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.

Contents

The metric [edit]

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 - \frac{1}{1 - r_\mathrm{S}/r + r_Q^2/r^2}\, dr^2 - r^2\, d\theta^2 - r^2 \sin^2 \theta \, d\phi^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, rS = 2GM/c2 is the Schwarzschild radius of the body, 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 [edit]

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. 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.[citation needed] Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.

The electromagnetic potential is

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]

See also [edit]

Notes [edit]

  1. ^ Landau 1975.

References [edit]

  • Reissner, H. (1916). "Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie". Annalen der Physik (in German) 50: 106–120. Bibcode:1916AnP...355..106R. doi:10.1002/andp.19163550905. 
  • Nordström, G. (1918). "On the Energy of the Gravitational Field in Einstein's Theory". Verhandl. Koninkl. Ned. Akad. Wetenschap., Afdel. Natuurk., Amsterdam 26: 1201–1208. 
  • Adler; Bazin, M.; Schiffer, M. (1965). Introduction to General Relativity. New York: McGraw-Hill Book Company. pp. 395–401. ISBN 978-0-07-000420-7. 
  • Wald, Robert M. (1984). General Relativity. Chicago: The University of Chicago Press. pp. 158,312–324. ISBN 978-0-226-87032-8. Retrieved 27 April 2013. 

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