The Hawking temperature calculator above treats a black hole as a thermodynamic object. Give it a mass and it returns the four quantities that describe the horizon as a thermal system: its temperature, its entropy, the power it radiates into the surrounding vacuum, and the time it would take to radiate itself away entirely. Those four numbers together are the content of black hole thermodynamics, and they are far stranger than the geometry that produced them.
Arb Digital builds free physics calculators that each own one job. This page owns the thermodynamics. The live Schwarzschild radius calculator owns the geometry — horizon size, horizon area and mean density for a given mass — and reports a temperature and a lifetime alongside them as supporting figures. This page starts from the temperature instead, adds the Bekenstein-Hawking entropy and the radiated power, and works through the regimes where the picture applies and where it stops applying. If you want to know how big the horizon is, use that page. If you want to know how hot it is and what that implies, use this one.
What This Hawking Temperature Calculator Does
In 1974 Stephen Hawking showed that quantum field theory in the curved spacetime around a black hole predicts a thermal flux escaping to infinity, with a genuine temperature set purely by the mass. Nothing about the material that formed the black hole survives in that temperature. A horizon made from collapsing stars and a horizon made from collapsing textbooks radiate identically if their masses match.
The calculator computes that temperature and then the quantities that follow from it. Entropy comes from the horizon area rather than from any internal volume, which is the observation that started the whole holographic line of thinking. Radiated power comes from treating the horizon as a blackbody of that temperature and that area. Lifetime comes from integrating the mass loss as the black hole shrinks and heats.
It also compares the Hawking temperature to whatever ambient temperature you set. That comparison decides the black hole's fate: colder than its surroundings and it absorbs more than it emits and grows, hotter and it shrinks. For every black hole known to astronomy the first case applies overwhelmingly, and the calculator says so in words.
How to Use It
- Enter the mass and choose a unit. Solar masses suit astrophysical black holes; kilograms suit the hypothetical primordial ones, which are the only ones for which the temperature is large.
- Leave the ambient temperature at the microwave background. 2.725 kelvin is the temperature of the present-day universe, and it is what an astrophysical black hole is bathed in.
- Leave the greybody factor at one unless you are deliberately exploring how emission efficiency changes the lifetime. It scales power up and lifetime down in exact proportion.
- Read the temperature against the ambient figure. That is the only comparison on the page that decides whether a black hole is growing or shrinking.
- Compare the lifetime to the age of the universe, which is about 1.38 × 1010 years. Everything longer than that has, in effect, not started evaporating.
The Formula: How Hawking Temperature Is Calculated
For a Schwarzschild black hole of mass M, the Hawking temperature is
TH = ℏc³ ÷ (8πGMkB)
where ℏ is the reduced Planck constant, whose 2022 CODATA value of 1.054571817 × 10−34 J·s is exact and published on the NIST CODATA page for the reduced Planck constant. The gravitational constant is 6.67430 × 10−11 m³/kg·s² from the NIST CODATA page for the Newtonian constant of gravitation, and the Boltzmann constant is exactly 1.380649 × 10−23 J/K.
Collect the constants and the whole expression reduces to TH = 1.2268 × 1023 ÷ M kelvin with the mass in kilograms. The horizon radius is rs = 2GM ÷ c², the Bekenstein-Hawking entropy in units of the Boltzmann constant is S/kB = 4πGM² ÷ (ℏc), the radiated power is P = ℏc⁶ ÷ (15360πG²M²), and the evaporation lifetime is t = 5120πG²M³ ÷ (ℏc⁴).
Work the default by hand. One solar mass is 1.98892 × 1030 kg, so TH = 1.2268 × 1023 ÷ 1.98892 × 1030 = 6.168 × 10−8 K. That is sixty nanokelvin, roughly forty million times colder than the microwave background. The horizon radius is 2 × 6.6743 × 10−11 × 1.98892 × 1030 ÷ 8.98755 × 1016 = 2,954 metres, just under three kilometres. The entropy works out at 1.049 × 1077 in units of kB, the radiated power at 9.0 × 10−29 watts, and the lifetime at 6.62 × 1074 seconds, which is 2.1 × 1067 years. NASA's black holes overview gives the astronomical context for the objects these numbers describe.
