60 nanokelvin: the empty sky is 44 million times hotter than a black hole

Here's a temperature: 60 nanokelvin, give or take.

A nanokelvin is a billionth of a kelvin, and kelvin temperatures count up from absolute zero, the coldest anything can be. Stephen Hawking found that black holes should glow, and his rule puts a black hole with the Sun's mass at about 62 of them (this page's arithmetic). The title rounds it to 60.

The trouble is the sky around it, which is far warmer. So not one black hole we know of has evaporated even a gram yet: to evaporate, a black hole has to glow away more than it takes in. This story is about why, how long they have to wait, and how physicists test the idea without a black hole to hand.

Chapter 1

A very cold glow

Black holes are famous for swallowing everything, light included. So it was a surprise when, in 1974, Stephen Hawking found that they should glow. His short letter in the journal Nature had a question for a title: "Black hole explosions?"

His answer came from quantum physics, the rules of the very small. Near a black hole's edge, its horizon, those rules make it give off particles as if it were a warm object. And the heavier the black hole, the colder its glow.

For a black hole with the Sun's mass, the glow works out at about 62 billionths of a kelvin (this page's arithmetic). Physicists call it the Hawking temperature.

The Sun itself will never become a black hole: "the Sun's mass" is just a handy yardstick, and every real black hole we know of is heavier (chapter 3).

Now the twist. Space is filled with faint microwaves left over from the hot early universe, at 2.7 kelvin: about 44 million times warmer than the black hole. And anything colder than its surroundings soaks up more than it gives off.

A Sun-mass black hole≈ 62 nK

The microwave sky2.7 K

the sky is 44 million × warmer

One caveat belongs at the start. Nobody has ever seen this glow from a real black hole. It's a prediction, built on theory most physicists trust here, and something like it has been seen only in lab imitations: horizons for sound and for light (chapters 12 and 13).

Chapter 2

Hawking's surprise

1971A rule about size

The road to that temperature began with a rule about size. In 1971 Hawking showed that, in Einstein's theory of gravity, the total surface area of black holes can never shrink. Black holes can grow and merge, but their combined area only goes up.

That sounded oddly familiar. One of the great rules of heat says the same about entropy, roughly a measure of how mixed up things are (chapter 4 says more): in total, it never goes down.
1973Entropy for a black hole

Jacob Bekenstein took the likeness seriously. Drop a box of hot gas into a black hole, and where does its entropy go? If it simply vanished, the rule would break. So in 1973 he argued that a black hole carries entropy of its own, more the bigger its area.
1973Four laws, as an analogy

The same year, James Bardeen, Brandon Carter and Hawking wrote down four laws for black holes that mirror the four laws of heat. But they treated it as an analogy: a real temperature would mean a real glow, and nothing was supposed to come out of a black hole.
1974The glow: the match is real

Then Hawking's calculation, mixing quantum physics with Einstein's gravity, found the glow. The match was real: a black hole has a true temperature, and a true entropy. And the glow is the exception to the 1971 area rule: a black hole that gives out more than it takes in shrinks, and its area falls.

Why trust it? Quantum gravity, the still-missing rules that would join quantum physics and gravity, should matter on tiny scales, and around a black hole as heavy as a star, gravity is gentle on those scales. So the calculation's prediction of how brightly a black hole glows, and with what mix of particles, is expected to hold until the black hole has shrunk almost to nothing (chapter 11).

Chapter 3

Heavier means colder

Everything about a black hole that doesn't spin follows from its mass. Double the mass and you halve its temperature.

Its size goes the other way: a black hole with the Sun's mass would be about 5.9 kilometres across, and twice the mass is twice as wide. The time it would take to evaporate grows far faster still, roughly as the mass times itself times itself again: double the mass and it takes about eight times as long.

Slide the mass, or pick a real object, and watch all three lines move 👇

No real black hole is as light as the Sun. When a dying star's core is lighter than a certain limit, it becomes a neutron star instead, a city-sized ball of packed neutrons, and the heaviest neutron stars measured weigh about 2.1 to 2.4 Suns. No black hole lighter than about 2.5 Suns is confirmed.

So the warmest black holes we know of glow at around 20 billionths of a kelvin. The lightest so far, probably a black hole, was half of a pair caught in 2023 (GW230529) by its gravitational waves, ripples in space itself. It was about 17 billionths of a kelvin before it merged with a neutron star. The giant at the centre of our galaxy, Sagittarius A*, about 4.3 million Suns, glows at about 1.4 × 10⁻¹⁴ kelvin (all this page's arithmetic).

From the sky down to the black holes

2.7 Kthe microwave sky ≈ 62 nKa Sun-mass black hole ≈ 20 nKthe warmest black holes we know of ≈ 1.4 × 10⁻¹⁴ KSagittarius A*

← coldereach tick: × 10warmer →

A Sun-mass black hole's glow would be radio waves, strongest at about 40 kilometres long (this page's arithmetic).

Run it the other way. A black hole just as warm as today's sky would weigh 61% of the Moon and be about a hair's width across (this page's arithmetic). Anything heavier is colder than the sky. No black hole that light is known.

Spin changes things. A spinning black hole is colder than a still one of the same mass, yet it gives off more, because it also sheds its spin. That matters in chapter 9; everywhere else, the numbers are for black holes that don't spin.

Chapter 4

As much entropy as a space can hold

For ordinary things, entropy counts the ways their parts could be arranged: for a gas, all the ways its molecules could sit and move while it looks, from outside, exactly the same.

A black hole has entropy too, set by the area of its horizon. For a Sun-mass black hole it comes to about 10⁷⁷, in the units physicists use (this page's arithmetic). What that entropy counts, and where, is still an open question, and this page leaves it open.

≈ 10⁷⁷A Sun-mass black hole's entropy, in the units physicists use: a 1 followed by 77 zeros.

The number is enormous. Crush the Sun into a black hole and its entropy would rise four to eight billion billion times (this page's estimate).

One Sun-mass black hole holds more entropy than all the stars and gas of the Milky Way put together, by millions of times, even counting generously (this page's estimate). That leaves out the Milky Way's own black holes. The giant at its centre alone holds about 3.6 times as much entropy as all the leftover microwave light in the observable universe, all the space whose light has had time to reach us (this page's arithmetic).

And there's a ceiling. In 1981 Bekenstein proposed a limit on how much entropy anything of a given size and energy could hold. A black hole sits exactly on that limit: it holds as much as anything its size and energy can. Nº 2 used the same limit to put a ceiling on what a brain could store.

Chapter 5

Hotter as it shrinks

Here's the strangest part. Take energy away from a black hole and it gets hotter, not colder.

It follows from chapter 3's rule: heavier means colder. A black hole's energy is its mass, so losing energy means losing mass, and a lighter black hole is a hotter one. Physicists say it has a negative heat capacity.

Black holes aren't alone in this. Stars and star clusters, held together by their own gravity, also heat up as they lose energy. A black hole that doesn't spin is the extreme case.

Negative heat capacity isn't the same as negative temperature. Nº 17 met tiny magnets at minus 750 picokelvin (trillionths of a kelvin), which warm up when you give them energy, like ordinary stuff. A black hole is the mirror case: a plus temperature, and it heats up as it loses energy.

That makes a black hole restless. Put an ordinary object in a bath kept at one temperature, nudge it a little warmer, and it cools back down to match. A black hole can't sit calmly in a bath that is just as warm as it is: a little too warm and it gives off heat, shrinks and gets hotter still; a little too cold and it soaks up heat, grows and gets colder still.

Everyday pictures catch only half of this.

A cold drink in a warm room

like a black holesoaks up heat,

unlike onebut it warms as it does.

A campfire that burns hotter as it shrinks

like a black holecatches the runaway,

unlike onebut real fires cool as they die.

Nudge the black hole off balance, then let go 👇

The bath here is a cartoon. The real sky isn't a bath at one fixed temperature: it's cooling, as the next chapters show.

Chapter 6

Bathed in a warm sky

Every black hole sits in a bath. All of space glows with light left over from the hot early universe, stretched by now into microwaves: the cosmic microwave background. Its temperature is 2.7 kelvin.

That glow is everywhere. Every cubic centimetre of space, about the size of a sugar cube, holds about 411 of its tiny packets of light, called photons.

A black hole catches the ones that pass close enough. A Sun-mass black hole soaks up about 2.3 kilowatts of microwaves, and gives back about 1.6 × 10⁻²⁸ watts of Hawking glow (this page's arithmetic). Put the two side by side and the black hole takes in about 10³¹ times more than it gives out: ten million trillion trillion times.

Heavier black holes are further behind still. Double the mass and the gap grows sixteenfold: the bigger black hole catches four times as much, and its glow is four times fainter.

Think of a bath with the tap running and the plug out. Microwaves pour in through the tap; Hawking's glow trickles down the drain. Today the tap wins, by 10³¹.

Your turn · the tap and the drain

In through the tap: about 2.3 kilowatts. Out down the drain: about 1.6 × 10⁻²⁸ watts. The tap wins by about 10³¹.

But this tap turns itself down. As the universe expands, the microwave sky cools, and one day the inflow will drop below the drain. Chapter 8 works out when. First, that gram.

Chapter 7

Not even a gram

The gram in the title is easy to check. In the universe's whole 13.8 billion years, a Sun-mass black hole would have glowed away about 8 × 10⁻²⁵ grams: less than a trillionth of a trillionth of a gram (this page's arithmetic). At that rate, its first whole gram would take 1.7 × 10³⁴ years.

