# General Relativity/Reissner-Nordström black hole

Reissner-Nordström black hole is a black hole that carries electric charge ${\displaystyle Q}$, no angular momentum, and mass ${\displaystyle M}$. 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:

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

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

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

The electromagnetic potential is

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

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