The area that is not allowed to shrink

On 14 September 2015 two black holes of about thirty-six and twenty-nine solar masses finished spiralling together and became one of about sixty-two. The arithmetic is not a mistake: three solar masses’ worth of energy left as gravitational waves, in a fifth of a second, at a peak power greater than the light of every star in the observable universe put together.

So the mass of the system went down. In a subject whose founding image is a thing nothing escapes from, that is worth pausing on, and it raises the obvious question of what — if anything — is the quantity that only goes one way.

The quantity with a direction #

Hawking proved in 1971 that the total area of black-hole horizons cannot decrease. Two holes may merge, matter may fall in, a hole may be spun up or slowed down, and through all of it the sum of the horizon areas goes up or stays the same.

The area of a rotating hole is

$$A = 8\pi\left(\frac{GM}{c^2}\right)^2\left(1 + \sqrt{1-\chi^2}\right)$$

with $\chi$ the spin as a fraction of the maximum the hole can have. At zero spin that is $16\pi (GM/c^2)^2$ — the sphere of the Schwarzschild radius, which the surface that only lets things in locates. At the maximum spin it is exactly half of that.

So area and mass are genuinely different quantities, because the area depends on the spin too. A hole can lose mass while gaining area, if it is slowed down in the process — and that is what a merger does.

What a spin costs and what it offers #

The dependence on spin is where the useful consequences are.

Define the irreducible mass as the mass a hole of the same area would have at rest: $M_{\text{irr}} = \sqrt{A/16\pi}$ in geometric units. The area theorem then says that the sum of the squares of the irreducible masses never decreases, and the difference between a hole’s actual mass and its irreducible mass is energy the theorem permits to be extracted.

That difference is not small. A maximally spinning hole has an irreducible mass of $1/\sqrt{2}$ of its total, so twenty-nine per cent of its mass-energy is available without anything crossing the horizon outwards. For comparison, hydrogen fusion releases seven-tenths of one per cent, and accretion onto a non-spinning hole about six per cent. The rotational energy of a black hole is the most concentrated energy store in the universe, and the theorem that forbids the horizon from shrinking is what says how much of it can be reached.

How the energy comes out #

Saying that twenty-nine per cent is available is not saying how to get it, and the mechanisms that do are worth naming because they are what the figure’s dashed curve is a budget for.

The Penrose process is the original. Inside the ergosphere — a region outside the horizon where nothing can stay still, because the hole’s rotation drags space around with it — a particle can have negative energy as measured from far away. Send a body in, split it in two so that one half takes a negative-energy orbit and falls through the horizon, and the other half comes out with more energy than the whole went in with. The hole pays, by losing exactly the angular momentum and mass the swallowed half carried, and its irreducible mass is unchanged if the split is done ideally and rises if it is not.

The astrophysically important version is the Blandford–Znajek mechanism, which does the same thing with a magnetic field instead of a projectile: field lines threading the hole are dragged round, the twist propagates outwards as a Poynting flux, and the hole spins down. It is the leading candidate for what powers a relativistic jet from an active galaxy, and its energy budget is the one this figure draws.

Both are limited by the same inequality, and neither can touch the irreducible mass. That is the practical content of the theorem: a black hole is a battery with a stated capacity, and the capacity is set by its spin rather than by its mass.

A ceiling on every merger #

The same inequality bounds what a merger can radiate, using nothing else at all.

The derivation is two lines. The final hole must have at least the combined area of the two that made it, and it has the smallest area for a given mass when it is not spinning, so $M_f \ge \sqrt{m_1^2 + m_2^2}$ — and everything above that is radiable in principle. For equal masses that permits $1 - 1/\sqrt{2}$, again twenty-nine per cent.

No orbital mechanics enters that bound. There is no post-Newtonian expansion, no numerical relativity, nothing about how the two holes actually approach each other. It is a conservation-law argument of the kind that bounds a chemical reaction by its enthalpies without knowing the mechanism, and like those bounds it is comfortably loose: real mergers radiate around five per cent, because the dynamics leaves most of the room unused.

