Reissner–Nordström Horizon and Surface Gravity Calculator

What this calculator tells you about a charged black hole

The Reissner–Nordström solution is the standard relativistic model for a black hole that has mass and electric charge but no rotation. In other words, it is the charged counterpart of the Schwarzschild black hole. That simple change introduces a surprisingly rich structure. Instead of one horizon, the geometry can contain an outer horizon r₊ and an inner horizon r₋. As the charge grows, those two radii move toward one another. At the extremal limit they meet, and beyond that limit the classical formula has no real horizon at all.

This Reissner–Nordström calculator turns that horizon structure into numerical results. Enter black-hole mass in solar masses and charge in coulombs to obtain the outer and inner radii, surface gravity κ, Hawking temperature T_H, outer-horizon area, classical extremal charge Q_max, signed charge ratio q, and electric potential Φ at the outer horizon. It is designed for exploring the idealized geometry, not for suggesting that astrophysical black holes normally retain enormous net charges.

If you have only worked with the neutral Schwarzschild case before, the most useful mental picture is this: mass sets the overall size scale, while charge changes the horizon structure. A neutral black hole has one event horizon. A charged one can have two horizons, and as the charge approaches the maximum allowed value, the gap between them shrinks. That shrinking gap also lowers the surface gravity and, through the Hawking relation, lowers the Hawking temperature. The calculator is therefore helpful for three common tasks: checking whether a horizon exists at all, comparing how the horizon radii shift as charge changes, and seeing how the thermodynamic quantities behave near extremality.

How to enter Reissner–Nordström mass and charge inputs correctly

For this Reissner–Nordström calculation, the form has only two physical inputs. Mass M is entered in solar masses, not kilograms; the script converts it to SI units using the solar-mass constant. Charge Q is entered directly in coulombs. The output textarea is not a third input: it displays the horizon and thermodynamic summary after you select Compute.

There are two interpretation details worth keeping in mind. First, the horizon radii depend on Q², so a positive charge and a negative charge with the same magnitude produce the same r₊ and r₋. That is why the extremal test compares |Q| with Q_max. The signed charge does affect the displayed ratio q and electric potential Φ. Second, the numeric scale of the charge can be surprisingly large in SI units. A charge that looks modest in everyday laboratory language is tiny compared with the extremal charge of a stellar-mass black hole, while a charge large enough to noticeably change the geometry is already far beyond what astrophysical objects are expected to hold for long.

When you test Reissner–Nordström values, use the result panel as a horizon check. A mass of zero or less is not physical in this model and is rejected. If the magnitude of the charge exceeds the extremal limit, the calculator says that no horizon forms. For accepted values, increasing mass makes the horizon scale larger; increasing charge pulls the outer horizon inward and pushes the inner horizon outward; and charge close to extremality drives the surface gravity and Hawking temperature downward.

Reissner–Nordström formulas used by the calculator

The calculator evaluates the Reissner–Nordström horizon equation in SI units from the mass M and electric charge Q you enter. The two roots of that equation are the outer and inner horizon radii:

The quantity under the square root determines the Reissner–Nordström horizon type. If it is positive, two distinct horizons exist. If it is zero, the black hole is extremal and the horizons merge. If it is negative, the horizon radii are not real, so the classical metric parameters correspond to a superextremal case rather than a black hole with an event horizon.

The Reissner–Nordström extremal bound is reported as Q_max. The maximum charge magnitude compatible with a horizon is

After finding the horizons, the calculator obtains the surface gravity from their separation and the outer radius, then obtains Hawking temperature from that surface gravity. It uses the outer horizon for both the area and the electric potential:

These relations show why the extremal boundary matters. The constants G, c, ε₀, ℏ, and k_B establish the physical scale, but the relative size of charge and Q_max controls the horizon separation. As that separation closes, κ and T_H tend to zero in the classical expression.

Worked example: a 10-solar-mass charged Reissner–Nordström black hole

Suppose you choose a mass of 10 solar masses. The calculator converts that to SI units and computes an extremal charge of about 1.71 × 10²¹ C. If you then choose a charge equal to 30% of that limit, the normalized charge is q ≈ 0.30. In that case the horizons are still well separated. The outer horizon comes out close to 2.89 × 10⁴ m, or about 28.9 km, while the inner horizon is only about 6.8 × 10² m. The area based on the outer horizon is roughly 1.0 × 10¹⁰ m².

This 10-solar-mass Reissner–Nordström example shows the qualitative behavior without pushing too close to the edge. Compared with the neutral Schwarzschild value for the same mass, the outer horizon is a little smaller, the inner horizon is no longer zero, and the Hawking temperature is slightly reduced. If you increase the charge while holding the mass fixed, the outer horizon keeps shrinking, the inner horizon keeps moving outward, and the two radii draw together. Near extremality, their separation becomes small and the temperature tends toward zero.

You do not need to calculate those Reissner–Nordström quantities by hand every time. This example supplies a reference picture for your own runs. A large mass with a small charge produces a nearly Schwarzschild result, whereas a charge near the extremal value crowds the two horizons together and sharply lowers the thermal outputs.

How to read the Reissner–Nordström output report

The Reissner–Nordström output report labels r_plus as the outer event-horizon radius and r_minus as the inner Cauchy-horizon radius. κ is the surface gravity in s⁻¹, and T_H is the Hawking temperature in kelvin. Horizon area uses the outer horizon, which is the relevant area for black-hole thermodynamics here. Q_max is the extremal threshold used for the horizon-existence check, q is charge divided by that threshold, and Φ is the electric potential at the outer horizon in volts.

A few Reissner–Nordström checks make the report easier to interpret. At zero charge, r₋ should be zero and the outer horizon should match the Schwarzschild radius. At fixed mass, increasing the magnitude of charge decreases the outer radius and increases the inner radius. As the charge approaches Q_max, the horizon separation, surface gravity, and Hawking temperature all decrease. Those trends are more useful checks than any isolated output value.

The textarea makes the calculated Reissner–Nordström summary convenient to copy, but the report remains an idealized model. This page assumes a stationary, non-rotating, charged black hole in the classical Reissner–Nordström framework. It does not include angular momentum, surrounding plasma, accretion physics, discharge processes, quantum-gravity corrections, or dynamical instabilities of the inner horizon. For educational comparisons, that deliberate simplification is often useful.

Reissner–Nordström assumptions, limits, and realistic expectations

The central Reissner–Nordström assumption is zero spin. Real astrophysical black holes are usually modeled with the Kerr or Kerr–Newman family rather than as perfectly non-rotating charged objects. This calculator sets rotation aside to isolate the geometric effect of charge. That makes it suitable for textbook exercises and conceptual comparisons among neutral, moderately charged, and near-extremal cases.

The second limit concerns astrophysical realism. Large, persistent net electric charges are generally not expected for real black holes because surrounding matter and plasma tend to neutralize them. An enormous charge entered here should therefore be treated as a probe of the mathematical solution, not as a claim about a common astronomical object. The solution remains valuable because it displays how horizons and thermodynamic quantities respond when charge becomes dynamically important.

Finally, a Reissner–Nordström report that says no horizon forms is not a software failure. It is the classical result when the chosen parameters violate the extremal bound. With those numbers, the metric would be superextremal. To return to a black-hole case with an event horizon, reduce the charge magnitude or increase the mass until |Q| ≤ Q_max.

Three concise checks summarize the Reissner–Nordström model: greater mass enlarges the system, greater charge narrows the gap between its horizons, and the extremal limit is where that gap closes. Keeping those relationships in mind makes the numerical report easier to assess.