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Physics 1975

Particle Creation by Black Holes

Stephen W. Hawking

Quantum theory forces a black hole to glow — and, ever so slowly, to evaporate away.

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In depth · the introduction

A black hole is supposed to let nothing escape — yet Hawking showed it must quietly glow, and over spans of time beyond imagining, evaporate away.

The idea, unpacked

In Einstein's classical picture a black hole only swallows: cross the edge, the event horizon, and nothing — not even light — comes back. So a black hole could only ever grow.

Quantum theory changes that. Empty space is never truly empty; it constantly froths with pairs of particles flickering into being and vanishing again. Right at the horizon, one of a pair can fall in while its partner escapes outward. To someone far away, that steady stream of escaping particles looks exactly like the warm glow of a hot object. The black hole therefore has a temperature — and shines.

Where it came from

In the early 1970s a young researcher, Jacob Bekenstein, made a heretical suggestion: black holes carry entropy — a measure of disorder — in proportion to the area of their horizon. Hawking thought this had to be wrong, because anything with entropy and a temperature must radiate, and a black hole was by definition something that couldn't.

So in 1973–74 he set out to disprove it using quantum mechanics — and, to his own astonishment, his calculation showed the radiation was real. He announced the result in a 1974 paper with the teasing title "Black hole explosions?", and laid out the full theory in this 1975 paper.

Why it mattered

For the first time, three great theories that almost never meet all described the same object and agreed. The temperature of a black hole contains Planck's constant (quantum theory), Newton's gravitational constant G (gravity), and Boltzmann's constant (heat), all at once. That a single formula needs all three is a giant signpost: it points straight at the still-missing theory of quantum gravity, and tells us black holes are where it will be found.

A precise picture

Picture the horizon as the lip of a waterfall, with space itself flowing over the edge into the hole. Bubbles keep forming in the water in pairs; near the lip, one bubble is swept over while its partner is left outside, free to drift away. Catch those escaping partners and they arrive with exactly the spread of energies you'd get from heat. And the smaller the hole, the steeper and faster the fall — so the smaller it is, the hotter it glows.

A black disk representing a black hole; a mass slider grows or shrinks the disk while changing a surrounding glow — heavy holes are large and dark and ice-cold, light holes are small with a fierce hot glow. A readout reports temperature, Schwarzschild radius and evaporation lifetime.

Where it sits

It completes a line that runs from Boltzmann's S = k log W — entropy as the counting of microscopic possibilities — and Planck's quantum glow of warm bodies, through Einstein's geometric gravity, to a single thermodynamic law for black holes. The black holes whose collision LIGO heard in 2016 are, by this result, also thermometers — the coldest large objects in the universe. And the puzzle it opened, the information paradox, is among the sharpest unsolved problems in physics today.

The original document
Original source text
S. W. Hawking · Communications in Mathematical Physics 43 (3), 199–220 · 1975
Abstract
In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature hκ/2πk ≈ 10⁻⁶(M⊙/M)°K where κ is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about 10¹⁵ g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon of a black hole cannot decrease, there remains a Generalized Second Law: S+1/4A never decreases where S is the entropy of matter outside black holes and A is the sum of the surface areas of the event horizons.
1 · Introduction
The paper applies quantum field theory to the curved spacetime of a body collapsing to form a black hole, and sets the calculation against the recently noticed analogy between the laws of black-hole mechanics and the laws of thermodynamics — in particular Bekenstein's proposal that the area of the event horizon is a measure of entropy.
2–4 · Quantum fields on a collapsing star
Hawking treats matter as quantum fields propagating on the classical geometry of the collapse. Because the geometry is time-dependent, the notion of "no particles" in the distant past differs from that in the distant future; expanding the late-time modes in terms of the early-time ones (the Bogoliubov coefficients) shows that the initial vacuum is seen, at late times, to contain particles.
The thermal spectrum
The decisive finding is that the expected number of emitted particles in each mode is exactly that of black-body radiation at the temperature κ/2π (in units with G = c = ħ = k = 1), where κ is the surface gravity — modulated only by frequency-dependent grey-body factors from the potential barrier outside the horizon. The emission is genuinely thermal, not a fixed signal.
[ … ]
Back-reaction, evaporation, and the second law
Carrying away energy makes the hole lose mass, so its temperature rises and the emission accelerates toward a final burst; and although a shrinking horizon violates the classical area theorem, the abstract's Generalized Second Law (entropy of matter outside, plus one quarter of the horizon area) is proposed to take its place.
Cambridge · 1975