The Charged Black Hole
Deriving the Reissner-Nordstrom solution from the Einstein-Maxwell equations
Introduction
The Reissner-Nordstrom (RN) solution, discovered independently by Hans Reissner (1916) and Gunnar Nordstrom (1918), describes the spacetime geometry of a static, spherically symmetric, electrically charged black hole. While astrophysical black holes are expected to be nearly electrically neutral (due to rapid discharge by accreting plasma), the RN solution is of profound theoretical importance. It provides the simplest example of a black hole with multiple horizons, an inner Cauchy horizon, and an extremal limit — features that carry over to the astrophysically relevant Kerr solution.
The RN solution is also central to string theory and the attractor mechanism, where extremal RN black holes serve as the prototypical examples for understanding microscopic entropy counting.
The Einstein-Maxwell System
The Reissner-Nordstrom solution arises from coupling gravity to electromagnetism. The total action is:
$$S = \int d^4x\,\sqrt{-g}\left[\frac{R}{16\pi G} - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right]$$
Einstein-Hilbert action plus the Maxwell field action
Variation with respect to the metric yields the Einstein equations with electromagnetic source:
$$G_{\mu\nu} = 8\pi G\,T_{\mu\nu}^{(\text{EM})}$$
where the electromagnetic stress-energy tensor is
$$T_{\mu\nu}^{(\text{EM})} = \frac{1}{4\pi}\left(F_{\mu\alpha}F_\nu^{\ \alpha} - \frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right)$$
Variation with respect to the gauge potential \( A_\mu \) gives Maxwell's equations in curved spacetime:
$$\nabla_\mu F^{\mu\nu} = 0, \qquad \nabla_{[\mu}F_{\nu\rho]} = 0$$
Maxwell's equations in curved spacetime (source-free exterior)
Deriving the Metric
We impose spherical symmetry and staticity. The most general static, spherically symmetric metric ansatz is:
$$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2\,d\Omega^2$$
where \( d\Omega^2 = d\theta^2 + \sin^2\theta\,d\phi^2 \) and \( f(r) \) is to be determined
For a purely electric charge, the electromagnetic field has only one independent component:\( F_{tr} = -F_{rt} = E(r) \). The Maxwell equation \( \nabla_\mu F^{\mu\nu} = 0 \)in the spherically symmetric background gives:
$$\frac{1}{\sqrt{-g}}\partial_r\left(\sqrt{-g}\,F^{tr}\right) = 0 \implies F^{tr} = \frac{Q}{r^2}$$
Coulomb's law in curved spacetime: the electric field of a point charge
Here \( Q \) is the electric charge, determined by Gauss's law:\( Q = \frac{1}{4\pi}\oint F^{tr}\,r^2\sin\theta\,d\theta\,d\phi \). Substituting into the electromagnetic stress-energy tensor, we find:
$$T^t_{\ t} = T^r_{\ r} = -\frac{Q^2}{8\pi r^4}, \qquad T^\theta_{\ \theta} = T^\phi_{\ \phi} = +\frac{Q^2}{8\pi r^4}$$
The EM stress-energy is traceless (\( T^\mu_{\ \mu} = 0 \)), as required for electromagnetism
The Einstein equation \( G^t_{\ t} = 8\pi G\,T^t_{\ t} \) becomes:
$$\frac{1}{r^2}\frac{d}{dr}\left[r(1 - f)\right] = \frac{GQ^2}{r^4}$$
A first-order ODE for the metric function \( f(r) \)
Integrating and imposing asymptotic flatness (\( f \to 1 \) as \( r \to \infty \)) with mass parameter \( M \) yields the Reissner-Nordstrom metric function:
$$f(r) = 1 - \frac{2GM}{r} + \frac{GQ^2}{r^2}$$
The Reissner-Nordstrom metric function (in Gaussian units with \( c = 1 \))
In geometric units (\( G = c = 1 \)), this simplifies to the elegant form:
$$\boxed{ds^2 = -\left(1 - \frac{2M}{r} + \frac{Q^2}{r^2}\right)dt^2 + \left(1 - \frac{2M}{r} + \frac{Q^2}{r^2}\right)^{-1}dr^2 + r^2\,d\Omega^2}$$
