Killing horizon
Encyclopedia
A Killing horizon is a null hypersurface on which there is a null Killing vector field (both are named after Wilhelm Killing
Wilhelm Killing
Wilhelm Karl Joseph Killing was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry....

).

Associated to a Killing horizon is a geometrical quantity known as surface gravity
Surface gravity
The surface gravity, g, of an astronomical or other object is the gravitational acceleration experienced at its surface. The surface gravity may be thought of as the acceleration due to gravity experienced by a hypothetical test particle which is very close to the object's surface and which, in...

, . If the surface gravity vanishes, then the Killing horizon is said to be degenerate.

## Black hole Killing horizons

Exact black hole metrics such as the Kerr-Newman metric
Kerr-Newman metric
The Kerr–Newman metric is a solution of the Einstein–Maxwell equations in general relativity, describing the spacetime geometry in the region surrounding a charged, rotating mass. It is assumed that the cosmological constant equals zero...

contain Killing horizons which coincide with their event horizon
Event horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...

s. It should be emphasized, however, that these two notions of horizon are independent. For this spacetime, the Killing horizon is located at

.

In the usual coordinates, outside the Killing horizon, the Killing vector field is timelike, whilst inside it is spacelike. The temperature of Hawking radiation
Hawking radiation is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to quantum effects. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974, and sometimes also after the physicist Jacob Bekenstein...

is related to the surface gravity
Surface gravity
The surface gravity, g, of an astronomical or other object is the gravitational acceleration experienced at its surface. The surface gravity may be thought of as the acceleration due to gravity experienced by a hypothetical test particle which is very close to the object's surface and which, in...

by .

## Cosmological Killing horizons

De Sitter space
De Sitter space
In mathematics and physics, a de Sitter space is the analog in Minkowski space, or spacetime, of a sphere in ordinary, Euclidean space. The n-dimensional de Sitter space , denoted dS_n, is the Lorentzian manifold analog of an n-sphere ; it is maximally symmetric, has constant positive curvature,...

has a Killing horizon at which emits thermal radiation at temperature .