Reissner–Nordström metric

In physics and astronomy, a Reissner-Nordström black hole, discovered by Gunnar Nordström and Hans Reissner, 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.

The black holes with $|Q|>M$ are believed not to exist in Nature because they would contain a naked singularity; their appearance would contradict Roger Penrose's cosmic censorship hypothesis which is generally believed to be true. Theories with supersymmetry usually guarantee that such "superextremal" black holes can't exist.

If magnetic monopoles are included into the theory, then a generalization to include magnetic charge $P$ is obtained by replacing $Q^{2}$ by $Q^{2}+P^{2}$ in the metric and including the term $P\cos \theta d\phi$ in the electromagnetic potential.

External links

• spacetime diagrams including Finkelstein diagram and Penrose diagram, by Andrew J. S. Hamilton