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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 mass${\displaystyle M}$. 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 with${\displaystyle r_Q\ll r_S}$ 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${\displaystyle 2r_Q=r_S}$, which corresponds to anextremal black hole. Black holes with${\displaystyle 2r_Q>r_S}$ have no event horizon (the term under the square root becomes negative) and would display anaked singularity.^[9] According to the weakcosmic censorship hypothesis, naked singularity cannot exist in nature.^[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 charge${\displaystyle P}$ is obtained by replacing${\displaystyle Q^2}$ by${\displaystyle Q^2+P^2}$ in the metric and including the term${\displaystyle P\cos \theta d\phi }$ 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 non-vanishing 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. For brevity andwithout loss of generality one may use${\displaystyle \theta }$ instead of${\displaystyle \phi }$. In dimensionless natural units of${\displaystyle G=M=c=K=1}$, the motion of an electrically charged particle with the charge${\displaystyle q}$ 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 as follows:

Here${\displaystyle \mathbf{k}}$ is aunit vector,${\displaystyle M}$ is the constant mass of the object,${\displaystyle Q}$ is the constant charge of the object, and${\displaystyle \eta }$ 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

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