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:

$d{s}^2=-(1-\frac{2M}{r}+\frac{Q^2}{r^2})d{t}^2+{(1-\frac{2M}{r}+\frac{Q^2}{r^2})}^{-1}d{r}^2+r^{2}d{\mathrm{\Omega }}^2$

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

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

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.