General Relativity/Reissner-Nordström black hole

<General Relativity

Reissner-Nordström black hole is a black hole that carries electric charge $Q$, no angular momentum, and mass $M$. General properties of such a black hole are described in the article charged black hole.

It is described by the electric field of a point-like charged particle, and especially by the Reissner-Nordström metric that generalizes the Schwarzschild metric of an electrically neutral black hole:

$ds^{2}=-\left(1-{\frac {2M}{r}}+{\frac {Q^{2}}{r^{2}}}\right)dt^{2}+\left(1-{\frac {2M}{r}}+{\frac {Q^{2}}{r^{2}}}\right)^{-1}dr^{2}+r^{2}d\Omega ^{2}$

where we have used units with the speed of light and the gravitational constant equal to one ($c=G=1$) and where the angular part of the metric is

$d\Omega ^{2}\equiv d\theta ^{2}+\sin ^{2}\theta \cdot d\phi ^{2}$

The electromagnetic potential is

$A=-{\frac {Q}{r}}dt$.

While the charged black holes with $|Q|<M$ (especially with $|Q|<<M$) are similar to the Schwarzschild black hole, they have two horizons: the event horizon and an internal Cauchy horizon. The horizons are located at $r=r_{\pm }:=M\pm {\sqrt {M^{2}-Q^{2}}}$. These horizons merge for $|Q|=M$ which is the case of an extremal black hole.