Lesson 2 · Gravity at its limit

Black Holes

Squeeze anything small enough and it becomes a black hole. Here is the number for each object, what happens to time near one, and how we know they are real.

If you could squeeze the Earth

9 millimetres

That is how small you would have to compress the whole planet, all 6 × 10²⁴ kg of it, before it became a black hole. About the size of a marble.

A black hole is not a vacuum cleaner and it is not a special kind of object. It is what any mass becomes if you compress it inside a certain radius. That radius is set by one tidy formula, and it is the first thing you will play with in this lesson.

Replace the Sun, right now, with a black hole of exactly the same mass. What happens to Earth's orbit?

Nothing. Gravity far from a mass depends only on that mass and your distance from it, not on what the mass is made of. Earth would keep orbiting exactly as before. We would freeze (the light would stop), but the orbit would not change. A black hole does not pull harder from far away. It is only close up that something extreme happens.

To leave Earth and never fall back, a rocket has to reach 11.2 km/s. That speed is called the escape velocity. Now imagine an object so dense that its escape velocity is the speed of light, 300,000 km/s, and nothing can go faster than light. Which object could that be?

Any object, if you are close enough to it. Escape velocity is v = √(2GM/r). Here M is the object's mass and r is your distance from its centre. Squeeze the same mass into a smaller space, so r gets smaller, and the escape speed goes up. Set v = c and solve for r, and you get the Schwarzschild radius, r_s = 2GM/c². Inside it, nothing that moves at or below light speed can escape.

Part 1

One formula, three centuries

In 1783 the English clergyman John Michell imagined a star so heavy that light could not escape it, using Newton's gravity. The formula he got happens to match the right answer. The real theory came in 1915–16: Karl Schwarzschild solved Einstein's equations of gravity (his rules for how mass bends space and time) for a single point of mass while serving in the German army on the Russian front, and died a few months later. The name “black hole” was made famous by the physicist John Wheeler in 1967.

In Einstein's picture, gravity is the curving of space and time. The edge of a black hole, the event horizon, is not a surface. There is nothing to touch. It is a one-way boundary: inside it, every possible path leads inward, as if “inward” had become “the future”. The singularity at the centre is less a place in space than a moment you cannot avoid. The equations break down there, which is a sign that our physics is missing something.

Part 2

Three sizes

Type	Mass	How it forms	Example
Stellar	About 3 to a few hundred M☉	Core collapse of a massive star, or a merger	Cygnus X-1 (about 21 M☉)
Intermediate	About 100 to 100,000 M☉	Unclear. Mergers in dense clusters?	Evidence only; one candidate in the cluster Omega Centauri
Supermassive	Millions to billions of M☉	Unknown	Sagittarius A* (4.3 million M☉), M87* (6.5 billion M☉)

The supermassive ones are a real puzzle. The James Webb Space Telescope keeps finding enormous black holes when the universe was less than 500 million years old, earlier than the standard growth models say is easy. Did they start huge? Is something about the early universe we do not understand? Nobody knows yet.

Part 3

How we know they are real

You cannot see a black hole directly, because it gives off no light of its own. We see what it does to everything around it.

Orbits. The star S2 swings round the centre of our galaxy every 16 years, coming within about 120 times the Earth–Sun distance, at about 2.5% of the speed of light. Only something with 4 million Suns' worth of mass in a tiny space can do that. Reinhard Genzel and Andrea Ghez shared the 2020 Nobel Prize for this work.
Glowing discs. Gas spiralling in is squeezed and heated to millions of degrees. Around supermassive holes this makes quasars, brighter than a thousand galaxies, visible across the universe.
The shadow. In 2019 the Event Horizon Telescope, a network of radio dishes across the Earth working as one planet-sized telescope, produced the first image of a black hole's shadow, in the galaxy M87. In 2022 it did the same for the black hole at the centre of our own Milky Way.
Gravitational waves. On 14 September 2015, the detector LIGO measured ripples in spacetime (space and time treated as one stretchy thing) from two black holes of about 36 and 29 Suns merging, 1.3 billion light-years away. They became one hole of about 62 Suns. The missing 3 Suns' worth of mass left as gravitational waves. For a fraction of a second, that was more power than all the stars in the observable universe combined. LIGO's arms changed length by less than a thousandth of the width of a proton.

