Reissner–Nordström metric

In spherical coordinates {{tmath|1= (t, r, \theta, \varphi) }}, the Reissner–Nordström metric (i.e. the line element) is

: $$

ds^2 $=\; c^2\backslash ,\; d\backslash tau^2$ $=$

\left( 1 - \frac{r_\text{s}}{r} + \frac{r_{\rm Q}^2}{r^2} \right) c^2\, dt^2 $-\backslash left(\; 1\; -\; \backslash frac\{r\_\backslash text\{s\}\}\{r\}\; +\; \backslash frac\{r\_Q^2\}\{r^2\}\; \backslash right)^\{-1\}\; \backslash ,\; dr^2$ $-\; ~\; r^2\; \backslash ,\; d\backslash theta^2$ $-\; ~\; r^2\backslash sin^2\backslash theta\; \backslash ,\; d\backslash varphi^2\; ,$

where

• $c$ is the speed of light
• $\backslash tau$ is the proper time
• $t$ is the time coordinate (measured by a stationary clock at infinity).
• $r$ is the radial coordinate
• $(\backslash theta,\; \backslash varphi)$ are the spherical angles
• $r\_\backslash text\{s\}$ is the Schwarzschild radius of the body given by $\backslash textstyle\; r\_\backslash text\{s\}\; =\; \backslash frac\{2GM\}\{c^2\}$
• $r\_Q$ is a characteristic length scale given by $\backslash textstyle\; r\_Q^2\; =\; \backslash frac\{Q^2\; G\}\{4\backslash pi\backslash varepsilon\_0\; c^4\}$
• $\backslash varepsilon\_0$ is the electric constant.

The total mass of the central body and its irreducible mass are related byThibault Damour: [http://lapth.cnrs.fr/pg-nomin/chardon/IRAP_PhD/BlackHolesNice2012.pdf#page=11 Black Holes: Energetics and Thermodynamics], S. 11 ff.{{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}}

: $M\_\{\backslash rm\; irr\}=\; \backslash frac\{c^2\}\{G\}\; \backslash sqrt\{\backslash frac\{r\_+^2\}\{2\}\}\; \backslash \; \backslash to\; \backslash \; M=\backslash frac\{Q\; ^2\}\{16\backslash pi\backslash varepsilon\_0\; G\; M\_\{\backslash rm\; irr\}\}\; +\; M\_\{\backslash rm\; irr\}.$

The difference between $M$ and $M\_\{\backslash rm\; irr\}$ is due to the equivalence of mass and energy, which makes the electric field energy also contribute to the total mass.

In the limit that the charge $Q$ (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 $r\_\backslash text\{s\}/r$ goes to zero. In the limit that both $r\_Q/r$ and $r\_\backslash text\{s\}/r$ go to zero, the metric becomes the Minkowski metric for special relativity.

In practice, the ratio $r\_\backslash text\{s\}/r$ is often extremely small. For example, the Schwarzschild radius of the Earth is roughly {{val|9|ul=mm}} (3/8 inch), whereas a satellite in a geosynchronous orbit has an orbital radius $r$ that is roughly four billion times larger, at {{val|42,164|ul=km}} ({{val|26,200|u=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.

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.{{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.}} As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component $g\_\{rr\}$ diverges; that is, where

$$1\; -\; \backslash frac\{r\_\{\backslash rm\; s\}\}\{r\}\; +\; \backslash frac\{r\_\{\backslash rm\; Q\}^2\}\{r^2\}\; =\; -\backslash frac\{1\}\{g\_\{rr\}\}\; =\; 0.$$

This equation has two solutions:

$$r\_\backslash pm\; =\; \backslash frac\{1\}\{2\}\backslash left(r\_\{\backslash rm\; s\}\; \backslash pm\; \backslash sqrt\{r\_\{\backslash rm\; s\}^2\; -\; 4r\_\{\backslash rm\; Q\}^2\}\backslash right).$$

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 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).Andrew Hamilton: [https://jila.colorado.edu/~ajsh/bh/rn.html The Reissner Nordström Geometry] (Casa Colorado) 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.{{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}} Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.

The electromagnetic potential is

$$A\_\backslash alpha\; =\; (Q/r,\; 0,\; 0,\; 0).$$

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 P cos θ dφ in the electromagnetic potential.{{clarify|date=January 2013}}

The gravitational time dilation in the vicinity of the central body is given by

$$\backslash gamma\; =\; \backslash sqrt$$

which relates to the local radial escape velocity of a neutral particle

$$v\_\{\backslash rm\; esc\}=\backslash frac\{\backslash sqrt\{\backslash gamma^2-1\}\}\{\backslash gamma\}.$$

