# Reissner–Nordström metric

General relativity :
$G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }$ :
Introduction
HistoryTimelineTests
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 :
EquationsFormalisms
Equations
Linearized gravity
Einstein field equations
Friedmann
Geodesics
Mathisson–Papapetrou–Dixon
Hamilton–Jacobi–Einstein
Formalisms
ADM
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
Lemaître–Tolman
Taub–NUT
Milne
Robertson–Walker
Oppenheimer–Snyder
pp-wave
van Stockum dust
Weyl−Lewis−Papapetrou
Hartle–Thorne :
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 :
vte :
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 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, Hermann Weyl, Gunnar Nordström and George Barker Jeffery independently.

## Metric

In spherical coordinates ⁠$(t,r,\theta ,\varphi )$⁠, the Reissner–Nordström metric (i.e. the line element) is
    $ds^{2}$ $=c^{2}\,d\tau ^{2}$ $=\left(1-{\frac {r_{\text{s}}}{r}}+{\frac {r_{\rm {Q}}^{2}}{r^{2}}}\right)c^{2}\,dt^{2}$ $-\left(1-{\frac {r_{\text{s}}}{r}}+{\frac {r_{Q}^{2}}{r^{2}}}\right)^{-1}\,dr^{2}$ $-~r^{2}\,d\theta ^{2}$ $-~r^{2}\sin ^{2}\theta \,d\varphi ^{2},$
where
* $c$ is the speed of light
* $\tau$ is the proper time
* $t$ is the time coordinate (measured by a stationary clock at infinity).
* $r$ is the radial coordinate
* $(\theta ,\varphi )$ are the spherical angles
* $r_{\text{s}}$ is the Schwarzschild radius of the body given by $\textstyle r_{\text{s}}={\frac {2GM}{c^{2}}}$
* $r_{Q}$ is a characteristic length scale given by $\textstyle r_{Q}^{2}={\frac {Q^{2}G}{4\pi \varepsilon _{0}c^{4}}}$
* $\varepsilon _{0}$ is the electric constant.
The total mass of the central body and its irreducible mass are related by
    $M_{\rm {irr}}={\frac {c^{2}}{G}}{\sqrt {\frac {r_{+}^{2}}{2}}}\ \to \ M={\frac {Q^{2}}{16\pi \varepsilon _{0}GM_{\rm {irr}}}}+M_{\rm {irr}}.$
The difference between $M$ and $M_{\rm {irr}}$ 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 ⁠$r_{Q}$⁠) goes to zero, one recovers the Schwarzschild metric. The classical Newtonian theory of gravity may then be recovered in the limit as the ratio $r_{\text{s}}/r$ goes to zero. In the limit that both $r_{Q}/r$ and $r_{\text{s}}/r$ go to zero, the metric becomes the Minkowski metric for special relativity.
In practice, the ratio $r_{\text{s}}/r$ 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 an orbital radius $r$ that is roughly four billion times larger, at 42164 km (26200 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 r$_Q$ ≪ r$_s$ are similar to the Schwarzschild black hole, they have two horizons: the event horizon and an internal Cauchy horizon. As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component $g_{rr}$ diverges; that is, where $1-{\frac {r_{\rm {s}}}{r}}+{\frac {r_{\rm {Q}}^{2}}{r^{2}}}=-{\frac {1}{g_{rr}}}=0.$
This equation has two solutions: $r_{\pm }={\frac {1}{2}}\left(r_{\rm {s}}\pm {\sqrt {r_{\rm {s}}^{2}-4r_{\rm {Q}}^{2}}}\right).$
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). 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. Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.
The electromagnetic potential is $A_{\alpha }=(Q/r,0,0,0).$
If magnetic monopoles are included in the theory, then a generalization to include magnetic charge P is obtained by replacing Q$^2$ by Q$^2$ + P$^2$ in the metric and including the term P cos θ dφ in the electromagnetic potential.

