The Kerr-Newman Electrovacuum

Kerr–Newman is the asymptotically flat stationary-axisymmetric Einstein–Maxwell family with mass, angular momentum, and electric or magnetic charge. Charge replaces $2mr$ by $2mr-q^2$ in the Boyer–Lindquist Killing block and adds $q^2$ to the horizon polynomial. The principal algebraic and separable structure survives, but the spacetime is no longer Ricci flat.

Required background. Frames, Coframes, and Anholonomy supplies adapted tetrads for the coupled metric and Maxwell field.

Helpful background. The Einstein Equation supplies Einstein–Maxwell dynamics; The Riemann Curvature Tensor supports Ricci and Weyl checks; Exact Solutions of Einstein’s Equations explains how to specify a spacetime and its matter fields; Causality and Global Structure supports extension language; Killing Fields and Isometry Groups supplies the symmetry generators.

Metric and charge normalization

Use Gaussian electromagnetic units as in the spherical electrovac chapter. On the branch parameterized by physical mass and angular momentum, take $M\ne0$ so that $a=J/M$ is defined, and set

$$m=GM, \qquad a=\frac{J}{M}, \qquad q^2=G(Q_e^2+Q_m^2),$$ $$\Sigma=r^2+a^2\cos^2\theta, \qquad \Delta=r^2-2mr+a^2+q^2.$$

The black-hole interpretation below further assumes $m\gt0$ and uses the asymptotically flat end $r\to+\infty$. The zero-mass Kerr limit is instead a flat oblate-coordinate degeneration and does not carry a physical ratio $J/M$.

The Boyer–Lindquist metric is

$$\begin{aligned} ds^2={}&-\left(1-\frac{2mr-q^2}{\Sigma}\right)dt^2 -\frac{2a(2mr-q^2)\sin^2\theta}{\Sigma}\,dt\,d\phi +\frac{\Sigma}{\Delta}dr^2+\Sigma\,d\theta^2 \\ &+\left[r^2+a^2+ \frac{a^2(2mr-q^2)\sin^2\theta}{\Sigma}\right] \sin^2\theta\,d\phi^2. \end{aligned}$$

An axis-string gauge for the full dyonic field is

$$A_0=-\frac{Q_e r}{\Sigma} \left(dt-a\sin^2\theta\,d\phi\right) +\frac{Q_m\cos\theta}{\Sigma} \left[a\,dt-(r^2+a^2)d\phi\right].$$

A single potential cannot be smooth on both pieces of the rotation axis when $Q_m\ne0$. Potentials regular at the north and south axes, respectively, are

$$A_N=A_0+Q_m\,d\phi, \qquad A_S=A_0-Q_m\,d\phi.$$

On their overlap,

$$A_N-A_S=2Q_m\,d\phi=d(2Q_m\phi), \qquad F=dA_N=dA_S.$$

Indeed, at $a=0$ their magnetic parts reduce to $Q_m(1-\cos\theta)d\phi$ and $-Q_m(1+\cos\theta)d\phi$. Thus $F$ approaches $Q_e\,dr\wedge dt/r^2+Q_m\sin\theta\,d\theta\wedge d\phi$, matching the chapter’s flux definitions. The dyonic Boyer–Lindquist potential and its principal field are given in Godazgar et al. 2021, Appendix E, printed pp. 40–41, Eqs. (E.1)–(E.9), Open PDF. The original charged rotating metric was presented in Newman et al. 1965, pp. 918–919.

The mechanics chapters use rationalized rather than Gaussian charge. Before comparing $Q$, $A$, or a horizon potential across that boundary, apply the complete Gaussian-to-rationalized Maxwell map; the invariant geometric parameter is $q^2=GQ_{\rm G}^2=GQ_{\rm R}^2/(4\pi)$.

Einstein–Maxwell verification

Choose the positive spacetime orientation $\epsilon_{tr\theta\phi}\gt0$ and let $*F$ denote the Lorentzian Hodge dual formed with that orientation and the site metric. The Maxwell equations $dF=0=d{*F}$ hold away from the curvature singular locus. With

$$T_{\mu\nu}=\frac{1}{4\pi} \left(F_{\mu\alpha}F_\nu{}^\alpha -\frac14g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right),$$

direct calculation gives $G_{\mu\nu}=8\pi G T_{\mu\nu}$. Since the Maxwell stress tensor is trace free, $R=0$, but $R_{\mu\nu}R^{\mu\nu}$ is nonzero when $q\ne0$. In an ordered principal null tetrad $(\ell,n,m,\bar m)$ normalized by $\ell\!\cdot n=-1$ and $m\!\cdot\bar m=1$, the Maxwell field has only its Coulomb scalar and the Weyl tensor has only its Coulomb scalar. Kerr–Newman is therefore Petrov type D at generic regular points. The qualification matters: with $\Psi_2=-C_{abcd}\ell^am^b\bar m^cn^d$, the nonzero candidate is

$$\Psi_2= \frac{m(r+ia\cos\theta)-q^2} {(r-ia\cos\theta)^3(r+ia\cos\theta)}.$$

When this scalar vanishes at a regular point, the Weyl tensor degenerates to type O there even though the Maxwell-sourced Ricci tensor need not vanish. For $m\gt0$ and $q\ne0$, the zero condition is $r=q^2/m$ together with $a\cos\theta=0$. This is the convention-translated form of Godazgar et al. 2021, Appendix E, printed p. 41, Eqs. (E.7)–(E.9), Open PDF; the scalar-sign translation is detailed on the principal-null-directions page.