Meanwhile, it's eating. From the microwave sky alone, a Sun-mass black hole gains about 0.8 milligrams a year today. The sky was hotter in the past, so over its life it has gained tens to thousands of tonnes, depending on how old it is (this page's arithmetic). Inside a galaxy, starlight and gas add more.

The gram, weighed: a Sun-mass black hole

≈ 8 × 10⁻²⁵ gout: glowed away in 13.8 billion years

tens to thousands of tonnesin: from the microwave sky, over its life

So it's gaining mass, not losing it. Black holes can lose mass in other ways, though, which is why the title chooses its words with care. A spinning black hole can give up some of its spin energy, for instance to power jets, beams of hot gas shot out, some at nearly the speed of light. And when two black holes merge, some of their mass leaves as gravitational waves: about three Suns' worth in GW150914, the first black-hole merger ever detected.

Even then the total area of the horizons grows, as Hawking's 1971 rule says it must: an exceptionally loud merger, GW250114, confirmed it in 2025. None of that is evaporation. Counting only the mass lost to Hawking's glow, no known black hole has lost a gram.

Chapter 8

Waiting for the sky to cool

As space expands, the microwaves in it are stretched, and the sky cools. Double the size of the universe and the sky's temperature halves.

To fall below a Sun-mass black hole that doesn't spin, the sky has to get about 44 million times colder. How long will that take?

It depends on dark energy, whatever is making the expansion speed up. If it stays the same, the universe settles into steady growth, and the sky gets about 2.7 times colder every 17.4 billion years. Cooling 44 million-fold takes about 17.6 of those steps.

The sky's staircase, if dark energy stays the same

2.7 K the sky, now

each step: 2.7 times colder, every 17.4 billion years

about 17.6 steps down, the sky falls below a Sun-mass black hole (≈ 62 nK)

Work it through properly and the sky falls below the black hole in about 300 billion years (305 billion from now, this page's arithmetic, if dark energy stays the same). Pick a black hole and run the clock 👇

The mass barely matters. A bigger black hole is colder, so the sky has further to fall, but each step of cooling covers a lot of ground. A black hole of 2.5 Suns, about the lightest known, waits about 321 billion years, the giant at our galaxy's centre about 571 billion, and the heaviest one whose mass has been measured from the stars moving around it, in the galaxy Holm 15A, about 730 billion (all this page's arithmetic). None waits a trillion years, for the microwave sky alone.

But the cold has a floor. In a universe whose expansion keeps speeding up, the far horizon, the distance beyond which light sent towards us now can never arrive, has a faint glow of its own, as Gary Gibbons and Hawking showed in 1977. For our universe it comes to about 2.2 × 10⁻³⁰ kelvin (this page's arithmetic). The sky reaches that floor in roughly 1.2 trillion years (Nº 3), after every known black hole's crossing.

So if dark energy stays the same, every black hole we know of will, in time, get its chance to evaporate.

Chapter 9

The small print

That 300 billion years has a narrow meaning: the microwave sky alone, a black hole that doesn't spin, and dark energy staying the same. It's the same wherever the black hole sits, but it isn't the day any real black hole starts to evaporate. Here's what it leaves out.

Starlight and gas

A Sun-mass black hole out between the stars in our part of the Galaxy would soak up about 4.8 kilowatts of starlight (this page's estimate), twice what it gets from the microwave sky. Inside a galaxy, starlight, gas and dark matter (unseen matter whose gravity helps hold galaxies together) keep feeding it for as long as the galaxy holds together, and stars keep shining for about 10¹⁴ years.

keeps it fed for longer

Leftover neutrinos

The Big Bang also left a sea of ghostly particles called neutrinos (Nº 4). Most are slow today, and a black hole gathers slow particles from far beyond its edge. On this page's estimate they feed it more than the microwave sky does. For how long, this page hasn't worked out.

feeds it too: for how long, not worked out

Spin

A spinning black hole is colder but gives off more, so it can start to lose mass on balance while the sky is still warmer than it. That brings the date earlier, by up to a few tens of billions of years (this page's rough estimate). So it's "about 300 billion years", never "at least".

brings the date earlier

Dark energy

The date assumes dark energy never changes. A big galaxy survey called DESI hints that it might, but as of October 2026 the hint is short of what physicists need to call it a discovery (Nº 1). If dark energy weakens, the wait grows, enormously if the speeding-up stops. If it turns negative, space could fall back in: in one published model the universe ends in a crunch about 20 billion years from now, and no black hole made by a star ever starts to evaporate (Nº 3).

if it changes: later, or never

Tiny black holes

Some physicists wonder whether tiny black holes were made in the early universe. None has been found, but any with the mass of a small asteroid would be hotter than the sky and evaporating now. That's why the title says "known".

why the title says "known"

Chapter 10

A very long fade

A colder sky is needed, but it isn't enough on its own. Only once a black hole's glow carries away more than it takes in from the sky and everything else does it start to evaporate.

Then the strange rule of chapter 5 takes over. Losing mass makes it hotter, so it glows faster and loses mass faster: a runaway, though a very slow one at first.

Guess first

How long would a Sun-mass black hole take to vanish?

Show the answer

How slow? A Sun-mass black hole would take about 10⁶⁷ years to vanish (this page's arithmetic). That's a 1 followed by 67 zeros.

The exact figure depends on which particles it can give off, a mix Don Page worked out in 1976. A cold black hole can only make particles that weigh almost nothing, so one as heavy as a star gives off light, gravitons (the particles of gravitational waves) and at most one kind of neutrino, if the lightest is light enough. Nobody knows its mass yet. If it weighs nothing at all, the black hole fades about three times faster: still about 10⁶⁷ years.

Heavier black holes last far longer. One of 10 Suns: about 10⁷⁰ years. The giant at our galaxy's centre: about 10⁸⁷. The heaviest whose mass has been measured from the stars moving around it, in Holm 15A: about 10⁹⁹ years (all this page's arithmetic).

Next to that, the 300-billion-year wait is a blink: about 3 × 10⁻⁵⁶ of a Sun-mass black hole's lifetime (this page's arithmetic).

Chapter 11

The last second

As a black hole shrinks and heats up, it starts making heavier particles: first neutrinos, then electrons, then muons (heavier cousins of electrons), then quarks and gluons, the building blocks of protons. Near the very end it's hot enough to give off every kind of particle we know.

Slide towards the end and watch it speed up 👇

In the standard calculation, a black hole starts its last second with about 1,300 tonnes left, at roughly 10¹⁷ kelvin. In that second it turns all of it into energy: about 1.2 × 10²³ joules, as much as the Sun gives out in a third of a millisecond (this page's arithmetic).

A Sun-mass black hole won't get there for a very long time. But a tiny black hole from the early universe, if one exists, might be finishing now: one born with about 5 × 10¹¹ kilograms, the mass of a rocky asteroid some 700 metres across.

Telescopes have looked for such final flashes. The tightest limit we have found, from the LHAASO observatory in China, says there are fewer than about 180 a year in each cubic parsec around us (a cube about three light-years on a side). As of October 2026, none has been seen.

And the very end? At about 22 micrograms, the Planck mass (Nº 10), the calculation stops being trustworthy. Whether a black hole then vanishes completely or leaves something behind, nobody knows.

Chapter 12

Black holes for sound

If no known black hole can be caught evaporating, can the idea be tested at all? In 1981 Bill Unruh showed that it can be imitated.

Picture a river speeding up towards a waterfall, with fish that can only swim so fast. Past the point where the current beats the fish, none can get back upstream. Sound in a fluid that flows faster than sound is trapped the same way. That point is a horizon for sound, a "dumb hole" ("dumb" as in silent), and by Hawking's argument it should glow, with sound.

A horizon for sound, drawn as a river

the river speeds up towards a waterfall

fish can only swim so fast

past the dashed line, none can get back upstream

In 2011 a team led by Silke Weinfurtner saw water waves at a horizon behave as Hawking's formula predicts, though only when triggered by waves they sent in.

The glow with nothing sent in came from Jeff Steinhauer's group in Haifa, first reported in 2016. In 2019 the group made a flowing cloud of about 8,000 rubidium atoms, so cold that they behaved as one quantum fluid. Over 7,400 repeats, the sound from its horizon matched the predicted temperature: about 0.35 nanokelvin, colder than a Sun-mass black hole. In 2021 they showed that the glow holds steady over time.

These are horizons for sound, not black holes. They test what happens to waves at a horizon, not gravity.

Chapter 13

A horizon made of light

Light can make a horizon for light, too.

A horizon in a glass fibre, step by step

Send a strong, short flash of light down a glass fibre and it changes the glass as it passes, slowing other light inside it.
Now send weaker probe light after it, just a little faster.
It catches up, slows as it runs into the flash's back edge, and can't get past it.
For that light the back edge works like a white hole's horizon, which nothing can enter, and the front like a black hole's horizon, which light inside can't leave.

Think of a slow lorry on a motorway: a car that catches up can't pass. But the light slows by only about 0.1%, and no gravity is involved. The first fibre horizon was made in 2008; in 2019 another light experiment showed a Hawking-like glow, triggered by light the team sent in.

In July 2026, Procopio and colleagues reported in the journal Nature that the Hawking-like light comes from a single direct step, not a chain of them. As in Hawking's picture it comes in pairs, and the team measured the partners, in the ultraviolet: light too blue for our eyes to see.

