29%: more than a quarter of a spinning black hole is energy you could mine

Here is a number with a catch: 29%.

The catch is two small words: "up to". The 29% is for a black hole spinning as fast as theory allows, and no real one is expected to get quite that far.

The idea behind it is simple. A black hole's mass comes in two parts. One part is locked in for good. The other is the energy of its spin, and that part could, in principle, be taken back out.

Nobody has done it. But by the end you'll know how much there is, how you might get it out, and three numbers that are easy to mix up: 29%, 20.7% and 42%.

Part IA flywheel with a floor

Chapter 1

Up to 29%

Store gaugeup to 29%the fastest spin theory allows

For a real black hole, two things matter: its mass and how fast it spins. In 1970 Demetrios Christodoulou showed that part of a spinning hole's mass is the energy of its spin, and that this part, and only this part, can in principle be taken back out.

For the fastest spin theory allows, it is 29% of the mass. Not 30%, and not a third.

"Up to" does a lot of work. A black hole fed by a disc of gas is expected to stop a little short, at a spin of 0.998 on a scale where 1 is the fastest, and there the share is 27%. Slower holes hold much less.

In everyday units: a black hole ten times the Sun's mass, at full spin, holds about 5 × 10⁴⁷ joules of spin energy. That's about four thousand times all the light the Sun gives out in its ten billion years as an ordinary star (this page's arithmetic). ☀️

One ten-Sun hole at full spin

≈ 4,000×

One dot: all the light the Sun gives out in its ten billion years as an ordinary star (this page's arithmetic). Each block: 100 dots.

It isn't something for nothing: it is the hole's own energy, and taking it out slows the hole down and makes it lighter. And nobody has taken energy from a real black hole.

The ideas came fast: Roger Penrose showed in 1969 that the energy can be taken out, Christodoulou found how much, and by 1977 others had added waves, a "black-hole bomb" and a way for magnetic fields to tap the spin to power jets, the narrow beams that shoot out from near some black holes.

1963Kerr: the spinning black hole
1969Penrose: the energy can be taken out
1970Christodoulou: how much
1971Hawking's area rule
1971Zel'dovich: the wave version
1972Press and Teukolsky: the black-hole bomb
1974Thorne: the usual ceiling, 0.998
1977Blandford and Znajek: jets

Chapter 2

Two numbers

Store gauge0%spin 0: no spin, nothing to take out

In 1963 Roy Kerr found the exact shape of space and time round a spinning black hole. It takes just two numbers: the mass, and the spin, which runs from 0 to 1, the most theory allows. (A black hole could also carry electric charge, but real ones are expected to be almost neutral.)

Spin changes the horizon, the surface from inside which nothing escapes. Compare a hole at full spin with a non-spinning one of the same mass. The spinning one's horizon has half the area, but its equator is just as long: for ten Suns' worth of mass, 186 kilometres round either way (this page's arithmetic). The horizon isn't shrunk. It's flattened.

Same mass, two horizons

No spinequator 186 km round

Full spinthe same equator, half the area

Ten Suns' worth of mass (this page's arithmetic). A sketch, not to scale.

It also turns. At full spin, the horizon of a ten-Sun hole goes round about 1,600 times a second, as seen from far away (this page's arithmetic).

Spin even cools a black hole. Hawking found in 1974 that every black hole glows very faintly, as if it had a temperature: about 62 billionths of a degree above absolute zero, the coldest possible temperature, for one with the Sun's mass and no spin (Nº 18). Spin lowers it, all the way to zero at full spin.

Yet a spinning hole gives off more power, because it sheds its spin too, so it fades away sooner, though by less than a factor of three (Nº 18).

Chapter 3

The floor

Guess first

A hole spins at 0.5, half the fastest spin theory allows. How much of its mass is spin energy?

See the answer

Only about 3%. The curve creeps up at first and climbs steeply near the end.

Christodoulou asked: by sending things into a spinning black hole, how far can you lower its mass? He found a part that no such process can remove, the irreducible mass.

It is set by the area of the horizon; it isn't the mass of anything inside. It can grow, but leaving quantum effects (the physics of the very small) aside, it can never shrink, because the area can't. Everything above it is spin energy.

Only a perfect, gentle process gets all of it out; anything real raises the floor as it goes. The one thing known to lower the floor is Hawking's quantum glow, far too slow to matter for any black hole today (Nº 18).

Half the spin doesn't give half the store. At a spin of 0.5 the share is only about 3%; at 0.9 it is 15%, about half the maximum. Going from 0.8 to full spin adds about two-thirds of the 29%, but going up from 0.98 adds only 23% of it (this page's arithmetic). Slide the spin and watch the bar

Store gauge15%spin 0.9: about half the maximum

At the usual ceiling, 0.998, the bar reads 27.1%. The two presets from black-hole mergers, spinning at 0.68 and 0.78, hold 6.9% and 9.8% (this page's arithmetic).

Hawking's 1971 rule makes the floor real: leaving quantum effects aside, and for ordinary matter, the total area of black-hole horizons never goes down. From the same area maths he found that two equal, non-spinning black holes that merge can give off at most 29% of their mass: the same number, but only for that case, and a ceiling, not a forecast.

So a spinning black hole is like a flywheel (a heavy wheel that stores energy as it spins and slows down as you draw it off), except that at most 29% of it is flywheel. The rest is a floor, and it rises whenever energy is taken out clumsily.

A spinning black hole is a flywheel with a floor.

Chapter 4

How fast can it spin?

Store gauge27%spin 0.998: fed by a disc of gas, the usual ceiling

The full 29% needs a spin of exactly 1, and real black holes aren't expected to get there.

In 1970 James Bardeen worked out that gas falling in from a thin disc would spin a hole all the way up to 1. In 1974 Kip Thorne added the light. The hot inner disc glows, and the hole swallows some of that light. Light heading against the spin is caught more easily, so it brings in spin the wrong way. The two balance at 0.998: the usual ceiling, and the 27%.

Magnetic gas may stop the spin lower: one set of computer models settled near 0.9. In other simulations, powerful jets even spin holes down (chapter 12).

And when two equal black holes with no spin merge, the new hole spins at about 0.69 (in detailed computer simulations), holding about 7% of its mass as spin energy (this page's arithmetic).

Part IIGetting it out

Chapter 5

Nowhere to stand still

Store gauge15%spin 0.9: the hole in the figure below

To get spin energy out, something has to happen outside the horizon, where things can still escape, in a region called the ergosphere.

A spinning black hole drags space round with it. Far away the drag is tiny, and you can hover in place. Inside the ergosphere it is so strong that nothing can stand still, as seen from far away. Even light sent backwards against the spin starts off going forwards.

Seen from the side

the horizon
the ergosphere: nothing can stand still here
the equator

A sketch, not to scale.

Things can still get out: the ergosphere lies outside the horizon, hugging it at the poles and bulging out round the equator. The same dragging turns a gyroscope (a spinning wheel that keeps pointing the same way) at the Earth's North Pole by about a fifth of an arcsecond a year (an arcsecond is a 3,600th of a degree; Nº 16). Here it can't be resisted at all. Try to hover

The figure opens inside the ergosphere of a ten-Sun hole spinning at 0.9, 26.6 kilometres from the centre (a label for the place, not a measured distance). Anything held there must go round between 215 and 1,091 times a second (this page's arithmetic).

Why it matters: count a thing's energy from far away, and wherever it could hover, that energy must be positive. Inside the ergosphere, and only there, something with a backwards twist (its own twist points the other way from the hole's spin, even though space still carries it round, just more slowly) can have negative energy, as measured from far away. Up close, every energy and mass is still positive. But that loophole is all you need.

Chapter 6

Penrose's rock

In 1969 Penrose spotted what the ergosphere allows.

InSend a rock in and let it split in two there, so that one piece, given a backwards twist, has negative energy as measured from far away and falls into the hole.
OutEnergy still adds up. So the other piece flies out with more energy than the whole rock brought in.
The hole paysIts mass drops by exactly what the escaping piece gained, and because the fallen piece carried its backwards twist into the hole, the hole spins a little slower too.

Note"Negative energy" isn't negative mass: someone right next to the fallen piece would measure an ordinary, positive energy.

The floor never drops: at best it stays put, and in any real split it rises. This Penrose process takes spin energy, and only spin energy.

It's a bit like a space probe swinging round a moving planet: it leaves faster than it came, and the planet slows down by a tiny amount to pay. But that slingshot needs no ergosphere and no negative energy.

This is the same Roger Penrose whose very different idea about gravity and quantum physics is tested in Nº 10, and who turns up in Nº 3. So how much can one split win? Less than 29%.

Chapter 7

One throw

Store gauge27%spin 0.998: before the first throw

Take the textbook version: drop a rock from far away, in the plane of the hole's equator, and at its deepest point it splits into two equal pieces. The deeper the split, the bigger the win: a rock that splits into two flashes of light right at the horizon of a hole at full spin would bring back 20.7% more energy than went in.

But a dropped rock can't get that deep. The deepest point it can reach, where it turns round, lies inside the ergosphere only if the spin is above about 0.83. Below that, it can't win at all.

So the best single throw wins 3.7% at a spin of 0.9, 14.3% at 0.99 and 17.7% at 0.998 (this page's arithmetic). The 20.7% is reached only as the spin gets ever closer to full.

The best single throw: extra energy back

as the spin gets ever closer to full20.7%
spin 0.99817.7%
spin 0.9914.3%
spin 0.93.7%
spin below about 0.83: it can't win at all0

More than went in, for a rock dropped from far away (the numbers below full spin are this page's arithmetic).

