View source for Reissner–Nordström metric

{{Short description|Spherically symmetric metric with electric charge}}
{{more footnotes needed|date=January 2013}}
{{General relativity sidebar |solutions}}

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''. The analogous solution for a charged, rotating body is given by the [[Kerr–Newman metric]].

The metric was discovered between 1916 and 1921 by [[Hans Reissner]],<ref>{{cite journal |last=Reissner |first=H. |date=1916 |title=Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie |url=https://zenodo.org/record/1447315 |journal=Annalen der Physik |language=en |volume=355 |issue=9 |pages=106–120 |bibcode=1916AnP...355..106R |doi=10.1002/andp.19163550905 |issn=0003-3804}}</ref> [[Hermann Weyl]],<ref>{{cite journal |last=Weyl |first=Hermann |date=1917 |title=Zur Gravitationstheorie |url=https://zenodo.org/record/1424330 |journal=Annalen der Physik |language=en |volume=359 |issue=18 |pages=117–145 |bibcode=1917AnP...359..117W |doi=10.1002/andp.19173591804 |issn=0003-3804}}</ref> [[Gunnar Nordström]]<ref>{{cite journal |last=Nordström |first=G. |date=1918 |title=On the Energy of the Gravitational Field in Einstein's Theory |journal=Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings |volume=20 |issue=2 |pages=1238–1245 |bibcode=1918KNAB...20.1238N}}</ref> and [[George Barker Jeffery]]<ref>{{cite journal |last=Jeffery |first=G. B. |date=1921 |title=The field of an electron on Einstein's theory of gravitation |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |language=en |volume=99 |issue=697 |pages=123–134 |bibcode=1921RSPSA..99..123J |doi=10.1098/rspa.1921.0028 |issn=0950-1207 |doi-access=free}}</ref> independently.<ref>{{cite web |last=Siegel |first=Ethan |date=2021-10-13 |title=Surprise: the Big Bang isn't the beginning of the universe anymore |url=https://bigthink.com/starts-with-a-bang/big-bang-beginning-universe/ |access-date=2024-09-03 |website=Big Think |language=en-US}}</ref>

== Metric ==
In [[spherical coordinates]] {{tmath|1= (t, r, \theta, \varphi) }}, the Reissner–Nordström metric (i.e. the [[line element]]) is
: <math>
ds^2</math> <math>= c^2\, d\tau^2</math> <math> = 
\left( 1 - \frac{r_\text{s}}{r} + \frac{r_{\rm Q}^2}{r^2} \right) c^2\, dt^2</math> <math> -\left( 1 - \frac{r_\text{s}}{r} + \frac{r_Q^2}{r^2} \right)^{-1} \, dr^2</math> <math> - ~ r^2 \, d\theta^2</math> <math> - ~ r^2\sin^2\theta \, d\varphi^2 ,</math>
where
* <math>c</math> is the [[speed of light]]
* <math>\tau</math> is the proper time
* <math>t</math> is the time coordinate (measured by a stationary clock at infinity).
* <math>r</math> is the radial coordinate
* <math>(\theta, \varphi)</math> are the spherical angles
* <math>r_\text{s}</math> is the [[Schwarzschild radius]] of the body given by <math>\textstyle r_\text{s} = \frac{2GM}{c^2}</math>
* <math>r_Q</math> is a characteristic length scale given by <math>\textstyle r_Q^2 = \frac{Q^2 G}{4\pi\varepsilon_0 c^4}</math>
* <math>\varepsilon_0</math> is the [[electric constant]].

The total mass of the central body and its irreducible mass are related by<ref>[[Thibault Damour]]: [http://lapth.cnrs.fr/pg-nomin/chardon/IRAP_PhD/BlackHolesNice2012.pdf#page=11 Black Holes: Energetics and Thermodynamics], S. 11 ff.</ref><ref name="quadir">{{cite journal |last=Qadir |first=Asghar |date=December 1983 |title=Reissner-Nordstrom repulsion |journal=Physics Letters A |language=en |volume=99 |issue=9 |pages=419–420 |bibcode=1983PhLA...99..419Q |doi=10.1016/0375-9601(83)90946-5}}</ref>
: <math>M_{\rm irr}= \frac{c^2}{G} \sqrt{\frac{r_+^2}{2}} \ \to \ M=\frac{Q ^2}{16\pi\varepsilon_0 G M_{\rm irr}} + M_{\rm irr}.</math>

The difference between <math>M</math> and <math>M_{\rm irr}</math> is due to the [[mass–energy equivalence|equivalence of mass and energy]], which makes the [[Electric field#Energy in the electric field|electric field energy]] also contribute to the total mass.

