The hole that outlives everything and then does not

The surface that only lets things in is the classical statement, and Hawking’s calculation overturns it: a horizon radiates, at a temperature inversely proportional to the mass. What that temperature implies once it is allowed to act for a while is a longer story than the temperature itself, and almost all of it follows from one sign.

A negative heat capacity #

An ordinary body cools as it loses energy. A black hole heats up.

That is the content of the temperature falling as one over the mass: radiate energy, lose mass, get hotter, radiate faster. There is no equilibrium anywhere in it, and no stable state to settle into. A system with a negative heat capacity cannot sit in thermal contact with anything for long — put it with a heat bath and either it runs away to nothing or it swallows the bath.

That single fact organises everything below. The lifetime, the shape of the decline, the character of the ending and the impossibility of a black hole in equilibrium are all consequences of a minus sign in one exponent.

It also explains a fact that would otherwise seem like an odd coincidence. Every black hole in the universe is currently growing. The microwave background is at 2.7 kelvin, a stellar-mass hole is at sixty nanokelvin, and a body far colder than its surroundings absorbs. The crossing point is at about $4.5\times10^{22}$ kilograms — a hundredth of the Moon — and nothing of that mass has ever been observed.

Evaporation is not something happening now. It is something that will begin when the universe has cooled far enough, which for a stellar-mass hole is in about $10^{12}$ years, and it will then take another $10^{67}$.

The cube law #

The lifetime follows from the Stefan–Boltzmann law applied to the horizon.

The luminosity of a black body is its area times the fourth power of its temperature. The area goes as the square of the mass and the temperature as its inverse, so the luminosity goes as $M^2 \cdot M^{-4} = M^{-2}$. Integrating $\mathrm{d}M/\mathrm{d}t \propto -M^{-2}$ gives a lifetime proportional to $M^3$.

That cube is a large exponent, and it does what large exponents do: it turns the forty decades of mass in the universe into a hundred and twenty decades of lifetime. A hole of a hundred thousand tonnes lasts a few seconds. One of asteroid mass lasts the age of the universe. One of stellar mass lasts $10^{67}$ years, which is not a long time so much as a number with no physical circumstances left to happen in.

The mass in the middle is the interesting one. A hole weighing about $1.7\times10^{11}$ kilograms — an asteroid squeezed to something smaller than a proton — would be finishing about now if it had formed in the first instants of the universe. That number is what every search for primordial black holes is aimed at, and the figure computes it rather than quoting it.

What a lifetime that long means #

The stellar-mass number deserves a sentence of its own, because $10^{67}$ years is not a duration anybody has intuition about and the comparison is instructive.

The universe is about $1.4\times10^{10}$ years old, so the ratio is $10^{57}$. Every other timescale in astrophysics is closer to today than that number is to itself: the Sun’s remaining life is $10^{-57}$ of it, the decay of a proton — if protons decay at all — is $10^{-33}$ of it, and the time for the last stars to burn out is $10^{-53}$.

What that means in practice is that the evaporation of stellar-mass holes is the last thing that happens. Long before it, the stars go out, the galaxies disperse, whatever matter is left decays or does not, and the universe consists of black holes and radiation in an otherwise empty space. Only then does the cube law begin to matter, and it then runs for longer than everything that preceded it by fifty-seven orders of magnitude.

A power law with an exponent of three, applied across the masses that exist, produces a hierarchy of timescales with nothing in the middle. The interesting holes are either finishing now or will not finish for effectively ever, and there is no observational window between the two.

The end #

Because the mass goes as the cube root of the time remaining, the decline is almost invisible until the very end.

A hole that has used ninety per cent of its life has lost about half its mass. One that has used ninety-nine per cent has lost about eighty per cent. The last per cent of the life takes it from a fifth of its mass to nothing, and the luminosity in that stretch rises without bound.

The final second is the striking part, and it is completely independent of the hole’s history. However massive it began, when one second of life remains a black hole weighs about two hundred and thirty tonnes, and it releases the whole of that as energy in that second: some $2\times10^{22}$ joules, a few million megatons, emitted as an increasingly hard spectrum of gamma rays.

