# 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).[1]

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

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

The square of the norm of ${\displaystyle V}$ is

${\displaystyle g(V,V)=x^{2}-t^{2}=(x+t)(x-t).}$

Therefore, ${\displaystyle V}$ is null only on the hyperplanes of equations

${\displaystyle x+t=0,{\text{ and }}x-t=0,}$

that, taken together, are the Killing horizons generated by ${\displaystyle V}$.[2]

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

${\displaystyle r=r_{e}:=M+{\sqrt {M^{2}-Q^{2}-a^{2}\cos ^{2}\theta }}.}$

In the usual coordinates, outside the Killing horizon, the Killing vector field ${\displaystyle \partial /\partial t}$  is timelike, whilst inside it is spacelike. The temperature of Hawking radiation is related to the surface gravity ${\displaystyle c^{2}\kappa }$  by ${\displaystyle T_{H}={\frac {\hbar c\kappa }{2\pi k_{B}}}}$  with ${\displaystyle k_{B}}$  the Boltzmann constant.

## Cosmological Killing horizons

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

## References

1. ^ Reall, Harvey (2008). black holes (PDF). p. 17. Archived from the original (PDF) on 2015-07-15. Retrieved 2015-07-15.
2. ^ Chruściel, P.T. "Black-holes, an introduction". In "100 years of relativity", edited by A. Ashtekar, World Scientific, 2005.