And the split has to be explosive: each piece must shoot away from the rock at more than 70% of the speed of light. That's why this version isn't thought to power anything in the sky. Make the throw, then press Mine it

Store gauge · after 703 throws≈ a fifth outof the starting store, and the floor rose

about two-fifths still spinning
about a fifth taken out
about two-fifths newly locked in

Mine it repeats the best throw from a spin of 0.998, each throw a thousandth of the hole's mass. After 703 throws the spin is just below 0.83, and the throws stop paying.

By then about a fifth of the starting spin energy has come out. About two-fifths is still spinning, out of this method's reach. And about two-fifths is locked in for good, because every throw made the horizon a little bigger (this page's arithmetic). From a spin of 0.9, only about 2% of its spin energy comes out.

Mining by throws raises the floor under your feet.

Chapter 8

Horizons only grow

Store gauge6.9%spin 0.68: GW250114's new hole (this page's arithmetic)

Hawking's area rule guarantees the floor, and in 2025 it was tested on real black holes.

On 14 January 2025 the LIGO detectors caught GW250114: two black holes of 33.6 and 32.2 times the Sun's mass, with little spin, merging into one.

As of October 2026 GW250114 is the loudest gravitational-wave signal published so far. (Gravitational waves are ripples in space and time, sent out here by the merging pair.) Its signal-to-noise ratio, a measure of how strongly it stood out from the detectors' noise, was 80, against 26 for the first detection in 2015. How detectors hear a merger is in Nº 7.

The teams estimated the horizons' areas before and after, and the final area was bigger, with high confidence (4.4 sigma, in physicists' terms). That's a strong confirmation of Hawking's rule (which assumes ordinary matter and leaves quantum effects aside), not a proof.

Horizon area, before and after the merger

≈ 237,000to373,000

square kilometres: 57% more horizon after the merger (this page's arithmetic)

By this page's arithmetic, the two old horizons had about 237,000 square kilometres between them, and the new one has 373,000: 57% more. The merger gave off about 3.1 Suns' worth of energy, and the new hole, spinning at 0.68, stores 4.3 Suns' worth as spin: more than the merger gave off.

Chapter 9

Waves that come back stronger

Store gauge29%full spin: the hole that gives the best bounce

The Penrose process has a wave version. In 1971 Yakov Zel'dovich, inspired by Penrose, argued that any spinning thing that absorbs waves will make some of them stronger.

A wave can have a pattern that goes round. If the body spins faster than the pattern turns, then seen from the body the pattern runs backwards, and an absorber that would normally take energy from the wave gives energy to it instead. (Think of a carousel: to a rider, a pattern turning more slowly than the platform seems to go backwards. That shows the backwards part, not the energy bookkeeping.)

Nothing about this is special to black holes: Zel'dovich worked it out for a spinning metal cylinder. Hold that thought for chapter 11. But a black hole fits too: its horizon swallows whatever crosses it, and it turns. This is superradiance, the wave version of the Penrose process.

At full spin, the best single bounce adds about 0.4% to the simplest kind of wave and 4.4% to light, but 138% to a gravitational wave: it can more than double its energy.

One bounce off a hole at full spin, at best

Each tick along the line is ten times more than the one before.

Zel'dovich also argued that a spinning absorber should make the quantum vacuum itself glow. (The quantum vacuum is empty space, which in quantum physics is never perfectly still.) The idea is usually told as a step on the road to Hawking radiation, the faint glow of chapter 2.

Chapter 10

The black-hole bomb

Store gauge9.8%spin 0.78: the hole heard merging in 2024, still spinning (this page's arithmetic)

A 4.4% gain isn't much. But send the wave back, again and again.

In 1972 William Press and Saul Teukolsky imagined a mirror round a spinning black hole. A wave bounces between the two, gaining a little at the hole and losing a little at the mirror. If the mirror loses less than the hole adds, the wave grows and grows until the mirror gives way: the black-hole bomb. Or it grows until the hole has given up enough spin that the boost stops.

One round trip, again and again

For the simplest kind of wave the mirror must keep more than 99.6% of the energy at each bounce, for light more than 95.8%, and for a gravitational wave just over 42% (this page's arithmetic). It's a thought experiment. Pick what bounces and press Bounce

It opens on light and a mirror that loses 1% a bounce: each round trip multiplies the energy by 1.034, so it grows tenfold every 70 trips (this page's arithmetic). Lose more than about 4% and the light dies away.

Real black holes have no mirrors, but nature may supply one. If there is a kind of particle with a tiny mass, its slow waves can't escape, so they stay trapped round the hole and do the mirror's job, and the hole pours its spin into a cloud of them.

So black holes seen still spinning rule some of these hypothetical particles out: one hole heard merging in 2024, spinning at 0.78, rules out a range of masses for them (which range depends on the kind of particle and on how old the hole is). That excludes some particles, not all.

Part IIIIs anyone mining it?

Chapter 11

A bomb on a bench

Store gaugeno holea metal cylinder, with no black hole involved

Zel'dovich's rule needs no black hole. By 2024 labs had seen spinning things strengthen water waves, sound and a "fluid" of light, none of them with real gravity.

On 5 November 2025, Marion Cromb and four colleagues in Southampton, Glasgow and Trento reported in Science Advances an electromagnetic analogue of the black-hole bomb (a stand-in built from magnets and electric circuits that follows the same rule), with no black hole involved.

An aluminium cylinder 40 millimetres across, spun by its own motor, sat inside the ring of coils from an electric motor. The coils make a magnetic pattern that turns round about 590 times a second, and their circuits hold the field like a leaky mirror.

The cylinder's speed, in turns a second

Absorbs
Slower than the pattern, the cylinder just soaks up energy.
about 590 the pattern's speedFeeds
Faster, it starts feeding the circuits.
about 640 beats the lossesGrows
Above about 640 turns a second it beat their losses too: a signal grew out of noise while the cylinder slowed down.

With the motor forced to hold its speed, the runaway burnt out a resistor, one of the circuit's parts. 💥

The rim moved at only about 80 metres a second (this page's arithmetic). The trick is that the field swings only about 1,200 times a second, so a spinning cylinder can outrun its pattern. Light swings far faster: to outrun it, a cylinder would have to spin faster than any object can survive. Spin up the motor

At 400 turns a second the field runs forwards as seen from the cylinder, which only absorbs. At the paper's 643-turn run it runs backwards, and the cylinder feeds it.

The honest caveat

The signal grew from room-temperature noise. The warmth of the room alone gives each mode (each way the circuit's field can ring) about five billion quanta, the smallest packets of its energy, against the quantum vacuum's half a quantum, and any other noise only adds to it (this page's arithmetic). So the vacuum version, Zel'dovich's glow, is still unseen.

Since then, a September 2026 preprint, a paper not yet checked by other scientists as of October 2026, reports waves on a whirlpool of superfluid helium (helium so cold it flows without friction) that grew and then settled into what the authors call the bomb's end state.

Chapter 12

Jets that may run on spin

Store gauge · assumed15%M87's hole, if it spins at 0.9, which is an assumption

The one place nature may be mining spin on a grand scale is in jets. In 1977 Roger Blandford and Roman Znajek showed that magnetic field lines threading a spinning horizon act as a brake: the turning horizon drags them round, and they carry spin energy outwards. (They thread the horizon; they don't come out of the hole.)

In computer simulations of strongly magnetic gas, the brake can be strong: round a hole spinning at 0.99, jets and winds carried out about 140% of the energy of the gas falling in. More out than in can only come from the spin, but so far that's only in simulations, and not every jet need tap spin.

Quasar light, from the brilliant centres of some distant galaxies, is a different account. Gas spiralling in through a disc glows before it plunges: round a hole at full spin it can turn about 42% of its mass into light, 32% at 0.998 and 5.7% with no spin (this page's arithmetic). That energy comes from the falling gas, not from the spin store. Quasars shine on falling gas; only jets may run on spin.

Your turn · spin or falling gas?

Where does the energy come from?

Quasar light from a disc round a hole at full spin, up to 42%That energy comes from the falling gas, not from the spin store.
Jets and winds in simulations, about 140% of the falling gas's energyMore out than in can only come from the spin, but so far that's only in simulations.
Penrose's rock, the piece that flies out with more than went inThis Penrose process takes spin energy, and only spin energy.

The best-studied jet is that of the galaxy M87, from a hole of 6.5 billion Suns. In the computer models of the Event Horizon Telescope team, who made the first picture of a black hole, holes without spin don't make powerful enough jets, and in the models that do, the jet runs on spin energy.

So spin energy is the best explanation in their models, not a measurement.

If M87's hole spins at 0.9, which is an assumption, its store is about 1.8 × 10⁵⁶ joules: enough for a jet of 10³⁷ watts for some 560 billion years (this page's arithmetic). That measures the store; it isn't a forecast.

Chapter 13

Weighing the spin

Store gauge · disputednot agreedour galaxy's hole: spin 0.1 or less, or 0.9

Spins are much harder to measure than masses.

No agreed measurement

M87's hole has no agreed spin measurement: a 2020 claim of 0.9 and a 2025 estimate of about 0.8 are both unconfirmed. Other clues say it spins, without saying how fast: its jet wobbles round every 11 years or so, which one team reads as the dragging of a spinning hole.

Disputed

Our galaxy's hole, 4.3 million times the Sun's mass, has a disputed spin: one team finds 0.1 or less from the orbits of nearby stars, another 0.9 from its outflows (the matter it blows out).

Argued over

Cygnus X-1, the most studied of the smaller, star-sized black holes, is argued over too: one team measured 21.2 times the Sun's mass in 2021, a 2025 analysis gives 12.7 to 17.8, and published spins run from almost the maximum down to zero, each depending on its model.