## Gravitational time dilation

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

## Christoffel symbols

The Christoffel symbols $\Gamma ^{i}_{jk}=\sum _{s=0}^{3}\ {\frac {g^{is}}{2}}\left({\frac {\partial g_{js}}{\partial x^{k}}}+{\frac {\partial g_{sk}}{\partial x^{j}}}-{\frac {\partial g_{jk}}{\partial x^{s}}}\right)$ with the indices $\{0,\ 1,\ 2,\ 3\}\to \{t,\ r,\ \theta ,\ \varphi \}$ give the nonvanishing expressions ${\begin{aligned}\Gamma ^{t}_{tr}&={\frac {Mr-Q^{2}}{r(Q^{2}+r^{2}-2Mr)}}\\[6pt]\Gamma ^{r}_{tt}&={\frac {(Mr-Q^{2})\left(r^{2}-2Mr+Q^{2}\right)}{r^{5}}}\\[6pt]\Gamma ^{r}_{rr}&={\frac {Q^{2}-Mr}{r(Q^{2}-2Mr+r^{2})}}\\[6pt]\Gamma ^{r}_{\theta \theta }&=-{\frac {r^{2}-2Mr+Q^{2}}{r}}\\[6pt]\Gamma ^{r}_{\varphi \varphi }&=-{\frac {\sin ^{2}\theta \left(r^{2}-2Mr+Q^{2}\right)}{r}}\\[6pt]\Gamma ^{\theta }_{\theta r}&={\frac {1}{r}}\\[6pt]\Gamma ^{\theta }_{\varphi \varphi }&=-\sin \theta \cos \theta \\[6pt]\Gamma ^{\varphi }_{\varphi r}&={\frac {1}{r}}\\[6pt]\Gamma ^{\varphi }_{\varphi \theta }&=\cot \theta \end{aligned}}$
Given the Christoffel symbols, one can compute the geodesics of a test-particle.

## Tetrad form

Instead of working in the holonomic basis, one can perform efficient calculations with a tetrad. Let ${\bf {e}}_{I}=e_{\mu I}$ be a set of one-forms with internal Minkowski index ⁠$I\in \{0,1,2,3\}$⁠, such that ⁠$\eta ^{IJ}e_{\mu I}e_{\nu J}=g_{\mu \nu }$⁠. The Reissner metric can be described by the tetrad
    ${\bf {e}}_{0}=G^{1/2}\,dt$
    ${\bf {e}}_{1}=G^{-1/2}\,dr$
    ${\bf {e}}_{2}=r\,d\theta$
    ${\bf {e}}_{3}=r\sin \theta \,d\varphi$
where ⁠$G(r)=1-r_{s}r^{-1}+r_{Q}^{2}r^{-2}$⁠. The parallel transport of the tetrad is captured by the connection one-forms ⁠${\boldsymbol {\omega }}_{IJ}=-{\boldsymbol {\omega }}_{JI}=\omega _{\mu IJ}=e_{I}^{\nu }\nabla _{\mu }e_{J\nu }$⁠. These have only 24 independent components compared to the 40 components of ⁠$\Gamma ^{\lambda }_{\mu \nu }$⁠. The connections can be solved for by inspection from Cartan's equation ⁠$d{\bf {e}}_{I}={\bf {e}}^{J}\wedge {\boldsymbol {\omega }}_{IJ}$⁠, where the left hand side is the exterior derivative of the tetrad, and the right hand side is a wedge product.
    ${\boldsymbol {\omega }}_{10}={\frac {1}{2}}\partial _{r}G\,dt$
    ${\boldsymbol {\omega }}_{20}={\boldsymbol {\omega }}_{30}=0$
    ${\boldsymbol {\omega }}_{21}=-G^{1/2}\,d\theta$
    ${\boldsymbol {\omega }}_{31}=-\sin \theta G^{1/2}d\varphi$
    ${\boldsymbol {\omega }}_{32}=-\cos \theta \,d\varphi$
The Riemann tensor ${\bf {R}}_{IJ}=R_{\mu \nu IJ}$ can be constructed as a collection of two-forms by the second Cartan equation ${\bf {R}}_{IJ}=d{\boldsymbol {\omega }}_{IJ}+{\boldsymbol {\omega }}_{IK}\wedge {\boldsymbol {\omega }}^{K}_{J},$ which again makes use of the exterior derivative and wedge product. This approach is significantly faster than the traditional computation with ⁠$\Gamma ^{\lambda }_{\mu \nu }$⁠; note that there are only four nonzero ${\boldsymbol {\omega }}_{IJ}$ compared with nine nonzero components of ⁠$\Gamma ^{\lambda }_{\mu \nu }$⁠.

