This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(January 2013) (Learn how and when to remove this message)

General relativity
${\displaystyle G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu }}$
Introduction History Timeline Tests Mathematical formulation
Fundamental concepts Equivalence principle Special relativity World line Pseudo-Riemannian manifold
Phenomena Kepler problem Gravitational lensing Gravitational redshift Gravitational time dilation Gravitational waves Frame-dragging Geodetic effect Event horizon Singularity Black hole Spacetime Spacetime diagrams Minkowski spacetime Einstein–Rosen bridge
Equations Formalisms Equations Linearized gravity Einstein field equations Friedmann Geodesics Mathisson–Papapetrou–Dixon Hamilton–Jacobi–Einstein Raychaudhuri Formalisms ADM NP BSSN Post-Newtonian Advanced theory Kaluza–Klein theory Quantum gravity
Solutions Schwarzschild (interior) Reissner–Nordström Einstein–Rosen waves Wormhole Gödel Kerr Kerr–Newman Kerr–Newman–de Sitter Kasner Kantowski-Sachs Lemaître–Tolman Wahlquist Taub–NUT Milne Robertson–Walker Oppenheimer–Snyder pp-wave van Stockum dust Hartle–Thorne Vaidya Peres De Sitter-Schwarzschild McVittie Weyl Kerr-Newman-de-Sitter
Scientists Einstein Lorentz Hilbert Poincaré Schwarzschild de Sitter Reissner Nordström Weyl Eddington Friedmann Milne Zwicky Lemaître Oppenheimer Gödel Wheeler Robertson Bardeen Walker Kerr Chandrasekhar Ehlers Penrose Hawking Raychaudhuri Taylor Hulse van Stockum Taub Newman Yau Thorne others
Physics portal  Category
v t e

Inphysics andastronomy, theReissner–Nordström metric is astatic solution to theEinstein–Maxwell field equations that corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of massM. The analogous solution for a charged, rotating body is given by theKerr–Newman metric.

The metric was discovered between 1916 and 1921 byHans Reissner,^[1]Hermann Weyl,^[2]Gunnar Nordström^[3] andGeorge Barker Jeffery^[4] independently.^[5]

Inspherical coordinates⁠${\displaystyle (t,r,\theta ,\phi )}$⁠, the Reissner–Nordström metric (i.e. theline element) is

${\displaystyle d{s}^2}$${\displaystyle =c^{2}d{\tau }^2}$${\displaystyle =\left(1-\frac{r_{\text{s}}}{r}+\frac{r_{\mathrm{Q}}^2}{r^2}\right)c^{2}d{t}^2}$${\displaystyle -{\left(1-\frac{r_{\text{s}}}{r}+\frac{r_Q^2}{r^2}\right)}^{-1}d{r}^2}$${\displaystyle -r^{2}d{\theta }^2}$${\displaystyle -r^2{\sin }^2\theta d{\phi }^2,}$

where

• ${\displaystyle c}$ is thespeed of light
• ${\displaystyle \tau }$ is the proper time
• ${\displaystyle t}$ is the time coordinate (measured by a stationary clock at infinity).
• ${\displaystyle r}$ is the radial coordinate
• ${\displaystyle (\theta ,\phi )}$ are the spherical angles
• ${\displaystyle r_{\text{s}}}$ is theSchwarzschild radius of the body given by${\displaystyle r_{\text{s}}=\frac{2GM}{c^2}}$
• ${\displaystyle r_Q}$ is a characteristic length scale given by${\displaystyle r_Q^2=\frac{Q^{2}G}{4\pi {\epsilon }_0{c}^4}}$
• ${\displaystyle {\epsilon }_0}$ is theelectric constant.

The total mass of the central body and its irreducible mass are related by^[6][7]

${\displaystyle M_{\mathrm{i}\mathrm{r}\mathrm{r}}=\frac{c^2}{G}\sqrt{\frac{r_{+}^2}{2}}\to M=\frac{Q^2}{16\pi {\epsilon }_{0}G{M}_{\mathrm{i}\mathrm{r}\mathrm{r}}}+M_{\mathrm{i}\mathrm{r}\mathrm{r}}.}$

The difference between${\displaystyle M}$ and${\displaystyle M_{\mathrm{i}\mathrm{r}\mathrm{r}}}$ is due to theequivalence of mass and energy, which makes theelectric field energy also contribute to the total mass.