Making it also pushes back on the flash: some of the flash's own light reappears just beside those partners. On the authors' reading, that push-back shows the same temperature as the Hawking-like light, to within 2%. Change the probe light's speed and strength 👇

In May 2026 a team in Mexico, reporting in Nature Communications, showed that a single photon, one particle of light, is enough to start the effect, and what comes out still arrives one photon at a time.

The caveats, in plain words. In both experiments the light was triggered by light shone in, not spontaneous. The first team says it can't yet rule out other effects making a similar signal. Neither team measured the quantum link, called entanglement, between the Hawking-like light and its partner, and the second didn't measure the partner at all.

Philosophers disagree about how much such experiments can tell us about real black holes. They test waves at horizons, not gravity, and they don't show a black hole losing mass.

Chapter 14

Adding it up

Here's the whole chain. A black hole should glow, more coldly the heavier it is: about 62 billionths of a kelvin for a Sun-mass one, less for every real one. The sky is far warmer, so every black hole we know of takes in more than it gives out, and none of them has glowed away a gram.

If dark energy stays the same, the cooling sky drops below a Sun-mass black hole that doesn't spin in about 300 billion years, and below even the giants in under a trillion, though starlight, gas and leftover neutrinos keep real ones fed for longer. Only once a black hole gives out more than it takes in does the long fade begin: about 10⁶⁷ years for a Sun-mass one, ending, in the standard calculation, in a last second of about 10²³ joules.

And nobody has yet seen the glow from a real black hole, only its imitations in the lab.

Let's add it up.

Forecast for a Sun-mass black hole

Its glowthis page's arithmetic62 nK

The microwave skyFixsen, 20092.7 K

How much warmer the sky isratio44 million ×

Heat soaked up from the skythis page's arithmetic2.3 kW

Hawking glow given outthis page's arithmetic1.6 × 10⁻²⁸ W

The outlook

So far8 × 10⁻²⁵ gGlowed away in 13.8 billion yearsthis page's arithmetic
Next≈ 300 billion yearsWait for the sky to cool below itconstant dark energy
Later≈ 10⁶⁷ yearsThen, once more goes out than comes in, to evaporatethis page's arithmetic
Last≈ 10²³ JIts last secondstandard calculation

Lab "black hole" of soundHaifa, 20190.35 nK

Grams any known black hole has evaporatedso farnone yet

colder than the sky ✦ heavier is colder ✦ waiting on the universe

Somewhere dark, a black hole sits ringed by the faint, warm glow of the oldest light.

It takes in ten million trillion trillion times more warmth than it gives off.

The universe keeps stretching, and the old glow keeps cooling.

One day, hundreds of billions of years from now, the sky will be colder than the hole.

Only around then can it begin, very slowly, to shine its mass away.

Black holes should glow. They're just colder than the sky, and they have to wait for the sky.

Part IA temperature for a black hole

Chapter 1Sixty-two billionths of a kelvin

In 1974 Stephen Hawking published a short letter in Nature whose title was a question: "Black hole explosions?" Its central result was not a question at all. Quantum effects near the horizon, he found, make a black hole emit particles as if it were a warm body, with a temperature that falls as its mass rises.¹ The full calculation followed the next year.² For a black hole that neither spins nor carries electric charge, the temperature is

T_H = ħc³ / (8πGMk_B)ħ is the reduced Planck constant, c the speed of light, G Newton's constant, k_B Boltzmann's constant and M the hole's mass. The subscript H is for Hawking. Mass sits in the denominator: the heavier the hole, the colder it is.

Put in one solar mass and the answer is tiny. Using the International Astronomical Union's nominal value of the product GM_☉, which astronomers measure far more precisely than G or the Sun's mass separately, and the CODATA 2022 values of the other constants,^3,4 T_H comes out at 6.1703 × 10⁻⁸ K, a value known to about nine digits: 61.7 nanokelvin, about 62 billionths of a kelvin (this page's arithmetic). Nº 17 quoted the same 6.2 × 10⁻⁸ K.

The Sun itself will never become a black hole. "A black hole with the Sun's mass" is a reference point, and Chapter 3 explains why no real black hole we know of is that light.

What makes the number interesting is what surrounds it. Space is filled with the cosmic microwave background, the cooled afterglow of the hot early universe, at 2.72548 ± 0.00057 K.⁵ That is about 44 million times warmer than 62 nK. A body colder than its surroundings takes in more radiation than it gives out, and a black hole is no exception. So the claim of this page is:

Every black hole we know of absorbs far more energy than it gives off as Hawking radiation. Not one has yet lost a gram to Hawking radiation, and none can start to evaporate until the expanding universe has cooled the microwave sky below roughly the temperature of a non-spinning hole of the same mass.

Two words in the title carry weight. Known: primordial black holes, if any exist, could be hotter than the sky (Chapter 9). Evaporated: black holes can lose mass by giving up spin energy or by merging (Chapter 7). This page is about the mass a hole loses to Hawking radiation.

One caveat belongs at the start. No telescope has ever seen Hawking radiation from a real black hole; it has been seen only in laboratory analogues, the horizons for sound and light of Chapters 12 and 13.⁶ The formula rests on quantum field theory in curved spacetime, which most physicists trust in this regime (Chapter 2).

Chapter 2Area, entropy and a glow

The route to a black hole's temperature began with a theorem about area. In 1971 Hawking showed that in classical general relativity the total area of black-hole horizons can never decrease: holes can grow and merge, but the sum of their horizon areas only goes up.⁷ The second law of thermodynamics says the same of entropy.

Jacob Bekenstein took the resemblance seriously. If you drop a box of hot gas into a black hole, its entropy seems to vanish from the universe, breaking the second law. Bekenstein argued in 1973 that the second law survives only if the hole itself carries entropy, in proportion to its horizon area, so that the area the hole gains pays for the entropy that fell in.⁸

The same year James Bardeen, Brandon Carter and Hawking set out four laws of black-hole mechanics that mirror the four laws of thermodynamics.⁹ The horizon's surface gravity κ, roughly the pull at the horizon as measured from far away, plays the role of temperature: it is uniform over a settled hole's horizon, and a change in mass is tied to a change in area as energy is tied to entropy. The three authors treated the parallel as formal: a real temperature would mean a real glow, and classically nothing comes out of a black hole.

Hawking's calculation removed the objection. In the semiclassical approximation, with spacetime treated as classical and only the fields on it as quantum, he found that a distant observer sees the hole emit a thermal spectrum at a temperature set by its surface gravity:^1,2

T_H = ħκ / (2πck_B) = ħc³ / (8πGMk_B)The first form holds for any stationary black hole; the second is the non-spinning, uncharged case, with κ = c⁴/(4GM).

With the temperature known, the first law of thermodynamics, dE = T dS, fixes the entropy, and Bekenstein's proportionality acquires an exact coefficient:

S = k_BA / (4ℓ_P²), ℓ_P² = ħG/c³A is the horizon area, 4πr_s² for a non-spinning hole of Schwarzschild radius r_s = 2GM/c². ℓ_P, the Planck length, is 1.616 × 10⁻³⁵ m. In words: a quarter of the horizon area, counted in Planck units.

So the analogy turned out to be exact: a black hole has a temperature and an entropy in the full thermodynamic sense. The area theorem is a classical statement, and Hawking radiation is the exception to it: a hole that radiates more than it absorbs shrinks, and its area falls.

Why trust a calculation that mixes quantum fields with classical gravity? For any hole of astronomical mass the curvature at the horizon is gentle compared with the Planck scale, where quantum gravity should matter: a Sun-mass hole's horizon is about 3 km in radius, some 10³⁸ Planck lengths, and for the rate and spectrum of the emission the approximation is expected to fail only near the Planck mass. At 62 nK against a 2.7 K sky, it cannot be tested on any black hole we know of; only a hot primordial hole, if one exists, could show it directly (Chapter 11).

Chapter 3Heavier is colder

Everything about a non-spinning black hole follows from its mass. The temperature goes as 1/M: double the mass, halve the temperature. The Schwarzschild radius goes as M, 2.953 km per solar mass, so a Sun-mass hole would be 5.9 km across. The entropy goes as M² (Chapter 4) and the lifetime nearly as M³ (Chapter 10). On a grid of powers of ten, temperature, width and lifetime run at slopes of −1, +1 and about +3.

T_H = 61.7 nK × (M_☉/M), 2r_s = 5.91 km × (M/M_☉)This page's arithmetic. M_☉ = GM_☉/G = 1.98841 × 10³⁰ kg.

Figure 1 draws those three lines on one chart: drag the mass, or pick a real object.