Every Real Black Hole Is Growing, Not Evaporating
The popular picture of black holes slowly boiling away is, for every black hole anyone has observed, exactly backwards. A stellar-mass black hole is at sixty nanokelvin and the universe around it is at 2.725 kelvin. It absorbs microwave background photons vastly faster than it emits Hawking quanta, so its mass increases.
The crossover mass is easy to find: set the Hawking temperature equal to 2.725 K and you get about 4.5 × 1022 kg, which is around six tenths of the mass of the Moon. Anything heavier than that is currently colder than the sky and is gaining mass. Anything lighter is losing it. Every black hole ever detected is many orders of magnitude above that line.
Evaporation only becomes the dominant process after the universe has expanded enough for the background to cool below the black hole's temperature, which for a solar-mass object means waiting until the microwave background falls under sixty nanokelvin. That is far into the future, and only then does the 1067-year clock start running properly. Quoting a lifetime without that caveat is the most common misstatement in popular accounts of Hawking radiation.
Negative Heat Capacity And The Violent Ending
Temperature goes as one over mass, so losing energy makes a black hole hotter. That is a negative heat capacity, and it is thermodynamically pathological: a normal object cools as it radiates and settles into equilibrium, while a black hole heats as it radiates and runs away from equilibrium.
Because the lifetime scales as the cube of the mass, the evaporation is enormously back-loaded. A black hole spends almost the whole of its life at very nearly its starting mass, and the final stages take a vanishing fraction of the total. Halving the mass uses seven eighths of the remaining time; the last one per cent of mass takes about one part in a million of the lifetime.
The equilibrium that this implies is unstable in both directions. A black hole in a bath at exactly its own temperature is balanced on a knife edge: absorb one photon too many and it cools, absorbs faster still, and grows without limit; emit one too many and it heats, emits faster, and runs to the endpoint. There is no stable thermal equilibrium between a Schwarzschild black hole and an infinite reservoir at all.
Entropy On A Surface, Not In A Volume
The entropy result is the most consequential number this page produces. For every ordinary system entropy scales with volume, because it counts states distributed through the interior. Black hole entropy scales with the horizon area, and the constant of proportionality is one quarter in units of the Planck area.
The magnitude is difficult to overstate. A solar-mass black hole has an entropy around 1077 in units of the Boltzmann constant, which is roughly twenty orders of magnitude more than the Sun itself has. The supermassive black holes at galactic centres dominate the entropy budget of the observable universe by a wide margin, and they do it with horizons that are geometrically tiny compared with the galaxies around them.
Area scaling is the seed of the holographic principle: it suggests that the information content of a region is bounded by the area of its boundary rather than its volume, which is not how any other physical system behaves. Entropy in more ordinary settings is handled by the entropy change calculator, and the general thermal-radiation relation behind the power figure is covered by the blackbody radiation calculator.
Where This Calculation Stops Being Trustworthy
Everything here is the Schwarzschild case: no spin, no charge. Real black holes rotate, sometimes near the maximum allowed, and a rotating Kerr black hole is cooler than a Schwarzschild one of the same mass. An extremal black hole, spinning as fast as the geometry permits, has zero Hawking temperature, so the numbers on this page are an upper bound on temperature and a lower bound on lifetime for a spinning object.
The blackbody estimate is also idealised. Real emission passes through the gravitational potential outside the horizon, which reflects part of it back, and the spectrum is greybody rather than perfectly thermal. As the black hole heats, more particle species become light enough to be emitted, which raises the effective emission and shortens the lifetime below the pure-photon estimate. The greybody field on this page lets you scale for that, but it is a crude stand-in for a proper species-dependent calculation.
