The entropy that lives on a surface

Take a cup of tea and drop it through a horizon. The tea had entropy; it is now inside, where nothing outside can measure it, and the outside world’s entropy has fallen. Repeat with anything at all and the second law of thermodynamics can be violated as often as desired, by an experiment requiring nothing but patience and a supply of teacups.

That was the state of the argument around 1970, and there were two ways out. Either the second law is a statement about the outside world that ceases to apply when horizons are present, or a horizon has an entropy of its own that goes up by at least as much as the entropy that went in.

What the tea argument actually shows #

It is worth being precise about the violation, because the loose version — “entropy went inside, so it is hidden” — is not a violation of anything. Physics has no objection to entropy being somewhere inconvenient.

The problem is sharper. The second law is a statement about the entropy of a closed system, and after the teacup has crossed the horizon the exterior is not one: something left it. So the honest statement is that the exterior’s entropy has decreased and the only region that could account for the difference is one from which no measurement can ever return. A law whose bookkeeping depends on a quantity nobody can ever check is not a law any more; it is an accounting convention.

Bekenstein’s move was to insist that the bookkeeping be completable from outside. If the horizon’s own entropy is a function of quantities measurable from outside — and for a stationary horizon, mass, angular momentum and charge are all there is — then the sum can be evaluated by an external observer at every moment, and the law can be tested.

That requirement is what forces the answer to be geometric. There is nothing else available. The horizon has no composition, no temperature gradient, no internal structure that an external measurement can reach; what it has is an area, and the area is the only candidate with the right monotonic behaviour.

The quantity that only goes up #

Bekenstein’s route to the answer, as a graduate student in 1972, was to look for something about a horizon that behaves the way entropy behaves. There is exactly one candidate.

Hawking had proved a theorem the year before: in any classical process, the total area of horizons never decreases. Throw something in and the area grows; merge two and the result has more area than the sum of the parts. It cannot be reduced by any classical means.

A quantity attached to a system that never decreases, in a theory with nothing thermodynamic in it, is a very suspicious object. Bekenstein proposed that the area is the entropy, up to a constant, and that the second law should be restated in a generalised form: the sum of the outside world’s entropy and a constant times the horizon area never decreases.

What an entropy is a count of is the number of microscopic arrangements consistent with what is known about the system, with a logarithm taken so that the quantity adds. That definition is what makes a black hole’s entropy a genuinely startling number: it says a hole has that many internal arrangements, and general relativity says a hole has three parameters and no internal structure at all.

The proposal was immediately unpopular, and for a good reason: if a horizon has an entropy and an energy, then it has a temperature, and something with a temperature radiates. A classical black hole cannot radiate — that is what the word means. Hawking set out to show that Bekenstein was wrong.

The temperature that followed #

What Hawking found in 1974, by considering quantum fields on the background of a collapsing body, was that a horizon does radiate, with a thermal spectrum at

$$T = \frac{\hbar c^3}{8\pi G M k_B}.$$

That fixed Bekenstein’s constant. Thermodynamics requires $\mathrm{d}E = T\,\mathrm{d}S$ with $E = Mc^2$, and integrating with the temperature above gives

$$S = \frac{k_B c^3 A}{4 G \hbar} = \frac{k_B A}{4\ell_P^2},$$

a quarter of the area in units of the Planck length squared. Nobody chose those units; they arrive because the expression contains $G$, $\hbar$ and $c$ together, and that combination is a Planck area whether anyone was looking for one or not.

Two features of that expression deserve attention before its consequences. The first is that it contains four constants — $\hbar$, $c$, $G$ and $k_B$ — one from each of quantum mechanics, relativity, gravitation and thermodynamics. No other formula in physics does that, and it is the reason the result is treated as a clue about how the four fit together rather than as a curiosity about one kind of object.

