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§ Phase xv · Stellar Quanta · Chapter 05

Hawking radiation

For half a century after Schwarzschild's solution, every textbook said the same thing: nothing leaves a black hole. Then in early 1974 a 32-year-old at Cambridge, already a decade into living with motor-neurone disease, ran the calculation that broke the rule. Quantum fields on a curved background, he showed, will pour a faint thermal glow out into space. The hole heats up, the hole shrinks, and eventually the hole disappears.

In the final winter of 1915, while millions of men were freezing into the mud of the Eastern Front, a forty-two-year-old German artillery officer named Karl Schwarzschild sat in a field tent with a copy of the Sitzungsberichte and a sharp pencil. He had just received the November issue of the Berlin Academy proceedings, which carried Einstein’s freshly published field equations of general relativity. Schwarzschild, who was the director of the Potsdam Observatory in civilian life, had volunteered for the army at the start of the war out of patriotic duty and was now computing artillery trajectories for the German military. In his spare hours, between barrages, he worked through Einstein’s equations and found, within weeks, an exact solution. It described the geometry of empty spacetime outside a single, isolated, non-rotating spherical mass. He posted the manuscript to Einstein from the front lines in December.

Einstein read the paper in disbelief. “I had not expected that one could formulate the exact solution of the problem in such a simple way,” he wrote back. He presented Schwarzschild’s manuscript to the Prussian Academy on Schwarzschild’s behalf in January 1916. By then Schwarzschild had contracted pemphigus, a rare and at that time incurable autoimmune blistering disease, on the front. He was discharged in March, returned home to Potsdam, and died on May 11. He was forty-two. Buried in the metric he had derived was a singular feature: at a particular radius, now called the Schwarzschild radius and equal to 2GM/c² for a mass M, his coordinates broke down. For an everyday star this surface sat well inside the body of the star itself, where the vacuum solution did not apply, and Schwarzschild himself thought no more of it. It would take another four decades before physicists understood that this surface, for a sufficiently compressed object, was a one-way door in spacetime. The name “black hole” did not appear in print until John Wheeler used it in a 1967 lecture at the Goddard Institute. The thing itself had been waiting in Einstein’s equations since 1916.

For the next half-century the Schwarzschild solution was a beautiful curiosity. Most physicists, Einstein included, suspected real stars would never compress hard enough to produce one. Chandrasekhar showed in 1931 that white dwarfs above 1.4 solar masses could not support themselves against gravity. Oppenheimer and Snyder showed in 1939 that the collapse of a sufficiently massive star, taken seriously in general relativity, would simply continue, producing a region cut off from the outside universe in finite proper time. Even then the idea was treated as a mathematical pathology. When the first X-ray binaries with unseen compact partners were discovered in the late 1960s, and when Roger Penrose proved the singularity theorems showing that gravitational collapse inevitably produces a singularity, the community at last began to take black holes physically. By 1971 Cygnus X-1 was on the table as a strong candidate. By 1973 the textbook law of black hole mechanics, written down by James Bardeen, Brandon Carter and Stephen Hawking, listed four rules that looked uncannily like the laws of thermodynamics. Energy was conserved. The horizon area never decreased. The surface gravity was constant over the horizon. Each had a thermodynamic partner: first law, second law, zeroth law, third law. The analogy was clean and, everyone agreed, a coincidence. A black hole had no temperature. It absorbed everything and emitted nothing. Calling its area “entropy” was, at most, a useful mnemonic.

The dissenting voice was Jacob Bekenstein, a young Israeli graduate student at Princeton working under Wheeler. In 1972 and 1973 Bekenstein published a sequence of papers arguing that a black hole really must carry entropy, proportional to the area of its event horizon, and that the analogy of the four laws was not a coincidence but a statement of fact. His argument was thermodynamic. Imagine dropping a cup of hot tea into a black hole. Before, the universe contained some entropy in the tea. After, the tea is gone and the black hole is slightly heavier. If the black hole has no entropy of its own, the total entropy of the universe has decreased and the second law of thermodynamics has been violated. The only escape, Bekenstein argued, is for the black hole’s area to play the role of entropy. The increase in area when the tea fell in must be at least the entropy the tea carried. To everyone but Wheeler this seemed nuts. Hawking, in particular, wrote a paper in late 1973 trying to refute it. To refute it properly, he ran the calculation of what happens when quantum fields are placed on the curved background of a forming black hole, expecting to find nothing. He found something instead.

