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| {{more footnotes|date=January 2013}}
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| {{General relativity|cTopic=[[Exact solutions in general relativity|Solutions]]}}
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| In [[physics]] and [[astronomy]], the '''Reissner–Nordström metric''' is a [[Static spacetime|static solution]] to the [[Einstein-Maxwell equations|Einstein-Maxwell field equations]], which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass ''M''.
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| The metric was discovered by [[Hans Reissner]] and [[Gunnar Nordström]].
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| These four related solutions may be summarized by the following table:
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| {| class="wikitable" style="margin: 1em auto"
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| | Non-rotating (''J'' = 0)
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| | Rotating (''J'' ≠ 0)
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| |-
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| | Uncharged (''Q'' = 0)
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| | [[Schwarzschild metric|Schwarzschild]]
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| | [[Kerr metric|Kerr]]
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| |-
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| | Charged (''Q'' ≠ 0)
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| | Reissner–Nordström
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| | [[Kerr–Newman metric|Kerr–Newman]]
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| |}
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| where ''Q'' represents the body's [[electric charge]] and ''J'' represents its spin [[angular momentum]].
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| ==The metric==
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| In [[spherical coordinates]] (''t'', ''r'', θ, φ), the [[line element]] for the Reissner–Nordström metric is
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| :<math>
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| ds^2 =
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| \left( 1 - \frac{r_\mathrm{S}}{r} + \frac{r_Q^2}{r^2} \right) c^2\, dt^2 - \frac{1}{1 - r_\mathrm{S}/r + r_Q^2/r^2}\, dr^2 - r^2\, d\theta^2 - r^2 \sin^2 \theta \, d\phi^2,</math>
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| where ''c'' is the [[speed of light]], ''t'' is the time coordinate (measured by a stationary clock at infinity), ''r'' is the radial coordinate, ''r''<sub>S</sub> = 2''GM''/''c''<sup>2</sup> is the [[Schwarzschild radius]] of the body, and ''r<sub>Q</sub>'' is a characteristic length scale given by
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| :<math>
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| r_{Q}^{2} = \frac{Q^2 G}{4\pi\varepsilon_{0} c^4}.
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| </math>
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| Here 1/4πε<sub>0</sub> is [[Coulomb's law|Coulomb force constant]].<ref name="landau_1975" >Landau 1975.</ref>
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| In the limit that the charge ''Q'' (or equivalently, the length-scale ''r''<sub>''Q''</sub>) 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''<sub>S</sub>/''r'' goes to zero. In that limit that both ''r<sub>Q</sub>''/''r'' and ''r''<sub>S</sub>/''r'' go to zero, the metric becomes the [[Minkowski metric]] for [[special relativity]].
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| In practice, the ratio ''r''<sub>S</sub>/''r'' is often extremely small. For example, the Schwarzschild radius of the [[Earth]] is roughly 9 [[millimeter|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 [[kilometer|km]] (26,200 [[mile]]s). 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 hole]]s and other ultra-dense objects such as [[neutron star]]s.
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| ==Charged black holes==
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| Although charged black holes with ''r<sub>Q</sub>'' ≪ ''r''<sub>S</sub> are similar to the [[Schwarzschild black hole]], they have two horizons: the [[event horizon]] and an internal [[Cauchy horizon]].<ref>{{cite book |last=Chandrasekhar |first=S. |authorlink=Subrahmanyan Chandrasekhar |title=The Mathematical Theory of Black Holes |year=1998 |publisher=Oxford University Press |isbn=0-19850370-9 |edition=Reprinted |url=http://www.oup.com/us/catalog/general/subject/Physics/Relativity/?view=usa&ci=9780198503705 |accessdate=13 May 2013 |page=205 |quote=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.}}</ref> As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component ''g<sup>rr</sup>'' diverges; that is, where
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| :<math> 0 = 1/g^{rr} = 1 - \frac{r_\mathrm{S}}{r} + \frac{r_Q^2}{r^2}.</math>
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| This equation has two solutions:
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| :<math>
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| r_\pm = \frac{1}{2}\left(r_{s} \pm \sqrt{r_{s}^2 - 4r_{Q}^2}\right).
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| </math>
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| These concentric [[event horizon]]s become [[Degenerate energy level|degenerate]] for 2''r<sub>Q</sub>'' = ''r''<sub>S</sub>, which corresponds to an [[extremal black hole]]. Black holes with 2''r<sub>Q</sub>'' > ''r''<sub>S</sub> 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.{{citation needed|date=January 2013}} Theories with [[supersymmetry]] usually guarantee that such "superextremal" black holes cannot exist.
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| The [[electromagnetic potential]] is | |
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| :<math>A_{\alpha} = \left(Q/r, 0, 0, 0\right).</math>
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| If magnetic monopoles are included in the theory, then a generalization to include magnetic charge ''P'' is obtained by replacing ''Q''<sup>2</sup> by ''Q''<sup>2</sup> + ''P''<sup>2</sup> in the metric and including the term ''P''cos θ ''d''φ in the electromagnetic potential.{{clarify|date=January 2013}}
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| ==See also==
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| *[[Black hole electron]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{cite journal | last=Reissner | first=H. | year=1916 | title=Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie | journal=Annalen der Physik | volume=50 | pages=106–120 | doi=10.1002/andp.19163550905 | bibcode=1916AnP...355..106R | language=German}}
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| *{{cite journal | last=Nordström | first=G. | authorlink=Gunnar Nordström | year=1918 | title=On the Energy of the Gravitational Field in Einstein's Theory | journal=Verhandl. Koninkl. Ned. Akad. Wetenschap., Afdel. Natuurk., Amsterdam | volume=26 | pages=1201–1208}}
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| *{{cite book | last1=Adler | first2=R. | last2=Bazin | first2=M. | last3=Schiffer | first3=M. | year=1965 | title=Introduction to General Relativity | publisher=McGraw-Hill Book Company | location=New York | isbn=978-0-07-000420-7 | pages=395–401}}<!--{{LCCN|64|0|16476}}--><!-- Note: 2nd edition, 1975: 978-0-07-000423-8 -->
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| *{{cite book |last=Wald |first=Robert M. |authorlink=Robert Wald |year=1984 |title=General Relativity |publisher=The University of Chicago Press |location=Chicago |isbn=978-0-226-87032-8 |pages=158,312–324 |url=http://press.uchicago.edu/ucp/books/book/chicago/G/bo5952261.html |accessdate=27 April 2013}}
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| ==External links==
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| *[http://casa.colorado.edu/~ajsh/rn.html spacetime diagrams] including [[Finkelstein diagram]] and [[Penrose diagram]], by Andrew J. S. Hamilton
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| * "[http://demonstrations.wolfram.com/ParticleMovingAroundTwoExtremeBlackHoles/ Particle Moving Around Two Extreme Black Holes]" by Enrique Zeleny, [[The Wolfram Demonstrations Project]].
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| {{Black holes}}
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| {{Relativity}}
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| {{DEFAULTSORT:Reissner-Nordstrom metric}}
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| [[Category:Exact solutions in general relativity]]
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| [[Category:Black holes]]
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| [[Category:Metric tensors]]
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