Mostly slow

Merging black holes mostly spin slowly. A 2026 preprint, also not yet checked by other scientists as of October 2026, finds a group with at least one hole at about 0.7, which fits holes born in earlier mergers. Chapter 10's hole at 0.78 was probably one.

X-ray studies of holes feeding on a nearby star, like Cygnus X-1, mostly find high spins, and merging holes low ones: one study suggests the two kinds may simply differ.

Chapter 14

What 29% means

Store gaugeup to 29%idealised: full spin, a perfect process

Your turn · three numbers, easy to mix up

1The whole spin store of an idealised hole at full spin
Which number?
29%, reachable only by a perfect process.
2The most one split of a dropped rock brings back, more than went in, as the spin gets ever closer to full
Which number?
20.7%, and 17.7% at 0.998.
3Light from a disc of falling gas round a hole at full spin
Which number?
42%: light from falling gas, not spin.

The 29% is the whole spin store of an idealised hole at full spin, reachable only by a perfect process. Holes fed by discs of gas top out near 27%, and a merger of two equal, non-spinning holes leaves about 7%.

One perfect throw of a rock dropped from far away gets back at most about a fifth more than went in, and only as the spin nears full (17.7% at 0.998). A disc's 42% is light from falling gas, not spin. And nobody has mined a real black hole: jets may run on spin, and the lab results are analogues.

As of 25 October 2026, Zel'dovich's glow from the vacuum is still unseen, the spins of M87's hole, our galaxy's and Cygnus X-1's are unsettled, and Hawking's area rule has passed a strong test.

Let's add it all up.

Assay certificate

Spin energy, itemised

Fastest spin theory allowsChristodoulou 1970up to 29%

Usual ceiling, spin 0.998Thorne 197427%

After two equal, non-spinning holes mergespin 0.697%

Ten-Sun hole, full spinthis page's arithmetic5 × 10⁴⁷ J

Compared with the Sun's 10 billion yearsthis page's arithmetic≈ 4,000×

Best single throwrock dropped from far away, full spin20.7% more than went in

Speed the pieces needrock dropped from far away> 70% of light

Light from a disc of falling gasfull spin; not spin energyup to 42%

Gravitational wave, one bounceenergy, at bestup to +138%

Lab cylinder, turns a second2025643

Room-temperature noise against the vacuumheat alone, this page's arithmetic5 billion to ½

Energy mined from a real black holeso farnone

spin is a store ✦ no ordinary process lowers the floor ✦ nothing is free

Far away, a black hole turns in the dark, dragging space round with it.

Most of its mass is locked in for good. The rest is its spin, and that part could, in principle, come back out.

Somewhere else, a galaxy's jet may already be running on it.

And in a lab, a metal cylinder hums inside a ring of coils, a little faster than the field.

Nobody has mined a black hole. But now we know how much is there.

29% of a black hole at full spin is energy you could take back out. The rest is a floor no ordinary process can lower.

Part IA flywheel with a floor

Chapter 1Up to twenty-nine per cent

For real black holes two properties matter: mass and spin. Part of the mass of a spinning hole is energy of rotation, and Demetrios Christodoulou showed in 1970 that this part, and only this part, can in principle be taken back out.¹ How much depends on how fast the hole spins. For the fastest spin the theory allows, the share is

1 − 1/√2 = 0.29289…The extractable share of the mass of a Kerr black hole at maximal spin, χ = 1; Chapter 3 derives it. Rounded, 29%: not 30%, and not a third.

So up to 29% of a maximally spinning hole's mass is spin energy, and "up to" carries the weight. Maximal spin is an idealisation: a hole spun up by a thin disc of gas should stop near χ = 0.998, as Kip Thorne showed in 1974,² and there the share is 27.1%. Slower holes hold much less (Chapter 3).

In joules: one solar mass, as energy, is M_☉c² = 1.78709 × 10⁴⁷ J, from the IAU's nominal solar mass parameter and the CODATA 2022 value of G.³ A hole of ten solar masses at maximal spin therefore holds 0.29289 × 10 × 1.78709 × 10⁴⁷ J = 5.2 × 10⁴⁷ J of extractable spin energy, and 4.8 × 10⁴⁷ J at χ = 0.998 (this page's arithmetic). In the model of Klaus-Peter Schröder and Robert Connon Smith the Sun's luminosity grows from 0.70 of today's value at its birth to 1.84 times it at ten billion years;⁴ added up, that main-sequence output is 1.36 × 10⁴⁴ J. The hole's store is about 3,900 times as much (3,600 times at χ = 0.998): about four thousand times what the Sun radiates in its ten billion years as an ordinary star (this page's arithmetic). Counting the Sun's later giant phases too, the ratio is roughly a thousand (this page's estimate: the Sun ends as a white dwarf of 0.54 solar masses, and the estimate counts the burning of all the hydrogen in that material, then most of its helium).

Nobody has taken energy from a real black hole; Part III asks whether nature does. And it is not something for nothing: it is the hole's own rotational energy, and taking it out slows the hole down and makes it lighter. The ideas came within a few years. Roger Penrose showed in 1969 that energy can be taken out of a spinning black hole;⁵ Christodoulou found in 1970 how much, through what he called the irreducible mass;¹ Stephen Hawking proved in 1971 that, classically, the area of a horizon never decreases;⁶ Yakov Zel'dovich found in 1971 that any spinning body that absorbs waves can also amplify them, the wave version of the same effect;⁷ William Press and Saul Teukolsky imagined the black-hole bomb in 1972;⁸ and Roger Blandford and Roman Znajek showed in 1977 how magnetic fields could tap the spin to power jets.⁹ In November 2025 a team reported a laboratory analogue of the bomb: a spinning aluminium cylinder inside a ring of coils (Chapter 11).¹⁰

The three numbers of the opening each come with a condition. The 29% is the whole store, reachable only by an ideal, reversible process at maximal spin. The 20.7% is the most that one Penrose split of a body dropped from far away can return over what went in, again only as the spin approaches its maximum (Chapter 7). The 42% is the share of rest mass that gas spiralling onto a maximally spinning hole can radiate before it falls in: accretion energy, a different account (Chapter 12).

Chapter 2Two numbers and a flattened horizon

In 1963 Roy Kerr found the exact geometry of a spinning black hole.¹¹ It has two parameters, the mass M and the angular momentum J. (Kerr's solution carries no electric charge; astrophysical holes are expected to be almost neutral, so it is the one that matters.) The spin is usually written as a pure number:

χ = cJ/(GM²), a = J/(Mc) = χGM/c²χ runs from 0 (no spin) to 1, the most the theory allows for a black hole. With G = c = 1, as inside every formula on this page, lengths are in units of the gravitational radius GM/c² (1.476625 km per solar mass) and a, the angular momentum per unit mass, is a length: a = χM.

In Boyer–Lindquist coordinates, the standard way of labelling points round a Kerr hole, the horizon sits at

r₊ = M + √(M² − a²), A = 4π(r₊² + a²) = 8πMr₊, Ω_H = a/(2Mr₊)r₊ is a coordinate value: 2M without spin, M at χ = 1. A is the horizon's area, which every observer agrees on. Ω_H is the angular velocity of the horizon as seen from far away.¹²

The halving of r₊ does not mean the hole shrinks to half its size: a coordinate radius is a label, not a length. The area falls from 16πM² to 8πM²: it halves. But the horizon's equatorial circumference is 4πGM/c² at every spin, because there g_φφ, the part of the Kerr metric that measures distance round the spin axis, equals r₊² + a² + 2Ma²/r₊ = 4M². Same equator, less area: the horizon is flattened. For ten solar masses the equator is 186 km round at any spin, while the radius of a sphere of the same area falls from 29.5 km to 20.9 km and the coordinate radius from 29.5 km to 14.8 km (this page's arithmetic). The comparison is between holes of equal mass, not one hole spinning up.

The horizon also turns. Written with χ, its rate is

f_H = Ω_H/2π = χ / [2(1 + √(1 − χ²))] × c³/(2πGM)As seen from far away. The rate scales as 1/M; GM_☉/c³ = 4.925491 µs.

For ten solar masses that is 1,013 turns a second at χ = 0.9 and 1,616 at the maximum.

Spin changes the hole's temperature too. Hawking found in 1974, with the full calculation in 1975, that a black hole radiates as if it had a temperature,^13,14 about 6.2 × 10⁻⁸ K for a non-spinning hole of one solar mass (Nº 18). For a Kerr hole it is multiplied by a factor that falls with spin:

T/T_Schw = 2√(1 − χ²) / [1 + √(1 − χ²)]0.85 at χ = 0.68, 0.61 at 0.9, 0.12 at 0.998 and 0 at χ = 1 (this page's arithmetic). T_Schw = ħc³/(8πGMk_B) = 6.17 × 10⁻⁸ K × M_☉/M.

Colder does not mean slower to evaporate. Don Page calculated in 1976 that, at a given mass, a spinning hole radiates more power, because it sheds its spin as well: as a share, its angular momentum falls several times faster than its mass. A hole born spinning fast therefore dies sooner, though the starting spin changes the lifetime by a factor of only 2.0 to 2.7.¹⁵ Why no black hole has started to evaporate yet is the subject of Nº 18, which also looks at what spin does to the date.

Chapter 3The floor: irreducible mass

Christodoulou asked how far the mass of a Kerr hole can be lowered by sending things into it, and found a quantity no such process can decrease: the irreducible mass.¹ It is set by the horizon's area:

M_irr = √(A/16π), M² = M_irr² + J²/(4M_irr²)With G = c = 1. A non-spinning hole has M_irr = M; for a given irreducible mass, more angular momentum means more mass.