## Equations of motion

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 ${\ddot {x}}^{i}=-\sum _{j=0}^{3}\ \sum _{k=0}^{3}\ \Gamma _{jk}^{i}\ {{\dot {x}}^{j}}\ {{\dot {x}}^{k}}+q\ {F^{ik}}\ {{\dot {x}}_{k}}$ which yields ${\ddot {t}}={\frac {\ 2(Q^{2}-Mr)}{r(r^{2}-2Mr+Q^{2})}}{\dot {r}}{\dot {t}}+{\frac {qQ}{(r^{2}-2mr+Q^{2})}}\ {\dot {r}}$ ${\ddot {r}}={\frac {(r^{2}-2Mr+Q^{2})(Q^{2}-Mr)\ {\dot {t}}^{2}}{r^{5}}}+{\frac {(Mr-Q^{2}){\dot {r}}^{2}}{r(r^{2}-2Mr+Q^{2})}}+{\frac {(r^{2}-2Mr+Q^{2})\ {\dot {\theta }}^{2}}{r}}+{\frac {qQ(r^{2}-2mr+Q^{2})}{r^{4}}}\ {\dot {t}}$ ${\ddot {\theta }}=-{\frac {2\ {\dot {\theta }}\ {\dot {r}}}{r}}.$
All total derivatives are with respect to proper time ⁠${\dot {a}}={\frac {da}{d\tau }}$⁠.
Constants of the motion are provided by solutions $S(t,{\dot {t}},r,{\dot {r}},\theta ,{\dot {\theta }},\varphi ,{\dot {\varphi }})$ to the partial differential equation $0={\dot {t}}{\dfrac {\partial S}{\partial t}}+{\dot {r}}{\frac {\partial S}{\partial r}}+{\dot {\theta }}{\frac {\partial S}{\partial \theta }}+{\ddot {t}}{\frac {\partial S}{\partial {\dot {t}}}}+{\ddot {r}}{\frac {\partial S}{\partial {\dot {r}}}}+{\ddot {\theta }}{\frac {\partial S}{\partial {\dot {\theta }}}}$ 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 ${\frac {\partial S}{\partial r}}-{\frac {2}{r}}{\dot {\theta }}{\frac {\partial S}{\partial {\dot {\theta }}}}=0$ immediately yields the constant relativistic specific angular momentum $S_{2}=L=r^{2}{\dot {\theta }};$ 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) $S_{3}=E={\frac {{\dot {t}}(r^{2}-2Mr+Q^{2})}{r^{2}}}+{\frac {qQ}{r}}.$
Substituting $S_{2}$ and $S_{3}$ into $S_{1}$ yields the radial equation $c\int d\tau =\int {\frac {r^{2}\,dr}{\sqrt {r^{4}(E-1)+2Mr^{3}-(Q^{2}+L^{2})r^{2}+2ML^{2}r-Q^{2}L^{2}}}}.$
Multiplying under the integral sign by $S_{2}$ yields the orbital equation $c\int Lr^{2}\,d\theta =\int {\frac {L\,dr}{\sqrt {r^{4}(E-1)+2Mr^{3}-(Q^{2}+L^{2})r^{2}+2ML^{2}r-Q^{2}L^{2}}}}.$
The total time dilation between the test-particle and an observer at infinity is $\gamma ={\frac {q\ Q\ r^{3}+E\ r^{4}}{r^{2}\ (r^{2}-2r+Q^{2})}}.$
The first derivatives ${\dot {x}}^{i}$ and the contravariant components of the local 3-velocity $v^{i}$ are related by ${\dot {x}}^{i}={\frac {v^{i}}{\sqrt {(1-v^{2})\ |g_{ii}|}}},$ which gives the initial conditions ${\dot {r}}={\frac {v_{\parallel }{\sqrt {r^{2}-2M+Q^{2}}}}{r{\sqrt {(1-v^{2})}}}}$ ${\dot {\theta }}={\frac {v_{\perp }}{r{\sqrt {(1-v^{2})}}}}.$
The specific orbital energy $E={\frac {\sqrt {Q^{2}-2rM+r^{2}}}{r{\sqrt {1-v^{2}}}}}+{\frac {qQ}{r}}$ and the specific relative angular momentum $L={\frac {v_{\perp }\ r}{\sqrt {1-v^{2}}}}$ of the test-particle are conserved quantities of motion. $v_{\parallel }$ and $v_{\perp }$ are the radial and transverse components of the local velocity-vector. The local velocity is therefore $v={\sqrt {v_{\perp }^{2}+v_{\parallel }^{2}}}={\sqrt {\frac {(E^{2}-1)r^{2}-Q^{2}-r^{2}+2rM}{E^{2}r^{2}}}}.$

## Alternative formulation of metric

The metric can be expressed in Kerr–Schild form like this: ${\begin{aligned}g_{\mu \nu }&=\eta _{\mu \nu }+fk_{\mu }k_{\nu }\\[5pt]f&={\frac {G}{r^{2}}}\left[2Mr-Q^{2}\right]\\[5pt]\mathbf {k} &=(k_{x},k_{y},k_{z})=\left({\frac {x}{r}},{\frac {y}{r}},{\frac {z}{r}}\right)\\[5pt]k_{0}&=1.\end{aligned}}$
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.