Part 4

Which one is gentler to fall into?

Surprisingly, the big ones. Falling feet-first into a stellar black hole, gravity at your feet is so much stronger than at your head that you are stretched out long before the horizon. Physicists call this spaghettification. The stretching comes from a tidal force: gravity pulls harder on the near side of an object than on the far side, and the difference pulls it apart. A supermassive horizon is so large that the pull changes slowly across it, so you would cross without feeling anything special. The lab on the next tab calculates exactly how big the stretching force is.

What the outside world sees. Someone watching you fall in would see you slow down, redden and fade as you approach the horizon, never quite seeing you cross. You, meanwhile, would cross in a finite time on your own watch. Both descriptions are correct. They disagree about time, which is the next point.

Part 5

Time is not the same everywhere

Gravity slows clocks. A clock held at radius r from a mass runs at a rate of √(1 − r_s/r) compared with a clock far away. Near a black hole this effect becomes dramatic. It is also in your pocket: GPS satellites orbit high enough that their clocks run about 45 microseconds a day faster than clocks on the ground because of weaker gravity, and about 7 microseconds slower because of their speed. The net 38 microseconds a day would add up to position errors of roughly 10 km a day if engineers did not correct for it.

Part 6

The glow

In 1974 Stephen Hawking showed that the strange rules of the very small, applied at the horizon, make a black hole very faintly glow, which is called Hawking radiation. The lighter the hole, the hotter it is. A Sun-mass black hole is about 60 billionths of a degree above absolute zero (the lowest temperature possible), which is colder than the faint 2.7 K glow left over from the Big Bang that fills all of space, so it currently absorbs more than it emits. Only in the very distant future, once the universe has cooled below that, could it start to shrink. It would take around 10⁶⁷ years to evaporate. The question of what happens to the information that fell in, the information paradox, is still an open problem.

Squeeze it

Pick a mass. The lab works out its Schwarzschild radius, r_s = 2GM/c², the density you would have at that size, and the stretching force across a 2-metre body at the horizon.

Mass (log scale)

Horizon radius r_s

That is about

Average density inside

Stretch across 2 m at horizon

Two clocks

One clock is far from a black hole. The other hovers at a distance r, measured in units of the horizon radius. Move closer and watch the hovering clock fall behind.

Distance (r / r_s)

Hovering clock runs at

1 hour here equals

1 day here equals

Hovering this close is not realistic: you would need a rocket with almost unlimited thrust, because everything inside the horizon, and nearly everything on its edge, is being pulled in. The formula applies to something held in place.

Check yourself

Five questions.

Say it out loud

Explain to someone why a supermassive black hole is easier to fall into than a small one, and why that sounds the wrong way round. Use the words horizon, gravity and tidal.

Rabbit holes

The information paradoxThink of information as the full record of what something was: what it was made of and how it was arranged. The rules of the very small say that record can never be truly erased. But if a book falls into a black hole and the hole later evaporates into plain heat, the record looks gone. Which of the two is wrong? This is one of the biggest open questions in physics.

Penrose processRoger Penrose showed that a spinning black hole has a region outside the horizon where you can, in principle, extract energy from its rotation. He shared the 2020 Nobel Prize for proving that black holes are a real prediction of general relativity.

Primordial black holesCould tiny black holes have formed in the Big Bang itself? If so, some could make up part of dark matter. Searches are ongoing and nothing is confirmed.

The Holographic principleA black hole's entropy (a measure of how much hidden detail it contains) is proportional to the area of its horizon, not its volume. This led some physicists to suspect that our three-dimensional world is, in some deep way, encoded on a two-dimensional surface.