In the limit that the charge <math>Q</math> (or equivalently, the length scale {{tmath|1= 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 <math>r_\text{s}/r</math> goes to zero. In the limit that both <math>r_Q/r</math> and <math>r_\text{s}/r</math> go to zero, the metric becomes the [[Minkowski metric]] for [[special relativity]].

In practice, the ratio <math>r_\text{s}/r</math> is often extremely small.  For example, the Schwarzschild radius of the [[Earth]] is roughly {{val|9|ul=mm}} (3/8&nbsp;[[inch]]), whereas a [[satellite]] in a [[geosynchronous orbit]] has an orbital radius <math>r</math> that is roughly four billion times larger, at {{val|42,164|ul=km}} ({{val|26,200|u=[[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=Subrahmanyan |author-link=Subrahmanyan Chandrasekhar |url=https://books.google.com/books?id=LBOVcrzFfhsC |title=The mathematical theory of black holes |date=2009 |publisher=Clarendon Press |isbn=978-0-19-850370-5 |edition=Reprinted |series=Oxford classic texts in the physical sciences |location=Oxford |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 <math>g_{rr}</math> diverges; that is, where
<math display=block> 1 - \frac{r_{\rm s}}{r} + \frac{r_{\rm Q}^2}{r^2} = -\frac{1}{g_{rr}} = 0.</math>

This equation has two solutions:
<math display=block>r_\pm = \frac{1}{2}\left(r_{\rm s} \pm \sqrt{r_{\rm s}^2 - 4r_{\rm 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> cannot exist in nature because if the charge is greater than the mass there can be no physical event horizon (the term under the square root becomes negative).<ref name="hamilton">Andrew Hamilton: ''[https://jila.colorado.edu/~ajsh/bh/rn.html The Reissner Nordström Geometry]'' (Casa Colorado)</ref> Objects with a charge greater than their mass can exist in nature, but they can not collapse down to a black hole, and if they could, they would display a [[naked singularity]].<ref>{{cite journal |last=Carter |first=Brandon |author-link=Brandon Carter |date=25 October 1968 |title=Global Structure of the Kerr Family of Gravitational Fields |journal=Physical Review |language=en |volume=174 |issue=5 |pages=1559–1571 |doi=10.1103/PhysRev.174.1559 |issn=0031-899X}}</ref> Theories with [[supersymmetry]] usually guarantee that such "superextremal" black holes cannot exist.

The [[electromagnetic potential]] is
<math display=block>A_\alpha = (Q/r, 0, 0, 0).</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''&nbsp;cos&nbsp;''θ''&nbsp;''dφ'' in the electromagnetic potential.{{clarify|date=January 2013}}

== Gravitational time dilation ==

The [[gravitational time dilation]] in the vicinity of the central body is given by
<math display=block>\gamma = \sqrt{|g^{t t}|} = \sqrt{\frac{r^2}{Q^2+(r-2 M) r}} ,</math>
which relates to the local radial escape velocity of a neutral particle
<math display=block>v_{\rm esc}=\frac{\sqrt{\gamma^2-1}}{\gamma}. </math>

== Christoffel symbols ==

The [[Christoffel symbols]]
<math display=block> \Gamma^i{}_{j k} = \sum_{s=0}^3 \ \frac{g^{is}}{2} \left(\frac{\partial g_{js}}{\partial x^k}+\frac{\partial g_{sk}}{\partial x^j}-\frac{\partial g_{jk}}{\partial x^s}\right)</math>
with the indices
<math display=block>\{ 0, \ 1, \ 2, \ 3 \} \to \{ t, \ r, \ \theta, \ \varphi \}</math>
give the nonvanishing expressions
<math display=block>
\begin{align}
\Gamma^t{}_{t r} & = \frac{M r-Q^2}{r ( Q^2 + r^2 - 2 M r ) } \\[6pt]
\Gamma^r{}_{t t} & = \frac{(M r-Q^2) \left(r^2-2Mr+Q^2\right)}{r^5} \\[6pt]
\Gamma^r{}_{r r} & = \frac{Q^2-M r}{r (Q^2 -2 M r+r^2)} \\[6pt]
\Gamma^r{}_{\theta \theta} & = -\frac{r^2-2Mr+Q^2}{r} \\[6pt]
\Gamma^r{}_{\varphi \varphi} & = -\frac{\sin ^2 \theta \left(r^2-2Mr+Q^2\right)}{r} \\[6pt]
\Gamma^\theta{}_{\theta r} & = \frac{1}{r} \\[6pt]
\Gamma^\theta{}_{\varphi \varphi} & = - \sin \theta \cos \theta \\[6pt]
\Gamma^\varphi{}_{\varphi r} & = \frac{1}{r} \\[6pt]
\Gamma^\varphi{}_{\varphi \theta} & = \cot \theta
\end{align}
</math>

