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 calculator turns that story into numbers you can inspect quickly. Enter the black hole mass in solar masses and the charge in coulombs, and the page reports the outer and inner horizon radii, the surface gravity κ, the Hawking temperature T_H, the horizon area, the classical extremal charge Q_max, a charge ratio q, and the page’s horizon-potential readout Φ. The goal is not to pretend that astrophysical black holes usually keep enormous net charge—they probably do not—but to let you explore the mathematics cleanly and see how close a given scenario sits to the extremal boundary.

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 the inputs correctly

The form below has only two fields, and that simplicity is a feature. Mass M is entered in solar masses, not kilograms. The script converts your value to SI units internally using the solar-mass constant. Charge Q is entered directly in coulombs. The output textarea is not a third input; it is the report area where the calculation summary appears after you click 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. 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 values, use the result panel as a quick sanity 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 correctly says that no horizon forms. For accepted values, you should expect the following trends: increasing mass makes the horizon scale larger, increasing charge pulls the outer horizon inward and pushes the inner horizon outward, and pushing the charge close to the extremal limit drives the surface gravity and Hawking temperature downward.

Formulas used by the calculator

At the most abstract level, any calculator maps a set of inputs to a set of outputs. The original page already included that general mathematical idea, and it is still a useful way to frame what happens here: the tool applies a fixed function to the mass and charge you supply, then formats the outputs in readable units.

For this specific problem, the function f is the Reissner–Nordström geometry. The two horizon radii come from solving the metric condition that marks horizon locations. In SI units, the calculator uses the same expressions already encoded in the script:

The quantity under the square root is the decisive part. 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 extremal limit itself is reported through Q_max. In the script, the maximum allowed charge magnitude for a horizon to exist is

Once the horizons are known, the page computes the surface gravity κ and then the Hawking temperature from κ. The horizon area comes from the outer horizon radius alone. Although the calculation is more geometric than a simple accounting model, it can still be helpful to remember the second generic MathML block from the original page: many calculators reduce complexity by combining a small number of weighted ingredients in a repeatable way.

Here the same modeling lesson applies in a more physical form: constants such as G, c, ε₀, ħ, and k_B set the unit conversions and physical scales, while your two inputs determine where the black hole sits inside the permitted parameter range.

Worked example

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.

That example is useful because it 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 now increase the charge further while keeping the mass fixed, the outer horizon keeps shrinking, the inner horizon keeps moving outward, and the two radii draw together. Near extremality, the separation becomes small and the temperature tends toward zero.

You do not need to calculate those numbers by hand every time. The point of the example is to give you a reference picture so that your own runs feel familiar instead of mysterious. If you enter a large mass and a small charge, you should see a nearly Schwarzschild result. If you deliberately enter a charge near the extremal value, you should see the two horizons crowd together and the thermal outputs fall sharply.

How to read the output report

The calculator prints the results in a compact report so you can copy them into notes or a problem set. r_plus is the outer event horizon radius and r_minus is the inner Cauchy horizon radius. κ is the surface gravity in s⁻¹. T_H is the Hawking temperature in kelvin. Horizon area is based on the outer horizon only, because that is the physically relevant horizon area for black-hole thermodynamics in this context. Q_max is the extremal charge threshold used for the horizon-existence test, and q is the reported charge ratio relative to that threshold.

A few quick interpretation checks make the output much easier to trust. If you set the charge to zero, r₋ should go to zero and the outer horizon should match the Schwarzschild radius. If you keep the mass fixed and increase the magnitude of the charge, the outer horizon should decrease while the inner horizon increases. If you drive the charge close to Q_max, the horizon separation should shrink, the surface gravity should drop, and the Hawking temperature should become even smaller than it already is for astrophysical black holes. Those trends matter more than memorizing any one number.

The textarea below also makes copying convenient, but remember that the copyable report is only as meaningful as the model behind it. 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 many educational or comparative uses that is exactly the right level of simplification, but it is still a simplification.

Assumptions, limits, and physically realistic expectations

The biggest assumption is the absence of spin. Real astrophysical black holes are usually modeled with the Kerr or Kerr–Newman family rather than with a perfectly non-rotating charged solution. This calculator intentionally sets rotation aside so you can isolate the effect of charge alone. That makes it excellent for textbook exercises and conceptual exploration, especially when you want to understand the difference between neutral, moderately charged, and near-extremal cases without mixing in angular-momentum effects.

The second important limit is astrophysical realism. Large, long-lived net electric charges are generally not expected for real black holes because surrounding matter and plasma tend to neutralize them. So if you use an enormous charge here, think of it as an exploration of the mathematical solution, not a claim that nature commonly produces such objects. The formula is still valuable because it reveals how horizons and thermodynamic quantities would respond if the charge were dynamically important.

Finally, be aware of what the page means when it says that no horizon forms. That output is not a software error. It is the correct classical response when the chosen parameters violate the extremal bound. In practical terms, the calculator is telling you that the Reissner–Nordström metric with those numbers would be superextremal. If you want a black hole with an event horizon, reduce the charge magnitude or increase the mass until |Q| ≤ Q_max.

Three short sanity checks summarize the entire model: a larger mass makes the system bigger, a larger charge compresses the horizon gap, and the extremal limit is where that gap closes. If you keep those three ideas in mind, the numerical output becomes much easier to interpret.