Figure 1⚖️ Try it

Weigh a black hole

Drag the mass

Mass

Things

Real black holes

Hawking temperatureHow warmly it glows

Width, 2r_sHow wide

Lifetime (this page's arithmetic)How long it takes to evaporate

Entropy, S/k_BEntropy (in physicists' units)

T_H = ħc³/(8πGMk_B) = 6.17 × 10⁻⁸ K × (M_☉/M) for a hole that neither spins nor carries charge; width 2r_s = 4GM/c² = 5.91 km × (M/M_☉); entropy S/k_B = 4πGM²/(ħc) = 1.05 × 10⁷⁷ (M/M_☉)² (IAU GM_☉, CODATA 2022). Each line counts powers of ten from its Sun-mass value, so the lines meet at the Sun's mass only because each is measured from there; each tick marks the mass at which a hole is as hot, as wide or as long-lived as the thing named. Lifetimes are this page's arithmetic, τ = ∫₀^M M′² dM′/α(M′), adding each species once k_BT passes about its rest energy divided by 4.4 to 6.0 (Page 1976; Ukwatta et al. 2016), quarks and gluons approximated. They assume the normal ordering and a nominal 1 meV for the lightest neutrino, whose mass is unknown; the toggle makes it massless, shortening stellar lifetimes about threefold. Holes below 5 × 10¹¹ kg are shown as if they existed now; one born that light would be gone. Bennu's mass, 7.33 × 10¹⁰ kg: Lauretta et al. (2019).¹⁰ GW230529: its primary before it merged. Hotter or colder is against the 2.72548 K microwave sky (Fixsen 2009); hotter is not the same as losing mass (Chapter 9).Slide the mass and follow the three lines: heavier black holes are colder (down), wider (up) and far longer-lived (the steep line, rising three times as fast). Each line counts from a Sun-mass black hole, the only reason they meet there. Each tick marks the mass at which a black hole is as hot, as wide or as long-lived as the thing named; "sky" is today's microwave background, and every real black hole on the list sits on its cold side. Lifetimes are this page's arithmetic. They assume the lightest neutrino weighs a tiny amount, still too much for a black hole as heavy as a star to give it off; nobody knows its mass yet. The button shows the case where it weighs nothing: black holes as heavy as stars then fade about three times faster.

No real black hole sits at the Sun's mass: a collapsing core lighter than the maximum neutron-star mass stays a neutron star, and the heaviest measured neutron stars weigh about 2.1 to 2.4 M_☉.^11,12 A few objects of 2.1 to 2.7 M_☉ could be either kind: GW190814's 2.59 M_☉ secondary and the companion of the pulsar PSR J0514−4002E.^13,14 No black hole lighter than about 2.5 M_☉ is confirmed, so the warmest known are near 20 nK: the lightest so far, the primary of GW230529 (2.5 to 4.5 M_☉, probably a black hole), was about 17 nK before it merged with a neutron star (this page's arithmetic at the median, 3.6 M_☉).¹⁵

Black hole	Mass (M_☉)	T_H
GW230529 primary, before merging¹⁵	2.5–4.5	17 nK
Gaia BH1, the nearest known¹⁶	9.62 ± 0.18	6.4 nK
Cygnus X-1¹⁷	21.2 ± 2.2	2.9 nK
Gaia BH3¹⁸	32.70 ± 0.82	1.9 nK
Sagittarius A*, the Milky Way's centre¹⁹	(4.297 ± 0.012) × 10⁶	1.4 × 10⁻¹⁴ K
M87*²⁰	(6.5 ± 0.7) × 10⁹	9.5 × 10⁻¹⁸ K
Holm 15A, the heaviest with a dynamically measured mass²¹	(4.0 ± 0.8) × 10¹⁰	1.5 × 10⁻¹⁸ K

Temperatures: this page's arithmetic for non-spinning holes at the central masses.

The hole's gravity filters its spectrum (greybody factors). For photons the power peaks at quanta of 6.04 k_BT,²² so a Sun-mass hole would glow in radio waves about 40 km long, some thirteen times its radius (this page's arithmetic; a perfect blackbody's power per unit energy would peak at about 80 km, so the hole's gravity pushes its glow to shorter waves).

Run the scaling the other way and a hole as warm as today's sky has 4.5 × 10²² kg, 61% of the Moon's mass, and a radius of 67 µm: about a hair's width across (this page's arithmetic). Anything heavier is colder than the sky. No hole that light is known.

Spin makes a hole colder. For a spinning (Kerr) hole with spin parameter χ, which runs from 0 for no spin to 1 for the fastest spin a hole can have,

T = [ħc³/(4πGMk_B)] × √(1 − χ²) / [1 + √(1 − χ²)]At χ = 0 this is the Schwarzschild value. Relative to it: 0.93 at χ = 0.5, 0.83 at 0.7, 0.61 at 0.9, 0.25 at 0.99, and zero at χ = 1 (this page's arithmetic).

Yet a spinning hole gives off more power, not less, because it also radiates away its rotation. Don Page's second paper of 1976 found the emission at fixed mass rising with spin "by factors of up to 13.35 for neutrinos, 107.5 for photons, and 26 380 for gravitons".²³ Colder yet brighter matters for the date in Chapter 9. Unless stated, every number here is for a hole that does not spin.

Chapter 4A quarter of the horizon

For ordinary matter, entropy is a count. Boltzmann's definition, S = k_B ln W, counts the number W of microscopic arrangements that look the same at large scale: for a gas, the positions and speeds its molecules can take at a given temperature and pressure. A black hole's entropy is fixed differently: by thermodynamics and Hawking's temperature (Chapter 2), not by a count. What it counts, and where, is unresolved; this page leaves that question open and gives only the number:

S/k_B = A/(4ℓ_P²) = 4πGM²/(ħc) = 1.05 × 10⁷⁷ × (M/M_☉)²This page's arithmetic. Entropies on this page are in units of Boltzmann's constant. A 10 M_☉ hole holds 1.05 × 10⁷⁹; Sagittarius A* 1.9 × 10⁹⁰.

A star like the Sun holds 11 to 21 k_B per baryon (proton or neutron), on Basu and Lynden-Bell's values as quoted by Chas Egan and Charles Lineweaver.²⁴ The Sun's 1.19 × 10⁵⁷ baryons therefore hold 1.3 to 2.5 × 10⁵⁸. Crush the Sun into a black hole and its entropy would rise four to eight billion billion times (4–8 × 10¹⁸; this page's estimate).

The Milky Way does no better. Its stars, 6.08 × 10¹⁰ M_☉,²⁵ hold (0.8–1.5) × 10⁶⁹ at the same entropy per baryon. For a generous bound on all its ordinary matter, take the Galaxy's total mass including dark matter, 1.54 × 10¹² M_☉,²⁶ count the cosmic baryon fraction of it, 0.158, as ordinary matter,²⁷ and give each of those 2.9 × 10⁶⁸ baryons 143 k_B, the highest value in Egan and Lineweaver's list (ionised intergalactic gas).²⁴ The total is 4.1 × 10⁷⁰. One Sun-mass black hole beats that bound about 2.5 million times, and the Galaxy's stars about 10⁸ times (this page's estimate). The comparison leaves out the Galaxy's own black holes, which hold far more: Sagittarius A* has about 10⁹⁰.

Egan and Lineweaver put the entropy of the observable universe at 3.1 (+3.0, −1.7) × 10¹⁰⁴ k_B, almost all in supermassive black holes, against 5.4 × 10⁸⁹ in microwave-background photons, 5.2 × 10⁸⁹ in relic neutrinos (the neutrinos left over from the first second after the Big Bang) and about 10⁸¹ in stars.²⁴ Sagittarius A* alone holds about 3.6 times the entropy of every microwave-background photon in the observable universe (this page's arithmetic).

There is a ceiling behind these numbers. Bekenstein also proposed, in 1981, a bound on the entropy any system of energy E that fits inside a sphere of radius R can hold, 2πk_BRE/(ħc).²⁸ Set R = r_s and E = Mc², and the bound equals the black hole's entropy exactly: 2π(2GM/c²)(Mc²)/(ħc) = 4πGM²/(ħc). A black hole holds the most information a region of its size and energy can hold. Nº 2 used the same bound to put a ceiling on what a brain could store.

Chapter 5Hotter as it shrinks

The formula T_H ∝ 1/M hides the strangest property of a black hole. A hole's energy is its mass-energy, E = Mc². Take energy away and M falls, so T_H rises. The heat capacity, the energy needed per kelvin of warming, is negative:

C = dE/dT = −Mc²/T_H = −2SFor one solar mass, −2.9 × 10⁵⁴ J/K, or −2.1 × 10⁷⁷ k_B (this page's arithmetic). The magnitude is twice the entropy.

A black hole is not the only thing that behaves this way. Stars and star clusters, held together by their own gravity, also heat up as they lose energy; Donald Lynden-Bell and Roger Wood worked out what that means for self-gravitating systems in 1968.²⁹ A black hole that does not spin is the extreme case of the same family, with a heat capacity that is negative at every mass.

Negative heat capacity is not negative temperature. Nº 17 described nuclear spins at −750 picokelvin: hotter than anything with a positive temperature, yet with a heat capacity that is positive on both sides of infinity. A black hole is the mirror case: a positive temperature, and a negative heat capacity.

The consequence is instability. Nudge an ordinary object warmer than the room and it loses heat until it matches again. A black hole in an endless bath at its own temperature cannot settle like that: slightly warmer than the bath, it radiates more than it absorbs, loses mass, heats up and runs away; slightly colder, it absorbs, gains mass, cools and absorbs faster still.

An ordinary object in a bath

Heat capacity: Positive
A little too warm: Loses heat, cools back to the bath
A little too cold: Gains heat, warms back to the bath
Balance: Stable

A black hole in a bath

Heat capacity: Negative, −2S
A little too warm: Radiates, shrinks, gets hotter
A little too cold: Absorbs, grows, gets colder
Balance: Unstable

Two everyday pictures each catch half of this. A cold drink in a warm room takes in heat from the room, as a black hole takes in light from the sky (Chapter 6); but the drink warms as it absorbs, while a black hole grows and gets colder. A campfire that burns hotter as it shrinks catches the runaway of Chapter 10; but real fires cool as they die. Figure 2 sets an ordinary object and a black hole side by side in the same abstract bath, each starting slightly off balance.