Stephen William Hawking was born in Oxford on January 8, 1942, exactly three hundred years to the day after the death of Galileo, a coincidence he liked to point out. He read Natural Sciences at University College, Oxford, and went to Cambridge in 1962 for a PhD on cosmology under Dennis Sciama. In his first year of postgraduate work he was diagnosed with amyotrophic lateral sclerosis, the same motor-neurone disease that killed Lou Gehrig. The doctors gave him two years. He outlived that prognosis by fifty-three. By 1974 he was already wheelchair-bound and speaking only with difficulty, but his mind worked at full speed. The calculation he produced that year would, more than any other single piece of theoretical physics in the second half of the twentieth century, change how physicists thought about gravity, quantum mechanics, and the texture of empty space.

The result is stated in a single equation. The temperature of a non-rotating, uncharged black hole of mass M is

T_H  =  ℏ c³ / (8 π G M k_B)

This is one of the most remarkable formulas in physics. On the right-hand side sit four of the fundamental constants of nature: Planck’s constant ℏ (the gauge of quantum mechanics), the speed of light c (the gauge of relativity), Newton’s constant G (the gauge of gravity), and Boltzmann’s constant k_B (the gauge of thermodynamics). All four meet in one equation. Plug in M equal to one solar mass, two times ten to the thirtieth kilograms, and out comes a temperature of about sixty billionths of a kelvin. Sixty nanokelvin. The cosmic microwave background, the leftover photon soup from the Big Bang that fills the present universe, sits at 2.7 K, around fifty million times hotter. A real solar-mass black hole, sitting in our universe today, absorbs vastly more CMB radiation than it emits. It grows, it does not shrink. Hawking radiation is a real effect, but for any astrophysical black hole in the present cosmic epoch it is fantastically slow.

The mechanism Hawking offered in his press summary, and the one that has appeared in every popular book since, runs as follows. Empty space is not really empty: the quantum vacuum is constantly fizzing with virtual particle-antiparticle pairs that appear and annihilate within times bounded by the energy-time uncertainty relation Δ E · Δ t ~ ℏ. Normally these pairs are unobservable. They appear out of nothing, recombine, and are gone. But near a black hole horizon, the gravitational field is steep enough that one partner can fall into the hole while its mate escapes outward. The infalling partner has, in a well-defined sense, negative energy as measured at infinity. When it crosses the horizon and merges with the black hole’s interior, the hole’s total mass goes down by that amount. The escaping partner shows up at infinity as a real, positive-energy quantum of radiation. The hole has emitted. Energy is conserved. Mass-energy bookkeeping demands that the radiation must, on average, carry away exactly what the hole loses. Pile up the bookkeeping over the whole horizon and the spectrum that emerges is exactly Planckian at temperature T_H. Wheeler called it “transcending Bohr.” It is the only place in known physics where quantum mechanics, gravity, and thermodynamics are forced to speak in the same sentence.

The Bekenstein entropy followed instantly. If a black hole has a temperature T_H and emits radiation, then the first law of thermodynamics applied to the hole, dE = T dS, where E is the rest energy M c² and dS is the change in entropy, fixes S in terms of M. Plug in T_H, integrate, and you get the Bekenstein-Hawking entropy:

S_BH  =  k_B · A / (4 ℓ_P²)

where A is the area of the horizon and ℓ_P = √(ℏ G / c³) is the Planck length, about 1.6 × 10⁻³⁵ metres. The entropy of a black hole is one quarter of its horizon area, measured in Planck units. For a solar-mass hole the area is about 110 square kilometres, and the entropy works out to about 10⁷⁷ k_B, dwarfing the thermodynamic entropy of the star that collapsed to form it. The numerical coefficient of one-fourth is uncannily specific. Bekenstein had argued for area-proportional entropy with an unknown coefficient. Hawking’s temperature pinned that coefficient. Any quantum theory of gravity that claims to be complete must, on Bekenstein-Hawking grounds, end up reproducing this one-quarter. String theory, in the 1996 work of Strominger and Vafa, did so for a particular class of extremal black holes by counting microstates of D-brane configurations. Loop quantum gravity reaches the same result by a different counting. The one-quarter is a milestone every candidate has to clear.

