Don Page (1976): How Fast Black Holes Evaporate, in Numbers

Black hole evaporation got its numbers in this 1976 paper by Don Page. Building on Hawking’s discovery that black holes radiate, Page calculated exactly what a non-rotating black hole emits, mostly neutrinos, then photons, then gravitons, and how long it takes to evaporate: a lifetime that grows as the cube of its mass.

The question before the paper

In 1974 Stephen Hawking showed that black holes glow and slowly evaporate. His result gave the temperature, but turning it into a practical timescale required much more: how much of the energy comes out as each kind of particle, and at what rate? Without those numbers, nobody could say precisely how long a black hole of a given mass survives.

What the paper found

• What comes out. For black holes much heavier than 10¹⁷ grams, about 81% of the emitted power is in neutrinos, 17% in photons and 2% in gravitons, given the particle physics known in 1976.
• A precise lifetime. Page derived the evaporation lifetime, which grows as the cube of the black hole’s mass. Double the mass and the black hole lasts eight times longer.
• Which black holes are gone. A primordial black hole, formed in the early universe, would have evaporated within the present age of the universe only if its initial mass was below about 5 × 10¹⁴ grams.

How he did it

Page treated the black hole as non-rotating and uncharged and computed the emission particle type by particle type for massless particles, which in 1976 meant neutrinos, photons and gravitons. Adding up the rates for all of them gave the total power radiated, and so the rate at which the black hole loses mass and the time it takes to disappear.

Why it mattered

Page turned Hawking’s prediction into numbers that can be used. Because the lifetime scales as mass cubed, stellar and supermassive black holes last unimaginably long, far longer than any star. His results set the timescale for the Black Hole Era, when evaporating black holes are the last energy sources in the universe, a phase built into Adams and Laughlin’s timeline of the far future.

What we know now

Hawking radiation has still not been observed from any astrophysical black hole, as of October 2026. The emission percentages depend on particle physics, which has moved on since 1976, but the scaling of lifetime with mass cubed remains the basis of every far-future timeline. A 2025 proposal by Falcke and colleagues extends Hawking-like evaporation to neutron stars and white dwarfs. See Deep Time and the future of the universe.

Who wrote it

Don Page wrote the paper alone as a student of Kip Thorne at Caltech, today home of Caltech astronomy; he later worked with Stephen Hawking. It has about 990 citations in Semantic Scholar as of October 2026.

Read the original

Page, D. N. (1976). “Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole.” Physical Review D 13, 198. doi:10.1103/PhysRevD.13.198. The journal version requires a subscription; the record is free at INSPIRE.