Reissner-Nordström metric

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General relativity

Failed to parse (Missing texvc executable; please see math/README to configure.): G_{\mu \nu} = {8\pi G\over c^4} T_{\mu \nu}\,

Einstein field equations

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In physics and astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M. Discovered by Gunnar Nordström and Hans Reissner, their metric can be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): c^2 {d \tau}^{2} = \left( 1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) c^{2} dt^{2} - \frac{dr^{2}}{1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}}} - r^{2} d\theta^{2} - r^{2} \sin^{2} \theta \, d\varphi^{2}

where

τ is the proper time (time measured by a clock moving with the particle) in seconds,

c is the speed of light in meters per second,

t is the time coordinate (measured by a stationary clock at infinity) in seconds,

r is the radial coordinate (circumference of a circle centered on the star divided by 2π) in meters,

θ is the colatitude (angle from North) in radians,

φ is the longitude in radians, and

r_s is the Schwarzschild radius (in meters) of the massive body, which is related to its mass M by

Failed to parse (Missing texvc executable; please see math/README to configure.): r_{s} = \frac{2GM}{c^{2}}

where G is the gravitational constant, and

r_Q is a length-scale corresponding to the electric charge Q of the mass

Failed to parse (Missing texvc executable; please see math/README to configure.): r_{Q}^{2} = \frac{Q^{2}G}{4\pi\epsilon_{0} c^{4}}

where 1/4πε₀ is Coulomb's force constant.^[1]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): c^{2} d\tau^{2} = c^{2} dt^{2} - dr^{2} - r^{2} d\theta^{2} - r^{2} \sin^{2} \theta d\phi^{2}

In practice, the ratio r_s is almost always extremely small. For example, the Schwarzschild radius r_s 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 Failed to parse (Missing texvc executable; please see math/README to configure.): r_{Q} \ll r_{s}

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

Failed to parse (Missing texvc executable; please see math/README to configure.): g^{00}= 1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} = 0

This quadratic equation for r has the solutions

Failed to parse (Missing texvc executable; please see math/README to configure.): r_\pm = \frac{r_{s} \pm \sqrt{r_{s}^2 - 4r_{Q}^2}}{2}.

These concentric event horizons become degenerate for Failed to parse (Missing texvc executable; please see math/README to configure.): 2r_{Q}=r_{s}

which corresponds to an extremal black hole.  Black holes with Failed to parse (Missing texvc executable; please see math/README to configure.): 2r_{Q}>r_{s}
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.

The electromagnetic potential is

Failed to parse (Missing texvc executable; please see math/README to configure.): A_{\alpha} = \left(\frac{Q}{r}, 0, 0, 0\right)

If magnetic monopoles are included into the theory, then a generalization to include magnetic charge Failed to parse (Missing texvc executable; please see math/README to configure.): P

is obtained by replacing Failed to parse (Missing texvc executable; please see math/README to configure.): Q^2
by Failed to parse (Missing texvc executable; please see math/README to configure.): Q^2 + P^2
in the metric and including the term Failed to parse (Missing texvc executable; please see math/README to configure.): P \cos \theta d \phi
in the electromagnetic potential.

References

^ Landau 1975.

Reissner, H (1916). "Über die Eigengravitation des elektrischen Feldes nach der Einstein'schen Theorie". Annalen der Physik 50: 106–120.

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, R; Bazin M, and Schiffer M (1965). Introduction to General Relativity. New York: McGraw-Hill Book Company, pp. 395–401. ISBN 978-0-07-000420-7.