Reissner–Nordström metric

Charged black holes

Although charged black holes with ${\displaystyle r_Q\ll r_S}$ 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 ${\displaystyle g_{rr}}$ 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 concentric event horizons become degenerate for ${\displaystyle 2r_Q=r_S}$, which corresponds to an extremal 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 a naked singularity. According to the weak cosmic censorship hypothesis, naked singularity cannot exist in nature. Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.

The electromagnetic 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.

Gravitational time dilation

The gravitational 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 $${\displaystyle v_{\mathrm{e}\mathrm{s}\mathrm{c}}=\frac{\sqrt{{\gamma }^2-1}}{\gamma }.}$$

Christoffel symbols

The Christoffel 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$${\displaystyle \begin{array}{cc}{\mathrm{\Gamma }}^t{}_{tr}& =\frac{Mr-Q^2}{r(Q^2+r^2-2Mr)}\\ {\mathrm{\Gamma }}^r{}_{tt}& =\frac{(Mr-Q^2)\left(r^2-2Mr+Q^2\right)}{r^5}\\ {\mathrm{\Gamma }}^r{}_{rr}& =\frac{Q^2-Mr}{r(Q^2-2Mr+r^2)}\\ {\mathrm{\Gamma }}^r{}_{\theta \theta }& =-\frac{r^2-2Mr+Q^2}{r}\\ {\mathrm{\Gamma }}^r{}_{\phi \phi }& =-\frac{{\sin }^2\theta \left(r^2-2Mr+Q^2\right)}{r}\\ {\mathrm{\Gamma }}^{\theta }{}_{\theta r}& =\frac{1}{r}\\ {\mathrm{\Gamma }}^{\theta }{}_{\phi \phi }& =-\sin \theta \cos \theta \\ {\mathrm{\Gamma }}^{\phi }{}_{\phi r}& =\frac{1}{r}\\ {\mathrm{\Gamma }}^{\phi }{}_{\phi \theta }& =\cot \theta \end{array}}$$

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 ${\displaystyle {\mathbf{e}}_I=e_{\mu I}}$ be a set of one-forms with internal Minkowski 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 ⁠${\displaystyle G(r)=1-r_s{r}^{-1}+r_Q^2{r}^{-2}}$⁠. The parallel transport of the tetrad is captured by the connection 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 the exterior derivative of the tetrad, and the right hand side is a wedge product.

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

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

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

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

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

The Riemann 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 ⁠${\displaystyle {\mathrm{\Gamma }}^{\lambda }{}_{\mu \nu }}$⁠.

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. For brevity and without 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 $${\displaystyle 0=\dot{t}{\displaystyle \frac{\mathrm{\partial }S}{\mathrm{\partial }t}}+\dot{r}\frac{\mathrm{\partial }S}{\mathrm{\partial }r}+\dot{\theta }\frac{\mathrm{\partial }S}{\mathrm{\partial }\theta }+\ddot{t}\frac{\mathrm{\partial }S}{\mathrm{\partial }\dot{t}}+\ddot{r}\frac{\mathrm{\partial }S}{\mathrm{\partial }\dot{r}}+\ddot{\theta }\frac{\mathrm{\partial }S}{\mathrm{\partial }\dot{\theta }}}$$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)$${\displaystyle S_3=E=\frac{\dot{t}(r^2-2Mr+Q^2)}{r^2}+\frac{qQ}{r}.}$$

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 total time 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 the contravariant 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 $${\displaystyle \dot{r}=\frac{v_{\parallel }\sqrt{r^2-2M+Q^2}}{r\sqrt{(1-v^2)}}}$$ $${\displaystyle \dot{\theta }=\frac{v_{\perp }}{r\sqrt{(1-v^2)}}.}$$

The specific orbital energy $${\displaystyle E=\frac{\sqrt{Q^2-2rM+r^2}}{r\sqrt{1-v^2}}+\frac{qQ}{r}}$$ and the specific 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 $${\displaystyle v=\sqrt{v_{\perp }^2+v_{\parallel }^2}=\sqrt{\frac{(E^2-1)r^2-Q^2-r^2+2rM}{E^2{r}^2}}.}$$

Alternative formulation of metric

The metric can be expressed in Kerr–Schild form as follows:$${\displaystyle \begin{array}{cc}{g}_{\mu \nu }& ={\eta }_{\mu \nu }+f{k}_{\mu }{k}_{\nu }\\ f& =\frac{G}{r^2}\left[2Mr-Q^2\right]\\ \mathbf{k}& =(k_x,k_y,k_z)=\left(\frac{x}{r},\frac{y}{r},\frac{z}{r}\right)\\ k_0& =1.\end{array}}$$

Here ${\displaystyle \mathbf{k}}$ is a unit vector, ${\displaystyle M}$ is the constant mass of the object, ${\displaystyle Q}$ is the constant charge of the object, and ${\displaystyle \eta }$ is the Minkowski tensor.