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Killing horizon

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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).

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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 (a Killing vector of the space-time)

V = x ∂ t + t ∂ 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 ,  and  x − t = 0 ,

that, taken together, are the Killing horizons generated by V .

Associated to a Killing horizon is a geometrical quantity known as surface gravity, κ . If the surface gravity vanishes, then the Killing horizon is said to be degenerate.

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Black hole Killing horizons

Exact black hole metrics such as the Kerr–Newman metric contain Killing horizons which coincide with their ergospheres. For this spacetime, the Killing horizon is located at

r = r e := M + M 2 − Q 2 − a 2 cos 2 ⁡ θ .

In the usual coordinates, outside the Killing horizon, the Killing vector field ∂ / ∂ t is timelike, whilst inside it is spacelike. The temperature of Hawking radiation is related to the surface gravity c 2 κ by T H = ℏ c κ 2 π k B with k B the Boltzmann constant.

Cosmological Killing horizons

De Sitter space has a Killing horizon at r = 3 / Λ which emits thermal radiation at temperature T = ( 1 / 2 π ) Λ / 3 .

References

Killing horizon Wikipedia
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