# Killing horizon

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). 

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)

$V=x\,\partial _{t}+t\,\partial _{x}.$ 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

$x+t=0,{\text{ and }}x-t=0,$ that, taken together, are the Killing horizons generated by $V$ . 

Associated to a Killing horizon is a geometrical quantity known as surface gravity, $\kappa$ . 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

$r=r_{e}:=M+{\sqrt {M^{2}-Q^{2}-a^{2}\cos ^{2}\theta }}.$ In the usual coordinates, outside the Killing horizon, the Killing vector field $\partial /\partial t$ is timelike, whilst inside it is spacelike. The temperature of Hawking radiation is related to the surface gravity $c^{2}\kappa$ by $T_{H}={\frac {\hbar c\kappa }{2\pi k_{B}}}$ with $k_{B}$ the Boltzmann constant.

## Cosmological Killing horizons

De Sitter space has a Killing horizon at $r={\sqrt {3/\Lambda }}$ which emits thermal radiation at temperature $T=(1/2\pi ){\sqrt {\Lambda /3}}$ .