Reissner–Nordström Horizon and Surface Gravity Calculator
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 produces a metric with two horizons: an outer event horizon and an inner Cauchy horizon. Their locations depend on both mass and charge. If the charge exceeds a critical value the horizons disappear, leaving a naked singularity and hinting at a violation of cosmic censorship. This calculator explores those features quantitatively by computing the horizon radii, the surface gravity at the outer horizon, and the associated Hawking temperature.
In geometric units where , the radii obey . Restoring units yields . The discriminant must be nonnegative; the maximal charge without exposing the singularity is .
The surface gravity sets the Hawking temperature through . In geometric units, , and restoring constants gives . The calculator evaluates these expressions directly in SI units, returning radii in meters, surface gravity in s⁻¹, and Hawking temperature in kelvins.
Charged black holes provide a laboratory for studying cosmic censorship, inner-horizon stability, and extremal remnants. Although astrophysical black holes likely neutralize quickly, miniature or primordial black holes could retain charge. As approaches , the surface gravity drops to zero and the Hawking temperature vanishes, highlighting possible stable remnants.
Dimensionless charge ratio
The ratio measures proximity to extremality. Values far below one indicate a nearly Schwarzschild object, while near unity signals a near-extremal hole with a dramatically reduced surface gravity. The calculator reports both and for quick intuition.
Horizon area also plays a central role in black hole thermodynamics. The outer area determines the Bekenstein–Hawking entropy. As charge increases the area shrinks, illustrating how extremal black holes carry minimal entropy.
Sample outputs
The table summarizes horizon radii, surface gravity, temperature, and area for a 10 M☉ black hole with varying charge. Charges exceeding ≈ 8.6×1020 C for this mass yield no horizon.
| 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 reshapes the near-horizon environment. Charged black holes evaporate by emitting both neutral and charged particles, gradually discharging toward the Schwarzschild case. The Reissner–Nordström metric also serves as a foundation for the Kerr–Newman solution that adds rotation.
Electric potential at the horizon, , governs how charged particles interact with the black hole. Combining with the surface gravity recovers the generalized first law, , which the calculator’s outputs let you check numerically.
Continue exploring spacetime physics with the Schwarzschild radius calculator, map rotating horizons using the Kerr–Newman horizon tool, or compare thermal emission via the Hawking Page transition calculator.