General relativity

G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}




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In physics and astronomy, the Reissner–Nordström metric is a static solution to the EinsteinMaxwell field equations, which corresponds to the gravitational field of a charged, nonrotating, 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:
where Q represents the body's electric charge and J represents its spin angular momentum.
Contents

The metric 1

Charged black holes 2

See also 3

Notes 4

References 5

External links 6
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 2sphere defined by

d\Omega^2_{(2)}=d\theta^2 + sin^2\theta d\phi^2
r_{S} is the Schwarzschild radius of the body given by

r_{s} = \frac{2GM}{c^2},
and r_{Q} is a characteristic length scale given by

r_{Q}^{2} = \frac{Q^2 G}{4\pi\varepsilon_{0} c^4}.
Here 1/4πε_{0} is Coulomb force constant.^{[1]}
In the limit that the charge Q (or equivalently, the lengthscale 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 that both r_{Q}/r and r_{S}/r go to zero, the metric becomes the Minkowski metric for special relativity.
In practice, the ratio r_{S}/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 ultradense objects such as neutron stars.
Charged black holes
Although charged black holes with r_{Q} ≪ r_{S} 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 g^{rr} 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 2r_{Q} = r_{S}, which corresponds to an extremal black hole. Black holes with 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 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 Q^{2} by Q^{2} + P^{2} in the metric and including the term Pcos θ dφ in the electromagnetic potential.
See also
Notes
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: McGrawHill Book Company. pp. 395–401.

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