The irreducible mass is not the mass of anything inside the hole, and it is not conserved: it can grow, but no classical process, that is, none that leaves quantum effects out, can shrink it, because the area cannot shrink (the area theorem, later in this chapter). Everything above it, M − M_irr, is spin energy that can in principle be taken out, all of it only by an ideal process that leaves the area unchanged, which Christodoulou called reversible. Any real process increases the area and leaves some spin energy behind for good.¹ The one thing known to lower the floor is Hawking's quantum evaporation, far too slow to matter for any black hole that exists today (Nº 18).

Putting in the Kerr area, 8πMr₊, gives the share of the mass that is spin energy:

f(χ) = 1 − M_irr/M = 1 − √[(1 + √(1 − χ²))/2]At χ = 1, M_irr/M = 1/√2 = 0.70711, so f = 1 − 1/√2 = 0.29289. For small spins f ≈ χ²/8.

That is the headline: up to 29.3%, for χ = 1. Half the spin does not give half the store: at χ = 0.5 the share is 3.4%, about a ninth of the maximum.

Spin χ	Extractable share	Spin energy, 10 M_☉	In Sun main-sequence outputs
0.5	3.4%	6.1 × 10⁴⁶ J	450
0.68 (GW250114's remnant)	6.9%	1.2 × 10⁴⁷ J	910
0.9	15.3%	2.7 × 10⁴⁷ J	2,000
0.998 (Thorne's limit)	27.1%	4.8 × 10⁴⁷ J	3,600
1 (idealised maximum)	29.3%	5.2 × 10⁴⁷ J	3,900

This page's arithmetic, with Chapter 1's inputs.

A hole at 0.9 holds 15.3%, about half the maximum, so most of the store does sit at high spin: spinning up from 0.8 to the maximum adds about two-thirds of the maximal 29.3%. But it is not packed into the last sliver: spinning up from 0.98 adds only 23% of it (this page's arithmetic). Figure 1 draws the curve for a hole of ten solar masses. Drag the spin and watch the bar: the spin-energy part barely grows at first, then climbs; the tape below it shows the horizon's area falling while its equator stays the same.

Figure 1🔋 Try it

How much of the mass is spin?

Spin it up

Spin χSpin

Limits

Measured mergers

Mass

More presets

Spin only

Cygnus X-1, published estimates disagree:

Extractable, 1 − M_irr/MShare you could mine

Spin energy (M − M_irr)c²Energy

In Sun main-sequence outputsSuns' ten-billion-year outputs

Horizon spin rate Ω_H/2πHow often the horizon turns

The share of a Kerr black hole's mass that is extractable spin energy, f = 1 − M_irr/M (Christodoulou 1970), against the spin χ. The bar splits the mass into the irreducible part and spin energy. The tape compares a spinning hole with a non-spinning one of the same mass: the horizon's equatorial circumference, 4πGM/c², never changes, its area falls to half at χ = 1, so the radius of a sphere with the same area falls to 70.7%; the Boyer–Lindquist radius r₊, in the line under the tiles, halves, but that is a coordinate, not a size. Energies use Chapter 1's inputs. The three curves share an axis but not a denominator: f is a share of the hole's mass; the disc overlay is the thin-disc efficiency for a disc turning with the hole, a share of the accreted gas's rest mass (accretion energy, a separate account); the throw overlay is the gain of the best single Penrose split of a body dropped from far away over the energy thrown in (Chapter 7). The merger and Cygnus X-1 presets are published values; the M87* preset sets χ = 0.2, below which one analysis marginally disfavours M87*'s spin; it is not a measurement, and neither M87* nor Sagittarius A* has an agreed spin (Chapter 13). This page's arithmetic.Slide the spin from none to the most theory allows. The coloured part of the bar is spin energy you could, in principle, take out; the grey part is locked in for good. The curve creeps up at first and climbs steeply near the end. The tape shows the horizon getting flatter: half the area, the same equator. Try the real black holes, and switch on the two extra lines. They are different kinds of number: the disc line is how much of the falling gas turns into light, and the throw line is the most extra energy one throw of a rock dropped from far away can bring back, as a share of what went in. Neither is a share of the hole's spin.

Drag to Thorne's limit and the bar reads 27.1%; the two measured-merger presets, GW250114's remnant at χ = 0.68 and GW241011's heavier hole at 0.78, land at 6.9% and 9.8% (this page's arithmetic).

Charge. For a maximally charged, non-spinning hole the irreducible mass is half the mass, so on paper up to 50% could be extracted (Christodoulou and Remo Ruffini, 1971).¹⁶ Real black holes are expected to carry almost no charge, so spin is the store that matters.

The area theorem. In 1971 Hawking proved that in classical general relativity, for ordinary matter (technically, matter that obeys the null energy condition) and assuming cosmic censorship (singularities stay hidden behind horizons), the total area of black-hole horizons never decreases.⁶ Penrose and R. M. Floyd had already found the area growing in the Penrose process (Chapter 6).¹⁷ So the irreducible mass of a single hole can never fall: the floor is real. In the same paper Hawking bounded the energy two merging holes can radiate. For two equal, non-spinning holes of mass m, the final area must be at least 2 × 16πm², so the final mass is at least √2 m, and at most 1 − 1/√2 of the total, 29%, can be radiated:⁶ the headline number again, from the same area mathematics. The coincidence holds only for that case. For two equal holes each spinning at χ = 1 the starting area is half as large, and the theorem allows up to 50% (this page's arithmetic). The bound is a ceiling, not a prediction of what any merger radiates.

Two analogies, and where they break. A spinning black hole is like a flywheel: energy can be drawn off its rotation, which slows it. But a flywheel can be stopped completely; at most 29% of a black hole is flywheel. The irreducible mass is like the charge left in a battery that can never be used. But a battery's floor does not move, and a black hole's rises whenever energy is taken out clumsily. A spinning black hole is a flywheel with a floor, and the floor can rise.

Chapter 4How fast can a black hole spin?

The 29% needs χ = 1, and real black holes are not expected to get there. In 1970 James Bardeen worked out that gas accreted from a thin disc brings in enough angular momentum to spin a hole all the way up to χ = 1.¹⁸ Thorne added the light. The hot inner disc radiates, and the hole swallows some of that radiation; photons going against the spin are captured more readily than those going with it, so they bring in angular momentum of the wrong sign. In Thorne's 1974 calculation the two effects balance at χ ≈ 0.998.² That is the usual ceiling, and the reason this page quotes 27% beside 29%.

Magnetised gas may stop the spin lower. Charles Gammie, Stuart Shapiro and Jonathan McKinney followed the spin of holes in magnetohydrodynamic simulations (computer models of magnetised, flowing gas) of accretion and found an equilibrium near χ ≈ 0.9 for one sequence of their models.¹⁹ Powerful jets can do more than stop the spin-up. In simulations of strongly magnetised, "magnetically arrested" discs, in which the field near the hole is strong enough to hold the gas back, Ramesh Narayan and colleagues found that the jets carry away enough angular momentum to spin holes down.²⁰ In those simulations the jets are drawing on the spin store, the subject of Chapter 12.

Mergers give another benchmark. When two equal black holes with no spin merge, the orbital angular momentum left at the end makes one hole spinning at χ = 0.686, with 95.2% of the original mass, in the high-accuracy simulations of Mark Scheel and colleagues.²¹ At that spin 7.1% of the remnant's mass is spin energy (this page's arithmetic). Unequal masses leave less spin.

For the rest of this page, χ = 1 is the idealised limit and χ = 0.998 the practical one.

Part IIGetting it out

Chapter 5Nowhere to stand still

To get spin energy out, something has to happen outside the horizon, where things can still escape, in a region called the ergosphere. Its outer boundary, the static limit, is the surface inside which nothing can stay still. It lies at

r_E(θ) = M + √(M² − a²cos²θ)Boyer–Lindquist coordinates; θ is the angle from the spin axis. On the equator r_E = 2M for any spin above zero; at the poles r_E = r₊, so the boundary touches the horizon there.¹²

The ergosphere is the region between the static limit and the horizon. Its equatorial "radius" of 2M is again a coordinate label. The circumference of the static limit round the equator, which every observer agrees on, is 2π√(4 + 2χ²) GM/c², which for ten solar masses grows from 186 km for a barely spinning hole to 227 km at χ = 1 (this page's arithmetic).

On the equator the Kerr metric has a term g_tφ = −2a/r that couples time to rotation, and a body held at fixed r can turn round the hole, as seen from far away, only at an angular velocity between two limits, the rates at which light sent sideways against and with the spin sets off:

Ω_± = ω ± √(ω² − g_tt/g_φφ), ω = −g_tφ/g_φφEquatorial Kerr, Boyer–Lindquist coordinates, G = c = M = 1: g_tt = −(1 − 2/r), g_φφ = r² + a² + 2a²/r. On the equator the square root is √Δ/g_φφ, with Δ = r² − 2r + a². ω is the frame-dragging rate.¹²

Outside the static limit Ω₋ is negative: a body can hover, or circle against the spin. On the static limit Ω₋ = 0. Inside, both limits are positive: even light aimed against the rotation starts off going round with it. At the horizon the two meet at Ω_H. These rates bound how fast something can start round; light sent sideways does not stay at that radius.

Figure 2 puts a probe on the equator of a ten-solar-mass hole. Drag it out beyond 2GM/c² and press Try to hover: there it can hold still. Bring it back inside the static limit and the slowest allowed rate turns positive: the probe is swept round with the spin. Flash light both ways to see the two limits set off.

