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.Big Think The 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, and r_\text{s} is the Schwarzschild radius of the body given by r_\text{s} = \frac{2GM}{c^2}, and r_Q is a characteristic length scale given by r_Q^2 = \frac{Q^2 G}{4\pi\varepsilon_0 c^4}.
Here, \varepsilon_0 is the electric constant.
The total mass of the central body and its irreducible mass are related byThibault Damour: Black Holes: Energetics and Thermodynamics, S. 11 ff.Ashgar Quadir: The Reissner Nordström Repulsion 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}.
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 42,164 km (26,200 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 rQ ≪ rs 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 2rQ = rs, which corresponds to an extremal black hole.
Black holes with 2rQ > rs can not 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: 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.Carter, Brandon.
Global Structure of the Kerr Family of Gravitational Fields, Physical Review, page 174 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 Q2 by Q2 + P2 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^{t t}|} = \sqrt{\frac{r^2}{Q^2+(r-2 M) 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_{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) with the indices \{ 0, \ 1, \ 2, \ 3 \} \to \{ t, \ r, \ \theta, \ \varphi \} 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}{Q^2 r-2 M r^2+r^3} \\[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: The Theoretical Minimum: Geodesics and Gravity, (General Relativity Lecture 4, timestamp: 34m18s)Eva Hackmann, Hongxiao Xu: Charged particle motion in Kerr–Newmann space-times 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^i_{j k} \ {\dot x^j} \ {\dot x^k} + q \ {F^{i k}} \ {\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^2r-Q^2L^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^2r-Q^2L^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^2-2 r+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_{i i}|}}, 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{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.
See also
Black hole electron
Notes
References
External links
spacetime diagrams including Finkelstein diagram and Penrose diagram, by Andrew J. S. Hamilton
"Particle Moving Around Two Extreme Black Holes" by Enrique Zeleny, The Wolfram Demonstrations Project.