Given the Christoffel symbols, one can compute the geodesics of a test-particle.<ref>[[Leonard Susskind]]: [http://theoreticalminimum.com/courses/general-relativity/2012/fall/lecture-4 ''The Theoretical Minimum: Geodesics and Gravity''], (''General Relativity Lecture 4'', timestamp: [https://www.youtube.com/watch?v=YdnLcYNdTzE&t=34m18s&index=4&list=PL9peWTxCcrBLLE89-Pwab3Qc1uoaKEH0e 34m18s])</ref><ref>{{cite journal |last1=Hackmann |first1=Eva |last2=Xu |first2=Hongxiao |date=2013 |title=Charged particle motion in Kerr-Newmann space-times |journal=Physical Review D |language=en |volume=87 |issue=12 |page=124030 |doi=10.1103/PhysRevD.87.124030 |arxiv=1304.2142 |issn=1550-7998}}</ref>

== Tetrad form ==
Instead of working in the holonomic basis, one can perform efficient calculations with a [[Tetrad formalism|tetrad]].<ref>{{cite book |last=Wald |first=Robert M. |title=General relativity |date=2009 |publisher=Univ. of Chicago Press |isbn=978-0-226-87033-5 |edition=Repr. |location=Chicago}}</ref> Let <math> {\bf e}_I = e_{\mu I} </math> be a set of [[one-form]]s with internal [[Minkowski space|Minkowski index]] {{tmath|1= I \in\{0,1,2,3\} }}, such that {{tmath|1= \eta^{IJ} e_{\mu I} e_{\nu J} = g_{\mu\nu} }}. The Reissner metric can be described by the tetrad
: <math> {\bf e}_0 = G^{1/2} \, dt </math>
: <math> {\bf e}_1 = G^{-1/2} \, dr </math>
: <math> {\bf e}_2 = r \, d\theta </math>
: <math> {\bf e}_3 = r \sin \theta \, d\varphi </math>
where {{tmath|1= G(r) = 1 - r_sr^{-1} + r_Q^2r^{-2} }}. The [[parallel transport]] of the tetrad is captured by the [[connection form|connection one-forms]] {{tmath|1= \boldsymbol \omega_{IJ} = - \boldsymbol \omega_{JI} = \omega_{\mu IJ} = e_{I}^\nu \nabla_\mu e_{J\nu} }}. These have only 24 independent components compared to the 40 components of {{tmath|1= \Gamma^\lambda{}_{\mu\nu} }}. The connections can be solved for by inspection from Cartan's equation {{tmath|1= d{\bf e}_I = {\bf e}^J \wedge \boldsymbol \omega_{IJ} }}, where the left hand side is the [[exterior derivative]] of the tetrad, and the right hand side is a [[wedge product]].
: <math> \boldsymbol \omega_{10} = \frac12 \partial_r G \, dt</math>
: <math> \boldsymbol \omega_{20} = \boldsymbol \omega_{30} = 0</math>
: <math> \boldsymbol \omega_{21} = - G^{1/2} \, d\theta</math>
: <math> \boldsymbol \omega_{31} = - \sin \theta G^{1/2} d \varphi</math>
: <math> \boldsymbol \omega_{32} = - \cos \theta \, d\varphi</math>

The [[Riemann curvature tensor|Riemann tensor]] <math> {\bf R}_{IJ} = R_{\mu\nu IJ} </math> can be constructed as a collection of two-forms by the second Cartan equation <math>{\bf R}_{IJ} = d \boldsymbol \omega_{IJ} + \boldsymbol \omega_{IK} \wedge \boldsymbol \omega^K{}_J,</math> which again makes use of the exterior derivative and wedge product. This approach is significantly faster than the traditional computation with {{tmath|1= \Gamma^\lambda{}_{\mu\nu} }}; note that there are only four nonzero <math>\boldsymbol \omega_{IJ} </math> compared with nine nonzero components of {{tmath|1= \Gamma^\lambda{}_{\mu\nu} }}.

