General Relativity/Reissner-Nordström black hole

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.

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