In the limit that the charge${\displaystyle Q}$ (or equivalently, the length scale⁠${\displaystyle r_Q}$⁠) goes to zero, one recovers theSchwarzschild metric. The classical Newtonian theory of gravity may then be recovered in the limit as the ratio${\displaystyle r_{\text{s}}/r}$ goes to zero. In the limit that both${\displaystyle r_Q/r}$ and${\displaystyle r_{\text{s}}/r}$ go to zero, the metric becomes theMinkowski metric forspecial relativity.

In practice, the ratio${\displaystyle r_{\text{s}}/r}$ is often extremely small. For example, the Schwarzschild radius of theEarth is roughly9 mm. 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 toblack holes and other ultra-dense objects such asneutron stars.

Although charged black holes withr_Q ≪r_s are similar to theSchwarzschild black hole, they have two horizons: theevent horizon and an internalCauchy horizon.^[8] As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component${\displaystyle g_{rr}}$ diverges; that is, where$${\displaystyle 1-\frac{r_{\mathrm{s}}}{r}+\frac{r_{\mathrm{Q}}^2}{r^2}=-\frac{1}{g_{rr}}=0.}$$

This equation has two solutions:$${\displaystyle r_{\pm }=\frac{1}{2}\left(r_{\mathrm{s}}\pm \sqrt{r_{\mathrm{s}}^2-4r_{\mathrm{Q}}^2}\right).}$$

These concentricevent horizons becomedegenerate for 2r_Q =r_s, which corresponds to anextremal 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).^[9] 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 anaked singularity.^[10] Theories withsupersymmetry usually guarantee that such "superextremal" black holes cannot exist.

Theelectromagnetic potential is$${\displaystyle A_{\alpha }=(Q/r,0,0,0).}$$

If magnetic monopoles are included in the theory, then a generalization to include magnetic chargeP is obtained by replacingQ² byQ² +P² in the metric and including the termP cos θ dφ in the electromagnetic potential.^{[clarification needed]}

Thegravitational time dilation in the vicinity of the central body is given by$${\displaystyle \gamma =\sqrt{|g^{tt}|}=\sqrt{\frac{r^2}{Q^2+(r-2M)r}},}$$which relates to the local radial escape velocity of a neutral particle$${\displaystyle v_{\mathrm{e}\mathrm{s}\mathrm{c}}=\frac{\sqrt{{\gamma }^2-1}}{\gamma }.}$$

TheChristoffel symbols$${\displaystyle {\mathrm{\Gamma }}^i{}_{jk}=\sum _{s=0}^3\frac{g^{is}}{2}\left(\frac{\mathrm{\partial }{g}_{js}}{\mathrm{\partial }{x}^k}+\frac{\mathrm{\partial }{g}_{sk}}{\mathrm{\partial }{x}^j}-\frac{\mathrm{\partial }{g}_{jk}}{\mathrm{\partial }{x}^s}\right)}$$with the indices$${\displaystyle \{0,1,2,3\}\to \{t,r,\theta ,\phi \}}$$give the nonvanishing expressions

Given the Christoffel symbols, one can compute the geodesics of a test-particle.^[11][12]

Instead of working in the holonomic basis, one can perform efficient calculations with atetrad.^[13] Let${\displaystyle {\mathbf{e}}_I=e_{\mu I}}$ be a set ofone-forms with internalMinkowski index⁠${\displaystyle I\in \{0,1,2,3\}}$⁠, such that⁠${\displaystyle {\eta }^{IJ}{e}_{\mu I}{e}_{\nu J}=g_{\mu \nu }}$⁠. The Reissner metric can be described by the tetrad