A signature that does not depend on the source’s history is worth a great deal to an observer. It is why searches for evaporating black holes look for exactly this — a short gamma-ray burst whose spectrum hardens as it goes, with a total energy fixed by nothing but the constants of nature. None has been seen, and the non-detection is what bounds the abundance of primordial black holes in that mass range.

Where the temperature comes from #

The temperature has been used above without an account of it, and the account is worth having because it explains why the same formula turns up in a place with no gravity at all.

The shortest derivation is not Hawking’s. It is a statement about an observer: someone hovering just outside a horizon has to accelerate hard to stay there, and an accelerating observer in empty space finds themselves in a thermal bath at a temperature proportional to their acceleration. That is the temperature of an acceleration, and applying it at the horizon with the acceleration a hovering observer needs, then redshifting the result out to infinity, gives Hawking’s formula exactly.

That route makes the negative heat capacity less mysterious. A bigger hole has a gentler surface gravity — the tidal forces at the horizon of a supermassive hole are unremarkable — so a hovering observer accelerates less and the bath is cooler. The temperature falls with mass because the surface gravity does, and the surface gravity falls because the horizon is further out while the mass rises only linearly.

Two derivations from unrelated starting points give the same number, which is the usual reason to believe a result that cannot be tested. The Unruh effect has not been observed either, and the two stand or fall together.

The picture that does not survive #

The arithmetic above uses a black-body formula with the horizon area in it, and the last figure shows why that is bookkeeping rather than a description.

The peak wavelength of the emitted radiation is about sixteen times the horizon’s radius, and the ratio is the same at every mass, because the only length in the problem is the horizon and everything else is a pure number. A body radiating waves an order of magnitude larger than itself is not a hot surface in any useful sense: there is no small patch it comes from, no direction it leaves in, and no way to localise the emission on the horizon at all.

That is not a criticism of the result; it is a statement about what the derivation actually is. The radiation comes from the behaviour of quantum fields over a region several times the hole’s size, and the area appears because the horizon’s area is what fixes the geometry there, not because a surface is glowing.

The practical consequence is the grey-body factors. A wave much larger than an obstacle scatters off it rather than being absorbed, so the horizon is a poor absorber of exactly the wavelengths it emits most of, and the true luminosity is below the black-body value by factors of order unity that depend on the spin of the emitted particle. Every quantitative statement above is right to a factor of a few and no better.

What has been looked for, and not found #

The whole subject rests on a calculation with no direct evidence behind it, which makes the observational side worth setting out.

The gamma-ray bursts. A primordial hole finishing now would produce the burst described above, and its spectrum is distinctive: not merely hard but hardening, on a timescale of seconds, with a characteristic energy climbing through the TeV range. Air-shower arrays and space telescopes have searched for decades. The limit is that fewer than about one such event per cubic parsec per year happens locally, which bounds the local density of holes in that mass range to something like ten thousand per cubic parsec — a small number in absolute terms and a weak constraint on cosmology.

The diffuse background. Holes evaporating throughout cosmic history would leave a contribution to the gamma-ray background at a few hundred MeV. It is not seen above what other sources explain, and the limit is that primordial holes near $10^{14}$ kilograms make up less than about $10^{-8}$ of the dark matter.

And the analogues. Since the effect cannot be produced, several groups produce something with the same mathematics instead: a sonic horizon in a flowing fluid or a Bose–Einstein condensate, where the flow exceeds the speed of sound and phonons cannot escape upstream. Correlated phonon pairs across such a horizon have been measured, with a spectrum that matches the predicted thermal one.

An analogue is not evidence for the astrophysical claim. What it tests is the field-theoretic argument — that a horizon in a medium produces correlated pairs with a thermal spectrum — and that argument is the part of the derivation nobody doubted. The part in doubt concerns gravity, and no analogue has any.

Where the model stops #

The calculation is semiclassical. The field is quantum and the geometry is not, which is fine while the hole is much heavier than the Planck mass and fails at the very end. The last stage — the final Planck mass, over the last Planck time — is outside every theory anyone has, and whether it ends in a burst, a remnant, or something else is an open question rather than a detail.

