Reissner-Nordström black hole

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In physics and astronomy, a Reissner-Nordström black hole, discovered by Gunnar Nordström and Hans Reissner, is a black hole that carries mass $M$ , electric charge $Q$ , and no angular momentum. 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:

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

where we have used geometrized units with the speed of light, gravitational constant, and the Coulomb force constant equal to one ( $c = G = 1 / 4πε 0 = 1$ ) and where the angular part of the metric is

$\mathrm{d}\Omega = \mathrm{d}\theta^2 +\sin^2\theta\,\mathrm{d}\phi^2$

The electromagnetic potential is

$A_{\alpha} = \left(\frac{Q}{r}, 0, 0, 0\right)$ .

While the charged black holes with $| Q | < M$ (especially with $|Q| \ll M$ ) are similar to the Schwarzschild black hole, they have two horizons: the event horizon and an internal Cauchy horizon. As usual, the event horizons for the spacetime may be reliably located by analyzing the equation $g 00 = 0$ . A brief look at this condition produces a quadratic in $r$ whose solutions show the event horizons to be located at

$r_\pm = M \pm \sqrt{M^2-Q^2}.$

These horizons become degenerate for $| Q | = M$ which is the case of an extremal black hole. The near horizon metric of an extremal black hole approaches AdS₂ × S² with the radius of curvature being M.

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 $P cosθ d φ$ in the electromagnetic potential.