${\displaystyle {\mathbf{e}}_0=G^{1/2}dt}$

${\displaystyle {\mathbf{e}}_1=G^{-1/2}dr}$

${\displaystyle {\mathbf{e}}_2=rd\theta }$

${\displaystyle {\mathbf{e}}_3=r\sin \theta d\phi }$

where⁠${\displaystyle G(r)=1-r_s{r}^{-1}+r_Q^2{r}^{-2}}$⁠. Theparallel transport of the tetrad is captured by theconnection one-forms⁠${\displaystyle {\mathit{\omega }}_{IJ}=-{\mathit{\omega }}_{JI}={\omega }_{\mu IJ}=e_I^{\nu }{\mathrm{\nabla }}_{\mu }{e}_{J\nu }}$⁠. These have only 24 independent components compared to the 40 components of⁠${\displaystyle {\mathrm{\Gamma }}^{\lambda }{}_{\mu \nu }}$⁠. The connections can be solved for by inspection from Cartan's equation⁠${\displaystyle d{\mathbf{e}}_I={\mathbf{e}}^J\wedge {\mathit{\omega }}_{IJ}}$⁠, where the left hand side is theexterior derivative of the tetrad, and the right hand side is awedge product.

${\displaystyle {\mathit{\omega }}_{10}=\frac{1}{2}{\mathrm{\partial }}_{r}Gdt}$

${\displaystyle {\mathit{\omega }}_{20}={\mathit{\omega }}_{30}=0}$

${\displaystyle {\mathit{\omega }}_{21}=-G^{1/2}d\theta }$

${\displaystyle {\mathit{\omega }}_{31}=-\sin \theta G^{1/2}d\phi }$

${\displaystyle {\mathit{\omega }}_{32}=-\cos \theta d\phi }$

TheRiemann tensor${\displaystyle {\mathbf{R}}_{IJ}=R_{\mu \nu IJ}}$ can be constructed as a collection of two-forms by the second Cartan equation${\displaystyle {\mathbf{R}}_{IJ}=d{\mathit{\omega }}_{IJ}+{\mathit{\omega }}_{IK}\wedge {\mathit{\omega }}^K{}_J,}$ which again makes use of the exterior derivative and wedge product. This approach is significantly faster than the traditional computation with⁠${\displaystyle {\mathrm{\Gamma }}^{\lambda }{}_{\mu \nu }}$⁠; note that there are only four nonzero${\displaystyle {\mathit{\omega }}_{IJ}}$ compared with nine nonzero components of⁠${\displaystyle {\mathrm{\Gamma }}^{\lambda }{}_{\mu \nu }}$⁠.

^[14]

Because of thespherical 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 useθ instead ofφ. In dimensionless natural units ofG = M = c = K = 1 the motion of an electrically charged particle with the chargeq is given by$${\displaystyle {\ddot{x}}^i=-\sum _{j=0}^3\sum _{k=0}^3{\mathrm{\Gamma }}_{jk}^i{\dot{x}}^j{\dot{x}}^k+q{F}^{ik}{\dot{x}}_k}$$which yields$${\displaystyle \ddot{t}=\frac{2(Q^2-Mr)}{r(r^2-2Mr+Q^2)}\dot{r}\dot{t}+\frac{qQ}{(r^2-2mr+Q^2)}\dot{r}}$$$${\displaystyle \ddot{r}=\frac{(r^2-2Mr+Q^2)(Q^2-Mr){\dot{t}}^2}{r^5}+\frac{(Mr-Q^2){\dot{r}}^2}{r(r^2-2Mr+Q^2)}+\frac{(r^2-2Mr+Q^2){\dot{\theta }}^2}{r}+\frac{qQ(r^2-2mr+Q^2)}{r^4}\dot{t}}$$$${\displaystyle \ddot{\theta }=-\frac{2\dot{\theta }\dot{r}}{r}.}$$

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

Constants of the motion are provided by solutions${\displaystyle S(t,\dot{t},r,\dot{r},\theta ,\dot{\theta },\phi ,\dot{\phi })}$ to the partial differential equation^[15]$${\displaystyle 0=\dot{t}{\displaystyle \frac{\mathrm{\partial }S}{\mathrm{\partial }t}}+\dot{r}\frac{\mathrm{\partial }S}{\mathrm{\partial }r}+\dot{\theta }\frac{\mathrm{\partial }S}{\mathrm{\partial }\theta }+\ddot{t}\frac{\mathrm{\partial }S}{\mathrm{\partial }\dot{t}}+\ddot{r}\frac{\mathrm{\partial }S}{\mathrm{\partial }\dot{r}}+\ddot{\theta }\frac{\mathrm{\partial }S}{\mathrm{\partial }\dot{\theta }}}$$after substitution of the second derivatives given above. The metric itself is a solution when written as a differential equation$${\displaystyle S_1=1=\left(1-\frac{r_s}{r}+\frac{r_{\mathrm{Q}}^2}{r^2}\right)c^2{\dot{t}}^2-{\left(1-\frac{r_s}{r}+\frac{r_Q^2}{r^2}\right)}^{-1}{\dot{r}}^2-r^2{\dot{\theta }}^2.}$$