The looseness is worth noticing rather than apologising for. A ceiling that every event clears by a factor of six is a weak constraint on any particular event and a strong statement about the theory, because a single measurement above it would refute a theorem.

Where the loose ceiling is not loose #

There is one arrangement in which the bound is nearly tight, and finding it is a good check that the inequality has been understood.

Take two holes of equal mass, both spinning as fast as they can, with their spins anti-aligned to their orbit. Each starts with the smallest horizon its mass allows, so the initial area is as small as possible; and the merger can then produce a remnant with modest spin, whose area need not be much larger. Numerical relativity puts the radiated fraction for such a configuration above ten per cent, against a ceiling that the same configuration raises well beyond the twenty-nine per cent of the non-spinning case.

The opposite arrangement — a light hole falling into a much heavier one — has almost no room at all, and the reason is worth stating. A small hole contributes area in proportion to the square of its mass, so a companion a tenth the mass brings a hundredth of the area, and there is almost nothing to be gained by rearranging. Extreme mass ratios radiate a fraction of a per cent, and the theorem says so before any calculation.

The bound is loose where the dynamics has choices and tight where it does not, which is the usual behaviour of a constraint that comes from a conservation law rather than from an equation of motion.

Measuring an inequality #

The interesting question is whether the area increase can be observed rather than computed, and it can, because the two areas are encoded in different parts of the signal.

The inspiral is a chirp whose rate of sweep depends on the masses, and reading them off it is the same analysis the orbit that has to shrink describes. The ringdown is different physics entirely: a disturbed black hole settles by radiating at a discrete set of complex frequencies fixed by its mass and spin alone, so a measurement of one damped sinusoid’s frequency and decay time gives both.

Analysing the two halves separately and comparing them is a test, and it was carried out on this event in 2021, some six years after the detection — the delay being the time it took to develop the analysis that isolates the ringdown cleanly. The increase came out several times its uncertainty, which is what a first test of a theorem should look like: enough to have failed, not enough to be called precise.

The one assumption #

Hawking’s proof needs an energy condition. Roughly: no observer sees a negative energy density anywhere. Every form of matter anybody has handled satisfies it.

The outgoing radiation is paired with an ingoing flux of negative energy, which is what makes the hole lighter, and negative energy is what the theorem’s hypothesis excludes. Take the assumption away and the conclusion goes with it. There is nothing wrong with the proof; it is simply about a class of processes that does not include this one, and the hole that outlives everything and then does not is what happens in the class it excludes.

What survives is the generalised second law: the horizon area, in Planck units and divided by four, plus the ordinary entropy of everything outside, never decreases. During evaporation the area term falls and the second term rises faster, because the radiation carries away more entropy than the horizon gives up.

That is the strongest reason to read the area as an entropy rather than as an area. Two one-way quantities that fail separately and hold together are not two coincidences; they are one quantity written in two halves, and the entropy that lives on a surface is where that identification is worked out.

What else the ringdown says #

The ringdown measurement deserves a paragraph of its own, because it is the part of gravitational-wave astronomy that tests the theory rather than using it.

A settled black hole in general relativity is described by two numbers, its mass and its spin. Everything about it follows — including the complete set of frequencies and damping times at which it rings when disturbed. Those are not adjustable: measure one mode’s frequency and decay and the mass and spin are fixed; measure a second mode and the theory is being tested, because the second mode’s parameters were already predicted by the first.

That programme — black-hole spectroscopy — is at the edge of what current detectors can do, since the ringdown is brief and the second mode is weak. The area test is a cheaper version of the same idea: it needs only one mode, and it compares the answer against a quantity measured from the inspiral rather than against another mode. Both rest on the same claim, which is that a black hole has no properties beyond mass and spin, and what what the instrument actually hears can extract from a signal is what decides how well the claim can be checked.

Why the area and not something else #

It is worth asking why area, of all quantities, is the one that behaves this way.

