Hotter as it shrinks

At its defaults it draws hotter as it shrinks. The temperature of a horizon and its evaporation time, against mass, on logarithmic axes. A solar-mass horizon is at 6.17·10⁻⁸ K — far colder than the coldest thing anybody has made — and takes 2.1·10⁶⁷ years to evaporate. Both lines are straight, with slopes of exactly −1 and +3, and the first slope is the whole difficulty: losing mass makes the object hotter, so the process accelerates and ends in a burst rather than fading out.

horizon-thermo is one function in lib/figures/gravity.js — geometry, horizons, and the waves in it. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The temperature of a horizon and its evaporation time, against mass, on logarithmic axes. A solar-mass horizon is at 6.17·10⁻⁸ K — far colder than the coldest thing anybody has made — and takes 2.1·10⁶⁷ years to evaporate. Both lines are straight, with slopes of exactly −1 and +3, and the first slope is the whole difficulty: losing mass makes the object hotter, so the process accelerates and ends in a burst rather than fading out.

The books for GW150914: two black holes of 36 and 29 solar masses merging into one of 62, with a final spin of 0.67. 3.0 solar masses left as gravitational waves, so the mass fell by 4.6 per cent. The total horizon area rose, from 107417 to 168330 in units of the Sun's gravitational radius squared — an increase of 57 per cent. The two progenitors are taken as non-spinning, which is the assumption that makes the test hardest to pass: a spinning hole of the same mass has a smaller horizon, so any spin they actually had would only widen the gap. Mass is the quantity that behaves the way energy usually does and it is not the one with a direction. Area is, and it is the reason the area has been read as an entropy ever since.

The horizon area of a hole of fixed mass against how fast it spins, in units of the area it would have at rest, together with its irreducible mass — the mass it would have if all its spin were removed without changing its area. A maximally spinning hole has exactly half the horizon of a still one of the same mass, which the figure checks at both ends rather than reading off the curve. At a spin of 0.3 the area is 98 per cent and the irreducible mass 99 per cent; At a spin of 0.67 the area is 87 per cent and the irreducible mass 93 per cent; At a spin of 0.9 the area is 72 per cent and the irreducible mass 85 per cent. The gap between the mass and the irreducible mass is energy that can be taken out without shrinking the horizon, and the area theorem says it is the only energy that can be taken out at all. At the maximum spin it is 1 − 1/√2, or twenty-nine per cent of the whole mass — an enormous fraction by any other standard, and available in principle by slowing the hole down rather than by feeding it.

The largest fraction of its mass a merger of two non-spinning black holes could turn into gravitational waves, against the ratio of the two masses, with the ceiling coming entirely from the requirement that the horizon area not decrease. Equal masses have the most room: 29.3 per cent, which is 1 − 1/√2. A merger with a small companion has almost none, because a small hole brings almost no area to add. The points are measured events. GW150914 radiated 4.6 per cent against a ceiling of 28.9; GW151226 radiated 4.1 per cent against a ceiling of 26.0; GW170814 radiated 4.7 per cent against a ceiling of 29.0; GW190521 radiated 6.0 per cent against a ceiling of 28.7. Every one is comfortably under, by a factor of five or six, and the gap is not slack in the theorem — it is the difference between what a conservation law permits and what the dynamics actually does. A ceiling derived from a single inequality, with no orbital mechanics in it at all, still bounds every event ever recorded.

The horizon area of GW150914 before and after the merger, each with the uncertainty its own measurement carries. The two are read from different parts of one signal and are independent of each other: the slow spiral before the merger fixes the two masses, and the ringing afterwards fixes the remnant's mass and spin from the frequency and decay of the modes it settles into. Taking the areas as 107417 ± 8593 and 168330 ± 20200 in units of the Sun's gravitational radius squared, the increase is 60913, which is 2.8 times the uncertainty on the difference. That is what a test of the theorem looks like: not a demonstration that the area grew, which no single number can be, but two separately measured quantities and an inequality between them that could have come out the other way. It did not.

The horizon area of an isolated hole against time, in units of its lifetime, when the only thing happening to it is that it radiates. The area falls to zero, in flat contradiction with a theorem that says it cannot. Nothing is wrong with the theorem. Its proof requires an energy condition — roughly, that no observer measures a negative energy density — and Hawking radiation is precisely a case where that fails: the outgoing radiation is paired with an ingoing flux of negative energy, which is what makes the hole lighter. Take the assumption away and the conclusion goes with it. The flat line is the classical statement, which holds for every process that respects the condition, including everything that has ever been observed. What survives the quantum case is the generalised version: the horizon area plus the entropy of everything outside it, which grows even while the area shrinks, because the radiation carries away more entropy than the horizon loses.

physicscheck asserts something about horizon-thermo that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.