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Concept# Killing horizon

Summary

In physics, a Killing horizon is a geometrical construct used in general relativity and its generalizations to delineate spacetime boundaries without reference to the dynamic Einstein field equations. 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). 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 (without reference to the Einstein field equations) predicted that a black hole formed by collapse will emit thermal radiation, 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.
In Minkowski space-time, in pseudo-Cartesian coordinates with signature an example of Killing horizon is provided by the Lorentz boost (a Killing vector of the space-time)
The square of the norm of is
Therefore, is null only on the hyperplanes of equations
that, taken together, are the Killing horizons generated by .
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
In the usual coordinates, outside the Killing horizon, the Killing vector field is timelike, whilst inside it is spacelike.
Furthermore, considering a particular linear combination of and , 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, . If the surface gravity vanishes, then the Killing horizon is said to be degenerate.

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