Figure 2🌪️ Try it

Try to stand still

Move the probe out, then in

Spin χ

Probe distance r

Slowest allowed, Ω₋/2πSlowest you can turn

Fastest allowed, Ω₊/2πFastest

Frame dragging ω/2πSpace is dragged at

Probe isYou are

The equatorial plane of a Kerr black hole of 10 M_☉. A probe at Boyer–Lindquist radius r can move round the hole only at angular velocities between Ω₋ and Ω₊, the rates of light sent against and with the spin (Chapter 5's formula; ω is the frame-dragging rate). Outside the static limit (r = 2GM/c² on the equator) Ω₋ is negative, so the probe can hover; inside, in the ergosphere, it is positive; at the horizon both equal Ω_H. Rates are as seen from far away and scale as 1/M. The arrows show only how fast anything can start round (Chapter 5). The top view is drawn in Kerr–Schild-like coordinates (drawn radius √(r² + a²), as in Figure 3), the side view in Boyer–Lindquist polar coordinates, so the drawn shapes depend on the coordinates. This page's arithmetic.Look down on a spinning black hole of ten Suns. The dark disc is the horizon; the coloured ring is the ergosphere. Slide the probe out beyond the coloured ring and press Try to hover: out there it can hold still, but inside the ring space is dragged round so hard that it gets swept along. Flash a light both ways: inside the ring, even the light sent backwards starts off forwards. (The arrows show how fast each one starts round; the light does not stay on the circle.)

The figure opens inside the ergosphere, at χ = 0.9 and r = 1.80 GM/c² (26.6 km, a coordinate value): anything held there must turn between 215 and 1,091 times a second (this page's arithmetic).

The same frame dragging happens round the Earth, far more weakly: it turns a gyroscope at the North Pole by about 0.22 arcseconds a year relative to the stars (Nº 16). Inside an ergosphere it cannot be resisted at all.

The ergosphere matters to this page for one reason. The energy a distant observer assigns to a body, the quantity conserved along its path, is measured against hovering: wherever something can hover, it is the energy a hovering observer would find, scaled down for the climb out, and so positive. Inside the static limit nothing can hover, so that quantity is no longer forced to be positive: inside the ergosphere, and only there, a body whose angular momentum points against the spin (it is still carried round, only more slowly than the frame drags) can have negative energy as measured from far away.^12,22 Locally every energy and every mass is still positive. The next chapter turns that loophole into a mining method.

Chapter 6Penrose's split

In 1969, in a review of gravitational collapse, Roger Penrose pointed out what the ergosphere allows.⁵ Send a body into the ergosphere and let it split in two there. Arrange the split so that one piece, with angular momentum against the spin, follows an orbit of negative energy, as measured from far away, and falls through the horizon. Energy is conserved, so the other piece must leave with more energy than the whole body brought in. It escapes, and can be collected far away.

The hole pays. Swallowing a piece of negative energy lowers its mass by exactly the escaping piece's gain, and because the fallen piece's angular momentum was against the spin, it carries negative angular momentum into the hole, so the hole's spin falls as well.^5,17 In Christodoulou's bookkeeping the irreducible mass cannot go down while this happens: at best it stays the same, and in any real split it rises. A Penrose process takes spin energy, and only spin energy. In symbols, with every energy measured from far away:

ΔM = E₂ < 0, ΔJ = L₂ < 0, E₁ = E₀ − E₂ > E₀E₀: the incoming body's energy; E₁, E₂: the energies of the escaping and the fallen piece, all as measured from far away; L₂: the fallen piece's angular momentum. The first law of black-hole mechanics, with a horizon area that cannot shrink, requires ΔM ≥ Ω_HΔJ: each unit of angular momentum given up returns at most Ω_H of energy.

"Negative energy" needs care. The fallen piece's mass is positive, and an observer next to it measures a positive energy. Only its energy as defined from far away, the quantity that adds to the hole's mass when it falls in, is negative.²²

The usual analogy is a gravitational slingshot: a probe swinging round a moving planet leaves faster than it came, and the planet slows imperceptibly to pay. The analogy breaks in an instructive place. A slingshot needs no ergosphere and no negative energy; the Penrose process works only where frame dragging lets energy, measured from far away, turn negative.

This is the same Roger Penrose whose very different 1996 idea, that gravity collapses quantum superpositions, is tested in Nº 10, and who appears in Nº 3; on this page "the Penrose process" always means the 1969 idea. How much can one split win? Less than 29%.

Chapter 7What one throw can win

Take the textbook version of the Penrose process. A body of rest mass µ₀ falls from rest far away in the hole's equatorial plane, so its energy per unit rest mass is 1. At its turning point r₀ it splits into two equal fragments, which move off in opposite directions, each at speed v = √(1 − (2µ/µ₀)²) in the body's own frame, where µ is each fragment's rest mass. Their energies are

E_1,2 = (µ₀/2)[1 ± v√(2M/r₀)]Upper sign: the escaping piece; lower sign: the piece sent in. Energies as measured from far away (Chandrasekhar 1983; Brito, Cardoso and Pani 2020).^23,22

The fallen piece has negative energy when v > √(r₀/2M), and the deeper the split, the bigger the gain. If the fragments are flashes of light (v = 1), any split inside the ergosphere wins, and a split exactly on the static limit wins nothing. At the horizon of a maximal hole the formula gives ½(√2 − 1) = 20.7%.

But a body dropped from far away cannot get arbitrarily deep. Its deepest possible turning point is the marginally bound radius,

r_mb = 2M − a + 2√(M(M − a))Bardeen, Press and Teukolsky 1972, eq. 2.19.¹² No dropped body turns round any deeper: given the angular momentum a deeper turning point would need, it meets a barrier further out and bounces there first. r_mb lies inside the ergosphere, below 2M, only for χ > 2√2 − 2 ≈ 0.83.

So for a body dropped from far away the best single split wins nothing below χ ≈ 0.83, then 0.97% at 0.85, 3.7% at 0.9, 14.3% at 0.99 and 17.7% at 0.998; the 20.7% is reached only as χ → 1, where r_mb approaches the horizon (this page's arithmetic). The formula's value at the horizon of a χ = 0.998 hole, 18.6%, is out of reach: no dropped body splits there. Bodies thrown in, or splitting away from a turning point, are not limited in this way; this chapter and Figure 3 use only the simplest set-up.

Even the deepest split is a limit rather than a place. At r_mb the body would sit on the unstable marginally bound circular orbit, so a body aimed there approaches it ever more slowly and circles without end. The 17.7% is approached but never reached; Figure 3 counts the turns.

The split must be violent. For the dropped body each fragment must leave at more than √(r₀/2M) of the speed of light: at least 0.71c even at the horizon of a maximal hole, 0.74c at the deepest split for χ = 0.998 (this page's arithmetic). Robert Wald proved in 1974 that for any parent a fragment can have negative energy only if it leaves at more than Ẽ/√(Ẽ² + 1) of the speed of light, with Ẽ the parent's energy per unit rest mass; among circular orbits the lowest such speed, c/2, belongs to the most tightly bound one, the innermost stable orbit of a maximal hole.²⁴ Bardeen, Press and Teukolsky found the same c/2 for the relative speed of colliding particles.¹² That is why the split version of the Penrose process is not thought to power astrophysical sources; a collisional version near the horizon is sometimes called more promising, but whether it matters in nature is open.²²

Figure 3 lets you make the throw. Set the spin, the fragments' rest mass and the split depth (0% splits on the static limit, 100% at r_mb), then press Throw, or Mine it to repeat the throw until it stops paying.

Figure 3🎯 Try it

Penrose's throw

Throw, then mine

Spin χ

Split depth

Fragment rest mass 2µ/µ₀How heavy the pieces are

Gain, E₁/E₀ − 1Comes back with

Fallen piece, E₂/E₀The piece that falls in carries

Split speed / needed √(r₀/2M)Split speed / needed

Hole: ΔM, ΔJThe hole loses

A body of rest mass µ₀ falls from rest at infinity onto a Kerr black hole in its equatorial plane and splits at its turning point r₀ into two equal fragments, each moving off at speed v in the body's frame. Energies follow Chapter 7's split formula (Chandrasekhar 1983; Brito, Cardoso and Pani 2020) and angular momenta from a form that stays finite on the static limit. A body aimed at the deepest turning point, r_mb (Bardeen, Press and Teukolsky 1972), circles ever longer before it splits, so the gain shown there is the limit and the drawn whirl is cut short. The hole's mass and angular momentum change by the fallen piece's, and its area always grows. "Mine it" repeats throws of 0.1% of the hole's mass, treated as test particles; the strip shows what was taken, what still spins, and what the growing horizon locked into the irreducible mass. Paths are equatorial geodesics drawn in Kerr–Schild-like coordinates (drawn radius √(r² + a²)), on a radial scale compressed beyond 3GM/c². This page's model.Drop a rock towards a spinning black hole and let it split in two. Slide the spin, how deep the split happens and how heavy the pieces are. With the right settings one piece falls in carrying negative energy and the other flies out with more than the rock had. Then press Mine it: throw after throw the hole slows down, until the throws stop working long before its spin energy runs out, and part of it gets locked in for good.

The figure opens on the deepest split into flashes of light at χ = 0.998, which gains the 17.7% limit. Below χ ≈ 0.83 a dropped body cannot split inside the ergosphere at all; with fragments slower than √(r₀/2M) of the speed of light the inward piece carries positive energy, and the escaping one leaves with less than the body brought in.