Figure 2🔥 Try it

Hotter as it shrinks

Nudge it, then let go

Start the hole off balance

T_H / T_bathHole vs the bath

M / M_bIts size vs the balance point

Heat flows

Ordinary object ends atWhere the object ends up

A hole exactly as warm as today's microwave sky would weigh 4.5 × 10²² kg, 61% of the Moon (this page's arithmetic). A real hole that size also takes in starlight and relic neutrinos, so it is not a balance point (Chapter 9).A black hole exactly as warm as today's sky would weigh 61% of the Moon (this page's arithmetic). A real one that size would also soak up starlight and other things, so it wouldn't sit balanced.

Schematic. A black hole in an endless bath at temperature T_bath, with its mass m = M/M_b in units of the balance mass M_b at which T_H = T_bath. Absorption grows as M² and Hawking emission falls as 1/M², so dm/ds = m² − 1/m² in a scaled time s, solved exactly by s = F(m) − F(m₀) with F(m) = ¼ ln|(m − 1)/(m + 1)| + ½ arctan m. The direction of the flow is exact; the speeds are not to scale, and greybody factors, gravitons and every other intake are left out. The ordinary object's temperature θ, in units of T_bath, relaxes as θ = 1 + (θ₀ − 1)e^−s/0.3; both thermometers are magnified, spanning 0.85 to 1.15 T_bath. Because C = −Mc²/T_H = −2S is negative, the black hole's balance with the bath is unstable.A cartoon, not to scale. Both sit in a bath kept at one fixed temperature. The ordinary object, slightly too warm, cools down to match the bath. The black hole, slightly too warm, gives off heat, which makes it smaller and even hotter, so it runs away. Start it slightly too cold and it runs away the other way: it soaks up heat, grows and gets colder still.

The real sky is not a bath at a fixed temperature: it cools as the universe expands (Part II).

Part IIColder than the sky

Chapter 6A sky at 2.7 kelvin

Every black hole sits in a bath. The cosmic microwave background fills space at 2.72548 ± 0.00057 K,⁵ with 411 photons per cubic centimetre and an energy density of 4.17 × 10⁻¹⁴ J/m³, 0.26 eV per cubic centimetre (this page's arithmetic from the blackbody law).

Millimetre microwaves are far shorter than a hole kilometres wide, so the hole presents them with a simple target, its capture cross-section: a disc of radius √27 GM/c². Radiation arriving from every direction delivers energy density times c times that area:

P_in = aT₀⁴ c × 27π(GM/c²)² ≈ 2.3 kW × (M/M_☉)²a = 7.5657 × 10⁻¹⁶ J m⁻³ K⁻⁴ is the radiation constant and T₀ = 2.72548 K. For one solar mass the cross-section is 1.85 × 10⁸ m². This page's arithmetic.

A Sun-mass hole therefore absorbs 2.3 kilowatts of microwave light. What it gives back is far smaller. Hawking emission falls as P = αc²/M², where α grows with the number of kinds of particle the hole is hot enough to emit. A stellar hole is far too cold to emit electrons or the two neutrinos known to have mass, so it emits photons and gravitons, plus the lightest neutrino if that is massless or nearly so (Chapter 10). Leaving that neutrino out, α = 7.2 × 10¹⁵ kg³/s: 6.4 × 10¹⁵ for photons, from Ukwatta and colleagues' per-species rates, and 0.8 × 10¹⁵ for gravitons, from Page's shares of the power, 2% for gravitons against 17% for photons (Chapter 10).^22,30

P_out = αc²/M² ≈ 1.6 × 10⁻²⁸ W × (M_☉/M)², P_in/P_out ≈ 1.4 × 10³¹ × (M/M_☉)⁴This page's arithmetic, with α = 7.2 × 10¹⁵ kg³/s for photons and gravitons. The ratio rises as the fourth power of the mass: two powers from the absorbing area, two from the falling emission.

So a Sun-mass hole takes in about 10³¹ times more energy than it sends out, and heavier holes are further behind: 3 × 10³⁶ for Cygnus X-1 (this page's arithmetic).

A bathtub that drains while the tap runs is a fair picture: microwaves, and in a galaxy starlight and gas, coming in; Hawking radiation going out. Today the tap wins by 10³¹. Where the picture breaks is the interesting part: this tap turns itself down. As the universe expands the microwave sky cools, and one day the inflow will fall below the drain. Chapter 8 works out when. First, the gram in the title.

Chapter 7Not one gram

The gram is easy to check. At 1.6 × 10⁻²⁸ W, a Sun-mass hole radiates 5.8 × 10⁻³⁸ kg a year as Hawking radiation. Over the 13.8 billion years since the Big Bang that comes to about 8 × 10⁻²⁵ g, less than a trillionth of a trillionth of a gram, and less still for heavier holes, since the loss falls as 1/M². To radiate a first whole gram at today's rate would take 1.7 × 10³⁴ years (this page's arithmetic).

That is the gross loss. On balance the hole has gained: from the microwave sky alone, 0.8 mg a year today. The sky was hotter in the past, by a factor of 1 + z at redshift z, when the universe was 1 + z times smaller, so added up over the past the gain is 31 tonnes since redshift 1, 1,100 since redshift 6 and 3,300 since redshift 10: tens to thousands of tonnes, depending on its age (this page's arithmetic). Before the sky falls below it, the hole adds about three tonnes more, and inside a galaxy starlight and gas add more again (Chapter 9). The loss and the gain, side by side:

ΔM_Hawking(13.8 Gyr) ≈ 8 × 10⁻²⁵ g ≪ ΔM_microwaves ≈ 3 × 10⁴–3 × 10⁶ kgFor one solar mass, non-spinning. The second figure depends on when the hole formed. This page's arithmetic.

So the hole is gaining mass, not losing it. Black holes do lose mass by other routes, which is why the title's scope matters. A spinning hole's rotational energy can be taken out in two ways. In the Penrose process, a particle splits in two inside the ergosphere, a region outside a spinning hole's horizon where everything is dragged round with the spin: one piece falls in with negative energy, and the other leaves with more energy than the whole brought in.³¹ And magnetic fields threading the horizon can draw the energy out to power jets, as Blandford and Znajek showed in 1977.³² In 2011 Tchekhovskoy, Narayan and McKinney simulated magnetically arrested discs, in which the field near the hole grows strong enough to hold back the inflowing gas, and found jet efficiencies above 100%: more energy carried out than falls in, the difference drawn from the hole's spin.³³

Merging holes lose mass too: in GW150914, the first merger LIGO detected, about three solar masses (3.0 ± 0.5) left as gravitational waves.³⁴ Even then the total horizon area grows, as Hawking's theorem requires; the exceptionally loud merger GW250114 confirmed this in 2025.³⁵ Neither spin energy nor merger losses are evaporation. Counting only the mass lost as Hawking radiation, no known black hole has lost a gram.

Chapter 8Waiting for the sky to cool

The microwave sky cools as space expands: its photons are stretched with space, so T_sky = T₀/a, where the scale factor a is 1 today and doubles when distances double. To fall below a Sun-mass hole the sky must cool by a factor of T₀/T_H = 4.42 × 10⁷, about 44 million. Since e^17.6 ≈ 4.42 × 10⁷ (17.6 is its natural logarithm, ln), that is 17.6 e-folds, factors of e = 2.718 in the size of the universe (this page's arithmetic).

Once dark energy dominates, and if it stays constant, the universe grows by a factor of e every

τ_Λ = 1/H_∞ = 1/(H₀√Ω_Λ) = 17.41 billion yearsH_∞ is the rate the expansion settles to once dark energy dominates. Planck 2018 with baryon acoustic oscillations (a pattern in how galaxies cluster):²⁷ the Hubble constant H₀ = 67.66 ± 0.42 km/s/Mpc and dark energy's share of the cosmic density Ω_Λ = 0.6889 ± 0.0056. This page's arithmetic; Nº 3 quotes the same 17.4 billion years.

At that rate, 17.6 e-folds would take 306.5 billion years (this page's arithmetic). The exact answer is slightly shorter, because the universe still expands a little faster today than its final rate. It comes from the flat ΛCDM solution, matter plus a cosmological constant, with radiation neglected:

a(t) = (Ω_m/Ω_Λ)^1/3 sinh^2/3(3t/2τ_Λ), t(a) = (2τ_Λ/3) asinh(√(Ω_Λ/Ω_m) a^3/2)Matter's share Ω_m = 0.3111, Ω_Λ = 0.6889; a = 1 gives the age, 13.79 billion years.

Set a = 4.42 × 10⁷ and the crossing comes 319 billion years after the Big Bang: 305 billion years from now (this page's arithmetic; 308 billion with Planck's parameters alone). That is this page's "about 300 billion years": the date at which the microwave sky falls below a non-spinning hole of one solar mass, if dark energy stays constant. Figure 3 runs the clock.

