# 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). (Notice that in the case of flat space time Killing vectors are null all over the space time, so they do not define a hypersurface and we have no Killing horizon.)

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 event horizons. It should be emphasized, however, that these two notions of horizon are independent. For this spacetime, the Killing horizon is located at

$r = r_+ := M + \sqrt{M^2 - Q^2 - J^2/M^2}$.

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 by $T_H = \kappa / 2 \pi$.

## Cosmological Killing horizons

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