# Reissner–Nordström metric

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In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M.

The metric was discovered by Hans Reissner and Gunnar Nordström.

These four related solutions may be summarized by the following table:

 Non-rotating (J = 0) Rotating (J ≠ 0) Uncharged (Q = 0) Schwarzschild Kerr Charged (Q ≠ 0) Reissner–Nordström Kerr–Newman

where Q represents the body's electric charge and J represents its spin angular momentum.

## The metric

In spherical coordinates (t, r, θ, φ), the line element for the Reissner–Nordström metric is

${\displaystyle ds^{2}=\left(1-{\frac {r_{\mathrm {S} }}{r}}+{\frac {r_{Q}^{2}}{r^{2}}}\right)c^{2}\,dt^{2}-\left(1-{\frac {r_{\mathrm {S} }}{r}}+{\frac {r_{Q}^{2}}{r^{2}}}\right)^{-1}dr^{2}-r^{2}\,d\Omega _{(2)}^{2},}$

where c is the speed of light, t is the time coordinate (measured by a stationary clock at infinity), r is the radial coordinate, ${\displaystyle \textstyle d\Omega _{(2)}^{2}}$ is a 2-sphere defined by

${\displaystyle d\Omega _{(2)}^{2}=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}}$

rS is the Schwarzschild radius of the body given by

${\displaystyle r_{s}={\frac {2GM}{c^{2}}},}$

and rQ is a characteristic length scale given by

${\displaystyle r_{Q}^{2}={\frac {Q^{2}G}{4\pi \varepsilon _{0}c^{4}}}.}$

Here 1/4πε0 is Coulomb force constant.[1]

In the limit that the charge Q (or equivalently, the length-scale rQ) goes to zero, one recovers the Schwarzschild metric. The classical Newtonian theory of gravity may then be recovered in the limit as the ratio rS/r goes to zero. In that limit that both rQ/r and rS/r go to zero, the metric becomes the Minkowski metric for special relativity.

In practice, the ratio rS/r is often extremely small. For example, the Schwarzschild radius of the Earth is roughly 9 mm (3/8 inch), whereas a satellite in a geosynchronous orbit has a radius r that is roughly four billion times larger, at 42,164 km (26,200 miles). Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The ratio only becomes large close to black holes and other ultra-dense objects such as neutron stars.

## Charged black holes

Although charged black holes with rQ ≪ rS are similar to the Schwarzschild black hole, they have two horizons: the event horizon and an internal Cauchy horizon.[2] As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component grr diverges; that is, where

${\displaystyle 0={\frac {1}{g^{rr}}}=1-{\frac {r_{\mathrm {S} }}{r}}+{\frac {r_{Q}^{2}}{r^{2}}}.}$

This equation has two solutions:

${\displaystyle r_{\pm }={\frac {1}{2}}\left(r_{s}\pm {\sqrt {r_{s}^{2}-4r_{Q}^{2}}}\right).}$

These concentric event horizons become degenerate for 2rQ = rS, which corresponds to an extremal black hole. Black holes with 2rQ > rS 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.[citation needed] Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.

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

If magnetic monopoles are included in the theory, then a generalization to include magnetic charge P is obtained by replacing Q2 by Q2 + P2 in the metric and including the term Pcos θ dφ in the electromagnetic potential.[clarification needed]