Figure 3⏳ Try it

When the sky falls below the hole

Pick a black hole, run the clock

Black hole mass

Scrub through time

Microwave sky falls below it (this page's arithmetic)Sky colder than the hole (this page's arithmetic)

Sky at this momentThe sky then

The holeThe black hole

Sky ÷ holeHow much warmer the sky is

The microwave sky's temperature, 2.72548 K/a(t) (Fixsen 2009), in flat ΛCDM: a(t) = (Ω_m/Ω_Λ)^1/3 sinh^2/3(3t/2τ_Λ), τ_Λ = 1/H_∞ = 17.41 Gyr, Ω_m = 0.3111, Ω_Λ = 0.6889 (Planck 2018 + BAO; radiation neglected). The flat line is a non-spinning hole's Hawking temperature. The crossing, the same wherever the hole sits, is when the microwave sky falls below the hole, not the start of evaporation: on radiation alone net loss begins with the sky up to about 3% warmer, since the hole also emits gravitons; a fast spin could bring it a few tens of billions of years earlier (this page's rough estimate); starlight, gas and other matter push it later. The galaxy view is schematic, its starlight intake this page's estimate for a hole in the Sun's part of the Galaxy, away from any star. Dates assume constant dark energy; if it weakens the wait grows, and if it turns negative the wait may never end (Chapter 9; as of October 2026). The dashed floor is the cosmic horizon's Gibbons–Hawking temperature, 2.2 × 10⁻³⁰ K (Nº 3).The sky's line falls as the universe expands; the flat line is the black hole. Where they cross, the microwave sky becomes colder than the hole, wherever it is: this page's arithmetic, for a black hole that doesn't spin, with dark energy staying the same. Inside a galaxy, starlight keeps feeding it for far longer, and even far from galaxies leftover particles from the Big Bang feed it, for a time this page hasn't worked out. The dashed line at the bottom is the floor: the faint glow of the far horizon (chapter 8).

The wait barely depends on the mass: each factor of e in mass adds one e-fold, 17.4 billion years. For non-spinning holes, the microwave sky alone and constant dark energy (this page's arithmetic):

Black hole	Sky falls below it, years from now	e-folds
One solar mass (reference)	305 billion	17.6
2.5 M_☉, about the lightest known	321 billion	18.5
Gaia BH1, 9.62 M_☉	345 billion	19.9
30 M_☉	365 billion	21.0
Sagittarius A*	571 billion	32.9
M87*	699 billion	40.2
Holm 15A	730 billion	42.0

A forty-billion-fold range in mass, from one solar mass to Holm 15A, moves the date from 305 to 730 billion years, a factor of 2.4: the exponential cooling swamps the mass. No known hole waits a trillion years for the microwave sky alone.

The cold has a floor, though: with a positive cosmological constant the cosmic horizon, the distance beyond which nothing sent towards us now can ever arrive, has a temperature of its own, as Gary Gibbons and Hawking showed in 1977:³⁶

T_dS = ħH_∞ / (2πk_B) ≈ 2.2 × 10⁻³⁰ KThis page's arithmetic with 1/H_∞ = 17.41 billion years. The microwave sky falls to this floor after ln(2.72548/2.2 × 10⁻³⁰) ≈ 69 e-folds, roughly 1.2 trillion years from now, as Nº 3 puts it.

The floor arrives after every known hole's crossing and is about twelve orders of magnitude colder than the coldest hole listed. So if dark energy stays constant and positive, every known black hole will in time evaporate.

One refinement moves the date very little. The sky supplies photons but no bath of gravitons, while the hole emits both. Photon intake and output balance at equal temperatures, so on radiation alone the hole starts to lose mass on balance once its net intake of photons falls below what it gives off as gravitons:

(T_sky/T_H)⁴ = 1 + P_graviton/P_photon = 1 + 0.02/0.17 ⇒ T_sky ≈ 1.03 T_HPage's photon and graviton shares, with photon intake taken to scale as T_sky⁴. By then the sky's waves are tens of kilometres long (Chapter 3), and the intake rises faster than that, so 3% is the upper end: up to 0.03 e-folds, about half a billion years earlier. If the lightest neutrino is massless, the hole emits it too, and net loss can begin with the sky about 20% warmer, some 3 billion years earlier (this page's arithmetic).

Chapter 9What the date leaves out

The 300 billion years has a narrow meaning: the microwave sky alone, a hole that does not spin, constant dark energy. It does not depend on where the hole sits, but it is not the date on which any real black hole starts to evaporate. Six things stand between the two.

(a) Starlight, gas and dark matter. In the Sun's part of the Galaxy, away from any star, starlight carries about 0.54 eV per cubic centimetre, about twice the microwave background's energy density;³⁷ a Sun-mass hole here would take in 4.8 kW of it (this page's estimate). This supply does not thin out with expansion: a hole moving through its galaxy at a couple of hundred kilometres a second keeps meeting gas, starlight and any dark matter for as long as the galaxy holds together. Stars shine until star formation ends, after about 10¹⁴ years³⁸ (Nº 3).

(b) Relic neutrinos. At least two of the three neutrino masses are not zero, so most relic neutrinos from the early universe are slow today (Nº 4), and a hole captures slow particles from far beyond its own size: for speed v the cross-section is 16π(GM/c²)²(c/v)².³⁹ On this page's estimate they feed a Sun-mass hole a few megawatts today, against 2.3 kW from the microwave sky. How long that lasts depends on where the hole is; this page gives no date.

(c) A hole far from any galaxy. Accelerating expansion means a lone mass can only ever gather matter from within its zero-gravity radius, where its pull balances the cosmological constant's push:⁴⁰ (GM/H_∞²)^1/3, roughly 110 parsecs (about 360 light-years) for a Sun-mass hole (this page's arithmetic), and when that supply runs out is not computed here.

(d) Spin. A spinning hole is colder than a non-spinning one of the same mass but emits more power (Chapter 3), so it can begin to lose mass on balance while the sky is still warmer than the hole. That brings the date earlier, by up to a few tens of billions of years on this page's rough estimate from the factors in Page's second paper²³ (about 35 billion for a maximally spinning hole, an upper end, since real spins stop just short of the maximum). A proper number needs a greybody code such as BlackHawk,⁴¹ not run here. Because spin can only bring the date earlier, the date is "about 300 billion years", never "at least".

(e) Dark energy. The date assumes dark energy is a constant. The Dark Energy Spectroscopic Instrument (DESI) survey prefers dark energy that changes over time: in 2025 at 3.1 standard deviations (3.1σ) with the cosmic microwave background, and in its July 2026 analysis, a preprint, at 2.7 to 3.5σ, depending on the data combined.⁴² As of October 2026 both are below the 5σ discovery standard (Nº 1). If dark energy weakens, the wait grows, by orders of magnitude if acceleration stops. As an illustration, if from now on the universe expanded as one holding only matter does, a ∝ t^2/3, the same cooling would take (4.4 × 10⁷)^3/2 × 2/(3H₀), where 2/(3H₀) = 9.6 billion years: about 3 × 10²¹ years (this page's arithmetic). If it turns negative, space recollapses and the sky warms again: in one published model, by Luu, Qiu and Tye, the universe ends in a crunch about 20 billion years from now,⁴³ and no stellar black hole ever starts to evaporate (Nº 3).

(f) Primordial black holes. Black holes might also have formed in the early universe; some cosmologists consider such primordial holes a dark-matter candidate. None has been found. In the standard calculation most masses are ruled out as the whole of the dark matter,^44,45 but in an "asteroid-mass window", about 10¹⁴ to 10¹⁹ kg, they could still be all of it.⁴⁶ Holes there would be at 10⁴ to 10⁹ K, with Hawking emission exceeding every intake by many orders of magnitude, so if they exist they are evaporating now, wherever they sit (this page's arithmetic). Heavier ones, up to 4.5 × 10²² kg, would also be hotter than the microwave sky, but would drink starlight and relic neutrinos too; this page gives no mass at which the balance tips. Hence "known" in the title.

Part IIIHow it ends, and how we test it

Chapter 10Ten to the sixty-seven years

Once a hole at last gives out more than it takes in, the negative heat capacity of Chapter 5 drives a runaway: it loses mass, gets hotter, radiates faster. With the Hawking power α(M)c²/M² of Chapter 6,

dM/dt = −α(M)/M², τ = ∫₀^M M′² dM′/α(M′) ≈ M³/(3α)τ is the time to evaporate completely, ignoring any intake. α grows as the hole heats and emits new kinds of particle (Chapter 11), so this page integrates rather than using the approximation.

The lifetime grows as the cube of the mass; α depends on which particles the hole emits. In 1976 Page computed the emission from an uncharged, non-spinning hole in detail. With neutrinos assumed massless, he found that 81% of the power went into neutrinos, 17% into photons and 2% into gravitons.³⁰

Neutrinos have mass. Oscillation experiments fix the differences between the squares of the three masses: in the NuFIT 6.0 global fit, Δm²₂₁ = 7.49 × 10⁻⁵ eV² and |Δm²₃₁| = 2.51 × 10⁻³ eV²,⁴⁷ so in the normal ordering, in which the two masses closest together are the lightest, two of them weigh at least 8.7 and 50 meV; the lightest may be massless. KATRIN, which weighs neutrinos directly, puts their mass below 0.45 eV (90% confidence);⁴⁸ a July 2026 DESI analysis, a preprint, lowers the upper limit on the sum to 0.059 eV (95% confidence, assuming a cosmological constant), the smallest sum the normal ordering allows;⁴² and in September 2026 the JUNO experiment reported its first indication of the normal ordering, at about 2σ once accelerator data are added, also a preprint⁴⁹ (Nº 4). None of this can tell zero from about 10⁻¹¹ eV, the scale that matters here.

A hole emits a species efficiently once k_BT passes roughly its rest energy divided by 4.4 for a neutrino, and by up to 6.0 for other particles.²² A Sun-mass hole has k_BT = 5.3 × 10⁻¹² eV; the 8.7 meV neutrino switches on only below 5.4 × 10²¹ kg (this page's arithmetic). So a stellar hole emits photons, gravitons and at most the lightest neutrino, if that weighs less than about 10⁻¹¹ eV, in which case it carries about two-thirds of the power.