Mining by throws. Repeat the best throw and the hole spins down. Figure 3's "Mine it" throws 0.1% of the hole's mass at a time, as test particles, from χ = 0.998, splitting into flashes of light at the deepest point. After 703 throws the spin is 0.8282, just below 2√2 − 2 = 0.8284, the spin under which a dropped body can no longer split inside the ergosphere. About a fifth of the starting store (19.7%) has come out; about two-fifths (40.8%) is still spin energy, out of this method's reach; and two-fifths (39.5%) is gone for good, because every throw enlarged the horizon and the irreducible mass rose from 0.729 to 0.836 of the starting mass (this page's arithmetic). Remo Ruffini and colleagues found the same in 2025 for repeated splits round an initially maximal hole: the irreducible mass grows by far more than the energy taken out, and extraction stops while the hole still spins.²⁵ From χ = 0.9 only 2.3% of the store comes out. Mining by throws raises the floor under your feet.

Chapter 8Horizons only grow

Hawking's area theorem is what guarantees the floor, and in 2025 it was tested on real black holes. On 14 January 2025 the two LIGO detectors recorded GW250114, the merger of two holes of about 33.6 and 32.2 solar masses with little spin (each χ ≤ 0.26), at a network signal-to-noise ratio of 80, against 26 for the first detection, GW150914.²⁶ As of October 2026 it is the loudest gravitational-wave signal published so far (the Gravitational Wave Open Science Center's event list, accessed on 25 October 2026).²⁷ The remnant has 62.7 solar masses and spins at χ = 0.68 ± 0.01. How the detectors hear a merger is in Nº 7.

Because the signal was so clean, the LIGO, Virgo and KAGRA collaborations could estimate the horizon areas before the merger, from the inspiral, the signal of the two holes spiralling together, and after it, from the ringing of the remnant, and compare them. The final area exceeded the initial one at 4.4σ in their main analysis, at 3.4σ or more wherever they chose to end the inspiral part of the signal, and at 3.6σ using two tones of the ringing (Physical Review Letters, September 2025).²⁶ σ counts standard deviations from "no increase". It is a confirmation of Hawking's theorem with high confidence, under the stated assumptions,⁶ not a proof.

The numbers make the floor concrete (this page's arithmetic, from the published medians, with the initial spins taken as zero). The two initial horizons had a total area of about 237,000 km²; the final one has 373,000 km², 57% more. The merger radiated about 3.1 M_☉c² in gravitational waves, 4.7% of the starting mass. The remnant, at χ = 0.68, stores f(0.68) × 62.7 = 4.3 M_☉c², or 7.7 × 10⁴⁷ J, of extractable spin energy: more than the merger gave off.

One trap: area goes as the square of the irreducible mass, so what cannot fall is the sum of the squared irreducible masses, not the sum of the masses themselves. GW250114's final irreducible mass, 58.4 M_☉, is less than the two initial ones added, 65.8 M_☉, while its area is 57% larger.

Chapter 9Waves that come back stronger

The Penrose process has a wave version. In 1971 Yakov Zel'dovich, inspired by Penrose's idea, argued that any rotating body that absorbs waves will amplify some of them.⁷ A wave of angular frequency ω whose pattern repeats m times round the axis has a pattern turning at ω/m. If the body turns faster than the pattern,

ω < mΩZel'dovich's condition for a body rotating at angular velocity Ω. For a Kerr black hole Ω becomes the horizon's Ω_H.

then in the body's own rotating frame the wave's frequency, ω − mΩ, is negative: the pattern runs backwards across the surface, and an absorber that would normally take energy from the wave gives energy to it instead. In 1972 Zel'dovich worked the effect out for electromagnetic waves and a rotating metal cylinder.⁷ Nothing about it is special to black holes; it is a property of spinning absorbers.

Think of overtaking on a carousel: to a rider, a pattern turning more slowly than the platform seems to move backwards. (The analogy captures the sign change, not the energy bookkeeping, which needs the thermodynamics of the rotating absorber.) A black hole fits the picture: its horizon absorbs whatever crosses it and rotates at Ω_H. Alexei Starobinsky calculated in 1973 how waves are amplified on reflection from a rotating black hole,²⁸ and Saul Teukolsky and William Press computed the gains for electromagnetic and gravitational waves in 1974.²⁹ Waves with ω < mΩ_H scatter off the hole with more energy than they brought in, and the extra comes from the spin. This is superradiance, the wave version of the Penrose process.²²

The gains are modest except for gravitational waves. For a maximally spinning hole and the lowest modes, just below the threshold ω = mΩ_H, the largest amplification of a wave's energy in one scattering is about 0.4% for a massless scalar wave (the simplest kind of wave), 4.4% for electromagnetic waves and 138% for gravitational waves (Teukolsky and Press, as quoted by Richard Brito, Vitor Cardoso and Paolo Pani).^29,22 A spinning hole can more than double the energy of a gravitational wave that bounces off it, while the wave's amplitude grows by √2.38, about 1.54 times.

Towards Hawking radiation. Zel'dovich also argued that a spinning absorber should amplify the quantum vacuum itself, and so glow spontaneously. Starobinsky's calculation for rotating black holes followed, and then Hawking's 1974–75 discovery that every black hole radiates, spinning or not.^13,14 Zel'dovich's idea is usually told as a step on that road,²² not as the discovery itself; spin then shapes how a hole evaporates (Chapter 2).

For light, one scattering gains 4.4% at best. Send the amplified wave back, and back again, and the story changes.

Chapter 10The black-hole bomb

In 1972 Press and Teukolsky asked what happens if a spinning hole is surrounded by a mirror.⁸ A wave that satisfies the superradiance condition is amplified each time it meets the hole, reflected by the mirror, and sent back to be amplified again. If the mirror keeps a fraction R of the energy at each bounce and the hole adds a fraction Z at each pass, the energy after n round trips is

E_n = E₀[R(1 + Z)]ⁿ, growth if R(1 + Z) > 1An idealised bookkeeping per round trip; the threshold reflectivity is 1/(1 + Z).

Above the threshold the wave grows exponentially until the mirror gives way: the black-hole bomb. Or it grows until the hole has given up enough spin that the amplification stops, as Vitor Cardoso, Óscar Dias, José Lemos and Shijun Yoshida examined in 2004.³⁰ With the largest gains of Chapter 9, a scalar wave needs a mirror that keeps more than 99.6% of its energy at each bounce, light more than 95.8%, and a gravitational wave only more than 42% (this page's arithmetic). The bomb is a thought experiment.

Figure 4 runs this bookkeeping. Pick what bounces, set the mirror's loss and press Bounce. It opens on light and a mirror that loses 1% a bounce: each round trip multiplies the energy by 1.034, so it grows tenfold every 70 round trips (this page's arithmetic).

Figure 4💣 Try it

The black-hole bomb

Lose less than you gain

What bounces, in a lab analogueWhat bounces, in a lab stand-in

Round a black hole at maximal spin, just below thresholdRound a black hole at full spin, at its best

Mirror loss per bounce

Each round trip, R(1 + Z)Each trip multiplies the energy by

Round trips per tenfoldTrips to grow ten times

Until the mirror gives wayTrips until it breaks

Threshold reflectivity 1/(1 + Z)Most loss it can take

An idealised black-hole bomb (Press and Teukolsky 1972), with Chapter 10's bookkeeping: each round trip multiplies the trapped wave's energy by R(1 + Z). The black-hole presets are Chapter 9's largest single-scattering gains for a maximally spinning hole (Teukolsky and Press 1974, as quoted by Brito, Cardoso and Pani 2020). The 2025 cylinder preset is this page's arithmetic from Cromb et al.'s values (about 0.5% net per cycle at a total resistance of about −0.75 Ω); the helium preset uses the growth rate reported in a 2026 preprint (Švančara et al., not yet refereed as of October 2026) and ends by levelling off, as the preprint's waves did after about 80 s. Both lab presets already include their own losses, so the mirror-loss slider is off for them. The breaking point at a million times the seed is this page's choice; the levelling-off ending is a cartoon of spin-down or saturation, not a calculation.A wave bounces between a spinning black hole and a mirror. Each time it meets the hole it comes back a little stronger; each time it meets the mirror it loses a little. Pick what bounces and how leaky the mirror is, then press Bounce. If the mirror loses less than the hole adds, the wave grows until the mirror breaks, or, if you choose, levels off. The two lab presets are bench-top stand-ins that follow the same rule with no black hole: one published in 2025, one a 2026 preprint, a paper not yet checked by other scientists.

Raise the loss above about 4% and the light dies away; a gravitational wave keeps growing until the mirror loses more than 58%. Real black holes have no mirrors, but nature may supply one.

Nature's mirror. A field whose particles have mass cannot carry low-frequency waves away to infinity, so they stay trapped round the hole, and the trapping does the mirror's job. Thibault Damour, Nathalie Deruelle and Remo Ruffini found such trapped states in 1976 and suggested that round a rotating hole they would grow; Theodoros Zouros and Douglas Eardley showed in 1979 that they do, and Steven Detweiler computed how fast for the lightest fields in 1980.^31,32,33 The growth is fastest when the particle mass µ makes GMµ/(ħc) a number of order one; for a scalar field round a hole with χ = 0.99 the fastest growth comes at about 0.42 (Sam Dolan, 2007).³⁴ For a ten-solar-mass hole that means a particle of about 5.6 × 10⁻¹² eV (electronvolts: particle physicists quote masses as energies; this page's arithmetic).

If such ultralight bosons exist, a spinning hole would pour its spin into a cloud of them, so holes seen still spinning rule out ranges of the particles' masses. Asimina Arvanitaki and colleagues made the argument in 2010, in proposing the "string axiverse",³⁵ and she and Sergei Dubovsky worked it out in 2011.³⁶ The heavier hole of GW241011, spinning at χ = 0.78, rules out ultralight bosons of roughly 10⁻¹³ to 3 × 10⁻¹² eV at 90% credibility; the exact range depends on whether the particle is a scalar or a vector and on how old the hole is assumed to be.³⁷ That is an exclusion of some particles, not a proof that none exist.

