Reissner–Nordström Horizon Anatomy

Electric charge changes Schwarzschild horizon anatomy before it changes any symmetry: Reissner–Nordström remains static and spherically symmetric, but its horizon equation can have two, one, or no positive roots. The outer root is the equilibrium event horizon, the inner root is a Cauchy horizon in the idealized maximal extension, and their merger marks extremality. These labels encode different causal roles and should not be collapsed into “two event horizons.”

Helpful background. Exact Solutions of Einstein’s Equations supplies the Einstein–Maxwell family and its extension; Causality and Global Structure supplies event and Cauchy horizons; The Einstein Equation fixes the gravitational dynamics; Killing Fields and Isometry Groups supplies static symmetry; and Black-Hole Geometry and Mechanics orients the horizon distinctions.

The charged horizon polynomial

Work in four-dimensional, asymptotically flat Einstein–Maxwell theory with Maxwell Lagrangian density $-F_{\mu\nu}F^{\mu\nu}/4$. Define the electric charge by

$$Q=\int_{S^2_\infty} *F,$$

so that a purely electric potential may be chosen as $A=-Q\,dt/(4\pi r)$. With $G$ explicit, the static line element has

$$f(r)=1-\frac{2GM}{r}+\frac{GQ^2}{4\pi r^2}.$$

It is useful to define the length-squared parameter

$$q^2\equiv\frac{GQ^2}{4\pi}.$$

The candidate Killing horizons are the roots of $f=0$, equivalently

$$r^2-2GMr+q^2=0, \qquad r_\pm=GM\pm\sqrt{G^2M^2-q^2}.$$

The discriminant organizes three regimes:

• If $G^2M^2\gt q^2$, then $0\lt r_-\lt r_+$ and the black hole is nonextremal.
• If $G^2M^2=q^2$, the double root $r_-=r_+=GM$ is extremal.
• If $G^2M^2\lt q^2$, this exact solution has no real horizon root and its singularity is not hidden by an event horizon.

The last statement classifies the Reissner–Nordström family. It does not prove that regular collapse can dynamically produce an overcharged naked singularity. That is a separate global evolution question.

The geometry depends on $Q^2$, so reversing the electric charge leaves the metric and both root locations unchanged. The electromagnetic field itself does change sign, as does its electrostatic potential. Moreover, the Maxwell stress tensor is tracefree, so the spacetime Ricci scalar vanishes away from the source even though the Ricci tensor does not. A zero Ricci scalar is therefore not evidence that the charged exterior is vacuum. Curvature invariants remain finite at nonzero horizon roots and diverge at $r=0$.

Outer event horizon and inner Cauchy horizon

In the nonextremal maximal extension, the region $r\gt r_+$ is static because $f\gt0$ and the asymptotically normalized Killing field $\xi=\partial_t$ is timelike. The future surface $r=r_+$ bounds the causal past of the chosen future null infinity, so it is both an event horizon and a Killing horizon. The domain of outer communications is the connected exterior associated with that asymptotic end.

Between the roots, $f\lt0$. The areal-radius gradient is timelike, and every future-directed causal curve in the black-hole sector moves toward smaller $r$. At $r=r_-$ the static Killing field is null again, but this surface is not the outer boundary of escape to the same infinity. In the exact maximal solution it is a Cauchy horizon. Data on a suitable partial Cauchy hypersurface determine the fields only within its domain of dependence; its inner null boundary is the future Cauchy horizon, and those data do not fix an extension beyond it. This predictability statement is summarized explicitly by Hollands, Wald, and Zahn 2020, §1, pp. 1–3.

Calling $r_-$ an “inner horizon” is therefore descriptive, whereas calling it a Cauchy horizon states its predictability role. Its smooth appearance in the stationary exact metric is also not a stability claim. Perturbations can be strongly blueshifted there, and the dynamical interior belongs to black-hole dynamics and perturbation theory rather than this equilibrium anatomy page. The distinction between stationary horizon data and a theorem about complete evolution is central to the electrovacuum discussion in Chruściel, Costa, and Heusler 2012, §§2.2 and 3.2.