With the lightest neutrino at a nominal 1 meV, too heavy for the hole to emit, a Sun-mass hole takes 1.15 × 10⁶⁷ years to evaporate; with the lightest massless, 3.6 × 10⁶⁶ (this page's arithmetic): about 10⁶⁷ years either way. Page's original, which took the neutrino species then known as massless, was 2.2 × 10⁶⁶; the often-quoted 2 × 10⁶⁷ is the photon-only blackbody formula 5120πG²M³/(ħc⁴). Heavier holes last far longer:

Black hole	Lifetime, years	Lightest neutrino massless
One solar mass	1.15 × 10⁶⁷	3.6 × 10⁶⁶
10 M_☉	1.15 × 10⁷⁰	3.6 × 10⁶⁹
Sagittarius A*	9.2 × 10⁸⁶	2.9 × 10⁸⁶
M87*	3.2 × 10⁹⁶	1.0 × 10⁹⁶
Holm 15A, the heaviest with a dynamically measured mass²¹	7.4 × 10⁹⁸	2.3 × 10⁹⁸
10¹¹ M_☉ (no hole this heavy has a measured mass)	1.15 × 10¹⁰⁰	3.6 × 10⁹⁹

This page's arithmetic: the integral above with the piecewise α of Figures 1 and 4, normal ordering, a nominal 1 meV for the lightest neutrino in the middle column, non-spinning holes.

The heaviest dynamically measured hole lasts about 10⁹⁹ years and a hole of 10¹¹ Suns about 10¹⁰⁰, Nº 3's "around 10¹⁰⁰ years". Against these spans the wait of Chapter 8 is nothing: about 3 × 10⁻⁵⁶ of a Sun-mass hole's lifetime (this page's arithmetic).

Chapter 11The last second

As a hole shrinks and heats, it starts to emit heavier particles. In the standard calculation the three neutrinos switch on one by one between about 5 × 10²² and 10²¹ kg (the lightest at a nominal 1 meV), electrons and positrons below about 9 × 10¹³ kg, muons below 4.5 × 10¹¹ kg, and quarks and gluons once k_BT reaches a few hundred MeV, where the calculation is approximate; above about 50 GeV the hole emits every particle of the Standard Model, the established theory of the known particles. There Ukwatta and colleagues give α_SM = 8.0 × 10¹⁷ kg³/s, about a hundred times the photon-and-graviton value.²² Figure 4 runs the whole life as a countdown.

Figure 4⏱️ Try it

The countdown

Slide to the last second

Time left

Mass left

Temperature (k_BT)How hot

PowerGlowing at

Energy still to come

The standard (semiclassical) evaporation curve, Figure 1's lifetime line read the other way round: mass left against time left, both logarithmic, τ(M) = ∫₀^M M′² dM′/α(M′) with the same piecewise α as Figure 1 (photons and gravitons 7.2 × 10¹⁵ kg³/s; neutrinos, electrons and muons switching on in turn; quarks and gluons approximated between 0.3 and 1 GeV; the Standard Model rate 8.0 × 10¹⁷ kg³/s above 50 GeV, Ukwatta et al. 2016). Every hole, whatever its starting mass, runs down this one line; its mass sets only where it joins. Temperature T_H = 6.17 × 10⁻⁸ K × M_☉/M; power αc²/M²; energy still to come Mc², given as the time the Sun takes to emit it, at 3.828 × 10²⁶ W (IAU). Only confirmed Standard Model particles are counted. The line stops at the Planck mass, 2.18 × 10⁻⁸ kg, where the calculation is no longer trustworthy. Dvali and colleagues' unconfirmed memory-burden proposal would make the curve depart from Hawking's rate, at the latest once a hole has lost half its starting mass (see the text).Slide towards the end. These are the same lifetimes as the first figure, now as a countdown. A black hole loses mass faster and faster, gets hotter and hotter, and gives off more kinds of particles as it heats. Every black hole follows this same line; a bigger one just joins it further back. This is the standard calculation; what happens in the very last instant is unknown.

Near the end, with α at the Standard Model value, the mass left τ seconds before the end and the temperature then are

M ≈ 1.3 × 10⁶ kg × (τ/1 s)^1/3, k_BT ≈ 7.8 TeV × (τ/1 s)^−1/3Ukwatta et al. (2016), for k_BT above about 1 TeV; this page's arithmetic with α_SM gives 7.9 TeV, or 9.2 × 10¹⁶ K, at one second.

So in the standard calculation a hole begins its last second with 1.3 × 10⁶ kg, some 1,300 tonnes, at about 8 TeV (10¹⁷ K), and turns it all into 1.2 × 10²³ J: what the Sun emits in a third of a millisecond, or 29 million megatons of TNT (this page's arithmetic). It comes out as every kind of particle, gamma rays among them, but it is not a gamma-ray burst: those come from collapsing massive stars and merging neutron stars.

A Sun-mass hole will not reach this stage for about 10⁶⁷ years after it starts to give out more than it takes in. A primordial hole might be reaching it now: in the standard calculation one born with about 5 × 10¹¹ kg, the mass of a rocky asteroid some 700 metres across, would be finishing today (Page; Ukwatta and colleagues; this page's integral gives 5.1 × 10¹¹ kg).^30,22 Gamma-ray observatories have looked for such final bursts. The tightest limit we have found is from the LHAASO observatory in China, published in 2025 from data taken in 2021 to 2024: fewer than about 180 bursts per cubic parsec per year near us (99% confidence; the best of its limits for bursts lasting 10, 20 and 100 seconds).⁵⁰ H.E.S.S. had put the rate below about 2,000 in 2023 (95% confidence, 120-second bursts),⁵¹ and HAWC below 3,400 in 2020 (99% confidence, from 959 days of data).⁵² As of October 2026, no final burst has been seen.

The curve has to stop somewhere. Near the Planck mass, √(ħc/G) = 2.18 × 10⁻⁸ kg, about 22 micrograms (Nº 10),⁴ Hawking's calculation stops being trustworthy. Whether a hole then vanishes or leaves a remnant is unknown.

Two open questions hang over the end. The first is information: if the radiation is exactly thermal, the information about what fell in seems lost. In 1993 Page argued that the entanglement entropy of the radiation, a measure of how strongly it is correlated with what remains in the hole, should rise and then fall back to zero over the hole's life, as it must if the information comes out. That rise and fall is the Page curve.⁵³ Since 2019 semiclassical calculations have reproduced it, and how the information escapes is still being worked out.⁵⁴

The second is a proposal by Gia Dvali and colleagues that the information a hole stores resists its decay, a "memory burden" that would make evaporation depart drastically from Hawking's rate at the latest once a hole has lost half its mass.⁵⁵ It has been used since 2024 to reopen primordial-hole masses below about 10⁶ kg as dark-matter candidates.⁵⁶ It is unconfirmed. If it holds, the last second and the "finishing now" mass change; nothing about today's holes or their wait does. Hawking's rate remains the standard calculation.

Chapter 12Horizons for sound

If no known black hole can be seen to evaporate, can the physics be tested? In 1981 Bill Unruh pointed out that it can be imitated. Sound in a fluid that flows faster than sound obeys the same wave equation as a field near a black hole's horizon. Where the flow crosses the speed of sound, sound waves cannot travel back upstream, so the crossing is a horizon for sound, a "dumb hole", which by Hawking's argument should glow with thermal phonons, quanta of sound.⁵⁷

The usual picture is a river speeding up towards a waterfall, with fish that can swim only so fast: past the point where the current beats the fish, none can get back upstream. The analogy fails in instructive ways. The water is not spacetime, the quanta are phonons, and at very short wavelengths the fluid stops being smooth and becomes atoms.

That last point is a feature. Hawking's argument seems to lean on wavelengths shorter than any physics we know: traced back towards the horizon, the outgoing radiation's wavelength shrinks far below the Planck length. A fluid has a known short-wavelength cut-off, and calculations by Unruh and Ralf Schützhold showed that the emission survives changes in how the medium behaves there, within limits.⁵⁸ That 2005 result is a theoretical analysis, not an experiment.

In 2011 a team led by Silke Weinfurtner saw water waves at an analogue white-hole horizon, the time reverse of a black hole's, undergo stimulated emission, following the thermal relation Hawking's formula predicts.⁵⁹ The emission was triggered by waves sent in, not spontaneous.

The spontaneous effect came from Jeff Steinhauer's group in Haifa, which reported it in 2016.⁶⁰ In 2019 Muñoz de Nova, Golubkov, Kolobov and Steinhauer made a flowing Bose–Einstein condensate, about 8,000 rubidium-87 atoms behaving as one quantum fluid, with a horizon where the flow became supersonic. Over 7,400 repetitions the correlations between phonons on either side of the horizon were thermal, at the temperature predicted from the flow: about 0.35 nK, colder than a Sun-mass black hole.⁶¹ In 2021 Kolobov and colleagues showed that the emission is stationary, unchanging over time, observing it at six times during the horizon's life.⁶²

These are horizons for sound, not black holes. They test what happens to waves at a horizon, not gravity.

