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 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
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

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^[1] and Gunnar Nordström.^[2]

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.^[3] 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

↑ 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.
↑ 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

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
Types	Schwarzschild Rotating Charged Virtual Binary
Size	Micro Extremal Electron Stellar Intermediate-mass Supermassive Quasar Active galactic nucleus Blazar Large quasar group
Formation	Stellar evolution Gravitational collapse Neutron star Related links Compact star Quark Exotic Tolman–Oppenheimer–Volkoff limit White dwarf Related links Supernova Related links Hypernova Gamma-ray burst
Properties	Thermodynamics Schwarzschild radius M–sigma relation Event horizon Quasi-periodic oscillation Photon sphere Ergosphere Hawking radiation Penrose process Blandford–Znajek process Bondi accretion Spaghettification Gravitational lens
Models	Gravitational singularity Penrose–Hawking singularity theorems Primordial black hole Gravastar Dark star Dark-energy star Black star Eternally collapsing object Magnetospheric eternally collapsing object Fuzzball White hole Naked singularity Ring singularity Immirzi parameter Membrane paradigm Kugelblitz Wormhole Quasi-star
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
Lists	Black holes Most massive Quasars
Related	Timeline of black hole physics Rossi X-ray Timing Explorer Hypercompact stellar system Gamma-ray burst progenitors Gravity well Black hole starship

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
Consequences	Time dilation Relativistic mass Mass–energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Relativistic disks
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 Penrose 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 Raychaudhuri equation Hamilton–Jacobi–Einstein equation Ernst 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

Scientists

Einstein field equations:

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

and their analytical solution by Ernst equation:

\displaystyle \Re (u)(u_{rr}+u_{r}/r+u_{zz})=(u_{r})^{2}+(u_{z})^{2}.

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