Reissner–Nordström Black Hole

The Reissner–Nordström family is the static, spherically symmetric electrovacuum black hole of four-dimensional Einstein–Maxwell theory. It is indispensable as a charged and extremal benchmark, but macroscopic astrophysical black holes are expected to neutralize efficiently. Its exact mathematical status should therefore be distinguished from its limited direct astrophysical realism.

Reissner–Nordström electrovacuum

Property	Value
Object class	Exact solution family
Theory	Four-dimensional Einstein–Maxwell theory
Parameters	ADM mass $M$ and electric/magnetic flux charge
Asymptotics	Asymptotically flat
Horizon topology	$S^2$
Physical status	Exact electrovacuum solution; mainly a charged benchmark

To avoid hiding Maxwell-normalization conventions, define the geometrized charge length $q$ directly. With $m=GM$,

$$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_2^2, \qquad f(r)=1-\frac{2m}{r}+\frac{q^2}{r^2}.$$

For a purely electric representative, one may take $F=(Q/r^2)\,dr\wedge dt$ in Gaussian normalization, where $q^2=GQ^2$. With the global orientation this sign makes $Q=(4\pi)^{-1}\int_{S^2}{*F}$; other action conventions move factors such as $4\pi$ between $Q$ and $q$. The invariant statement is that the metric depends on the sum of squared electric and magnetic flux charges. The charged family and its role in exact-solution theory are reviewed by Bičák 2000, § 3, printed pp. 31–41.

This page uses Gaussian variables throughout. The linked horizon-anatomy and mechanics discussions use rationalized variables; the complete Gaussian-to-rationalized Maxwell map preserves $q^2$ and $\Phi_H\,dQ$ when converting between these conventions.

Two horizons and extremality

The lapse factorizes as

$$f(r)=\frac{(r-r_+)(r-r_-)}{r^2}, \qquad r_\pm=m\pm\sqrt{m^2-q^2}.$$

For $0\lt q^2\lt m^2$, the outer root $r_+$ is the event horizon and $r_-$ is an inner Cauchy horizon. At the neutral endpoint $q=0$, $r_-=0$ coincides with the curvature singularity and is not an inner horizon. At $q^2=m^2$ the two positive roots coincide at $r=m$, producing an extremal horizon. If $q^2\gt m^2$, both roots leave the real domain; the exact electrovacuum metric is then superextremal and its $r=0$ singularity is naked, so it is not a black hole.

Ingoing time $v=t+\int dr/f$ gives $ds^2=-f\,dv^2+2\,dv\,dr+r^2d\Omega_2^2$, regular at either simple root in an appropriate patch. Polynomial curvature invariants are finite at $r_\pm$ and diverge at $r=0$. The inner horizon is nevertheless dynamically delicate: blueshift and mass-inflation effects mean that the analytic stationary extension is not a robust prediction for a perturbed interior.

Mechanics and coupled perturbations

For the outer horizon,

$$A_H=4\pi r_+^2, \qquad \kappa=\frac{r_+-r_-}{2r_+^2}, \qquad \Phi_H=\frac{Q}{r_+}$$

in the Gaussian electric convention and a gauge vanishing at infinity. The mechanical first law is

$$dM=\frac{\kappa}{8\pi G}\,dA_H+\Phi_H\,dQ,$$

with a corresponding magnetic term if magnetic charge varies. Extremality gives $\kappa=0$, but the horizon area remains finite. A near-horizon scaling yields AdS$_2\times S^2$ with equal radii in the simplest normalization, the Bertotti–Robinson throat.

Gravitational and electromagnetic perturbations couple because the background stress tensor is nonzero. They can be reorganized into gauge-invariant parity sectors and decoupled master variables, as developed by Moncrief 1975, complete article, printed pp. 1526–1537. This exact decoupling does not make the Cauchy horizon nonlinearly stable.

Flux, energy, and physical diagnostics

In Gaussian normalization the charges are sphere fluxes,

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

Source-free Maxwell equations make them independent of the chosen homologous sphere. A magnetic potential requires overlapping gauge patches, but the field strength and metric remain regular on the overlap. This bundle issue is not a spacetime singularity.

The sign of the charge term has an energy check. Maxwell energy outside radius $r$ falls as $Q^2/(2r)$ in Gaussian units, so the gravitational mass enclosed at finite radius approaches the ADM mass from below. A negative $q^2/r^2$ term would contradict this accounting and the Einstein equations. Although the metric depends only on $Q_e^2+Q_m^2$, electrically and magnetically charged probes distinguish the two sectors.

For realistic plasmas, a large electric charge attracts opposite charges and is rapidly screened or neutralized. The family remains indispensable for extremal throats, charged wave equations, Cauchy-horizon dynamics, and the conditional uniqueness results reviewed by Chruściel, Costa, and Heusler 2012, §§ 2–3. Those uses follow from exact theory status, not from claiming that an observed black hole is electromagnetically near extremality.

Controlled limits

Neutral limit. Taking $q\to0$ at fixed $M$ gives $r_+\to2m$, $r_-\to0$, $F\to0$, and the Schwarzschild geometry. The inner root collapses onto the curvature singularity rather than becoming a second Schwarzschild horizon.

Extremal limit. Taking $q^2\to m^2$ from below gives $f=(1-m/r)^2$ and $\kappa\to0$. The proper spatial distance down the exterior throat diverges on a static slice; the throat geometry requires a scaled limit.

Weak-charge limit. For $q^2/m^2\ll1$, $r_+=2m-q^2/(2m)+O(q^4/m^3)$ and $r_-=q^2/(2m)+O(q^4/m^3)$. Charge decreases the outer horizon radius at fixed ADM mass.

The superextremal regime is physically distinct: it has no event horizon. Quantum or matter effects may forbid or alter attempts to realize it, but that question is not decided by the stationary metric.

Connections to the detailed theory

The exact theory contains an electromagnetic field everywhere outside the horizon; “electrovacuum” does not mean $F=0$. A detailed derivation and flux normalization appear in The Reissner–Nordstrom Solution. The causal and mechanical root data are organized in Reissner–Nordström Horizon Anatomy. Gauge-invariant dynamics are treated in Charged Spherical Black Holes and Coupled Sectors.

References

• Bičák, Jiří. “The Role of Exact Solutions of Einstein’s Equations in the Developments of General Relativity and Astrophysics.” Lecture Notes in Physics 540 (2000): 1–126. arXiv.
• Chruściel, Piotr T., João Lopes Costa, and Markus Heusler. “Stationary Black Holes: Uniqueness and Beyond.” Living Reviews in Relativity 15 (2012): 7. DOI.
• Moncrief, Vincent. “Odd-Parity Stability of a Reissner–Nordström Black Hole.” Physical Review D 12 (1975): 1526–1537. DOI.