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physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, a Killing horizon is a geometrical construct used in
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
and its generalizations to delineate spacetime boundaries without reference to the dynamic
Einstein field equations In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
. Mathematically a Killing horizon is a null hypersurface defined by the vanishing of the norm of a Killing vector field (both are named after
Wilhelm Killing Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. Life Killing studied at the University of M ...
). It can also be defined as a null hypersurface generated by a Killing vector, which in turn is null at that surface. After Hawking showed that
quantum field theory in curved spacetime In theoretical physics, quantum field theory in curved spacetime (QFTCS) is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime. This theory uses a semi-classical approach; it treats spacetime as a fixed ...
(without reference to the Einstein field equations) predicted that a black hole formed by collapse will emit
thermal radiation Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
, it became clear that there is an unexpected connection between spacetime geometry (Killing horizons) and thermal effects for quantum fields. In particular, there is a very general relationship between thermal radiation and spacetimes that admit a one-parameter group of isometries possessing a bifurcate Killing horizon, which consists of a pair of intersecting null hypersurfaces that are orthogonal to the Killing field.


Flat spacetime

In Minkowski space-time, in pseudo-Cartesian coordinates (t,x,y,z) with signature (+,-,-,-), an example of Killing horizon is provided by the
Lorentz boost In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...
(a Killing vector of the space-time) : V = x \, \partial_t + t \, \partial_x. The square of the norm of V is : g(V,V)=x^2-t^2=(x+t)(x-t). Therefore, V is null only on the hyperplanes of equations x+t=0, \text x-t=0, that, taken together, are the Killing horizons generated by V .


Black hole Killing horizons

Exact black hole metrics such as the Kerr–Newman metric contain Killing horizons, which can coincide with their ergospheres. For this spacetime, the corresponding Killing horizon is located at r = r_e := M + \sqrt. In the usual coordinates, outside the Killing horizon, the
Killing vector In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a pseudo-Riemannian manifold that preserves the metric tensor. Killing vector fields are the infinitesimal generators of isom ...
field \partial / \partial t is timelike, whilst inside it is spacelike. Furthermore, considering a particular linear combination of \partial / \partial t and \partial / \partial \phi , both of which are Killing vector fields, gives rise to a Killing horizon that coincides with the event horizon. Associated with a Killing horizon is a geometrical quantity known as
surface gravity The surface gravity, ''g'', of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experi ...
, \kappa. If the surface gravity vanishes, then the Killing horizon is said to be degenerate. The temperature of
Hawking radiation Hawking radiation is black-body radiation released outside a black hole's event horizon due to quantum effects according to a model developed by Stephen Hawking in 1974. The radiation was not predicted by previous models which assumed that onc ...
, found by applying
quantum field theory in curved spacetime In theoretical physics, quantum field theory in curved spacetime (QFTCS) is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime. This theory uses a semi-classical approach; it treats spacetime as a fixed ...
to black holes, is related to the
surface gravity The surface gravity, ''g'', of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experi ...
c^2\kappa by T_H = \frac with k_B the Boltzmann constant and \hbar the reduced Planck constant.


Cosmological Killing horizons

De Sitter space In mathematical physics, ''n''-dimensional de Sitter space (often denoted dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canonical Rie ...
has a Killing horizon at r = \sqrt, which emits thermal radiation at temperature T = \frac 1 \sqrt.


Further details

The term "Killing horizon" originates from the Killing vector field, a concept in differential geometry. A Killing vector field, in a given spacetime, is a vector field that preserves the metric. In the context of black holes, a Killing horizon is often associated with the event horizon. However, they are not always the same. For instance, in a rotating black hole (a Kerr black hole), the event horizon and the Killing horizon do not coincide. The concept of a Killing horizon is significant in the study of Hawking radiation. This is a theoretical prediction that black holes should emit radiation due to quantum effects near the event horizon. The Killing horizon also plays a role in the study of cosmic censorship hypotheses, which propose that singularities (points where quantities become infinite) are always hidden inside black holes, and thus cannot be observed from the rest of the Universe.Penrose, Roger (1969). "Gravitational collapse: The role of general relativity". ''Rivista del Nuovo Cimento''. 1: 252–276. . .


References

{{Reflist General relativity Mathematical physics