Part IIIIs anyone mining it?

Chapter 11A bomb on a bench

Zel'dovich's condition needs no black hole, only a spinning absorber, and by 2024 laboratory analogues had shown rotational superradiance in four systems: surface waves on water amplified by a draining vortex (Torres, Weinfurtner and colleagues, 2017);³⁸ sound carrying orbital angular momentum amplified by up to about 30% by a spinning absorbing disc (Cromb, Faccio and colleagues, 2020);³⁹ Penrose superradiance in a fluid of light (Braidotti and colleagues, 2022);⁴⁰ and a spinning metal cylinder that acted as a negative resistance, cutting the losses of a low-frequency circuit without yet beating them (Braidotti and colleagues, 2024).⁴¹ None involves real gravity: there is no black hole, and in the fluid systems the flow only makes the waves behave as if space were curved.

A year later the cylinder beat the losses. On 5 November 2025 Marion Cromb, Maria Chiara Braidotti, Andrea Vinante, Daniele Faccio and Hendrik Ulbricht, a Southampton–Glasgow–Trento team, reported in Science Advances the creation of a black-hole bomb instability in an electromagnetic system.¹⁰ A solid aluminium cylinder 40 mm across, spun by its own motor at up to 900 turns a second, sat inside the stator of a three-phase induction motor: the fixed ring of coils, whose quadrupole field pattern (m = 2: two north and two south lobes) turns at half the circuit frequency. Each phase's resonant circuit, near f = 1,181 Hz, stores the field like a leaky mirror.

The cylinder gives energy to the field when it turns faster than the pattern, F > f/m: here, above 590.5 turns a second. That is not yet growth: the cylinder must also beat the circuit's own losses. With low resistance and no input, a signal turning with the cylinder grew exponentially out of noise in runs at F = 643 and 660 turns a second, while the cylinder slowed down; the runaway set in above about 640 turns a second, the circuit's total resistance during growth was about −0.75 Ω, and with the motor forced to hold its speed the runaway blew a resistor.¹⁰ From the paper's values, the circuits gained about 0.5% of their stored energy per cycle in the runaway (this page's arithmetic, with the coil inductance of 0.121 H that the resonance implies).

In a lossier setting (a variable resistor of about 24 Ω) there was no net growth up to 700 turns a second. At 643 turns a second the rim moved at only about 80 m/s (this page's arithmetic). The trick is the field's low frequency: light, at about 10¹⁴ Hz or more, would need a spin no object survives.

Figure 5 sets up the experiment: spin the cylinder past the turning field pattern, or watch from the cylinder, where the field runs backwards once F > f/m. The presets include the paper's two runs and its lossy setting.

Figure 5🧲 Try it

Outrun the field

Spin past the pattern

Cylinder spin F

Field frequency f

Lobe pairs mNorth–south pairs

Circuit

Watch from

Pattern turns, f/mField pattern turns

Cylinder turns, FCylinder turns

Rotating-frame frequency f − mFSeen from the cylinder, the field runs

Rim speed

Zel'dovich's condition for a rotating absorber, as used by Cromb and colleagues (Science Advances, November 2025): a field pattern with m pairs of lobes oscillating at f turns at f/m, and a cylinder turning at F amplifies it when F > f/m, that is, when its rotating-frame frequency f − mF is negative. In the experiment m = 2 and f ≈ 1,181 Hz (Chapter 11). The cylinder starts to give energy above 590.5 turns a second; the low-loss circuits grew from noise only above about 640, and with the variable resistor at 24 Ω the total resistance only approached zero near 680. Net-growth marks apply only to the experiment's circuit. The growth was seeded by room-temperature noise, not by the quantum vacuum. This page's model of the paper's numbers.A metal cylinder spins inside a ring of coils whose magnetic field pattern turns round. The lab's pattern had two north–south pairs, so it turned at half the rate the field swings. While the cylinder turns slower than the pattern it just soaks up energy, like any metal in a changing field. Spin it past the pattern and, seen from the cylinder, the field runs backwards: now the cylinder feeds it. In the lab's low-loss circuit, above about 640 turns a second, the feeding beat the losses and a signal grew: the 2025 bench-top analogue of the black-hole bomb, with no black hole involved. The signal grew from room-temperature noise, not from the quantum vacuum.

At the figure's opening setting of 400 turns a second, the field still runs forwards as seen from the cylinder, f − mF = +381 Hz, and the cylinder only absorbs. At the 643-turn run it runs backwards, at −105 Hz, and the cylinder feeds it.

The caveat. The amplifier ran at room temperature, so the growth was, in the authors' words, "seeded by noise that dominates vacuum fluctuations by many orders of magnitude", and the vacuum-seeded Zel'dovich effect remains to be observed.¹⁰ The paper does not say what kind of noise. Thermal noise alone, at 1,181 Hz and 293 K, puts on average k_BT/(hf) ≈ 5 × 10⁹ quanta into each mode, against the vacuum's half a quantum, and any other noise only adds to it (this page's arithmetic).

2026. Two later results go further, one of them still a preprint. In a September 2026 preprint, not yet refereed as of October 2026, a Nottingham-led team (Patrik Švančara, Silke Weinfurtner and colleagues) watched surface waves on a draining vortex of superfluid helium grow exponentially and then, through nonlinear mixing, settle into a long-lived steady state.⁴² The sloshing mode at 3.47 Hz grew in amplitude at 0.081 per second, about 4.8% of its energy per wave period (this page's arithmetic), and saturated after about 80 s. The authors describe this as reaching the bomb's end state, which they say earlier experiments had not; the 2025 cylinder's growth also stopped, in Chapter 10's two ways: the cylinder slowed, or a resistor failed. And in July 2026 Hadiseh Nasari, Andrea Alù and colleagues at the City University of New York reported in Nature the amplification of waves circulating in a ring of resonators (the paper's "orbital waves"), modulated one after another so that the ring acts as if it were spinning, with no moving parts.⁴³

Chapter 12Jets that may run on spin

The one place where nature may be mining spin on a large scale is in jets. In 1977 Blandford and Znajek showed that magnetic field lines threading the horizon of a spinning hole act as a brake: the turning horizon drags them round, and they carry spin energy outwards as electromagnetic energy.⁹ The power rises with the magnetic flux Φ through the horizon and with the horizon's spin rate:

P_BZ ∝ Φ²Ω_H²To leading order: at a given flux, twice the horizon's spin rate gives four times the power. The constant depends on the geometry of the field. The field lines thread the horizon; they do not come out of the hole.

Simulations suggest the brake can be strong. Around strongly magnetised, "magnetically arrested" discs, Alexander Tchekhovskoy, Narayan and McKinney found in 2011 that for a hole with χ = 0.99 the jets and winds carry out about 140% of the rest-mass energy falling in (30% at χ = 0.5).⁴⁴ More out than in can only come from the spin; so far it has been seen only in simulations.

Whether the Blandford–Znajek process is a Penrose process depends on definitions: Jean-Pierre Lasota and colleagues show that it meets Penrose's condition, absorption of negative energy and angular momentum at the horizon, and call it a generalised Penrose process;⁴⁵ Serguei Komissarov stresses the differences.⁴⁶ And not every jet need tap spin. Fields anchored in the disc can launch jets too (Blandford and David Payne, 1982),⁴⁷ and studies of stellar-mass black holes disagree on whether jet power tracks spin: Narayan and Jeffrey McClintock reported a correlation,⁴⁸ Rob Fender, Elena Gallo and David Russell found no evidence for one.⁴⁹

Quasar light is a different account. A quasar shines with gas in a thin disc spiralling inwards, which radiates its binding energy, given up as it sinks, before it reaches the innermost stable circular orbit and plunges. For a disc turning with the hole, the share of rest mass the disc radiates is 5.7% at χ = 0, 32% at 0.998 and 42.3% at χ = 1 (this page's arithmetic from Bardeen, Press and Teukolsky's orbits);¹² at 0.998 the hole swallows some of that light, and Thorne found that about 30% gets out.² Spin raises it, by moving the innermost orbit inwards, but the energy comes from the falling gas, not from the spin store. Quasars shine on accretion; only jets may run on spin. Figure 1's disc-light overlay draws this curve beside the spin store.

The spin store

What it is: the hole's own rotational energy, M − M_irr
At most: 29.3% of the hole's mass at χ = 1; 27.1% at 0.998
Taking it: slows the hole down and makes it lighter

Accretion light

What it is: the binding energy of gas falling in
At most: 42.3% of the gas's rest mass at χ = 1; 32% at 0.998
Taking it: the gas falls in and the hole gets heavier

M87. The best-studied case is the jet of the galaxy M87, whose central hole, M87*, has 6.5 billion solar masses, with a systematic uncertainty of about 0.7 billion.⁵⁰ The Event Horizon Telescope (EHT) team found that, in their library of simulations, models of non-spinning holes "do not produce sufficiently powerful jets", and that in the models that do, the jet is powered by the extraction of spin energy through mechanisms akin to Blandford and Znajek's.⁵¹ The 2021 images in polarised light favour magnetically arrested flows that also make a jet of sufficient power.^52,53 So spin energy is the best explanation of M87's jet in the EHT team's models, not a measurement. If M87* spins at χ = 0.9, an assumption (Chapter 13), its store is about 1.8 × 10⁵⁶ J, enough to run a jet of 10³⁷ W for some 560 billion years (this page's arithmetic): a measure of the store, not a forecast for the jet.

