Charged Black-Hole Spacetimes

Reissner–Nordström and Kerr–Newman are the static spherical and stationary rotating black-hole families of four-dimensional, asymptotically flat Einstein–Maxwell theory. This record fixes one Gaussian-unit charge normalization and shows how mass, charge, and angular momentum determine the Killing horizons, stationary-limit surfaces, and admissible parameter regimes. These exact-family statements do not establish stability, formation, or observational relevance.

Helpful background. Lorentzian Metrics and Causal Cones supplies causal classification and chart regularity; Exact Solutions of Einstein’s Equations supplies the metric-field and domain-check method.

Einstein–Maxwell and charge normalization

Use $c=1$ with $G$ explicit, vanishing cosmological constant, and Gaussian electromagnetic units:

$$G_{ab}=8\pi G T_{ab}, \qquad T_{ab}=\frac{1}{4\pi} \left( F_{ac}F_b{}^c-\frac14g_{ab}F_{cd}F^{cd} \right),$$

with $dF=0=d{*F}$ away from sources. Define the electric and magnetic charges by flux at an enclosing two-sphere,

$$Q_e=\frac{1}{4\pi}\int_{S^2}{*F}, \qquad Q_m=\frac{1}{4\pi}\int_{S^2}F,$$

and set

$$Q^2=Q_e^2+Q_m^2, \qquad m=GM, \qquad q^2=GQ^2, \qquad a=\frac{J}{M}.$$

Assume $M\gt0$ and take $q\geq0$. Thus $m$, $q$, and $a$ have dimensions of length. A convention that absorbs $G$ or $4\pi$ into charge must translate the stress tensor, flux definition, metric coefficient, and potential together; replacing only the printed $Q^2$ term is inconsistent.

The Black-Hole Geometry mechanics pages use rationalized charge. The complete Gaussian-to-rationalized Maxwell map shows why both conventions give the same $q^2$, stress tensor, and first-law work term when the entire package is rescaled.

Reissner–Nordström

In areal radius $r$, the dyonic field and metric are

$$F=\frac{Q_e}{r^2}\,dr\wedge dt +Q_m\sin\theta\,d\theta\wedge d\phi,$$ $$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega_2^2, \qquad f(r)=1-\frac{2m}{r}+\frac{q^2}{r^2}.$$

The electric gauge $A=-(Q_e/r)dt$ vanishes at infinity. When $Q_m\ne0$, the magnetic potential must instead be given on overlapping angular patches; a string in one potential is not a singularity of $F$.

The radial roots are

$$r_\pm=m\pm\sqrt{m^2-q^2}.$$

For $0\lt q^2\lt m^2$ they are two simple Killing horizons, for $q^2=m^2$ they merge into one double horizon, and for $q^2\gt m^2$ no real horizon root exists. At the neutral endpoint $q=0$, only $r_+=2m$ is a horizon: the formal value $r_-=0$ is the Schwarzschild curvature singularity, where $f$ is undefined. In the charged subextremal maximal extension, $r_+$ is the outer event and Killing horizon and $r_-$ is an inner Cauchy and Killing horizon. The event-horizon statement uses that global extension; it does not follow from $f=0$ alone.

With $dv=dt+dr/f$, the metric becomes

$$ds^2=-f\,dv^2+2\,dv\,dr+r^2d\Omega_2^2.$$

Its radial block has determinant $-1$ at either regular horizon root, and the electric part of $F$ is $Q_e r^{-2}dr\wedge dv$. The advanced chart therefore crosses a selected future horizon without a metric or field singularity. Bičák compares the three root regimes and their analytic blocks in Bičák 2000, §3, printed pp. 31–41, PDF.

The complete metric-field derivation is The Reissner–Nordström Solution; chart domains and extension cautions are treated in Reissner–Nordström Horizons and Extensions.

Kerr–Newman

For the rotating family, define

$$\Sigma=r^2+a^2\cos^2\theta, \qquad \Delta=r^2-2mr+a^2+q^2.$$

The Boyer–Lindquist metric can be written compactly as

$$\begin{aligned} ds^2={}&-dt^2+\frac{\Sigma}{\Delta}dr^2+\Sigma\,d\theta^2 +(r^2+a^2)\sin^2\theta\,d\phi^2\\ &+\frac{2mr-q^2}{\Sigma} \left(dt-a\sin^2\theta\,d\phi\right)^2. \end{aligned}$$

For a purely electric member, use the gauge

$$A=-\frac{Q_e r}{\Sigma} \left(dt-a\sin^2\theta\,d\phi\right),$$

which vanishes at the chosen asymptotically flat end. Magnetic charge can be added at the field-strength level by electromagnetic duality, but its potential remains patchwise. Newman and collaborators give the defining rotating charged metric in Newman et al. 1965, pp. 918–919.