The complete Reissner-Nordstrom metric in geometric units
The Two Horizons
The horizons occur where \( f(r) = 0 \), i.e., where \( g_{tt} = 0 \)(equivalently, where \( g^{rr} = 0 \)). Solving the quadratic equation\( r^2 - 2Mr + Q^2 = 0 \):
$$r_\pm = M \pm \sqrt{M^2 - Q^2}$$
\( r_+ \): outer (event) horizon, \( r_- \): inner (Cauchy) horizon
The metric function can be factored as:
$$f(r) = \frac{(r - r_+)(r - r_-)}{r^2}$$
Making manifest the zero structure of \( f(r) \)
Surface Gravity and Temperature
The surface gravity \( \kappa \) at each horizon is computed from the general formula\( \kappa^2 = -\frac{1}{2}(\nabla_\mu\xi_\nu)(\nabla^\mu\xi^\nu) \) evaluated at the horizon, where \( \xi = \partial_t \) is the timelike Killing vector. For a metric of the form \( f(r) \), this reduces to:
$$\kappa_\pm = \frac{1}{2}\left|f'(r_\pm)\right| = \frac{r_+ - r_-}{2r_\pm^2} = \frac{\sqrt{M^2 - Q^2}}{r_\pm^2}$$
Surface gravity at the outer and inner horizons
By the Hawking effect, each horizon has an associated temperature:
$$T_\pm = \frac{\kappa_\pm}{2\pi} = \frac{\sqrt{M^2 - Q^2}}{2\pi\,r_\pm^2} = \frac{r_+ - r_-}{4\pi\,r_\pm^2}$$
Hawking temperature of each horizon (in natural units \( \hbar = k_B = 1 \))
Three Qualitative Cases
The character of the RN solution depends crucially on the ratio \( |Q|/M \):
Case 1: Sub-extremal (\( |Q| < M \))
Two distinct horizons exist at \( r_+ > r_- > 0 \). The outer horizon \( r_+ \)is the event horizon — the point of no return for external observers. The inner horizon\( r_- \) is the Cauchy horizon, a surface beyond which the future is no longer uniquely determined by initial data. Between the horizons,\( f(r) < 0 \), so \( r \) becomes timelike and \( t \) becomes spacelike. The singularity at \( r = 0 \) is timelike (unlike Schwarzschild where it is spacelike), meaning infalling observers can in principle avoid it.
Note: For \( Q = 0 \), we recover Schwarzschild with \( r_+ = 2M \) and \( r_- = 0 \).
Case 2: Extremal (\( |Q| = M \))
The two horizons merge into a single degenerate horizon at \( r_+ = r_- = M \). The metric function becomes a perfect square:
$$f(r) = \left(1 - \frac{M}{r}\right)^2$$
The surface gravity vanishes (\( \kappa = 0 \)), so the Hawking temperature is zero. The extremal RN black hole does not radiate thermally. This is intimately connected to the third law of black hole thermodynamics and to supersymmetric (BPS) states in string theory. The near-horizon geometry of extremal RN is \( \text{AdS}_2 \times S^2 \), a fact of enormous importance in the attractor mechanism.
Case 3: Super-extremal (\( |Q| > M \))
No horizons exist (\( M^2 - Q^2 < 0 \) gives complex roots). The singularity at\( r = 0 \) is globally naked — visible to all observers. The metric function\( f(r) > 0 \) everywhere, so there is no horizon to shield the singularity. This scenario is believed to be forbidden by the weak cosmic censorship conjecture (Penrose, 1969), which states that singularities forming from generic gravitational collapse are always hidden behind event horizons.
The super-extremal RN solution does not represent a black hole but rather a naked singularity with a repulsive gravitational core at small \( r \).
The interplay between charge and mass in the RN solution illustrates a deep principle: the electrostatic repulsion of the charge fights against gravitational collapse, and at\( |Q| = M \) the two forces exactly balance. This balance is a manifestation of the BPS (Bogomolny-Prasad-Sommerfield) bound in supergravity, where extremal black holes saturate a mass-charge inequality imposed by supersymmetry.