The lifetime formula has the same one-trick simplicity. The radiated power, by the Stefan-Boltzmann law applied to the horizon area, scales as T⁴ · A, and since T ∝ 1/M and A ∝ M² the power scales as 1/M². The mass loss rate is then dM/dt ∝ -1/M², which integrates to give an evaporation time

t_evap  =  (5120 π G² / ℏ c⁴) · M³

The cube of the mass. For a solar-mass black hole this comes out to roughly 2 × 10⁶⁷ years, an absurdity beside the current age of the universe at 1.4 × 10¹⁰ years. For a black hole the mass of Earth, 6 × 10²⁴ kg, the evaporation time is 5 × 10⁵⁰ years, still impossibly long. For a primordial black hole of mass around 1.7 × 10¹¹ kg (forty thousand tonnes per cubic Schwarzschild radius, the mass of a small asteroid compressed inside a horizon the size of a proton), the evaporation time is exactly the present age of the universe. Such objects, if they were created in primordial density fluctuations during the first second of cosmic time, would be evaporating in their final paroxysm now. Their final seconds would be visible as a burst of gamma-rays carrying about 10²² joules, the energy of a small nuclear bomb but spread over a few seconds and across a wide spectrum. Several gamma-ray observatories have hunted for the signature. None has been seen.

There is a second consequence that took longer to dawn on the community and that has been the dominant question in fundamental physics ever since. If a black hole forms from a collapsing cloud of well-specified quantum information (the wavefunctions of every particle that fell in) and then evaporates into a thermal soup of Hawking radiation, what happens to the information? A thermal spectrum, by definition, encodes only its temperature. Two black holes formed from totally different infalling matter, but with the same final mass and charge and angular momentum, will emit indistinguishable Hawking radiation. If that radiation is really thermal in the strict sense and the hole eventually disappears, then the quantum state of the early matter has been mapped onto a mixed thermal state. Pure quantum evolution has been broken. This is the black hole information paradox, posed by Hawking in 1976. He argued the universe would have to abandon unitary evolution at the quantum level when black holes evaporated, and most of the rest of the theoretical community pushed back. By the 1990s the consensus had swung toward unitarity, partly because of arguments by Don Page about how information should leak back out gradually in the radiation, and partly because Maldacena’s AdS/CFT correspondence (1997) gave a concrete realisation in which black hole formation and evaporation in a certain class of spacetimes is dual to perfectly unitary evolution on a boundary quantum field theory. In 2004 Hawking publicly conceded the bet he had made with John Preskill on the matter; the bet, oddly, was for an encyclopaedia, and Hawking presented Preskill with a baseball encyclopaedia at a Dublin conference. The paradox itself is still not fully resolved, but the current best guess, fed by holography and by recent calculations of the entropy of Hawking radiation by Penington, Almheiri, Engelhardt, Maxfield, Marolf, Maldacena, and others, is that information does come out, very slowly and very subtly, encoded in correlations that an asymptotic observer can in principle reconstruct.

κ  =  c⁴ / (4 G M)

T_U  =  ℏ α / (2 π k_B c)

T_H  =  ℏ κ / (2 π k_B c)

T_H  =  ℏ c³ / (8 π G M k_B)

It is worth saying plainly what Hawking changed. Before 1974, black holes were objects of relativity, with no place inside quantum theory. After 1974, every theory of quantum gravity has had to reproduce the Bekenstein-Hawking entropy, the thermal spectrum at temperature ℏ c³/(8πGM k_B), and the eventual evaporation. The horizon stopped being a one-way classical membrane and became the thermodynamic surface of a quantum system with a finite number of internal microstates equal to the exponential of S_BH/k_B. That number, for a stellar-mass black hole, is the largest finite integer that ever shows up in physics: roughly e^(10⁷⁷). It is the count of distinct ways the inside of the hole could have been arranged for the same outside appearance. The information paradox is the question of how, when the hole evaporates, those microstates are recovered from the outgoing radiation. The Maldacena duality and the more recent work on entanglement wedges suggest a path, but the path passes through a country no one has yet completely mapped. Hawking himself never lived to see the puzzle settled. He died on March 14, 2018, the anniversary of Einstein’s birth. He was 76.