Charged Spherical Black Holes and Coupled Sectors

Electric charge changes more than the background potential. Because the Reissner–Nordström electromagnetic field is nonzero, a metric perturbation changes Maxwell’s stress tensor at first order and an electromagnetic perturbation changes the linearized Einstein equation. Gravitational and electromagnetic amplitudes of the same $(\ell,m)$ and parity therefore form a coupled system. Suitable gauge-invariant linear combinations diagonalize that system, but they are mixtures—not pure spin-two and spin-one fields—whenever the background charge is nonzero.

The comparison axes throughout are coupling, potential, low-multipole content, and horizon behavior, with the neutral Schwarzschild limit serving as the independent normalization check.

Required background. PDEs and Green Functions for Gravitation supplies coupled hyperbolic systems, conserved currents, and horizon boundary conditions.

Helpful background. Black-Hole Geometry and Mechanics fixes charged horizons and their parameter range; Formulations of General Relativity separates the Einstein–Maxwell constraints from propagation; Killing Fields and Isometry Groups organizes the harmonic and parity blocks.

Reissner–Nordström as a coupled background

For this sustained calculation use geometrized units $G=c=1$. Write the mass and charge lengths as $\mu$ and $q$; in ordinary units $\mu=GM$, while the map from physical electric charge to $q$ depends on the normalization of the Maxwell action. The background is

$$ds^2=-f\,dt^2+f^{-1}\,dr^2+r^2\,d\Omega^2, \qquad f=1-\frac{2\mu}{r}+\frac{q^2}{r^2},$$

with an electric field proportional to $q/r^2$. For the subextremal family, $|q|<\mu$ and

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

The exterior is $r>r_+$. The extremal limit is not obtained uniformly in every late-time or near-horizon calculation, so it is not silently included. The background is parity invariant: modes of different $\ell,m$, or parity still decouple. What no longer decouples is the gravitational/electromagnetic pair within a fixed block.

Let $h_{\mu\nu}$ and $a_\mu$ perturb the metric and electromagnetic potential.

The local vector $\xi$ below is the inverse-identification generator, $\xi=-\xi_{\text{Ch. 1}}$; this accounts for the minus sign in the metric rule relative to Chapter 1’s active-plus convention.

There are two gauge freedoms,

$$h_{\mu\nu}\mapsto h_{\mu\nu}-2\nabla_{(\mu}\xi_{\nu)}, \qquad a_\mu\mapsto a_\mu-\mathcal L_\xi A_\mu+\nabla_\mu\chi.$$

The Lie-derivative term is essential: a diffeomorphism moves the nonzero background potential as well as the metric. A combination invariant only under $a_\mu\mapsto a_\mu+\nabla_\mu\chi$ is not yet an Einstein–Maxwell gauge invariant.

Coupled parity sectors

After harmonic projection and constraint elimination, each parity sector for $\ell\ge2$ has two gauge-invariant canonical channels. In odd parity, choose the canonically normalized Moncrief channel vector

$$\boldsymbol\Psi =\begin{pmatrix}\Psi_\gamma\\ \Psi_g\end{pmatrix}, \qquad \left[-\partial_t^2+\partial_{r_*}^2-\mathbf V^{(-)}_\ell(r)\right] \boldsymbol\Psi=0, \qquad \frac{dr_*}{dr}=f^{-1}.$$

Here $\Psi_\gamma$ and $\Psi_g$ are canonical gauge-invariant coordinates, not raw electromagnetic and metric components; their radial kinetic operator is the identity. Their signs can be rephased, but the following convention fixes an explicit dictionary. Set

$$\Lambda_\ell=(\ell-1)(\ell+2), \qquad \mathbf K^{(-)}_\ell= \begin{pmatrix} 0&-2q\sqrt{\Lambda_\ell}\\ -2q\sqrt{\Lambda_\ell}&6\mu \end{pmatrix}.$$

Then

$$\mathbf V^{(-)}_\ell(r) =f\left[ \left(\frac{\ell(\ell+1)}{r^2}+\frac{4q^2}{r^4}\right)\mathbf 1 -\frac{\mathbf K^{(-)}_\ell}{r^3} \right].$$

The off-diagonal entries are proportional to $q$, so a gravitational wave can source an electromagnetic response through the background field and the reverse conversion also occurs. This follows directly by varying $G_{\mu\nu}=8\pi T^{\mathrm{EM}}_{\mu\nu}$: because $\bar F_{\mu\nu}\ne0$, $\delta T^{\mathrm{EM}}$ contains terms linear in both $h$ and $\delta F$. Moncrief first constructed the canonical reduction Moncrief 1975. An equation-level derivation of the standard odd branches is given in Liu and Zeng 2025, § III.A, eqs. (41)–(49), while the compact canonical matrix above and its spectral projectors are recorded in Takahara and Kitahara 2026, eqs. (1) and (55)–(59). That source writes the mostly-minus metric. Reversing the overall metric sign and expressing its radial equation with the mostly-plus wave operator used here leaves the displayed potential unchanged. The odd-parity phase choice fixes the minus signs in the off-diagonal entries; rephasing either canonical channel flips both entries without changing the eigenvalues or fluxes.