The proof is geometric. The horizon is generated by light rays that never quite escape, and a theorem about how bundles of light rays behave under gravity — the focusing theorem — says that gravity focuses such a bundle and never defocuses it, provided the energy condition holds. A converging bundle of horizon generators would meet, and generators that meet leave the horizon, which cannot happen on a surface defined as the boundary of what escapes. So the generators can only spread, and a surface whose generators spread has a growing area.

Nothing in that argument mentions thermodynamics, and the resemblance to the second law was noticed as a formal analogy before it was believed to be more. What made it more was Bekenstein’s observation that the analogy has to be exact or the second law of ordinary thermodynamics can be broken — drop a box of hot gas into a hole and the entropy of the outside world falls — followed by Hawking’s discovery that a hole radiates at a temperature which makes the analogy dimensionally consistent.

A quantity that had to be an entropy for another law to survive turned out to be one. That sequence — an analogy, a paradox that forces it to be taken literally, and then a mechanism — is how a good deal of physics arrives.

The number that is not small #

It is worth putting the entropy of a black hole beside something familiar, because the size of it is the reason the identification is not a formality.

The area in Planck units, divided by four, is the entropy in bits — and a Planck area is about $10^{-70}$ square metres. A hole of one solar mass has a horizon some three kilometres across, so its area is around $10^{8}$ square metres and its entropy is of order $10^{77}$ in units of Boltzmann’s constant.

The Sun’s entropy, computed as an ordinary ball of hot gas, is around $10^{58}$. Collapsing it to a black hole would therefore multiply its entropy by nineteen orders of magnitude. There is nothing else in physics that does that: the entropy of matter is roughly the number of its particles, and the entropy of a horizon is the number of Planck areas on it, and for anything of stellar size the second number is unimaginably larger.

That gap is why the generalised second law is not in danger from anything ordinary. Throwing matter into a hole destroys its entropy from the outside world’s point of view, and the horizon grows by so much more that the sum is never close.

Where the model stops #

The holes here are Kerr, and isolated, and stationary. The area formula belongs to a hole that has settled down. During a merger there is no stationary horizon at all and the area is defined by a construction that only agrees with the simple formula before and long after.

The progenitor spins are taken as zero. They are poorly measured for most events, which is why the conservative choice is made; it weakens the test rather than biasing it, since any real spin makes the initial area smaller and the increase larger.

The uncertainties are illustrative. The published analysis propagates full posterior distributions for four correlated parameters rather than symmetric error bars on two numbers, and the resulting statement is a probability that the area increased rather than a number of standard deviations.

And nothing here says the theorem is true. It says that one event is consistent with it at a stated strength. A theorem in a classical theory is proved rather than measured; what a measurement can do is test the theory the proof is carried out in, and one event tests it once.

What the pictures cannot show #

The merger figure draws two bars for the mass and two for the area, as though the merger were a transaction between two states. It lasts about two-tenths of a second and during it neither quantity is defined in the way the figure uses: there is one distorted horizon, changing shape violently, and only numerical relativity can say what its area is at any moment. The bars are the endpoints of a process the figure omits entirely.

Nor does anything here show the waveform. The measurement described in the fourth figure is a fit of a model to a strain time series buried in noise, and the two “independent measurements” are independent in the sense of using different stretches of data and different physics — not in the sense of being separate experiments.

Where the ladder goes next #

The horizon ladder began with the surface that only lets things in, went through two clocks that disagree about the fall and the horizon that nothing marks, where the surface turns out to be locally unremarkable, past the circle light cannot leave, and reached evaporation. This rung asks what quantity the whole subject conserves or increases, and finds one that is neither the mass nor the energy but a geometric area.

The rung after it is the rest of the analogy: a horizon has a temperature and an area, so it should have a first law relating a change in mass to a change in area and a change in spin — and it does, with the surface gravity playing the part of the temperature. The habit worth carrying is the one this rung is built on: when the obvious conserved quantity turns out not to be monotone, look for the one that is, because a one-way quantity usually means an entropy and an entropy usually means something being counted.