At large radius, the dyonic potential has

$$A_t=-\frac{Q_e}{r}+\frac{Q_ma\cos\theta}{r^2}+O(r^{-3}).$$

Its leading term fixes electric flux, while $g_{tt}=-1+2m/r-q^2/r^2+O(r^{-3})$ fixes the mass and charge normalization. In the purely electric family $Q_m=0$, the rotational component $A_\phi=Q_ea\sin^2\theta/r+O(r^{-3})$ gives magnetic dipole moment $Q_ea$.

The Boyer–Lindquist potential is a gauge choice as well as a coordinate expression. Transforming to ingoing coordinates introduces a radial term proportional to $dr/\Delta$. Its divergent part is exact and can be removed by a gauge transformation on a horizon-crossing patch. The gauge-invariant field $F$ and its components in a regular tetrad remain finite at a regular horizon. Declaring a Maxwell singularity from one divergent component of $A$ would therefore repeat the same error as declaring a gravitational singularity from $g_{rr}$.

For the dyonic north/south potentials, contraction with the horizon generator gives a useful regularity and gauge check. Define

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

Then

$$\chi\mathbin{\lrcorner}A_N\big|_{r_+}=-\Phi_e+\Omega_HQ_m, \qquad \chi\mathbin{\lrcorner}A_S\big|_{r_+}=-\Phi_e-\Omega_HQ_m.$$

The apparent $\theta$ dependence cancels because $1-a\Omega_H\sin^2\theta=\Sigma/(r_+^2+a^2)$ and $a-\Omega_H(r_+^2+a^2)=0$. On stationary exterior patches, horizon gauges satisfying $\chi\mathbin{\lrcorner}A^{(H)}_{N,S}=0$ are

$$\begin{aligned} A_N^{(H)}&=A_N+(\Phi_e-\Omega_HQ_m)\,dt,\\ A_S^{(H)}&=A_S+(\Phi_e+\Omega_HQ_m)\,dt. \end{aligned}$$

Their transition remains exact,

$$A_N^{(H)}-A_S^{(H)} =2Q_m(d\phi-\Omega_Hdt) =d\!\left[2Q_m(\phi-\Omega_Ht)\right],$$

and its gauge function is constant along $\chi$. These potential choices leave the common field strength unchanged; horizon regularity is ultimately a statement about $F$ in a regular frame, not about one Boyer–Lindquist component of $A$.

Domains, roots, and limits

The radial roots are

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

Relative to the $r\to+\infty$ end with $m\gt0$, the black-hole regime is $a^2+q^2\le m^2$; equality is extremal, and larger combined spin-charge produces no real horizon root. When $0\lt a^2+q^2\lt m^2$, both roots are positive and simple. At the neutral nonrotating endpoint $a=q=0$, $r_-=0$ is the Schwarzschild curvature singularity rather than an inner horizon. Ingoing coordinates use the same differential pattern as Kerr,

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

and remove the Boyer–Lindquist pole at a selected horizon. Identifying the outer Killing horizon with an event horizon still uses the global asymptotically flat extension.

Three limits recover familiar solutions. $q\to0$ gives Kerr; $a\to0$ gives dyonic Reissner–Nordström; $a,q\to0$ gives Schwarzschild. The algebraic form of the inverse metric continues to split into radial and angular pieces, which is why Hamilton–Jacobi and scalar-wave separability persist. That fact does not imply perturbative stability or observational uniqueness.

Exercise

At fixed $M$ and $J$, find the maximum allowed squared flux charge for a Kerr–Newman black hole and verify the spherical limit.

Solution

Real horizon roots require $m^2-a^2-q^2\ge0$. Thus a charge bound exists at fixed $(M,J)$ only if

$$|J|\le GM^2.$$

When this feasibility condition holds,

$$(Q_e^2+Q_m^2)_{\max} =GM^2-\frac{J^2}{GM^2}.$$

If $\lvert J\rvert\gt GM^2$, no choice of real charge produces a Kerr–Newman black hole; charge only decreases the discriminant. At $J=0$, the bound becomes $Q_e^2+Q_m^2\le GM^2$, and the roots reduce to $GM\pm\sqrt{G^2M^2-G(Q_e^2+Q_m^2)}$, as required for Reissner–Nordström.

References

• Godazgar, Hadi, Mahdi Godazgar, Ricardo Monteiro, David Peinador Veiga, and C. N. Pope. “Asymptotic Weyl Double Copy.” arXiv:2109.07866v2 [hep-th] (2021). 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: 10.1063/1.1704351.