Charged Black Holes in General Relativity

The Reissner–Nordström solution extends the familiar Schwarzschild black hole to include electric charge. In Einstein’s field equations coupled to electromagnetism, a spherically symmetric mass with charge Q produces a metric characterized by two horizons: an outer event horizon and an inner Cauchy horizon. The locations of these horizons depend on both the mass and the charge of the black hole. If the charge exceeds a critical value, the horizons disappear entirely, leaving a naked singularity and signaling a potential violation of cosmic censorship. This calculator allows users to explore these features quantitatively, computing the horizon radii, the surface gravity associated with the outer horizon, and the corresponding Hawking temperature.

In geometric units where $G=c=1$, the Reissner–Nordström metric for a black hole of mass M and charge Q yields horizon radii $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. When translated to SI units, the expression becomes $r_{\pm }=\frac{\mathrm{GM}}{c^2}\pm \sqrt{{\frac{\mathrm{GM}}{c^2}}^2-\frac{G}{4\pi {\epsilon }_0}{Q}^2}$. The term under the square root must remain nonnegative; otherwise the solution represents a naked singularity. The maximal charge without exposing the singularity is $Q{max}_{}=\sqrt{4\pi {\epsilon }_{0}G$ M_2$).\; Our\; script\; evaluates\; these\; expressions\; directly\; in\; SI\; units,\; taking\; mass\; in\; solar\; masses\; and\; charge\; in\; coulombs,\; providing\; results\; in\; meters.}$

The surface gravity κ associated with the outer horizon governs the gravitational acceleration experienced by an observer hovering just above it and sets the black hole’s Hawking temperature via $T=\frac{ħ\kappa }{2\pi k\mathrm{_B}}$. For the Reissner–Nordström metric the surface gravity in geometric units is $\kappa =\frac{r_{+}}{-}$. Converting to SI units introduces factors of G and c: $\kappa =\frac{c^4}{2GM{r_{+}}^2(r_{+}-r_{-})}$. The Hawking temperature follows immediately by multiplying κ by ħ/(2π k_B). For an uncharged Schwarzschild black hole, Q = 0, the familiar results for r_s = 2GM/c² and κ = c⁴/(4GM) are recovered.

Charged black holes offer a theoretical laboratory for exploring concepts such as cosmic censorship, mass inflation, and the stability of inner horizons. Although astrophysical black holes are expected to be nearly neutral due to charge neutralization by surrounding plasma, hypothetical mini black holes or primordial remnants might retain significant charge. In quantum gravity discussions, the extremal limit where Q approaches Q_max plays a special role because the Hawking temperature vanishes, hinting at stable remnants and connections to supersymmetric solutions. The calculator enables users to probe this extremal regime and see how the surface gravity drops to zero as Q approaches Q_max.

The dimensionless charge ratio $q=\frac{Q}{Q_{\max }}$ captures how close a black hole is to the extremal limit. Values much less than one indicate a nearly Schwarzschild object, while $q$ approaching unity signals a near-extremal hole with dramatically reduced surface gravity. Our calculator reports both $\mathrm{Q\_\{max\}}$ and $q$, providing immediate intuition about the charge regime for a given mass.

Horizon area also plays a central role in black hole thermodynamics. The outer horizon’s area $A=4\pi {r_{+}}^2$ determines the Bekenstein–Hawking entropy. As charge increases and $r_{+}$ shrinks, the horizon area drops, illustrating how extremal black holes possess minimal entropy.

To illustrate, the table below lists horizon radii, surface gravity, temperature, and area for a 10 M_☉ black hole with varying charge. The example demonstrates the shrinkage of the outer horizon and the decline in κ as charge increases. Values of Q exceeding Q_max ≈ 8.6×10²⁰ C for this mass yield no horizon, so such cases are physically excluded in classical general relativity.

Q (C)	r+ (km)	κ (s−1)	TH (K)	A (km²)
0	29.5	1.5×104	6×10−9	10,900
5×1020	25.2	9×103	4×10−9	8,000
8×1020	11.7	2×103	1×10−9	1,700

These figures emphasize how electric charge modifies the near-horizon environment. In semiclassical gravity, charged black holes evaporate by emitting both neutral and charged particles, gradually discharging and moving toward the Schwarzschild case. Nevertheless, understanding the charged solution remains important for theoretical consistency and for potential exotic scenarios where significant charge persists. The Reissner–Nordström metric also serves as a stepping stone toward more complex solutions such as the Kerr–Newman black hole, which includes both charge and rotation. By gaining intuition with this simpler case, one can better appreciate the rich structure of spacetime in general relativity.

Electric potential at the horizon $\Phi =\frac{Q}{r_{+}}$ governs how charged particles interact with the black hole. A high potential can accelerate particles to relativistic speeds as they fall in. Combining $\Phi$ with the surface gravity reveals the generalized first law of black hole mechanics, $+\Phi \; dQ$. The calculator’s extra outputs help students verify this relation numerically, bridging abstract theory with tangible numbers.

For researchers interested in astrophysical observations, even a tiny charge fraction could leave imprints on the dynamics of charged particles near the horizon or on gravitational wave signals from black hole mergers. By experimenting with different $q$ values and recording the copyable outputs, one can set up parameter studies for hypothetical charged black holes in numerical simulations or classroom exercises.