Black Hole Temperature Calculator
Enter a black hole mass in solar masses, kilograms, or Earth masses to instantly get its Hawking radiation temperature, Schwarzschild radius, Bekenstein-Hawking entropy, luminosity, and the time it will take to evaporate. You can also reverse-solve: enter a temperature to find the corresponding mass. All results update as you type.
Formula
Worked example
A stellar black hole of 10 solar masses: M = 10 x 1.989e30 = 1.989e31 kg. Hawking temperature T = (1.055e-34 x (3e8)^3) / (8pi x 6.674e-11 x 1.989e31 x 1.381e-23) = 6.17e-8 K. Schwarzschild radius r = 2 x 6.674e-11 x 1.989e31 / (3e8)^2 = 29,530 m (about 29.5 km). Evaporation time = 5120pi x (6.674e-11)^2 x (1.989e31)^3 / (1.055e-34 x (3e8)^4) = 2.1e74 years.
What is Hawking radiation?
In 1974, Stephen Hawking showed that black holes are not perfectly black. Quantum field theory in curved spacetime predicts that virtual particle-antiparticle pairs created near the event horizon can separate: one falls in while the other escapes, carrying energy away from the black hole. The net effect is that a black hole radiates as a blackbody at a temperature inversely proportional to its mass. The smaller the black hole, the hotter and brighter it is, and the faster it loses mass. This process is called Hawking radiation, and the characteristic temperature is called the Hawking temperature or Bekenstein-Hawking temperature.
How to use this calculator
Select "Temperature (from mass)" to compute the Hawking temperature and all derived quantities from a given mass. Enter the mass value and choose your preferred unit (solar masses for astrophysical black holes, kilograms for primordial black holes, or Earth masses for comparison). Switch to "Mass (from temperature)" to reverse the calculation and find what mass corresponds to a particular temperature. All six derived quantities update in real time. The steps panel shows the exact arithmetic using your numbers.
The Hawking temperature formula and its meaning
The Hawking temperature is T = hbar c^3 / (8 pi G M k_B). Every constant in the numerator is a quantum or relativistic fundamental: hbar is the reduced Planck constant (quantum scale), c is the speed of light (relativistic scale). The denominator mixes G (gravity) and k_B (thermodynamics). The formula is a landmark unification: it sits at the intersection of quantum mechanics, general relativity, and statistical thermodynamics. The inverse proportionality means that a black hole twice as massive is half as hot, and an evaporating black hole accelerates its own heating as it loses mass, ending in a runaway burst.
Schwarzschild radius, entropy, luminosity, and evaporation time
The Schwarzschild radius r_s = 2GM/c^2 is the radius of the event horizon for a non-rotating (Schwarzschild) black hole. A stellar-mass black hole of 10 solar masses has r_s of about 29.5 km, smaller than a city. The Bekenstein-Hawking entropy S = 4 pi G k_B M^2 / (hbar c) is proportional to the surface area of the event horizon, not its volume, a fact that underpins the holographic principle. The Hawking luminosity L = hbar c^6 / (15360 pi G^2 M^2) gives the radiated power; for a 10-solar-mass black hole this is about 1e-29 W, completely undetectable. The evaporation time tau = 5120 pi G^2 M^3 / (hbar c^4) scales with M^3, so doubling the mass increases the lifetime by a factor of 8. A primordial black hole formed in the early universe with mass around 5e14 kg would be completing its evaporation now.
Why stellar black holes do not evaporate
The cosmic microwave background fills the universe with thermal photons at 2.725 K. A 10-solar-mass black hole has a Hawking temperature of roughly 6e-8 K, far colder than the CMB. It therefore absorbs CMB radiation at a rate that vastly exceeds its own Hawking output, so it grows rather than shrinks under present cosmic conditions. Only black holes smaller than about 7e22 kg (a few percent of Earth mass) are currently hotter than the CMB and actively losing mass. Supermassive black holes like Sagittarius A* (4.1 million solar masses) have temperatures below 10^-14 K and evaporation times longer than 10^90 years.
Hawking temperature for reference black holes
| Black hole | Mass | Hawking temperature | Evaporation time |
|---|---|---|---|
| Stellar (10 M☉) | 2 × 10^31 kg | 6 × 10^-8 K | 2 × 10^74 years |
| Sagittarius A* (4.1 × 10^6 M☉) | 8.2 × 10^36 kg | 1.5 × 10^-14 K | 8 × 10^92 years |
| M87* (6.5 × 10^9 M☉) | 1.3 × 10^40 kg | 9.5 × 10^-18 K | 2 × 10^105 years |
| Evaporating primordial (5 × 10^14 kg) | 5 × 10^14 kg | 0.024 K | ~13.8 × 10^9 years |
| Planck mass black hole (2.18 × 10^-8 kg) | 2.18 × 10^-8 kg | 5.6 × 10^31 K | Planck time |
Approximate values for well-known or theoretically important black holes. Stellar and supermassive black holes are far colder than the CMB (2.725 K) and effectively do not evaporate.
Frequently asked questions
Has Hawking radiation ever been detected?
Not yet. For stellar-mass and larger black holes the Hawking temperature is billions of times colder than the CMB, making the radiation completely undetectable with current technology. Analogue experiments in condensed-matter systems (sonic black holes, Bose-Einstein condensates) have produced results consistent with Hawking radiation, but direct astrophysical detection remains an open challenge. Primordial black holes finishing their evaporation might be detectable as gamma-ray sources.
Why is a smaller black hole hotter?
The Hawking temperature is inversely proportional to mass (T = hbar c^3 / (8 pi G M k_B)), so as mass decreases temperature rises. Intuitively, a smaller event horizon has stronger spacetime curvature, which drives a faster rate of virtual-pair separation. As a black hole radiates and loses mass it gets hotter and radiates faster, creating a runaway process that ends in a final evaporative burst.
What is the Bekenstein-Hawking entropy?
The Bekenstein-Hawking entropy S = k_B A / (4 l_P^2) is proportional to the surface area A of the event horizon, where l_P is the Planck length (about 1.616e-35 m). This is remarkable because thermodynamic entropy usually scales with volume, not area. The area law suggests that all the information about what fell into the black hole is encoded on its two-dimensional horizon, a key insight behind the holographic principle.
What happens to information when a black hole evaporates?
This is the "black hole information paradox," one of the biggest unsolved problems in theoretical physics. Standard quantum mechanics requires that information is never destroyed, but the apparent thermal nature of Hawking radiation carries no information about what fell in. Many researchers now believe information is encoded in subtle correlations in the late-time radiation, but a fully satisfying resolution requires a theory of quantum gravity.
Can this calculator be used for rotating (Kerr) black holes?
This calculator implements the Schwarzschild (non-rotating) Hawking temperature. Rotating Kerr black holes have a modified temperature that depends on both mass and angular momentum; they can also have an ergosphere and superradiance effects. The Schwarzschild formula is a good approximation for the mass-temperature relationship when spin is not known.
What is the evaporation time for the Sun?
If the Sun were somehow compressed into a black hole without losing any mass (mass = 1 solar mass = 1.989e30 kg), its Hawking temperature would be about 6e-8 K and its evaporation time would be roughly 2e67 years, about 57 orders of magnitude longer than the current age of the universe.