Reissner-Nordström metric

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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

$c^2 {d \tau}^{2} = \left( 1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) c^{2} dt^{2} - \frac{dr^{2}}{1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}}} - r^{2} d\theta^{2} - r^{2} \sin^{2} \theta \, d\varphi^{2}$

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

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

where G is the gravitational constant, and

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

$r_{Q}^{2} = \frac{Q^{2}G}{4\pi\epsilon_{0} c^{4}}$

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

$c^{2} d\tau^{2} = c^{2} dt^{2} - dr^{2} - r^{2} d\theta^{2} - r^{2} \sin^{2} \theta d\phi^{2}\,$

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.

[edit] Charged black holes

Although charged black holes with $r_{Q} \ll r_{s}$ 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

$g^{00}= 1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} = 0$

This quadratic equation for r has the solutions

$r_\pm = \frac{r_{s} \pm \sqrt{r_{s}^2 - 4r_{Q}^2}}{2}.$

These concentric event horizons become degenerate for $2 r Q = r s$ which corresponds to an extremal black hole. Black holes with $2 r Q > r s$ 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

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

If magnetic monopoles are included into the theory, then a generalization to include magnetic charge $P$ is obtained by replacing $Q 2$ by $Q 2 + P 2$ in the metric and including the term $P cosθ d φ$ in the electromagnetic potential.

[edit] 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.