Reissner–Nordström metric

General relativity
$G_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }$
Introduction History Mathematical formulation Tests
Fundamental concepts Equivalence principle Special relativity World line Riemannian geometry
Phenomena Kepler problem Gravitational field Gravitational collapse Gravitational lensing Gravitational waves Gravitational redshift Gravitational time delay Gravitational time dilation Frame-dragging Geodetic effect Gravity well Event horizon Singularity Naked singularity Black hole White hole Spacetime Space Time Spacetime diagrams Minkowski spacetime Closed timelike curve (CTC) Wormhole Ellis wormhole
Equations Formalisms Equations Linearized gravity Einstein field equations Friedmann Geodesics Mathisson–Papapetrou–Dixon Hamilton–Jacobi–Einstein Formalisms ADM BSSN Post-Newtonian Advanced theory Kaluza–Klein theory Quantum gravity
Solutions Schwarzschild (interior) Reissner–Nordström Gödel Kerr Kerr–Newman Kasner Lemaître–Tolman Taub-NUT Milne Robertson–Walker pp-wave van Stockum dust Weyl−Lewis−Papapetrou
Scientists Einstein Lorentz Hilbert Poincaré Schwarzschild de Sitter Reissner Nordström Weyl Eddington Friedman Milne Zwicky Lemaître Gödel Wheeler Robertson Bardeen Walker Kerr Chandrasekhar Ehlers Penrose Hawking Raychaudhuri Taylor Hulse van Stockum Taub Newman Yau Thorne others

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.

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

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

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 1/4πε₀ is Coulomb force constant.

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 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 ultra-dense 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. As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component g^rr diverges; that is, where

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

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 Q² by Q² + P² in the metric and including the term Pcos θ dφ in the electromagnetic potential.

See also

• Black hole electron

References

• Adler, R.; 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. 

External links

• spacetime diagrams including Finkelstein diagram and Penrose diagram, by Andrew J. S. Hamilton
• "Particle Moving Around Two Extreme Black Holes" by Enrique Zeleny, The Wolfram Demonstrations Project.