Black Hole Parameters

Introduction

This calculator will calculate the properties of a black hole described by given parameters (mass, charge, angular momentum), or the mass of a black hole possessing given properties; update one of the values below, and the others will recalculate.

A black hole is described (exactly) by only its mass, charge, and angular momentum. The solutions to the Einstein-Maxwell equations of general relativity can be categorized by the assumptions made:

	Massless		Massive
	Non-Rotating	Rotating	Non-Rotating	Rotating
Uncharged	Minkowski		Schwarzschild	Kerr
Charged			Reissner-Nordström	Kerr-Newman

Clearly, Kerr-Newman is the most general (and is what this calculator calculates). However, since the universe appears to be electrically balanced (or nearly so), it is expected that most real black holes are Kerr (or nearly so).

For certain values of angular momentum or charge, a point is reached (an extremal black hole) where the equations break down (the event horizon calculation becomes complex-valued). By most interpretations, this results in a so-called naked singularity—a condition many think is impossible (although research into what it would mean continues). Consequently, when such a condition occurs, this calculator will automatically clamp the values to the feasible range.

This calculator has been validated against several reference solutions to Schwarzschild black holes, and the equations check out—dimensionally and otherwise. However, I haven't found examples given for charged or rotating black holes, worked out to actual numbers. Hence, although all calculations are done fully generally, results for non-Schwarzschild holes should be taken cautiously. Equations given elsewhere are also typically in some variety of dimensionless natural units, which makes getting real numbers and checking them more difficult. Your examples and corrections are always welcome.

Calculator

Parameters:

Mass:

	(mass \(M\) (mass units))
	(mass (energy units))

Angular Momentum:

	(angular momentum, \(J\))
	(velocity at Cauchy horizon, \(\Omega_-\)) ^[1]
	(velocity at event horizon, \(\Omega_+\)) ^[1]
	(Kerr parameter, \(a\))
	(dimensionless Kerr parameter, \(\chi\))

Charge:

	(charge, \(Q\))
	(extremal fraction)

Properties:

Schwarzschild radius ^[2]:

(radius, \(\rS\))

Cauchy Horizon ^[3]:

(radius, \(r_-\))

Event Horizon ^[3]:

	(radius, \(r_+\))
	(surface area, \(A_H\) ^[4])
	(surface gravity as measured at infinity, \(\kappa\) ^[5])

Ergosurfaces ^[6]:

(inner, \(r_{0,-}\))

(outer, \(r_{0,+}\), \(r_0\), Killing horizon, static limit)


-90°	0°	90°

(elevation angle (°) off ecliptic)

Tidal Forces ^[7]:

(distance from center)

(distance from center, Schwarzschild radii)

s⁻²

(acceleration gradient at distance)

Given an object whose base is that the above distance from the center:

	(object height)
	(tidal acceleration within object)

Entropy ^[8]:

(\(S_{BH}\))

Hawking spectrum ^[9]:

	(blackbody temperature, \(T_H\))
	(luminosity, semiclassical, \(L\))

Schwarzschild evaporation time ^[10,11]:

(\(t\))

Some Examples to Try:

Schwarzschild	Kerr	Reissner-Nordström	Kerr-Newman

	^[13]

^[14]

Definitions, Formulae, and Math:

Show Section

User-Specified Parameters:

Name	Symbol	Units
Total mass-energy	\(M\)	\(\text{(kg)}\)
Angular Momentum	\(J\)	\( \text{(kg} \cdot \text{m}^2 \cdot \text{s}^{-1} \text{)} \)
Charge	\(C\)	\(\text{(coulombs)}\)
Time	\(t\)	\(\text{(s)}\)
Latitude	\(\theta\)	\(\text{(rad)}\)
Radius	\(r\)	\(\text{(m)}\)

Constants:

Values marked with "\(=\)" are given by definition. Those marked with "\(\approx\)" are either derived from these and not representable in finite precision, or are measured empirically.

Name	Symbol	Value	Units
Gravitational Constant	\(G\)	\(\approx 6.67430 \cdot 10^{-11}\)	\( \text{(m}^3 \cdot \text{kg}^{-1} \cdot \text{s}^{-2} \text{)} \)
Vacuum Permittivity / Permittivity of Free Space / Electric Constant	\(\epsilon_0\)	\(\approx 8.8541878128 \cdot 10^{-12}\)	\( \text{(F} \cdot \text{m}^{-1} \text{)} \)
Speed of Light	\(c\)	\(=2.99792458 \cdot 10^{8}\)	\( \text{(m} \cdot \text{s}^{-1} \text{)} \)
Boltzmann Constant	\(k_B\)	\(=1.380649 \cdot 10^{-23}\)	\( \text{(J} \cdot \text{K}^{-1} \text{)} \)
Stefan-Boltzmann Constant (\(=(2 \pi^5 k_B^4)/(15 h^3 c^2)\))	\(\sigma_{sb}\)	\(\approx 5.670374 \cdot 10^{-8}\)	\( \text{(W} \cdot \text{m}^{-2} \cdot \text{K}^{-4} \text{)} \)
Coulomb's Constant	\(k_e\)	\(\approx 8.9875517923 \cdot 10^{9}\)	\( \text{(H} \cdot \text{m}^{-1} \text{)} \)
Planck Constant	\(h\)	\(= 6.62607015 \cdot 10^{-34}\)	\( \text{(J} \cdot \text{s} \text{)} \)
Reduced Planck Constant (\(=h/2\pi\))	\(\hbar\)	\(\approx 1.0545718 \cdot 10^{-34}\)	\( \text{(J} \cdot \text{s} \text{)} \)