== Equations of motion ==
<ref name="Nordebo">{{cite web |last1=Nordebo |first1=Jonatan |title=The Reissner-Nordström metric |url=https://www.diva-portal.org/smash/get/diva2:912393/FULLTEXT01.pdf |website=diva-portal |access-date=8 April 2021}}</ref>

Because of the [[Spherically symmetric spacetime|spherical symmetry]] of the metric, the coordinate system can always be aligned in a way that the motion of a test-particle is confined to a plane, so for brevity and without restriction of generality we use ''θ'' instead of ''φ''. In dimensionless natural units of ''G''&nbsp;=&nbsp;''M''&nbsp;=&nbsp;''c''&nbsp;=&nbsp;''K''&nbsp;=&nbsp;1 the motion of an electrically charged particle with the charge ''q'' is given by
<math display=block> \ddot x^i = - \sum_{j=0}^3 \ \sum_{k=0}^3 \ \Gamma^i_{j k} \ {\dot x^j} \ {\dot x^k} + q \ {F^{i k}} \ {\dot x_k} </math>
which yields
<math display=block>\ddot t =  \frac{ \ 2 (Q^2-Mr) }{r(r^2 -2Mr +Q ^2)}\dot{r}\dot{t}+\frac{qQ}{(r^2-2mr+Q^2)} \ \dot{r}</math>
<math display=block>\ddot r = \frac{(r^2-2Mr+Q^2)(Q^2-Mr) \ \dot{t}^2}{r^5}+\frac{(Mr-Q^2) \dot{r}^2}{r(r^2-2Mr+Q^2)}+\frac{(r^2-2Mr+Q^2) \ \dot{\theta}^2}{r} + \frac{qQ(r^2-2mr+Q^2)}{r^4} \ \dot{t}</math>
<math display=block>\ddot \theta = -\frac{2 \ \dot\theta \ \dot{r}}{r} .</math>

All total derivatives are with respect to proper time {{tmath|1= \dot a=\frac{da}{d\tau} }}.

Constants of the motion are provided by solutions <math>S (t,\dot t,r,\dot r,\theta,\dot\theta,\varphi,\dot\varphi) </math> to the partial differential equation<ref>{{cite journal |last=Smith |first=B. R. |date=December 2009 |title=First-order partial differential equations in classical dynamics |journal=American Journal of Physics |language=en |volume=77 |issue=12 |pages=1147–1153 |bibcode=2009AmJPh..77.1147S |doi=10.1119/1.3223358 |issn=0002-9505}}</ref>
<math display=block> 0=\dot t\dfrac{\partial S}{\partial t}+\dot r\frac{\partial S}{\partial r}+\dot\theta\frac{\partial S}{\partial\theta}+\ddot t \frac{\partial S}{\partial \dot t} +\ddot r \frac{\partial S}{\partial \dot r} + \ddot\theta \frac{\partial S}{\partial \dot\theta}  </math>
after substitution of the second derivatives given above. The metric itself is a solution when written as a differential equation
<math display=block>
 S_1=1 = 
\left( 1 - \frac{r_s}{r} + \frac{r_{\rm Q}^2}{r^2} \right) c^2\, {\dot t}^2 -\left( 1 - \frac{r_s}{r} + \frac{r_Q^2}{r^2} \right)^{-1} \, {\dot r}^2 - r^2 \, {\dot \theta}^2 .</math>

The separable equation
<math display=block> \frac{\partial S}{\partial r}-\frac{2}{r}\dot\theta\frac{\partial S}{\partial \dot\theta}=0  </math>
immediately yields the constant relativistic specific angular momentum
<math display=block> S_2=L=r^2\dot\theta;  </math>
a third constant obtained from
<math display=block>
\frac{\partial S}{\partial r}-\frac{2(Mr-Q^2)}{r(r^2-2Mr+Q^2)}\dot t\frac{\partial S}{\partial \dot t}=0
</math>
is the specific energy (energy per unit rest mass)<ref>{{cite book |last1=Misner |first1=Charles W. |title=Gravitation |last2=Thorne |first2=Kip S. |last3=Wheeler |first3=John Archibald |last4=Kaiser |first4=David |date=2017 |publisher=Princeton University Press |isbn=978-0-691-17779-3 |location=Princeton, N.J |pages=656–658 |oclc=on1006427790 |display-authors=etal}}</ref>
<math display=block>S_3=E=\frac{\dot t(r^2-2Mr+Q^2)}{r^2} + \frac{qQ}{r} .</math>

