole.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a Cauchy horizon is a light-like boundary of the domain of validity of a Cauchy problem
Cauchy problem
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions which are given on a hypersurface in the domain. Cauchy problems are an extension of initial value problems and are to be contrasted with boundary value problems...
(a particular boundary value problem
Boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions...
of the theory of partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s). One side of the horizon contains closed space-like geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
s and the other side contains closed time-like geodesics.
Waves traveling in Misner space approaching a Cauchy horizon would receive a frequency boost proportional to
each time they pass through the identification world line
World line
In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...
. As this happens infinitely many times while approaching the horizon, the stress-energy tensor
Stress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
diverges at the horizon. Presumably, this prevents spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
from developing closed time-like curves that would otherwise be feasible.
Under the averaged weak energy condition
Energy condition
In relativistic classical field theories of gravitation, particularly general relativity, an energy condition is one of various alternative conditions which can be applied to the matter content of the theory, when it is either not possible or desirable to specify this content explicitly...
(AWEC), Cauchy horizons are inherently unstable. However, cases of AWEC violation, such as the Casimir effect
Casimir effect
In quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field. The typical example is of two uncharged metallic plates in a vacuum, like capacitors placed a few micrometers apart, without any external electromagnetic field...
, do exist. If we can assume that the spacetime inside the Cauchy horizon violates AWEC, then the horizon becomes stable and frequency boosting effects would be canceled out by the tendency of the spacetime to act as a divergent lens. Were this conjecture to be shown empirically true, it would provide a counter-example to the strong cosmic censorship
Cosmic censorship hypothesis
The weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of singularities arising in general relativity....
conjecture.
The simplest example is the internal horizon of a Reissner-Nordström black hole.