Black Hole Evaporation Time Calculator
Hawking radiation and black hole lifetimes
In classical general relativity, a black hole is an “eternal” object: once something crosses the event horizon, it cannot return to the outside universe. Quantum field theory in curved spacetime changes that picture. In 1974, Stephen Hawking showed that a black hole should emit nearly thermal radiation, causing it to lose mass. This slow loss of mass is usually called black hole evaporation.
The key qualitative result is simple and counterintuitive: the smaller the black hole, the hotter it is. Because a hotter black hole radiates more power, it loses mass faster. That creates a runaway behavior where evaporation accelerates as the mass drops, ending (in the semiclassical model) in a rapid final stage.
What this calculator computes
Enter a black hole mass in solar masses (M☉). The calculator converts it to SI units (kilograms) and estimates:
- Hawking temperature (Kelvin), using the Schwarzschild (non-rotating, uncharged) result.
- Evaporation time (seconds and/or years), using the standard semiclassical lifetime formula.
These outputs are meant as order-of-magnitude physical estimates for isolated black holes in the semiclassical regime.
Formulas (Schwarzschild black hole)
The Hawking temperature for a non-rotating, uncharged black hole of mass M is:
T = ħc³ / (8π G M kB)
The commonly cited semiclassical evaporation time (ignoring detailed particle “greybody” effects and changes in degrees of freedom) is:
t ≈ 5120π G² M³ / (ħ c⁴)
These show the main scaling:
- T ∝ 1/M (smaller mass → higher temperature)
- t ∝ M³ (smaller mass → dramatically shorter lifetime)
Constants and units
This calculator uses standard SI constants:
- Gravitational constant: G = 6.67430×10−11 m3 kg−1 s−2
- Speed of light: c = 299,792,458 m/s
- Reduced Planck constant: ħ = 1.054571817×10−34 J·s
- Boltzmann constant: kB = 1.380649×10−23 J/K
- Solar mass (unit conversion): M☉ = 1.98847×1030 kg
Input mass is entered in solar masses and converted to kilograms internally. Temperature is reported in Kelvin. Evaporation time is computed in seconds and can be displayed in years for readability (1 year ≈ 365.25 days).
How to interpret the results
Hawking temperature tells you the characteristic thermal scale of the emitted radiation as measured by a distant observer. For very massive black holes, the temperature is far below the cosmic microwave background (CMB) temperature (~2.7 K), meaning they are effectively colder than the universe and would tend to gain energy from ambient radiation in realistic environments.
Evaporation time is the estimated time for the black hole to radiate away essentially all of its mass, assuming it is isolated and does not accrete. Because the lifetime scales like M³, changing the mass by a factor of 10 changes the lifetime by a factor of 1000.
Worked example (order of magnitude)
Suppose you enter 1 solar mass (M = 1 M☉). The calculator will convert this to M ≈ 1.988×1030 kg and evaluate:
- Temperature: extremely small, on the order of 10−8 K (tens of nanokelvin). That is far colder than the CMB.
- Evaporation time: enormously long, on the order of 1067 years, vastly exceeding the current age of the universe (~1010 years).
This sanity-check illustrates why Hawking evaporation is negligible for astrophysical black holes today, while hypothetical much smaller black holes (e.g., primordial black holes) could evaporate on cosmological timescales.
Mass scaling comparison
The table below summarizes how temperature and lifetime scale with mass. Values are approximate and meant for intuition; the exact numbers depend on the constant factors and modeling details.
| Mass (in M☉) | Relative temperature (T ∝ 1/M) | Relative lifetime (t ∝ M³) |
|---|---|---|
| 10 | ~0.1× (colder) | ~1000× (longer) |
| 1 | 1× | 1× |
| 0.1 | ~10× (hotter) | ~0.001× (shorter) |
| 10−12 | ~1012× (much hotter) | ~10−36× (much shorter) |
Assumptions & limitations
- Schwarzschild only: The formulas used assume a non-rotating, uncharged black hole. Real astrophysical black holes likely have spin, which changes the temperature and emission spectrum.
- Isolated black hole: The lifetime estimate assumes no accretion, no mergers, and no significant absorption from surrounding radiation fields. In many realistic settings, growth dominates evaporation.
- Semiclassical regime: Hawking’s derivation treats quantum fields on a classical spacetime background. For extremely small masses approaching the Planck scale, the semiclassical approximation may break down, and the final evaporation stage is unknown.
- Greybody factors and particle content: The simple lifetime constant (5120π…) is an idealized expression. Detailed calculations include frequency-dependent transmission (“greybody”) factors and changing numbers of particle species at high temperatures, which can shift lifetimes by factors of order unity to larger.
- Interpretation of “total evaporation time”: The result is best read as the timescale to radiate away most of the mass under the model assumptions, not a precisely predictable endpoint for a real black hole in an environment.