Derived Quantities:

Name	Definition	Units
Gravitational Radius	\[\rG := \frac{M G}{c^2}\]	\(\text{(m)}\)
Specific Charge ^[12]	\begin{align} \rQ, e :=& \frac{Q}{c^2} \cdot \sqrt{\frac{G}{4 \pi \epsilon_0}}\\ =& \frac{Q}{c^2} \cdot \sqrt{k_e G} \end{align}	\(\text{(m)}\)
Mass-energy	\[E := M c^2\]	\(\text{(J)}\)
Kerr Parameter	\[a := \frac{J}{M c}\]	\(\text{(m)}\)
Kerr Parameter (Dimensionless)	\begin{align} \chi :=& \frac{a c^2}{M G}\\ =& \frac{J c}{M^2 G} \end{align}	\(\text{(none)}\)

Formulae:

Generally speaking, these formulae are hard to invert for other quantities, and this calculator makes extensive use of numerical solvers.

Name	Equation	Units
Angular Velocity (For values at Cauchy (\(\Omega_-\)) and Event (\(\Omega_+\)) horizons, use \(r=r_-\) and \(r=r_+\), respectively.)	\[\Omega(r) = \frac{a}{a^2 + r^2} \cdot c\]	\(\text{(rad} \cdot \text{s}^{-1} \text{)}\)
Schwarzschild Radius	\[\rS = 2 \cdot \rG\]	\(\text{(m)}\)
Cauchy Horizon (\(r_-\)), Event Horizon (\(r_+\))	\[r_{\pm} = \rG \pm \sqrt{ \rG^2 - a^2 - \rQ^2 }\]	\(\text{(m)}\)
Event Horizon Surface Area	\begin{align} A_H &= 8 \pi \cdot \left( \rG^2 + \rG \sqrt{\rG^2 - a^2 - \rQ^2} - \frac{1}{2} \rQ^2 \right)\\ &= 4 \pi \cdot \left( r_+^2 + a^2 \right) \end{align}	\(\text{(m}^2\text{)}\)
Event Horizon Surface Gravity ^[5]	\[\kappa = \left(\frac{G M}{\rG}\right) \left(\frac{r_+ - r_-}{2 \cdot (r_+^2 + a^2)}\right)\]	\(\text{(m} \cdot \text{s}^{-2} \text{)}\)
Ergosurfaces	\[r_{0,\pm} = \rG \pm \sqrt{\rG^2 - a^2 \cos^2 (\theta) - \rQ^2}\]	\(\text{(m)}\)
Entropy	\[S_{BH} = \frac{(1/2)\ln(2)}{4\pi} \cdot \frac{c^3}{\hbar G} \cdot A_H \cdot k_B\]	\(\text{(J} \cdot \text{K}^{-1} \text{)}\)
Blackbody Temperature \((^{\circ}K)\)	\[T_H = \left(\frac{ \hbar c }{ 4 \pi k_B }\right) \left(\frac{ \sqrt{ \rG^2 - a^2 - \rQ^2 } }{ \rG^2 + \rG \sqrt{ \rG^2 - a^2 - \rQ^2 } - \rQ^2 }\right) \]	\(\text{(K)}\)
Semiclassical Luminosity	\[L = \sigma_{sb} \cdot T_H^4 \cdot A_H\]	\(\text{(W)}\)
Schwarzschild Evaporation Time	\[t = 5120 \pi \cdot \frac{G^2 M^3}{\hbar c^4}\]	\(\text{(s)}\)

Notes:

^[1] Velocities measured from perspective of observer at infinity. Note: the Cauchy horizon's velocity doesn't make sense when it is at radius 0. This happens exactly when the black hole is Schwarzschild, and is reported as NaN.
^[2] Given a mass, the Schwarzschild radius is the radius within which if that mass were packed, the escape velocity would be the speed of light.
^[3] A black hole has two horizons, or imaginary surfaces surrounding them. The inner horizon—or Cauchy horizon—bounds a region that contains closed time-like curves. The outer horizon—or event horizon—bounds the region where light (and therefore anything else) cannot escape.
^[4] For a Schwarzschild black hole, this is the familiar surface-area-of-a-sphere formula. Otherwise, it's more complex.
^[5] Newtonian acceleration is infinity at the event horizon; hence the only useful concept is the limiting value of local proper acceleration multiplied by the gravitational time dilation factor (or equivalently, the acceleration as measured by an observer at infinity). More discussion here. I couldn't find a formula for Kerr-Newman in SI units, so I took a guess, and confirmed it agrees with the SI formula for Schwarzschild.
^[6] The ergosurface bounds the region where acceleration, possibly due to frame dragging, is so extreme, just to remain "stationary" according to an outside observer requires traveling at the speed of light. Hence, inside this region, everything moves. For a non-Schwarzschild black hole, this is different from (and outside of) the event horizon. Coincides with the Killing horizon: where the Killing vector field becomes zero.
^[7] The difference in acceleration between at different distances from any object produces an internal force, called a tidal force, corresponding to the acceleration gradient (taking the units \(s^{-2}\), best-interpreted as acceleration (\(m s^{-2}\)) per meter (\(m^{-1}\))). All treatments of black holes' tidal forces derive it from Newtonian gravity, which seems deeply suspect to me, but perhaps it is correct in some proper frame.
^[8] Black holes have Bekenstein-Hawking entropy: the entropy that is required to make thermodynamics work. It's very hard to find a formula in SI units, but eq. 17 of the 1973 paper "Black Holes and Entropy" (here) gives the correct formula (listed in the equations section). My calculator is consistent with the solar-mass black hole example from that paper, but flatly contradicts the same calculation given here and another calculator here (and they disagree with each other, too). I strongly suspect the first confused themselves with natural units, and I don't know where the second got their formula, which is probably wrong.
^[9] Hawking radiation, emitted by black holes due to quantum effects, fits a blackbody radiation spectrum—a spectrum emitted by an idealized object at a specific temperature. The "temperature" of a black hole is the temperature that a blackbody emitter emitting the same radiation would be.
^[10] Hawking radiation depletes a black hole's mass. In this way a black hole "evaporates". This is the time until completion for a Schwarzschild black hole. For a non-Schwarzschild black hole, I have not been able to find a formula. Probably, one should redo the derivation in the Kerr-Newman metric, although I don't know if it exists in closed-form. If not, it could still probably be solved numerically.
^[11] To evaporate, a black hole must emit more Hawking radiation than falls in from the cosmic microwave background radiation (2.725 K). Hence, as you can see, black holes with more than ~4.503e22 kg mass actually grow until the universe cools further. This effect is not modeled.
^[12] Elsewhere, often the definition is given for \(\rQ^2\) instead, presumably because it avoids the square root.
^[13] From the film Interstellar. Mass of \(1 \cdot 10^8\) solar masses. Plot-wise, \(\chi:=0.999\) (secondary source: \(1-1\cdot 10^{-14}\)). Rendering-wise, \(\chi:=0.6\). The definition \(\chi:=0.999\) is used.
^[14] From the excellent book The Collapsium.

Other References:

Calculator: Hawking Radiation Calculator
Another JavaScript calculator, only valid for Schwarzschild black holes, but unlike this computes tidal forces. My calculator nearly agrees with it (it appears to have dated values for \(\hbar\), \(k_B\), \(G\), and \(\sigma_{sb}\)) for everything except the entropy (I don't know where that weird "\(1/\ln(10)\)" factor comes from, and by default I multiply by Boltzmann's constant to get \(\text{(J} \cdot \text{K}^1 \text{)}\) dimensions, though this can be changed).
Book Chapter: The Kerr spacetime: A brief introduction
Nice surfaces figure (fig. 1)
Angular velocity formula (eqn. 112)
Website: scholarpedia.org
Formula for and discussion of entropy (eqn. 1).
Formula for and discussion of Kerr-Newman surface area (eqn. 4)
Formula for and discussion of Kerr-Newman temperature (eqn. 11)
Paper: Surface properties of Kerr–Newman black holes
Appears to give lots of formulae in SI units, but paywalled so can't see how good it is.
Paper: ISCO, Lyapunov exponent and Kolmogorov-Sinai entropy for Kerr-Newman Black hole
Implicit definition of innermost stable circular orbit (eqn. 64).
Paper: Foundations of Black Hole Accretion Disk Theory
Formulae for orbital radii.
Calculator: Kerr Calculator V2
Calculates orbits.
Discussion of orbits.
Website: nus.edu
Plots of photon sphere orbits around Kerr black hole.
Book Chapter: Introduction to Black Hole Astrophysics (Chapter 2)
Lots of good diagrams.
Paper: Partition Function of the Schwarzschild Black Hole
Discussion of surface gravity (see pg. 6 in PDF).
Book: Black Holes: A Student Text
Discussion of surface gravity (see §2.8.2).
Website: Explain Kerr-Newmann Black Hole Spins in SI Units
Explanation of what the various angular parameters mean. Be sure to read comments for the corrections.
Website: What is the inner and outer horizon on a black hole?
Explanation of inner/outer horizons and related issues.
Paper: Mass and Angular Momentum of Sgr A*
Angular momentum of Sgr A*.

Help and TODO:

There are still some ways to break the calculator (for example, attempting to solve for a positive Cauchy horizon for a Schwarzschild hole, which can't happen). These should be fixed.
Formulae for orbits, in particular ISCO.
Formula for average power (mass-energy divided by evaporation time).
The formulae for surface gravity and gravitational tides should be checked.
The formula for angular velocities should be checked to ensure it is valid for Kerr-Newman; not just Kerr.
A diagram of the requested black hole's geometry would be nice.