The second is the sign of the mass dependence. Temperature goes as $1/M$, so the biggest objects are the coldest. That is backwards from everything ordinary: a large hot thing is hot because of what is in it, and here the temperature is set by the geometry, which gets gentler as the object gets larger. The same $1/M$ appears in the tidal stretching at the horizon and for a related reason — everything about a big horizon is mild.

Negative heat capacity, which is the whole trouble #

The slope of $-1$ on the temperature curve is the strangest thing here and its consequences are severe.

Every ordinary object cools as it loses energy. A horizon does the opposite: losing mass makes it hotter, which makes it radiate faster, which makes it lose mass faster. There is no equilibrium to settle into. The heat capacity is negative, which for any ordinary system would be a thermodynamic impossibility.

The consequence is the $M^3$ lifetime. Integrating the radiated power gives an evaporation time

$$t \sim \frac{5120\pi G^2 M^3}{\hbar c^4},$$

which for a solar mass is $2\times10^{67}$ years — a number so large that the process is irrelevant for anything of stellar size, and which falls to a second for a mass of about $2\times10^{5}$ kg. The end is not a fade but an acceleration: the last mass goes in a burst.

The ordinary thermodynamic ceiling is worth setting against this for contrast. Carnot’s argument assumes two reservoirs at fixed temperatures, and a reservoir is fixed precisely because it is large — adding heat to it does not warm it. A black hole does the opposite: adding energy makes it cooler, because its temperature falls as its mass rises. It is a system with a negative heat capacity, so it cannot serve as a reservoir and cannot come to equilibrium with an infinite bath.

How the radiation is supposed to arise #

The mechanism deserves a paragraph, with a warning attached, because the popular version of it is a picture rather than the calculation.

The picture: pairs of virtual particles appear everywhere in empty space and normally annihilate immediately; near a horizon one member of a pair can fall in while the other escapes, and the escaping one carries away energy, so the horizon loses mass. It is memorable, it gets the sign right, and it is not what the calculation says. The calculation is about how a quantum field’s notion of “no particles” differs between an observer far away and one falling in, on a spacetime where a horizon has formed — and the answer is that the distant observer’s vacuum is populated, thermally, at the temperature above. No pair is localised anywhere in that derivation, and the energy bookkeeping in the pair picture requires one member to have negative energy, which is not a thing that happens in flat space.

Two features of the real derivation are worth keeping. The spectrum is exactly thermal, which is what makes the temperature a temperature rather than a characteristic energy — and which is also the root of the information problem, because a thermal spectrum carries no information beyond its temperature. And the effect is not specific to gravity: an observer accelerating uniformly through empty flat space also sees a thermal bath, at a temperature proportional to the acceleration, which is the Unruh effect and is the same mathematics with the Rindler horizon in place of a gravitational one. Whatever this is, it is a property of horizons rather than of black holes.

An area, where every other entropy is a volume #

The deepest oddity is not the temperature. It is the scaling.

Ordinary entropy is extensive: double the volume of a gas at fixed density and the entropy doubles, because the number of microstates multiplies and its logarithm adds. Every thermodynamic argument in ordinary physics leans on that.

A horizon’s entropy goes as the area, so doubling the mass — which doubles the radius — quadruples the entropy rather than multiplying it by eight.

The radius is $2GM/c^2$, so the area is $16\pi G^2M^2/c^4$ and goes as the square of the mass — which is the whole of why the entropy is extensive in an unfamiliar way. The area is also the one geometrical quantity about a horizon that never decreases in any classical process, which is what made the identification with entropy irresistible before anybody knew what it was counting.

Follow the consequence through. Since a horizon is the most entropic object of its mass, the entropy that can be packed into any region of space is bounded by what would happen if it all collapsed — that is, by the area of the region’s boundary in Planck units. A cubic metre of space cannot hold more than about $10^{70}$ bits, and the bound is set by its surface rather than by its volume.

That is the holographic bound, and it says something uncomfortable about how much information a region of space can hold: the number of degrees of freedom does not grow with the volume, as every field theory says it should, but with the boundary.

