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.