The Christoffel symbols

$$\backslash Gamma^i\{\}\_\{j\; k\}\; =\; \backslash sum\_\{s=0\}^3\; \backslash \; \backslash frac\{g^\{is\}\}\{2\}\; \backslash left(\backslash frac\{\backslash partial\; g\_\{js\}\}\{\backslash partial\; x^k\}+\backslash frac\{\backslash partial\; g\_\{sk\}\}\{\backslash partial\; x^j\}-\backslash frac\{\backslash partial\; g\_\{jk\}\}\{\backslash partial\; x^s\}\backslash right)$$

with the indices

$$\backslash \{\; 0,\; \backslash \; 1,\; \backslash \; 2,\; \backslash \; 3\; \backslash \}\; \backslash to\; \backslash \{\; t,\; \backslash \; r,\; \backslash \; \backslash theta,\; \backslash \; \backslash varphi\; \backslash \}$$

give the nonvanishing expressions

\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}

Given the Christoffel symbols, one can compute the geodesics of a test-particle.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]){{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}}

Instead of working in the holonomic basis, one can perform efficient calculations with a tetrad.{{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}} Let $\{\backslash bf\; e\}\_I\; =\; e\_\{\backslash mu\; I\}$ be a set of one-forms with internal 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

: $\{\backslash bf\; e\}\_0\; =\; G^\{1/2\}\; \backslash ,\; dt$

: $\{\backslash bf\; e\}\_1\; =\; G^\{-1/2\}\; \backslash ,\; dr$

: $\{\backslash bf\; e\}\_2\; =\; r\; \backslash ,\; d\backslash theta$

: $\{\backslash bf\; e\}\_3\; =\; r\; \backslash sin\; \backslash theta\; \backslash ,\; d\backslash varphi$

where {{tmath|1= G(r) = 1 - r_sr^{-1} + r_Q^2r^{-2} }}. The parallel transport of the tetrad is captured by the 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.

: $\backslash boldsymbol\; \backslash omega\_\{10\}\; =\; \backslash frac12\; \backslash partial\_r\; G\; \backslash ,\; dt$

: $\backslash boldsymbol\; \backslash omega\_\{20\}\; =\; \backslash boldsymbol\; \backslash omega\_\{30\}\; =\; 0$

: $\backslash boldsymbol\; \backslash omega\_\{21\}\; =\; -\; G^\{1/2\}\; \backslash ,\; d\backslash theta$

: $\backslash boldsymbol\; \backslash omega\_\{31\}\; =\; -\; \backslash sin\; \backslash theta\; G^\{1/2\}\; d\; \backslash varphi$

: $\backslash boldsymbol\; \backslash omega\_\{32\}\; =\; -\; \backslash cos\; \backslash theta\; \backslash ,\; d\backslash varphi$

The Riemann tensor $\{\backslash bf\; R\}\_\{IJ\}\; =\; R\_\{\backslash mu\backslash nu\; IJ\}$ can be constructed as a collection of two-forms by the second Cartan equation $\{\backslash bf\; R\}\_\{IJ\}\; =\; d\; \backslash boldsymbol\; \backslash omega\_\{IJ\}\; +\; \backslash boldsymbol\; \backslash omega\_\{IK\}\; \backslash wedge\; \backslash boldsymbol\; \backslash omega^K\{\}\_J,$ 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 $\backslash boldsymbol\; \backslash omega\_\{IJ\}$ compared with nine nonzero components of {{tmath|1= \Gamma^\lambda{}_{\mu\nu} }}.

{{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}}

Because of the 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 = M = c = K = 1 the motion of an electrically charged particle with the charge q is given by

$$\backslash ddot\; x^i\; =\; -\; \backslash sum\_\{j=0\}^3\; \backslash \; \backslash sum\_\{k=0\}^3\; \backslash \; \backslash Gamma^i\_\{j\; k\}\; \backslash \; \{\backslash dot\; x^j\}\; \backslash \; \{\backslash dot\; x^k\}\; +\; q\; \backslash \; \{F^\{i\; k\}\}\; \backslash \; \{\backslash dot\; x\_k\}$$

which yields

$$\backslash ddot\; t\; =\; \backslash frac\{\; \backslash \; 2\; (Q^2-Mr)\; \}\{r(r^2\; -2Mr\; +Q\; ^2)\}\backslash dot\{r\}\backslash dot\{t\}+\backslash frac\{qQ\}\{(r^2-2mr+Q^2)\}\; \backslash \; \backslash dot\{r\}$$

$$\backslash ddot\; r\; =\; \backslash frac\{(r^2-2Mr+Q^2)(Q^2-Mr)\; \backslash \; \backslash dot\{t\}^2\}\{r^5\}+\backslash frac\{(Mr-Q^2)\; \backslash dot\{r\}^2\}\{r(r^2-2Mr+Q^2)\}+\backslash frac\{(r^2-2Mr+Q^2)\; \backslash \; \backslash dot\{\backslash theta\}^2\}\{r\}\; +\; \backslash frac\{qQ(r^2-2mr+Q^2)\}\{r^4\}\; \backslash \; \backslash dot\{t\}$$

$$\backslash ddot\; \backslash theta\; =\; -\backslash frac\{2\; \backslash \; \backslash dot\backslash theta\; \backslash \; \backslash dot\{r\}\}\{r\}\; .$$

