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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). ^[1]

In Minkowski space-time, in pseudo-Cartesian coordinates (t,x,y,z)displaystyle (t,x,y,z) with signature (+,−,−,−),displaystyle (+,-,-,-), an example of Killing horizon is provided by the Lorentz boost (a Killing vector of the space-time)

V=x∂t+t∂x.displaystyle V=x,partial _t+t,partial _x.

The square of the norm of Vdisplaystyle V is

g(V,V)=x2−t2=(x+t)(x−t).displaystyle g(V,V)=x^2-t^2=(x+t)(x-t).

Therefore, Vdisplaystyle V is null only on the hyperplanes of equations

x+t=0, and x−t=0,displaystyle x+t=0,text and x-t=0,

that, taken together, are the Killing horizons generated by Vdisplaystyle V. ^[2]

Associated to a Killing horizon is a geometrical quantity known as surface gravity, κdisplaystyle kappa . 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 contain Killing horizons which coincide with their ergospheres. For this spacetime, the Killing horizon is located at

r=re:=M+M2−Q2−a2cos2⁡θ.displaystyle r=r_e:=M+sqrt M^2-Q^2-a^2cos ^2theta .

In the usual coordinates, outside the Killing horizon, the Killing vector field ∂/∂tdisplaystyle partial /partial t is timelike, whilst inside it is spacelike. The temperature of Hawking radiation is related to the surface gravity c2κdisplaystyle c^2kappa by TH=ℏcκ2πkBdisplaystyle T_H=frac hbar ckappa 2pi k_B with kBdisplaystyle k_B the Boltzmann constant.

Cosmological Killing horizons

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

References

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