Finally, the derivation is semiclassical: quantum fields on a classical spacetime. That is a controlled approximation while the horizon is far larger than the Planck length, and it fails entirely in the last instants, when the remaining mass approaches the Planck mass of about 2.2 × 10−8 kg. Nobody knows what happens then, which is precisely the information-paradox question. For the relativistic-geometry neighbours, see the gravitational time dilation calculator and the escape velocity calculator.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Saying black holes evaporate today — every observed black hole is far colder than the microwave background and is therefore gaining mass. Evaporation only dominates once the universe cools below the hole's own temperature.
- Treating the lifetime as a countdown — the mass loss is negligible for almost the whole span and then abrupt. The number is an integrated total, not a rate you can extrapolate from.
- Expecting entropy to scale with volume — black hole entropy goes with horizon area, which is why doubling the mass quadruples the entropy rather than multiplying it by eight.
- Applying the Schwarzschild formula to a spinning black hole — spin lowers the temperature for the same mass, and an extremal one has none at all, so these figures are bounds rather than values.
- Trusting the result near the Planck mass — the semiclassical derivation assumes the horizon is huge compared with the Planck length, and the final instants of an evaporation lie outside any theory currently available.
Related Free Tools From Arb Digital
For the horizon geometry rather than the thermodynamics, use the Schwarzschild radius calculator. The gravitational time dilation calculator covers how clocks behave near a strong field, and the escape velocity calculator covers the Newtonian intuition the horizon radius happens to reproduce. For thermal radiation from ordinary objects, use the blackbody radiation calculator, and for entropy in ordinary thermodynamics the entropy change calculator. The gravitational force calculator, the Kepler's third law calculator and the black hole merger calculator cover the surrounding orbital physics. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
It is the thermal temperature a black hole horizon appears to have to a distant observer, arising from quantum field effects in curved spacetime. For a non-rotating black hole it depends only on the mass, and it works out at about 1.23 times ten to the twenty-third kelvin divided by the mass in kilograms.
About 6.2 times ten to the minus eight kelvin, which is sixty nanokelvin. That is roughly forty million times colder than the cosmic microwave background, so such a black hole absorbs far more energy than it emits and its mass grows rather than shrinks.
None that have been observed. A black hole only shrinks when its Hawking temperature exceeds its surroundings, which today means a mass below about 4.5 times ten to the twenty-second kilograms, roughly six tenths of a lunar mass. Every detected black hole is far heavier than that.
Because temperature is inversely proportional to mass, so losing energy raises the temperature. This is a negative heat capacity, and it means the process accelerates rather than settling down, which is why the final stages of an evaporation are sudden after an enormously long quiet period.
The Bekenstein-Hawking entropy equals one quarter of the horizon area in units of the Planck area, which is unlike every ordinary system where entropy scales with volume. That area scaling is the observation that led to the holographic principle and the idea that information content is bounded by a surface.
The lifetime scales as the cube of the mass, so a solar-mass black hole would need about two times ten to the sixty-seventh years while a black hole of roughly 1.7 times ten to the eleventh kilograms would have a lifetime comparable to the present age of the universe.
Yes. These formulas are for the Schwarzschild case with no spin and no charge. A rotating Kerr black hole is cooler than a Schwarzschild one of the same mass, and an extremal one has zero Hawking temperature, so the figures here are an upper bound on temperature.
Not from an astrophysical black hole. The predicted signal from any known black hole is many orders of magnitude below the microwave background it sits in, so it is undetectable in practice. Laboratory analogue systems have been used to study similar physics in other media.
This tool is provided for educational and study use. It implements the semiclassical Schwarzschild results for a non-rotating, uncharged black hole with an idealised blackbody emission model, and it does not account for spin, charge, greybody spectra, particle species thresholds or any quantum-gravitational effects near the endpoint, so treat its output as a teaching result rather than a prediction.