Chapter 13Weighing the spin

M87*. Spins are much harder to measure than masses, and for M87* there is no agreed measurement. Fabrizio Tamburini, Bo Thidé and Massimo Della Valle claimed χ = 0.90 ± 0.05 (about 95% confidence) in 2020, from the orbital angular momentum, the "twisted light", of the radiation in the 2017 EHT data;⁵⁴ the claim is unconfirmed. In 2025 Michael Drew and colleagues estimated about 0.8 from how much brighter one side of the ring is than the other, with a simple model of the disc, and take it to be probably a lower limit.⁵⁵ As of October 2026, simulation-based modelling of the EHT images leaves the size of the spin almost unconstrained. Other clues point to a spinning hole without fixing the number. The jet's direction wobbles with a period of about 11 years, which Yuzhu Cui and colleagues interpret as the sign of a tilted disc swung round by the frame dragging of a spinning hole (Lense–Thirring precession).⁵⁶ The asymmetry of the bright ring in the EHT images over three observing campaigns marginally disfavours spins below about 0.2 in magnetically arrested models,⁵⁷ and the ring's position angles strongly disfavour spin axes pointing towards the Earth.⁵⁸

Sagittarius A*. The hole at the centre of our galaxy has 4.30 million solar masses, known to about 0.25% from the orbits of stars round it.⁵⁹ Its spin is disputed. Giacomo Fragione and Abraham Loeb argued that the orbits of the closest stars, which a fast spin would make precess by frame dragging, limit it to χ ≲ 0.1;⁶⁰ Ruth Daly and colleagues found 0.90 ± 0.06 with the outflow method.⁶¹ Daly's team also computed this page's own quantity, the spin energy available for extraction: 15 ± 4% of the mass, matching the 15.3% Figure 1 gives at χ = 0.9. In August 2026 the GRAVITY+ team reported a star, S301, on an 8.7-year orbit whose motion is "directly sensitive to the spin of Sgr A*"; it has not yet given a measurement.⁶² As of October 2026, modelling of the EHT images leaves this spin unconstrained too.⁵⁸

Cygnus X-1. Even the most studied stellar black hole is argued over. James Miller-Jones and colleagues measured 21.2 ± 2.2 solar masses in 2021;⁶³ a 2025 analysis of the companion star's spectrum by V. Ramachandran and colleagues gives 12.7 to 17.8, depending on the assumed tilt of the orbit.⁶⁴ Published spins run from above 0.9985 (a 3σ lower limit from the disc's X-ray continuum, Xueshan Zhao and colleagues, 2021)⁶⁵ through about 0.98 from X-ray reflection seen by the XRISM X-ray satellite (Paul Draghis and colleagues, 2025)⁶⁶ and 0.87 (Andrzej Zdziarski and colleagues, 2024)⁶⁷ down to 0.0 (+0.17) with models in which hot electrons scatter the disc's X-rays (Comptonisation); Zdziarski and colleagues conclude that the spin "is strongly model dependent" (2026).⁶⁸ X-ray spins are often high, but each depends on its model.

Merging black holes. In the refereed population analysis of the GWTC-4.0 catalogue, 158 mergers, 90% of black holes have χ below about 0.6 (0.57 in the preprint version).⁶⁹ A 2026 preprint, not yet refereed as of October 2026, analysing 267 mergers in GWTC-5.0, finds 69–84% at χ ≤ 0.5 and, as the authors put it, "now confidently" a subpopulation with at least one hole spinning at χ ~ 0.7, consistent with holes that were themselves born in earlier mergers.⁷⁰ GW241011's heavier hole, 19.6 solar masses at χ = 0.78 ± 0.09, is probably such a second-generation hole.³⁷ GW231123, the most massive merger heard so far (as of October 2026: GWTC-5.0), has a larger hole very probably spinning fast, at about 0.9.^71,70 X-ray binaries mostly report high spins and merging binaries low ones; the two populations may simply differ (Maya Fishbach and Vicky Kalogera).⁷²

Chapter 14What 29% means

Six things 29% does not mean:

Every black hole: The 29% needs maximal spin, an idealisation: 27% at the usual ceiling of 0.998, about 7% after an equal-mass merger, 3% at half spin (Chapters 3 and 4).
One throw: One Penrose split of a body dropped from far away returns at most 20.7% more than went in, as χ → 1, and 17.7% at 0.998 (Chapter 7).
Something for nothing: It is the hole's own energy: taking it out slows the hole down and makes it lighter (Chapter 6).
Already mined: Nobody has taken energy from a real black hole. Jets may run on spin; the laboratory results are analogues (Chapters 11 and 12).
Quasar light: The 42% a thin disc can radiate round a maximally spinning hole is accretion energy from falling gas, a separate account (Chapter 12).
Half the size: At maximal spin the horizon has half the area of a non-spinning hole of the same mass, with the same equator: it flattens. The radius that halves is a coordinate (Chapter 2).

Where things stand, as of 25 October 2026: rotational superradiance had been seen in four kinds of analogue by 2024; in 2025 it grew into a runaway in a spinning-cylinder analogue of the black-hole bomb, seeded by room-temperature noise; and in 2026 a ring of resonators with no moving parts showed it too. The vacuum-seeded version is still unseen. The helium "end state" is a preprint. M87*'s spin has no agreed measurement, the spins of Sagittarius A* and Cygnus X-1 are disputed, and most merging black holes spin slowly. Hawking's area theorem has passed a 4.4σ test.

The thread is short. Part of a spinning black hole's mass is rotational energy that can, in principle, be taken out; the rest, the irreducible mass, is fixed by the horizon's area, and no classical process can lower it. Ways out exist: a split in the ergosphere, a wave that comes back stronger, magnetic field lines that brake the horizon. Each takes the hole's own spin, and every real one raises the floor as it goes, so the full 29% is a limit that no real process reaches.

A spinning black hole is a flywheel with a floor.

Glossary

Solar mass, M_☉c²: The Sun's mass, 1.98841 × 10³⁰ kg; as energy, M_☉c² = 1.787 × 10⁴⁷ J.
Accretion: Gas falling onto a black hole, usually through a disc.
Rest mass: A body's mass measured at rest; as energy, mc².
Kerr black hole: Roy Kerr's 1963 solution for an uncharged spinning black hole, fixed by its mass and spin.
Spin, χ: cJ/(GM²): 0 for no spin, 1 for the most a black hole can have.
Gravitational radius: GM/c², 1.48 km per solar mass; written M when G = c = 1.
Event horizon: The surface from inside which nothing escapes. Spin flattens it: less area, same equator.
Boyer–Lindquist coordinates: The usual Kerr coordinates. Their radius is a label, not a distance.
Kerr–Schild-like coordinates: Coordinates that stay well behaved at the horizon, used to draw Figures 2 and 3.
Horizon area: 8πMr₊, agreed by every observer; at maximal spin, half that of a non-spinning hole of the same mass.
Irreducible mass: √(A/16π): the part of the mass no classical process can remove. The rest is spin energy.
Classical: Leaving quantum effects out. No classical process lowers the irreducible mass; Hawking evaporation, a quantum effect, does.
Ergosphere, static limit: The region outside a spinning horizon where nothing can stay at rest relative to the distant stars, and its outer boundary.
Frame dragging: A spinning mass dragging everything near it, light included, round with it.
Negative energy (as measured from far away): Possible only in the ergosphere, for angular momentum against the spin. Locally, energy and mass stay positive.
Penrose process: A body splits in the ergosphere; one piece falls in with negative energy, the other escapes with more than the body had.
Test particle: A body small enough that its own gravity can be ignored.
Marginally bound radius: The deepest turning point of a body dropped from far away; inside the ergosphere only for χ above about 0.83.
Innermost stable circular orbit: The smallest stable orbit; inside it, gas plunges. Spin moves it inwards.
Thin-disc efficiency: The share of infalling gas's rest mass radiated as light: 5.7% without spin, 42.3% at χ = 1. Not spin energy.
Quasar: The brilliant centre of a distant galaxy, lit by gas falling onto its central black hole.
Area theorem: Hawking, 1971: classically, total horizon area never decreases.
First law of black-hole mechanics: dM = (κ/8π)dA + Ω_HdJ, κ the surface gravity; with dA ≥ 0 it gives ΔM ≥ Ω_HΔJ.
Superradiance: Amplification of waves by a spinning absorber: the wave version of the Penrose process.
Zel'dovich condition: ω < mΩ: the wave pattern turns more slowly than the absorber.
Scalar field: A field with one number at each point and no direction.
Black-hole bomb: Press and Teukolsky's thought experiment: a mirror traps superradiant waves, which grow until it fails.
Blandford–Znajek process: Magnetic field lines threading a spinning horizon brake it and carry its spin energy into a jet.
Magnetically arrested disc: An accretion flow so magnetised that the field near the hole holds the gas back; in simulations it makes powerful jets.
Electronvolt, eV: 1.6 × 10⁻¹⁹ J, the energy an electron gains across one volt; particle masses are quoted as energies in eV.
Ultralight boson: A hypothetical particle, around 10⁻¹² eV for a ten-solar-mass hole, that would drain a hole's spin into a cloud.
Hawking temperature: The temperature of a black hole's quantum radiation: 6.2 × 10⁻⁸ K for one solar mass without spin, lower with spin.
Quantum vacuum: A field's lowest state; each mode keeps half a quantum of energy even at zero temperature.
Analogue experiment: A laboratory system whose waves obey similar equations, with no black hole and no real gravity.
Negative resistance: A circuit element that feeds energy in instead of draining it.
X-ray binary: A black hole or neutron star feeding on a companion star, seen through the X-rays of the gas falling in.
Second-generation merger: One in which a hole was itself made in an earlier merger; such holes tend to spin fast.
Signal-to-noise ratio: How strongly a signal stands out from the detectors' noise.
σ: As in "4.4σ": standard deviations from "no effect".

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