The ingoing differentials

$$dv=dt+\frac{r^2+a^2}{\Delta}dr, \qquad d\widetilde\phi=d\phi+\frac{a}{\Delta}dr$$

remove the Boyer–Lindquist metric pole at a selected future horizon. In these coordinates the displayed electric potential acquires the radial term $Q_e r\,dr/\Delta$. That term is locally exact and can be removed by a gauge transformation, leaving

$$A_{\mathrm{in}} =-\frac{Q_e r}{\Sigma} \left(dv-a\sin^2\theta\,d\widetilde\phi\right),$$

whose field strength is regular at a regular horizon. A divergent component of $A$ in one gauge is therefore not evidence that $F$ is singular.

Horizons, ergosurfaces, and domains

The Kerr–Newman algebraic radial roots and stationary-limit surfaces are

$$r_\pm=m\pm\sqrt{m^2-a^2-q^2},$$ $$r_{\mathrm E\pm}(\theta) =m\pm\sqrt{m^2-a^2\cos^2\theta-q^2}.$$

Family	Black-hole parameter domain	Killing horizons	Stationary limit
Reissner–Nordström	$q^2\leq m^2$	$f=0$ at $r_\pm$ when $q\ne0$; at $q=0$, only $r_+=2m$ is a horizon	Coincides with $f=0$ because the geometry is static
Kerr–Newman	$a^2+q^2\leq m^2$	$\Delta=0$ at $r_\pm$ unless $(a,q)=(0,0)$, when $r_-=0$ is not a horizon	$g_{tt}=0$ at $r_{\mathrm E\pm}(\theta)$; for $a\ne0$, the outer surface meets $r_+$ only on the rotation axis, while for $a=0$ it coincides with $r_+$ at every angle

Equality gives a double root. Strict inequality with $a^2+q^2\gt0$ gives distinct outer and inner horizons; the joint neutral, nonrotating endpoint is Schwarzschild and has only the outer horizon. Beyond the stated bound, the formally superextremal electrovac metric still solves the local field equations away from its singular locus, but it is not a black-hole member of the family.

For the outer horizon, the normalized generator is

$$\chi=\partial_t+\Omega_H\partial_\phi, \qquad \Omega_H=\frac{a}{r_+^2+a^2}.$$

In the asymptotically vanishing electric gauge,

$$A_a\chi^a\big|_{r_+} =-\frac{Q_e r_+}{r_+^2+a^2},$$

independent of $\theta$. This constancy checks both the horizon generator and gauge regularity. It does not make the potential itself gauge invariant.

The four loci must remain distinct: $\Delta=0$ gives Boyer–Lindquist horizon candidates, $g_{tt}=0$ gives stationary limits, and $\sin\theta=0$ is the angular-coordinate axis. On the rotating branch $a\ne0$, $\Sigma=0$ is the ring curvature singularity; at $a=0$, it is instead the central locus $r=0$. The exact rotating metric, field, and limiting cases are developed in The Kerr–Newman Electrovacuum.

Checks and limitations

The Maxwell stress tensor is trace free in four dimensions, so every member obeys $R=0$. For nonzero charge the Ricci tensor is nevertheless nonzero: $R=0$ does not mean vacuum. The $1/r$ and $1/r^2$ asymptotic coefficients and the two flux integrals independently check the mass and charge normalizations.

Three limits connect the families: $q\to0$ gives Kerr, $a\to0$ gives Reissner–Nordström, and $(a,q)\to(0,0)$ gives Schwarzschild.

These are family maps; the coordinate and gauge choices must also have smooth limits. Analytic extensions of the exact subextremal solutions with $a^2+q^2\gt0$ contain ideal inner Cauchy horizons and additional blocks; the Schwarzschild endpoint does not. Their perturbative survival, dynamical formation, censorship status, mechanics, thermodynamics, and empirical relevance lie outside this lookup record.

References

• Bičák, Jiří. “The Role of Exact Solutions of Einstein’s Equations in the Developments of General Relativity and Astrophysics: Selected Themes.” In Einstein’s Field Equations and Their Physical Implications, edited by Bernd G. Schmidt, 1–126. Lecture Notes in Physics 540. Springer, 2000. Open PDF.
• Newman, Ezra T., Evelyn Couch, K. Chinnapared, A. Exton, A. Prakash, and Robert Torrence. “Metric of a Rotating, Charged Mass.” Journal of Mathematical Physics 6, no. 6 (1965): 918–919. DOI.