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{{more footnotes|date=January 2013}}
 
{{General relativity|cTopic=[[Exact solutions in general relativity|Solutions]]}}
In [[physics]] and [[astronomy]], the '''Reissner&ndash;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''.
 
The metric was discovered by [[Hans Reissner]] and [[Gunnar Nordström]].
 
These four related solutions may be summarized by the following table:
 
{| class="wikitable" style="margin: 1em auto"
|
| Non-rotating (''J'' = 0)
| Rotating (''J'' ≠ 0)
|-
| Uncharged (''Q'' = 0)
| [[Schwarzschild metric|Schwarzschild]]
| [[Kerr metric|Kerr]]
|-
| Charged (''Q'' ≠ 0)
| Reissner–Nordström
| [[Kerr–Newman metric|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&ndash;Nordström metric is
 
:<math>
ds^2 =
\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>
 
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>&nbsp;= 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
:<math>
r_{Q}^{2} = \frac{Q^2 G}{4\pi\varepsilon_{0} c^4}.
</math>
Here 1/4πε<sub>0</sub> is [[Coulomb's law|Coulomb force constant]].<ref name="landau_1975" >Landau 1975.</ref>
 
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]].
 
In practice, the ratio ''r''<sub>S</sub>/''r'' is often extremely small.  For example, the Schwarzschild radius of the [[Earth]] is roughly 9&nbsp;[[millimeter|mm]] (3/8&nbsp;[[inch]]), whereas a [[satellite]] in a [[geosynchronous orbit]] has a radius ''r'' that is roughly four billion times larger, at 42,164&nbsp;[[kilometer|km]] (26,200&nbsp;[[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.
 
==Charged black holes==
Although charged black holes with ''r<sub>Q</sub>''&nbsp;≪ ''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
 
:<math> 0 = 1/g^{rr} = 1 - \frac{r_\mathrm{S}}{r} + \frac{r_Q^2}{r^2}.</math>
 
This equation has two solutions:
 
:<math>
r_\pm = \frac{1}{2}\left(r_{s} \pm \sqrt{r_{s}^2 - 4r_{Q}^2}\right).
</math>
 
These concentric [[event horizon]]s become [[Degenerate energy level|degenerate]] for 2''r<sub>Q</sub>''&nbsp;= ''r''<sub>S</sub>, which corresponds to an [[extremal black hole]]. Black holes with 2''r<sub>Q</sub>''&nbsp;> ''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.
 
The [[electromagnetic potential]] is
 
:<math>A_{\alpha} = \left(Q/r, 0, 0, 0\right).</math>
 
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&thinsp;θ&nbsp;''d''φ in the electromagnetic potential.{{clarify|date=January 2013}}
 
==See also==
 
*[[Black hole electron]]
 
==Notes==
{{Reflist}}
 
==References==
*{{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}}
*{{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}}
*{{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 -->
*{{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}}
 
==External links==
*[http://casa.colorado.edu/~ajsh/rn.html spacetime diagrams] including [[Finkelstein diagram]] and [[Penrose diagram]], by Andrew J. S. Hamilton
* "[http://demonstrations.wolfram.com/ParticleMovingAroundTwoExtremeBlackHoles/ Particle Moving Around Two Extreme Black Holes]" by Enrique Zeleny, [[The Wolfram Demonstrations Project]].
{{Black holes}}
{{Relativity}}
 
{{DEFAULTSORT:Reissner-Nordstrom metric}}
[[Category:Exact solutions in general relativity]]
[[Category:Black holes]]
[[Category:Metric tensors]]

Revision as of 16:01, 31 January 2014

Template:More footnotes

Diving Coach (Open water ) Dominic from Kindersley, loves to spend some time classic cars, property developers in singapore house for rent (Source Webpage) and greeting card collecting. Finds the world an interesting place having spent 8 days at Cidade Velha. 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

ds2=(1rSr+rQ2r2)c2dt211rS/r+rQ2/r2dr2r2dθ2r2sin2θdϕ2,

where c is the speed of light, t is the time coordinate (measured by a stationary clock at infinity), r is the radial coordinate, rS = 2GM/c2 is the Schwarzschild radius of the body, and rQ is a characteristic length scale given by

rQ2=Q2G4πε0c4.

Here 1/4πε0 is Coulomb force constant.[1]

In the limit that the charge Q (or equivalently, the length-scale rQ) goes to zero, one recovers the Schwarzschild metric. The classical Newtonian theory of gravity may then be recovered in the limit as the ratio rS/r goes to zero. In that limit that both rQ/r and rS/r go to zero, the metric becomes the Minkowski metric for special relativity.

In practice, the ratio rS/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 rQ ≪ rS 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 grr diverges; that is, where

0=1/grr=1rSr+rQ2r2.

This equation has two solutions:

r±=12(rs±rs24rQ2).

These concentric event horizons become degenerate for 2rQ = rS, which corresponds to an extremal black hole. Black holes with 2rQ > rS 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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.

The electromagnetic potential is

Aα=(Q/r,0,0,0).

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.Template:Clarify

See also

Notes

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References

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    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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External links

Template:Black holes Template:Relativity

  1. Landau 1975.
  2. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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