Black holes, simply

A black hole is the ultimate black box: an interface through which things go in and nothing comes out. Anyone who spends time with programs whose insides they cannot see will find the idea familiar. Yet black holes are also among the simplest objects in physics. Once one has settled down, three numbers describe it completely: its mass, its spin and its electric charge. Classically, everything else about what fell in is forgotten. This post builds the idea from one school-level formula.

Throw a ball upward and gravity brings it back. Throw it harder and it goes higher. There is a speed beyond which it never returns, the escape velocity: v = sqrt(2GM/R), where M is the mass of the planet or star, R the distance from its centre and G the gravitational constant. For Earth's surface it is 11.2 km/s, for the Moon 2.4 km/s, and for the Sun's surface 618 km/s. Light travels at 299,792 km/s.

In 1783 John Michell, and in 1796 Pierre-Simon Laplace, asked the obvious question: what if a star were so massive and compact that its escape velocity exceeded the speed of light? Light would never get out. Set v equal to c and solve for R:

This is the Schwarzschild radius. Karl Schwarzschild found the same number in 1916 as an exact solution of Einstein's equations of general relativity, only weeks after they were published, while serving in the German army in the First World War. The Newtonian route gets the number right partly by luck; its real meaning comes from general relativity.

Python does the arithmetic:

So the Sun would have to be squeezed into a ball of 2.95 km radius, and Earth into one of 8.9 mm, the size of a pea. A good rule of thumb: about 3 km per solar mass. For you, it comes to 10^-25 m, billions of times smaller than a proton.

The radius grows with the mass, but the volume grows with its cube, so bigger black holes are less dense. The Sun squeezed into 2.95 km would have 1.8 x 10^19 kg/m3, some 80 times the density of an atomic nucleus. The average density inside M87*'s horizon is only about 0.4 kg/m3, a third of the density of air.

In general relativity, mass tells space-time how to curve, and curved space-time tells things how to move. The old picture is a rubber sheet with balls on it. It is a cartoon, but a useful one:

A planet makes a shallow dent, a star a deeper one. A black hole is a dent so deep that, within the event horizon at the Schwarzschild radius, every possible path leads inward. The horizon is not a surface made of matter. It is a boundary: events inside cannot send any signal to the outside. At its centre, classical general relativity predicts a singularity of infinite density. Most physicists read that as the theory breaking down and expect a quantum theory of gravity to replace it.

Two myths deserve correcting. First, black holes do not "suck". Replace the Sun with a black hole of the same mass and Earth keeps orbiting exactly as before, only in the dark and the cold. Gravity is extreme only near the horizon. Second, falling in is not equally deadly everywhere. Across your own height (2 m), the difference in pull at the horizon of a 10-solar-mass hole would be about 20 million times Earth's gravity: you would be torn apart before arriving. At Sagittarius A* it would be about 1/10,000 of Earth's gravity, and at M87* negligible, so you would cross without noticing, with no way back. Seen from outside, you would appear to slow down and fade, never quite reaching the horizon.

A massive star ends its life when its core can no longer hold itself up. If the remaining core exceeds roughly two to three solar masses, no known force stops the collapse and a stellar black hole forms. The Sun will not make one; it will end as a white dwarf, billions of years from now. The progenitor stars must be much heavier, very roughly 20 solar masses or more (the exact threshold is uncertain). Stellar black holes weigh from a few to some tens of solar masses; in 2020, gravitational waves revealed one of about 140 solar masses, described as the first clear "intermediate" case. At the centres of most large galaxies sit supermassive black holes, millions to billions of solar masses. How they grew so big so early is an open problem, since the James Webb telescope keeps finding massive ones in the young universe.

The best evidence for the one at the heart of our Milky Way came from tracking a star called S2. It circles an invisible point in 16 years and passes it at about 7,650 km/s, 2.5 percent of the speed of light. Kepler's laws turn that orbit into a mass of about 4 million Suns packed within the 120 AU of S2's closest approach. Reinhard Genzel and Andrea Ghez shared the 2020 Nobel Prize in Physics for this, together with Roger Penrose, who showed that black hole formation is a robust prediction of general relativity.

On 14 September 2015 the two LIGO detectors in the United States, each with arms 4 km long, recorded a signal later named GW150914. Two black holes of about 36 and 29 solar masses had spiralled together and merged, 1.3 billion light-years away. About 3 solar masses of mass were turned into gravitational waves in a fraction of a second. By the time they reached Earth they changed the length of each arm by about 4 x 10^-18 m, roughly 1/200 of a proton's radius.

The signal climbed from 35 to 250 Hz in 0.2 seconds, which in a loudspeaker sounds like a chirp. Announced on 11 February 2016, it was the first direct detection of gravitational waves and the first observation of a binary black hole. The 2017 Nobel Prize went to Rainer Weiss, Barry Barish and Kip Thorne. Since then the detectors have logged well over a hundred merger candidates, and the catalogue keeps growing.

The Event Horizon Telescope links radio dishes across the Earth into one virtual telescope observing at 1.3 mm wavelength. On 10 April 2019 it released the first image of a black hole, M87*, 55 million light-years away. On 12 May 2022 it followed with Sagittarius A*, 26,000 light-years away. Each shows a bright, lopsided ring around a dark centre. The rings measure about 42 and 52 micro-arcseconds across, the angle of an orange on the Moon (43 micro-arcseconds). The two look alike in size even though M87* is 1,600 times heavier, because it is also about 2,000 times farther away.

The ring is hot gas whose light is bent by gravity; the dark centre is the shadow, about 2.6 times the horizon's radius, because light passing too close is captured. So the picture is not a photo of the horizon itself. What matters is that the measured size agrees with general relativity's prediction for the known mass.

In 1974 Stephen Hawking showed that quantum effects near a horizon should make a black hole glow very faintly, as if it had a temperature, T = hbar c^3 / (8 pi G M k). For a solar-mass hole that is 6 x 10^-8 K, far colder than the 2.7 K microwave background, so every real black hole today absorbs more than it emits. A solar-mass hole would need some 2 x 10^67 years to evaporate, against a universe 1.4 x 10^10 years old. Only tiny hypothetical relics, below about 10^11 to 10^12 kg, could be finishing their evaporation now. None has been seen.

So Hawking radiation is a prediction, trusted because it links gravity, quantum theory and thermodynamics so neatly, but never observed. It also leads to the information paradox: if a black hole evaporates, what happens to what fell in? That is still unresolved.

Compute the Schwarzschild radius of a mountain with the snippet above. The same general relativity runs the whole cosmos in How we know the universe is expanding. Black holes in our galaxy can also be found by the wobble they cause in a companion star, as in Exoplanets: how we find other worlds, and A tour of the solar system gives a feel for what "128 AU" means.