Chapter 13A horizon made of light

Light can make a horizon for light. An intense, short pulse in an optical fibre raises the glass's refractive index as it passes (the Kerr effect). A weaker probe beam travelling slightly faster than the pulse catches up with it from behind, slows in the region of higher index, and cannot get through if its speed falls to the pulse's own. For that light the pulse's trailing edge acts as the analogue of a white-hole horizon, which light from behind cannot enter, and its front as a black-hole horizon, which light inside the pulse cannot leave.

A slow lorry on a motorway gives the idea: a car that catches up cannot pass, slowed to the lorry's speed. The real thing differs in two ways: the light slows by only about 0.1%, and no event horizon in the gravitational sense exists.

Philbin, Leonhardt and colleagues demonstrated the first fibre horizon in 2008: probe light shifted to shorter, bluer wavelengths as it caught up with the pulse's trailing, white-hole edge.⁶³ In 2019 Drori, Leonhardt and colleagues reported stimulated Hawking radiation in optics.⁶⁴

In July 2026 Procopio, Aguero-Santacruz, Bermudez and Leonhardt reported in Nature that the Hawking-like light comes from a single direct process, not a cascade, and that the emission measurably pushes back on the pulse that makes the horizon.⁶ As in Hawking's picture, the light comes in pairs: each Hawking wave has a partner, which for a black hole falls in. Here the Hawking wave is the probe's light shifted to the red, and the partners, which the team measured, are ultraviolet light near 233 nm. Some of the pulse's own light reappears 0.5 to 1 nm further into the ultraviolet than that partner line, and in their analysis this backreaction signal grows with the square of the probe power, while the partner light grows in proportion to it. Across five of their six probe wavelengths (the fits leave out the 1,100 nm probe), both signals fell on the straight lines a thermal spectrum gives, and the slopes of the two lines, which set the temperature, "agree with 2% accuracy", so on the authors' reading the backreaction shows the same temperature as the Hawking light.

The authors list the limits. The radiation was stimulated and classical, so they could not measure entanglement; without a systematic study of probe power they cannot exclude effects beyond a simple frequency shift; and "alternative nonlinear processes producing similar effects cannot be explicitly ruled out".⁶ Figure 5 draws the two horizons and the square law.

Figure 5💡 Try it

A horizon made of light

Change the probe's speed

How much faster the probe is than the pulseHow much faster the probe light is than the flash

slower010: the pulse's whole index bumpwhole bumpas much as the flash can slow itthe limit15

Probe power (relative)How strong the probe light is (relative)

Horizons

Front horizon (black-hole analogue)Where the front horizon sits

Hawking-partner lineHawking partner light

Backreaction signalPush-back on the flash

Schematic, in the pulse's frame, speeds exaggerated about a thousand times. The pulse raises the fibre's refractive index by δn(x) = δn₀ exp(−x²/w²), δn₀ ≈ 10⁻³ from the Kerr effect (Procopio et al. 2026), on a group index of about 1.47 (Malitson 1965).⁶⁵ A probe faster than the pulse by Δ = n_u − n_p, the pulse's group index minus the probe's, moves at a speed ∝ Δ − δn(x); if 0 < Δ < δn₀ it meets horizons at x = ±w√ln(δn₀/Δ). The front acts as an analogue black-hole horizon, as the 2026 paper draws it; the trailing edge is a white-hole horizon, where the first fibre horizon was seen (Philbin et al. 2008). Dispersion is not modelled. Inset heights are schematic: in the paper's analysis the Hawking-partner line near 233 nm grows with probe power and the backreaction signal, 0.5–1 nm to its ultraviolet side, with its square; a systematic probe-power study is still to be done, the authors note. Single-photon set-up: Felipe-Elizarraras et al. (Nature Communications, 2026). In both experiments the radiation is stimulated; neither tests gravity.A cartoon: the slowing is exaggerated about a thousand times, so you can see it. A strong flash of light in a glass fibre slows other light passing through it. Light that's only a little faster than the flash can't get past its edges: for that light, the edges work like horizons. It's a horizon for light, not a black hole: no gravity is involved.

A horizon exists only while the probe's speed mismatch is smaller than the index bump; push the probe a little faster and the horizons merge at the pulse's peak and vanish. The square law matters because it is how the backreaction can be told apart from the Hawking-like light itself.

In May 2026 a team in Mexico, Felipe-Elizarraras, Cruz-Ramirez, Garay-Palmett, U'Ren and Bermudez, reported in Nature Communications that a single photon is enough to stimulate the analogue effect.⁶⁶ In a single 90 cm fibre the same laser pulses did two jobs: they made photon pairs, so that detecting one photon announces, or heralds, the other, and they formed a horizon for the heralded photon. The output still came one photon at a time: heralded g⁽²⁾(0) as low as 0.106 ± 0.018 (at 0.72 mW of laser power), well below 0.5. They did not measure the Hawking partner or any entanglement.

Such experiments test the physics of waves at horizons, including the worry about impossibly short wavelengths. They do not test gravity or quantum gravity, and they do not show a black hole losing mass. Philosophers of science disagree about how much more they can do. Radin Dardashti, Karim Thébault and Eric Winsberg argue that diverse analogue experiments can in principle confirm Hawking radiation for real black holes;⁶⁷ Karen Crowther, Niels Linnemann and Christian Wüthrich argue that they cannot, because the inference assumes the modelling framework holds in the target, the very thing in question.⁶⁸ Both sides agree on what the experiments measure.

Chapter 14The balance sheet

The chain runs like this. A black hole glows at a temperature that falls as its mass rises: about 62 nK for a Sun-mass hole, less for every real one. The microwave sky is 44 million times warmer, so every known hole takes in far more energy than it radiates: it has lost far less than a gram to Hawking radiation, gained far more from the sky, and not begun to evaporate. If dark energy stays constant, the cooling sky falls below a non-spinning Sun-mass hole in about 300 billion years, and below the heaviest known holes in under a trillion, while starlight, gas and relic neutrinos keep a real hole fed for longer. Only once a hole gives out more than it takes in does it start to evaporate, which takes about 10⁶⁷ years for one solar mass and ends, in the standard calculation, with a last second of about 10²³ J.

Five things the title does not mean:

Black holes never lose mass: Spinning holes can give up spin energy, and merging holes radiate mass as gravitational waves (Chapter 7). The title is about evaporation.
A start date in 300 billion years: That is when the microwave sky alone falls below a non-spinning Sun-mass hole, if dark energy stays constant. Other intakes push a real start later; spin brings it a little earlier (Chapters 8 and 9).
Every black hole there is: Primordial holes, if any exist in the asteroid-mass window, would be evaporating now (Chapter 9).
Hawking radiation has been seen: Only in laboratory analogues, never from a real black hole (Chapters 12 and 13).
A black hole in the lab: The experiments make horizons for sound and light. They test waves at horizons, not gravity (Chapters 12 and 13).

Where things stand, as of 25 October 2026: the two 2026 fibre experiments are the newest analogue results here, each with limits its authors state; DESI's preference for evolving dark energy, roughly 3 to 3.5σ, is short of discovery; no final burst from a primordial black hole has been seen; and no telescope has seen Hawking radiation.

Colder than the sky, and waiting.

Glossary

Hawking temperature: The temperature at which a black hole radiates, about 62 nK for one solar mass.
Event horizon, Schwarzschild radius: The boundary from inside which nothing escapes; for a non-spinning hole a sphere of radius 2GM/c².
Planck length, Planck mass: 1.6 × 10⁻³⁵ m and 2.2 × 10⁻⁸ kg, the scales at which quantum gravity is expected to matter.
Entropy (in units of k_B): For ordinary matter, the logarithm of the number of microscopic arrangements. For a black hole, a quarter of the horizon area in Planck units.
Heat capacity: The energy needed per kelvin of warming. Negative for a black hole, which heats up as it loses energy.
Cosmic microwave background: The cooled afterglow of the hot early universe, 2.72548 K today.
Relic neutrinos: Neutrinos left over from the first second after the Big Bang; most are slow today because neutrinos have mass.
e-fold: Growth by a factor of e = 2.718.
Dark energy, cosmological constant: Whatever drives the accelerating expansion; a cosmological constant is dark energy whose density never changes.
Gibbons–Hawking temperature: The temperature of the cosmic horizon in an accelerating universe: the sky's floor.
Greybody factor: The filtering of a hole's emission by its own gravity.
Kerr black hole: A spinning black hole, named after Roy Kerr, who found the spinning-hole solution in 1963; not the Kerr of the Kerr effect.
Primordial black hole: A hypothetical black hole formed in the early universe rather than from a star. None has been found.
Standard Model: The established theory of the known particles and of every force except gravity.
Temperature as energy: k_BT in electronvolts: 1 eV corresponds to about 11,600 K, 1 TeV to about 10¹⁶ K.
Analogue horizon: A place in a fluid or fibre that waves cannot cross back over: a "dumb hole" for sound; black-hole and white-hole horizons for light.
Hawking partner: The other member of each pair that Hawking's process makes: for a black hole it falls in; in a fibre it appears as light at its own wavelength, here in the ultraviolet.
Kerr effect: A change in a material's refractive index caused by a field, here the light's own intensity; named after John Kerr, who found the electric-field version in 1875; a different Kerr.
Stimulated and spontaneous emission: Emission triggered by light or waves sent in, or emission with nothing sent in, as from a real black hole.
Backreaction: The effect of an emission on whatever emits it: here, the pulse losing some of its light as it makes Hawking-like light.
Heralded g⁽²⁾(0): A test of whether light comes one photon at a time, each heralded by detecting its twin from the same pair; below 0.5 means single photons.

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