The length that supplies the unit is found where a mass’s Compton wavelength crosses its Schwarzschild radius, and the horizon entropy is a quarter of the area measured in squares of it. That a quantity of thermodynamics should be measured in units built from $G$, $c$ and $\hbar$ together is the reason this subject is taken as evidence about quantum gravity rather than as a curiosity about black holes.

The parallel that was noticed before it was believed #

Before any of the physics was settled, there was a table. Bardeen, Carter and Hawking wrote down four laws of horizon mechanics in 1973 and observed that they had the same structure as the four laws of thermodynamics, term for term.

The zeroth law of thermodynamics says a body in equilibrium has one temperature throughout; the zeroth law of horizon mechanics says a stationary horizon has one surface gravity throughout. The first law relates a change in energy to a temperature times a change in entropy; the horizon version relates a change in mass to the surface gravity times a change in area, plus rotation and charge terms that correspond exactly to the work terms. The second law says entropy never decreases; the area theorem says area never decreases. The third says absolute zero cannot be reached in finitely many steps; the horizon version says a maximally rotating configuration, with zero surface gravity, cannot be reached either.

Their paper presented this as a formal analogy and said explicitly that the temperature was not a real temperature. It is worth noticing why that was the sensible position: a classical black hole absorbs everything and emits nothing, which is the behaviour of an object at absolute zero, and assigning it a non-zero temperature looked like an obvious contradiction.

What changed was not the table but the discovery that the analogy was an identity. The lesson, for anybody assembling a table of correspondences, is that a structural parallel this complete is unlikely to be a coincidence — and that the objection which makes it look impossible may be the thing that is wrong.

The acceleration that would be warm #

The remark that an accelerating observer in empty flat space sees a thermal bath was made in a clause, and it deserves a number, because the number explains why nobody has seen it.

The Unruh temperature is

$$T = \frac{\hbar a}{2\pi c k_B},$$

and the constant in front is about $4\times10^{-21}$ kelvin per unit of acceleration in SI. So standing on the Earth, at one gravity, corresponds to a temperature of $4\times10^{-20}$ kelvin — twenty orders of magnitude below anything a laboratory has reached, and nineteen below the temperature of empty space.

Reaching one kelvin requires an acceleration of $2.5\times10^{20}$ metres per second squared. That is not a number any macroscopic object survives, and it is why the effect has never been observed directly.

There are two places where it is nearly within reach and both are argued about. An electron circulating in a storage ring experiences, in its own instantaneous rest frame, a centripetal acceleration multiplied by two powers of its Lorentz factor — for a few-billion-electronvolt machine that reaches $10^{23}$ metres per second squared, corresponding to a temperature of around a thousand kelvin. Bell and Leinaas argued that this shows up in the incomplete spin polarisation such beams acquire, which is measured and which is slightly less than the standard theory predicts. Whether that discrepancy is an Unruh effect is a genuine open question with a literature.

The other is a strong laser. An electron in the field of a focused pulse at $10^{22}$ watts per square centimetre is accelerated at around $10^{25}$ metres per second squared, which is tens of thousands of kelvin — and a number of schemes have been proposed for detecting the resulting radiation against the enormous background the laser itself produces.

The conceptual content matters more than the prospect of a measurement, and it is the same lesson the horizon teaches. What the Unruh calculation says is that an accelerating detector clicks in a state that a static detector calls empty. So the question “how many particles are there in this region” has no observer-independent answer, any more than “what time is it there” does. That is a statement about quantum field theory in flat space, with no gravity anywhere in it, and it is the reason the horizon result is regarded as a fact about horizons rather than a fact about black holes.

The burst nobody has seen #

The essay notes that no Hawking radiation has been observed and that none is expected from anything of stellar mass. There is one class of object from which it would be observable, a programme of searches for it, and a result — the result being that nothing has been found and that the absence is itself useful.

Set the evaporation lifetime equal to the age of the universe and solve for the mass. The answer is about $5\times10^{11}$ kilograms: the mass of a small mountain, in a horizon the size of a proton. Anything that mass formed in the early universe would be finishing its evaporation now.