Areas, potentials, and signed surface gravities

Each spherical root has area

$$\mathcal A_\pm=4\pi r_\pm^2.$$

Using $r_++r_-=2GM$ and $r_+r_-=q^2$, factor the metric function as

$$f(r)=\frac{(r-r_+)(r-r_-)}{r^2}.$$

For the same asymptotically normalized generator $\xi=\partial_t$, the signed surface gravities are

$$\kappa_\pm=\frac12 f'(r_\pm) =\frac{r_\pm-r_\mp}{2r_\pm^2}.$$

Thus $\kappa_+\gt0$ and $\kappa_-\lt0$ in this convention. The negative inner value does not mean a negative local acceleration measured by every observer; it records the orientation and the use of one globally normalized Killing field at both components. Many applications quote $|\kappa_-|$, so a formula without its convention is incomplete.

In the gauge that vanishes at infinity, the electric potential entering stationary mechanics at the outer horizon is

$$\Phi_H=-\xi^\mu A_\mu\big|_{r_+}=\frac{Q}{4\pi r_+}.$$

Its numerical normalization changes if the Maxwell action or charge flux is normalized differently. The mass-variation role of $\Phi_H$ and the horizon area is developed by Bardeen, Carter, and Hawking 1973, pp. 164–166; the formulas here have been written in the rationalized charge convention stated above. The exact Gaussian-to-rationalized Maxwell map must be applied before comparing them with Gaussian-normalized exact-spacetime or Atlas formulas.

Limits that discriminate the anatomy

The neutral limit is subtler than merely setting $Q=0$ in every label. The roots become

$$r_+\longrightarrow 2GM, \qquad r_-\longrightarrow 0.$$

The outer data reduce smoothly to Schwarzschild: $\mathcal A_+\to16\pi G^2M^2$ and $\kappa_+\to1/(4GM)$. The inner root collapses onto the curvature singularity; it does not survive as a second Schwarzschild horizon.

At extremality, $r_+=r_-=GM$, the factorization has a double zero and $\kappa_+=0$. The exterior remains static, but the horizon is degenerate and the global extension is not the smooth limit of every nonextremal structure: in particular, the usual bifurcation two-sphere is absent. The limiting area is $4\pi G^2M^2$, one quarter of the Schwarzschild area at the same mass.

These checks distinguish root multiplicity, causal role, and normalization. A coordinate transformation can regularize metric components at either simple root, but it cannot turn the inner Cauchy boundary into the outer event horizon or create a root in the overcharged regime.

Check the three charge regimes

Let $q^2=GQ^2/(4\pi)$ and suppose first that $q^2=3G^2M^2/4$. Compute $r_\pm$ and the signs of $\kappa_\pm$. Then take the neutral and extremal limits. Why does the existence of real roots not by itself identify their global causal roles?

Solution

The square root is $GM/2$, so $r_+=3GM/2$ and $r_-=GM/2$. Therefore

$$\kappa_+=\frac{GM}{2(3GM/2)^2}=\frac{2}{9GM}\gt0, \qquad \kappa_-=-\frac{GM}{2(GM/2)^2}=-\frac{2}{GM}\lt0.$$

As $q\to0$, the roots approach $2GM$ and $0$, and only the outer one remains a regular Schwarzschild horizon. As $q^2\to G^2M^2$, both roots approach $GM$ and both signed surface gravities approach zero. Solving $f=0$ proves that the static Killing field becomes null on candidate hypersurfaces. Event-horizon and Cauchy-horizon labels additionally require the causal structure of the chosen global extension.

Kerr Horizon and Ergoregion Anatomy next replaces electric charge by rotation and separates the horizon from the static limit.

References

• Bardeen, James M., Brandon Carter, and Stephen W. Hawking. “The Four Laws of Black Hole Mechanics.” Communications in Mathematical Physics 31 (1973): 161–170. DOI.
• Chruściel, Piotr T., João Lopes Costa, and Markus Heusler. “Stationary Black Holes: Uniqueness and Beyond.” Living Reviews in Relativity 15 (2012): 7. DOI.
• Hollands, Stefan, Robert M. Wald, and Jochen Zahn. “Quantum Instability of the Cauchy Horizon in Reissner–Nordström–de Sitter Spacetime.” Classical and Quantum Gravity 37 (2020): 115009. DOI. Open manuscript.