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Black holes evaporate: the maths

Hawking's formulas for a non-rotating, uncharged black hole radiating as a black body, photons only. Computed live.

Equations

T = ħc³ / (8πGMk_B)

The black hole's temperature: smaller holes are hotter.

P = ħc⁶ / (15360 π G² M²)

Power radiated.

t_life = 5120 π G² M³ / (ħ c⁴)

Time to evaporate completely, from mass M.

r_s = 2GM / c²

Horizon radius.

Constants

ħ	1.054571817 × 10⁻³⁴ J·s	CODATA
G	6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻²	CODATA
kB	1.380649 × 10⁻²³ J/K	exact
M☉	1.989 × 10³⁰ kg	the Sun's mass

Worked examples ✓ checked on every change

Temperature of a Sun-mass black hole	6.17 × 10⁻⁸ K	the formula above — about 60 billionths of a degree
Lifetime of a Sun-mass black hole	2.1 × 10⁶⁷ years	the formula above
Page time, as a share of the lifetime	64.6%	when the hole has lost half its entropy: 1 − 2−3/2

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

• Photons only: counting neutrinos and gravitons shortens lifetimes by a factor of a few.
• The hole sits in empty space; today, any black hole heavier than about half the Moon is colder than the microwave background and grows instead.

Where it breaks

In the final instants, when the hole is near the Planck mass, where a theory of quantum gravity is needed.

Sources: S. W. Hawking, Nature 248, 30 (1974); D. N. Page, Phys. Rev. D 13, 198 (1976) and PRL 71, 3743 (1993).