Substituting <math>S_2</math> and <math>S_3</math> into <math>S_1</math> yields the radial equation 
<math display=block>c\int d\tau =\int \frac{r^2\,dr}{ \sqrt{r^4(E-1)+2Mr^3-(Q^2+L^2)r^2+2ML^2r-Q^2L^2 } } .</math>

Multiplying under the integral sign  by <math>S_2</math> yields the orbital equation
<math display=block>c\int Lr^2\,d\theta =\int \frac{L\,dr}{ \sqrt{r^4(E-1)+2Mr^3-(Q^2+L^2)r^2+2ML^2r-Q^2L^2 }  }.
</math>

The total [[time dilation]] between the test-particle and an observer at infinity is
<math display=block>\gamma= \frac{q \  Q  \ r^3 + E \ r^4}{r^2 \ (r^2-2 r+Q^2)} .</math>

The first derivatives <math>\dot x^i</math> and the [[Covariance and contravariance of vectors|contravariant]] components of the local 3-velocity <math>v^i</math> are related by
<math display=block>\dot x^i = \frac{v^i}{\sqrt{(1-v^2) \ |g_{i i}|}},</math>
which gives the initial conditions
<math display=block>\dot r = \frac{v_\parallel \sqrt{r^2-2M+Q^2}}{r \sqrt{(1-v^2)}}</math>
<math display=block>\dot \theta = \frac{v_\perp}{r \sqrt{(1-v^2)}} .</math>

The [[specific orbital energy]]
<math display=block>E=\frac{\sqrt{Q^2-2rM+r^2}}{r \sqrt{1-v^2}}+\frac{qQ}{r}</math>
and the [[specific relative angular momentum]]
<math display=block>L=\frac{v_\perp \ r}{\sqrt{1-v^2}}</math>
of the test-particle are conserved quantities of motion. <math>v_{\parallel}</math> and <math>v_{\perp}</math> are the radial and transverse components of the local velocity-vector. The local velocity is therefore
<math display=block>v = \sqrt{v_\perp^2+v_\parallel^2} = \sqrt{\frac{(E^2-1)r^2-Q^2-r^2+2rM}{E^2 r^2}}.</math>

== Alternative formulation of metric ==
The metric can be expressed in [[Kerr–Schild perturbations|Kerr–Schild form]] like this:
<math display=block>
\begin{align}
g_{\mu \nu} & = \eta_{\mu \nu} + fk_\mu k_\nu \\[5pt]
f & = \frac{G}{r^2}\left[2Mr - Q^2 \right] \\[5pt]
\mathbf{k} & = ( k_x ,k_y ,k_z ) = \left( \frac{x}{r} , \frac{y}{r}, \frac{z}{r} \right) \\[5pt]
k_0 & = 1.
\end{align}
</math>

Notice that '''k''' is a [[unit vector]].  Here ''M'' is the constant mass of the object, ''Q'' is the constant charge of the object, and ''η'' is the [[Minkowski space#Standard basis|Minkowski tensor]].

== See also ==
* [[Black hole electron]]

== Notes ==
{{reflist}}

== References ==
* {{cite book |last1=Adler |first1=R. |url=https://archive.org/details/introductiontoge0000adle |title=Introduction to General Relativity |last2=Bazin |first2=M. |last3=Schiffer |first3=M. |date=1965 |publisher=McGraw-Hill Book Company |isbn=978-0-07-000420-7 |location=New York |pages=395–401}}<!--{{LCCN|64|0|16476}}--><!-- Note: 2nd edition, 1975: 978-0-07-000423-8 -->
* {{cite book |last=Wald |first=Robert M. |author-link=Robert Wald |url=http://press.uchicago.edu/ucp/books/book/chicago/G/bo5952261.html |title=General Relativity |date=1984 |publisher=The University of Chicago Press |isbn=978-0-226-87032-8 |location=Chicago |pages=158, 312–324}}

==External links==
* [https://web.archive.org/web/20070707053358/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}}

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[[Category:Exact solutions in general relativity]]
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[[Category:Metric tensors]]