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:


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

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.