The separable equation$${\displaystyle \frac{\mathrm{\partial }S}{\mathrm{\partial }r}-\frac{2}{r}\dot{\theta }\frac{\mathrm{\partial }S}{\mathrm{\partial }\dot{\theta }}=0}$$immediately yields the constant relativistic specific angular momentum$${\displaystyle S_2=L=r^2\dot{\theta };}$$a third constant obtained from$${\displaystyle \frac{\mathrm{\partial }S}{\mathrm{\partial }r}-\frac{2(Mr-Q^2)}{r(r^2-2Mr+Q^2)}\dot{t}\frac{\mathrm{\partial }S}{\mathrm{\partial }\dot{t}}=0}$$is the specific energy (energy per unit rest mass)^[16]$${\displaystyle S_3=E=\frac{\dot{t}(r^2-2Mr+Q^2)}{r^2}+\frac{qQ}{r}.}$$

Substituting${\displaystyle S_2}$ and${\displaystyle S_3}$ into${\displaystyle S_1}$ yields the radial equation$${\displaystyle c\int d\tau =\int \frac{r^{2}dr}{\sqrt{r^4(E-1)+2M{r}^3-(Q^2+L^2)r^2+2M{L}^{2}r-Q^2{L}^2}}.}$$

Multiplying under the integral sign by${\displaystyle S_2}$ yields the orbital equation$${\displaystyle c\int L{r}^{2}d\theta =\int \frac{Ldr}{\sqrt{r^4(E-1)+2M{r}^3-(Q^2+L^2)r^2+2M{L}^{2}r-Q^2{L}^2}}.}$$

The totaltime dilation between the test-particle and an observer at infinity is$${\displaystyle \gamma =\frac{qQ{r}^3+E{r}^4}{r^2(r^2-2r+Q^2)}.}$$

The first derivatives${\displaystyle {\dot{x}}^i}$ and thecontravariant components of the local 3-velocity${\displaystyle v^i}$ are related by$${\displaystyle {\dot{x}}^i=\frac{v^i}{\sqrt{(1-v^2)|g_{ii}|}},}$$which gives the initial conditions$${\displaystyle \dot{r}=\frac{v_{\parallel }\sqrt{r^2-2M+Q^2}}{r\sqrt{(1-v^2)}}}$$$${\displaystyle \dot{\theta }=\frac{v_{\perp }}{r\sqrt{(1-v^2)}}.}$$

Thespecific orbital energy$${\displaystyle E=\frac{\sqrt{Q^2-2rM+r^2}}{r\sqrt{1-v^2}}+\frac{qQ}{r}}$$and thespecific relative angular momentum$${\displaystyle L=\frac{v_{\perp }r}{\sqrt{1-v^2}}}$$of the test-particle are conserved quantities of motion.${\displaystyle v_{\parallel }}$ and${\displaystyle v_{\perp }}$ are the radial and transverse components of the local velocity-vector. The local velocity is therefore$${\displaystyle v=\sqrt{v_{\perp }^2+v_{\parallel }^2}=\sqrt{\frac{(E^2-1)r^2-Q^2-r^2+2rM}{E^2{r}^2}}.}$$

The metric can be expressed inKerr–Schild form like this:

Notice thatk is aunit vector. HereM is the constant mass of the object,Q is the constant charge of the object, andη is theMinkowski tensor.

• Black hole electron

• 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.

• Spacetime diagrams includingFinkelstein diagram andPenrose diagram, by Andrew J. S. Hamilton
• "Particle Moving Around Two Extreme Black Holes" by Enrique Zeleny,The Wolfram Demonstrations Project.

• Exact solutions in general relativity
• Black holes
• Metric tensors
• Gravitational singularities

• Articles with short description
• Short description matches Wikidata
• Articles lacking in-text citations from January 2013
• All articles lacking in-text citations
• Pages using sidebar with the child parameter
• Wikipedia articles needing clarification from January 2013