Only massless particles were counted. Once the temperature exceeds the mass of a particle, that particle joins the emission and the luminosity jumps. A hole below about $10^{14}$ kilograms is hot enough to emit electrons, and one below $10^{11}$ emits everything in the standard model, so the effective coefficient in the cube law is not a constant across the whole range.

The universe is treated as static. The background temperature falls as the universe expands, so the crossing point in the first figure moves upward with time — a hole that is growing today will begin shrinking when the background drops below its own temperature, and the date at which that happens depends on the mass. The figure draws one instant of a moving boundary.

The horizon is treated as a surface with an area and nothing else. It has a shape, a spin can distort it, and light circles it at the circle light cannot leave rather than at the horizon itself — so the geometry the radiation actually samples is larger than the area in the formula.

Rotation and charge are absent. A spinning hole radiates preferentially in a way that spins it down, and a charged one discharges before it evaporates, so both approach the uncharged non-rotating case before the endgame. That is a real result and it is why the simple treatment is not as restrictive as it looks.

And the information question is untouched. The radiation as computed is exactly thermal, which means the hole’s whole history is erased, which is inconsistent with quantum mechanics being unitary. That is the information paradox, it is not resolved by anything in this essay, and it is the reason the calculation is still being argued about half a century later.

The entropy underneath it all #

There is a way of reading the whole subject that makes the temperature look inevitable rather than surprising, and it is the one the second law supplies.

Attach an entropy to a black hole proportional to its horizon area, and the classical theorem that the area never decreases becomes the second law. That was Bekenstein’s argument, and it was made before Hawking’s calculation and against his objection: a body with an entropy and an energy has a temperature, by thermodynamics alone, and the temperature that comes out of differentiating the area entropy with respect to the mass is exactly the one Hawking later derived from the field theory.

The consistency is the strongest reason to take the result seriously. One route is quantum field theory in a curved background and involves Bogoliubov coefficients and mode mixing; the other is a thermodynamic identity applied to a formula for entropy. Neither knows about the other and they agree on the coefficient.

The entropy also puts the numbers in perspective. A stellar-mass hole has an entropy of about $10^{78}$ in units of Boltzmann’s constant, which is some twenty orders of magnitude more than the star it formed from. Collapse is by an enormous margin the most entropy-producing process available — far beyond anything the second law with a probability attached counts in a laboratory —, and evaporation runs it backwards — the outgoing radiation carries more entropy still, which is how the second law survives the hole’s disappearance. Where the information goes is the separate and unresolved question.

Two derivations, one number, and a second law that holds throughout. That is why a result nobody can test is nevertheless treated as established.

What the pictures cannot show #

Every figure treats the hole as isolated. A real black hole is accreting — gas, dust, the microwave background, and starlight — and the balance between accretion and emission is what actually decides its mass. The evaporation curves describe a hole in an empty universe and are a limit rather than a history.

The runaway figure plots luminosity on its own scale because it diverges, and a divergence cannot be drawn. What the plot shows is the shape of the approach; the actual value at the end is bounded only by where the semiclassical treatment fails, which is not a place the drawing can mark.

A third omission is what the radiation is made of. Every figure treats the emission as a single black-body curve, and it is not: it is photons, neutrinos, gravitons and — once the hole is hot enough — every massive particle in turn, each with its own grey-body factor and its own threshold. The spectrum of a real evaporating hole is a sum of components that switch on one after another as it heats, and the hardening of the final burst is largely that switching rather than the smooth rise of a single temperature. A figure of the total luminosity hides the fact that the composition of the output changes completely over the last stages, which is the part an observer would actually see.

Where the ladder goes next #

The horizons ladder began with the surface that only lets things in, passed through two clocks that disagree about the fall and the horizon that nothing marks, and reached the circle light cannot leave. This rung takes the temperature seriously and follows it to the end. The rungs after it: the entropy and the area law, where the count of states is the horizon’s area in Planck units; the information paradox, which is the same calculation read as a problem; and the Page time, at which a hole has radiated half its entropy and the question of what the radiation carries becomes sharp.

The habit worth carrying away is that the sign of a heat capacity decides a system’s whole history. A body that gets hotter as it loses energy has no equilibrium available to it, and everything about a black hole’s fate — the runaway, the burst, the impossibility of one sitting quietly in a warm universe — is that one fact worked out.