Such objects cannot form by stellar collapse — nothing that light can collapse — but they could have formed from density fluctuations in the first fraction of a second, when the density was high enough that a sufficiently overdense region would collapse on its own. Whether any did is unknown.

If any are, they are visible. The last stages of an evaporation are a burst: the temperature climbs as the mass falls, the spectrum hardens, and the final seconds release around $10^{22}$ joules mostly in gamma rays of increasing energy. That is a distinctive signature — a short, hard, spectrally-rising flash — and several instruments have looked for it.

Nothing has been found. Air-Cherenkov arrays and space-based gamma-ray telescopes have searched their data for bursts of the right shape and duration, and the limits are now stringent: fewer than a few thousand such events per cubic parsec per year in the local neighbourhood. Independently, the absence of an excess in the diffuse gamma-ray background limits the total cosmological density of objects in that mass range to a tiny fraction of the dark matter.

Those are real constraints derived entirely from a theory nobody has confirmed, and they are worth noticing for that reason. Hawking radiation is used quantitatively — to convert a non-detection into a bound on how much of the universe is made of a particular thing — long before it has been observed at all.

There is also a window the argument leaves open. Objects of $10^{17}$ to $10^{22}$ kilograms are far too cold to have evaporated appreciably, emit essentially nothing, and are too small to lens or to disturb anything measurably. They remain a viable candidate for the dark matter, unexcluded, and they are unexcluded precisely because the temperature goes as one over the mass.

What it costs, and where the model stops #

The calculation is semiclassical. The gravitational field is treated classically while the matter fields are quantum, which is an approximation with no controlled error estimate. It is expected to be good while the horizon is large compared with the Planck length and to fail completely at the end of an evaporation, which is exactly the part everybody wants to know about.

Nothing has been counted. The formula gives a number, and an entropy is supposed to be the logarithm of a count of microstates. What those microstates are is not known in general. String theory has succeeded in counting them for certain highly idealised, extremal cases and obtains $A/4$ exactly, which is a genuine and impressive check; the general case is open.

Information appears to be destroyed, and that is a real problem. The radiation is thermal, which means it carries no information about what fell in. If the horizon then evaporates entirely, a pure quantum state has evolved into a mixed one, which quantum mechanics forbids. The information paradox has been argued about for fifty years, has produced several proposed resolutions and no consensus, and is one of the few places where two extremely well-tested theories give incompatible answers about a physical process.

None of it has been observed. No Hawking radiation has been detected, and none is expected from anything of stellar mass — the temperature is far below the temperature of empty space, so such an object absorbs far more than it emits and grows rather than shrinking. Laboratory analogues in fluids and optical systems reproduce the kinematics and are not the same thing.

The entropy of the infalling matter is not conserved into the horizon’s. The generalised law requires the sum not to decrease, and in every case examined the area gains far more than the matter brought — usually by tens of orders of magnitude. So the law is satisfied with enormous room to spare, which is comforting and also slightly suspicious: a conservation statement that is never close to being tight is one whose mechanism has not been understood.

The generalised second law is a hypothesis in good standing rather than a theorem. It has been proved in various restricted settings and is believed generally, and “believed generally” is the right phrase for it.

The ladder from here #

Later rungs on this anchor: the four laws of horizon mechanics, which parallel the four laws of thermodynamics term by term and were originally written down as a formal analogy before anyone thought they were literal; the derivation of the temperature, in whichever of its three or four routes is clearest; the information paradox stated precisely enough to see why it is hard; the holographic bound and what it says about field theory’s counting of degrees of freedom; and the microstate counting that has been achieved, along with what makes the general case resist.

The neighbouring ladders are entropy as a count, which is the rung this one stands on and the thing a horizon’s entropy has never been shown to be a count of, and the horizon itself, whose area turned out to be a thermodynamic quantity forty years after it was first written down.