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# Reissner–Nordström metric

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 Title: Reissner–Nordström metric Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Reissner–Nordström metric

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.

## Contents

• The metric 1
• Charged black holes 2
• Notes 4
• References 5

## The metric

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

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, \textstyle d\Omega^2_{(2)} is a 2-sphere defined by

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

rS is the Schwarzschild radius of the body given by

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

and rQ is a characteristic length scale given by

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

0 = 1/g^{rr} = 1 - \frac{r_\mathrm{S}}{r} + \frac{r_Q^2}{r^2}.

This equation has two solutions:

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. Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.

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.

## Notes

1. ^ Landau 1975.
2. ^

## References

• Reissner, H. (1916). "Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie". Annalen der Physik (in German) 50: 106–120.
• Adler, R.; Bazin, M.; Schiffer, M. (1965). Introduction to General Relativity. New York: McGraw-Hill Book Company. pp. 395–401.