Reissner–Nordström metric

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

Fundamental concepts

Phenomena

Spacetime
Kepler problem Gravitational lensing Gravitational waves Frame-dragging Geodetic effect Event horizon Singularity Black hole
Spacetime diagrams Minkowski spacetime Einstein–Rosen bridge

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

Scientists

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.

These four related solutions may be summarized by the following table:

	Non-rotating (J = 0)	Rotating (J ≠ 0)
Uncharged (Q = 0)	Schwarzschild	Kerr
Charged (Q ≠ 0)	Reissner–Nordström	Kerr–Newman

where Q represents the body's electric charge and J represents its spin angular momentum.

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.^[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 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.^[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={\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.

Notes

↑ Landau 1975.
↑ Chandrasekhar, S. (1998). The Mathematical Theory of Black Holes (Reprinted ed.). Oxford University Press. p. 205. ISBN 0-19850370-9. Retrieved 13 May 2013. And finally, the fact that the Reissner-Nordström solution has two horizons, an external event horizon and an internal 'Cauchy horizon,' provides a convenient bridge to the study of the Kerr solution in the subsequent chapters.

References

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, 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.

Black holes

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Issues	No-hair theorem Information paradox Cosmic censorship Alternative models Holographic principle Black hole complementarity firewall (physics) ER=EPR Final parsec problem

Metrics	Schwarzschild Kerr Reissner–Nordström Kerr–Newman

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Relativity

Special
relativity

Background	Principle of relativity Special relativity

Foundations	Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations

Formulation	Galilean relativity Galilean transformation Lorentz transformation

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Spacetime	Light cone World line Spacetime diagram Biquaternions Minkowski space

General
relativity

Background	Introduction Mathematical formulation

Fundamental concepts	Special relativity Equivalence principle World line Riemannian geometry Minkowski diagram

Phenomena	Black hole Event horizon Frame-dragging Geodetic effect Lenses Singularity Waves Ladder paradox Twin paradox Two-body problem

Equations	ADM formalism BSSN formalism Einstein field equations Geodesic equation Friedmann equations Linearized gravity Post-Newtonian formalism Hamilton–Jacobi–Einstein equation

Advanced theories	Brans–Dicke theory Kaluza–Klein Mach's principle Quantum gravity

Solutions	Schwarzschild (interior) Reissner–Nordström Gödel Kerr Kerr–Newman Kasner Taub–NUT Milne Robertson–Walker pp-wave van Stockum dust Weyl−Lewis−Papapetrou