Decoupled axial combinations and potentials

The diagonalization is now ordinary two-dimensional spectral algebra. Define

$$D_\ell=\sqrt{9\mu^2+4q^2\Lambda_\ell}, \qquad c_\pm=3\mu\pm D_\ell,$$ $$\mathbf\Pi_\pm =\frac12\left[ \mathbf 1\pm\frac{\mathbf K^{(-)}_\ell-3\mu\mathbf 1}{D_\ell} \right].$$

These constant matrices obey $\mathbf\Pi_\pm^2=\mathbf\Pi_\pm$, $\mathbf\Pi_+\mathbf\Pi_-=0$, and $\mathbf K^{(-)}_\ell\mathbf\Pi_\pm=c_\pm\mathbf\Pi_\pm$. For $q\ne0$, one explicit normalized choice is

$$\mathbf e_\pm =\frac{1}{\sqrt{4q^2\Lambda_\ell+c_\pm^2}} \begin{pmatrix}-2q\sqrt{\Lambda_\ell}\\ c_\pm\end{pmatrix}, \qquad Z_\pm=\mathbf e_\pm^{\mathsf T}\boldsymbol\Psi.$$

At $q=0$, define the limiting axes directly as $\mathbf e_+=(0,1)^{\mathsf T}$ and $\mathbf e_-=(1,0)^{\mathsf T}$; the sign of a normalized eigenvector is physically immaterial. Each scalar then obeys

$$\left[-\partial_t^2+\partial_{r_*}^2-V_\pm^{(-)}(r)\right]Z_\pm=0, \qquad V_\pm^{(-)}(r) =\frac{f}{r^2} \left[\ell(\ell+1)-\frac{c_\pm}{r}+\frac{4q^2}{r^2}\right].$$

The plus branch is gravitational-led. As $q\to0$, $c_+\to6\mu$ and $c_-\to0$, while the eigenvectors approach the gravitational and photon axes, respectively. The two potentials therefore become

$$V_{\mathrm{grav}}\to f\left[\frac{\ell(\ell+1)}{r^2} -\frac{6\mu}{r^3}\right], \qquad V_{\mathrm{em}}\to f\frac{\ell(\ell+1)}{r^2}.$$

This recovers the Regge–Wheeler and Maxwell operators independently and fixes which eigenbranch has which neutral interpretation. At finite $q$, neither $Z_+$ nor $Z_-$ is purely gravitational or electromagnetic. Even-parity invariants can also be diagonalized into two master equations; their potentials have a more complicated rational form and are related to the odd system by an intertwining structure. The photon/graviton labels here name four-dimensional field content. By contrast, the “scalar” and “vector” sectors of a higher-dimensional formalism classify harmonics on the symmetry space, not field spin Kodama and Ishibashi 2003, §§ 2–3.

Horizons, flux, and exceptional modes

Every displayed axial potential contains the factor $f$. It vanishes at both simple roots $r_\pm$ and is finite in regular horizon coordinates; the Schwarzschild $r_*$ chart itself diverges there. The exterior scattering problem uses the future event horizon $r_+$ and infinity. Algebraic regularity at $r_-$ does not establish bounded evolution at the Cauchy horizon, which is a distinct global-stability problem.

For real frequency, each decoupled equation has a conserved Wronskian. Because the canonical radial kinetic term is proportional to the identity and the projectors are orthogonal, the sum of the two channel fluxes equals the flux derived from the coupled Einstein–Maxwell symplectic current in this normalization.

Checking only one metric component can obscure energy exchanged between channels. Positivity and boundary orientation should instead be checked on the invariant pair.

Low multipoles again require separate treatment. The monopole changes mass and electric charge; the odd dipole changes angular momentum and includes the electromagnetic field of linearized Kerr–Newman. Sources can also support nonradiative multipoles. General rotating separation belongs to Kerr theory, and nonlinear charged evolution remains outside this chapter.

Exercise: perform the neutral-limit test

Take $q\to0$ in the two constants $c_\pm$ and show that the decoupled axial potentials recover one gravitational and one electromagnetic Schwarzschild sector. Why is this a stronger check than showing that the coupling matrix is diagonal at $q=0$?

Solution

At $q=0$,

$$\sqrt{9\mu^2+4q^2(\ell-1)(\ell+2)}\longrightarrow3\mu,$$

so $(c_+,c_-)\to(6\mu,0)$. The explicit $4q^2/r^2$ term also vanishes, while $f\to1-2\mu/r$. Substitution yields precisely the Regge–Wheeler potential for the $6\mu$ branch and the spin-one Maxwell potential for the zero branch.

A matrix can become diagonal yet have incorrect diagonal entries, swapped normalizations, or a wrong curvature term. Recovering both known potentials checks the coupling, the eigenvalues, the branch identification, and the radial normalization at once.

References

• Kodama, Hideo, and Akihiro Ishibashi. “A Master Equation for Gravitational Perturbations of Maximally Symmetric Black Holes in Higher Dimensions.” Progress of Theoretical Physics 110 (2003): 701–722. arXiv.
• Liu, Hui-Fa, and Ding-fang Zeng. “Master Functions of Reissner–Nordström Black Hole Perturbations and Their Darboux Transformation.” Physical Review D 112 (2025): 044025. DOI. Open PDF.
• Moncrief, Vincent. “Gauge-Invariant Perturbations of Reissner–Nordström Black Holes.” Physical Review D 12 (1975): 1526–1537. DOI.
• Takahara, Kento, and Teppei Kitahara. “On-Shell Amplitudes and Black-Hole Perturbations: Exact Reissner–Nordström Mixing.” arXiv preprint, version 1 (2026). arXiv:2608.11703.