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 $S\; (t,\backslash dot\; t,r,\backslash dot\; r,\backslash theta,\backslash dot\backslash theta,\backslash varphi,\backslash dot\backslash varphi)$ to the partial differential equation{{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}}

$$0=\backslash dot\; t\backslash dfrac\{\backslash partial\; S\}\{\backslash partial\; t\}+\backslash dot\; r\backslash frac\{\backslash partial\; S\}\{\backslash partial\; r\}+\backslash dot\backslash theta\backslash frac\{\backslash partial\; S\}\{\backslash partial\backslash theta\}+\backslash ddot\; t\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; \backslash dot\; t\}\; +\backslash ddot\; r\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; \backslash dot\; r\}\; +\; \backslash ddot\backslash theta\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; \backslash dot\backslash theta\}$$

after substitution of the second derivatives given above. The metric itself is a solution when written as a differential equation

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 .

The separable equation

$$\backslash frac\{\backslash partial\; S\}\{\backslash partial\; r\}-\backslash frac\{2\}\{r\}\backslash dot\backslash theta\backslash frac\{\backslash partial\; S\}\{\backslash partial\; \backslash dot\backslash theta\}=0$$

immediately yields the constant relativistic specific angular momentum

$$S\_2=L=r^2\backslash dot\backslash theta;$$

a third constant obtained from

\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

is the specific energy (energy per unit rest mass){{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}}

$$S\_3=E=\backslash frac\{\backslash dot\; t(r^2-2Mr+Q^2)\}\{r^2\}\; +\; \backslash frac\{qQ\}\{r\}\; .$$

Substituting $S\_2$ and $S\_3$ into $S\_1$ yields the radial equation

$$c\backslash int\; d\backslash tau\; =\backslash int\; \backslash frac\{r^2\backslash ,dr\}\{\; \backslash sqrt\{r^4(E-1)+2Mr^3-(Q^2+L^2)r^2+2ML^2r-Q^2L^2\; \}\; \}\; .$$

Multiplying under the integral sign by $S\_2$ yields the orbital equation

$$c\backslash int\; Lr^2\backslash ,d\backslash theta\; =\backslash int\; \backslash frac\{L\backslash ,dr\}\{\; \backslash sqrt\{r^4(E-1)+2Mr^3-(Q^2+L^2)r^2+2ML^2r-Q^2L^2\; \}\; \}.$$

The total time dilation between the test-particle and an observer at infinity is

$$\backslash gamma=\; \backslash frac\{q\; \backslash \; Q\; \backslash \; r^3\; +\; E\; \backslash \; r^4\}\{r^2\; \backslash \; (r^2-2\; r+Q^2)\}\; .$$

The first derivatives $\backslash dot\; x^i$ and the contravariant components of the local 3-velocity $v^i$ are related by

$$\backslash dot\; x^i\; =\; \backslash frac\{v^i\}\{\backslash sqrt\{(1-v^2)\; \backslash \; |g\_\{i\; i\}|\}\},$$

which gives the initial conditions

$$\backslash dot\; r\; =\; \backslash frac\{v\_\backslash parallel\; \backslash sqrt\{r^2-2M+Q^2\}\}\{r\; \backslash sqrt\{(1-v^2)\}\}$$

$$\backslash dot\; \backslash theta\; =\; \backslash frac\{v\_\backslash perp\}\{r\; \backslash sqrt\{(1-v^2)\}\}\; .$$

The specific orbital energy

$$E=\backslash frac\{\backslash sqrt\{Q^2-2rM+r^2\}\}\{r\; \backslash sqrt\{1-v^2\}\}+\backslash frac\{qQ\}\{r\}$$

and the specific relative angular momentum

$$L=\backslash frac\{v\_\backslash perp\; \backslash \; r\}\{\backslash sqrt\{1-v^2\}\}$$

of the test-particle are conserved quantities of motion. $v\_\{\backslash parallel\}$ and $v\_\{\backslash perp\}$ are the radial and transverse components of the local velocity-vector. The local velocity is therefore

$$v\; =\; \backslash sqrt\{v\_\backslash perp^2+v\_\backslash parallel^2\}\; =\; \backslash sqrt\{\backslash frac\{(E^2-1)r^2-Q^2-r^2+2rM\}\{E^2\; r^2\}\}.$$

The metric can be expressed in Kerr–Schild form like this:

\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}

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 tensor.

• Black hole electron

{{reflist}}

• {{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}}
• {{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}}

• [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

Category:Black holes

Category:Metric tensors