Black Holes lessons

Escape speed

Throw a ball up and it falls back down. Throw it at 11.2 km/s and it escapes the Earth for good, so that is the Earth's escape speed.

Escape speed: \textcolor{#0d9488}{v} = \sqrt{\frac{2G\textcolor{#8b5cf6}{M}}{\textcolor{#3b82f6}{r}}}

The Earth is squeezed smaller with its mass fixed; a plot of escape speed against radius, both in powers of ten, climbs until it meets the speed of light at 9 mm.

Here M is the mass, r is the distance from the centre, and G is the gravitational constant. Squeeze the same mass into a smaller ball and the escape speed goes up.

Once the escape speed reaches the speed of light, nothing can get out. That is a black hole.

Practice: Squeeze a planet

A world is squeezed or made heavier; type its new escape speed, using the square root of M divided by r.

How massive stars die

A star is a tug of war. Gravity pulls its gas inwards, while the heat from fusion in its core pushes outwards.

Four starting masses, four life stories in a flow chain; a gauge shows the dead core's mass against the white dwarf and neutron star limits.

When the fuel runs out, the core collapses. If what is left is more than about 2 to 3 times the mass of the Sun, nothing can stop it shrinking.

Only the heaviest stars, roughly 20 or more times the Sun's mass, are likely to end this way. Our Sun will become a white dwarf instead.

Practice: The life of a star

Put the stages of a star's life and death in order, for a massive star, its final second, and a Sun-like star.

The event horizon

The event horizon is a sphere around the black hole. Inside it, even light moving outwards is dragged back in.

Schwarzschild radius: \textcolor{#f871b9}{r_s} = \frac{2G\textcolor{#8b5cf6}{M}}{c^2}

Its radius is the Schwarzschild radius, r_s. It is where the escape speed equals the speed of light, c, so it depends only on the mass.

The horizon of the Earth, the Sun, 10 Suns, Sgr A* and M87*, each drawn to scale beside a ruler, the Sun or Neptune's orbit.

1. r_s = \frac{2GM}{c^2} (for the Sun)
2. r_s = \frac{2 \times 6.67 \times 10^{-11} \times 1.99 \times 10^{30}}{\left(3.00 \times 10^8\right)^2} (put in G, the Sun's mass and c)
3. r_s = 2950\ \text{m} \approx 3\ \text{km} (about 3 km for every Sun of mass)

Far from the horizon, a black hole pulls like any other object of the same mass. Swap the Sun for a black hole of one Sun and the Earth would keep the same orbit.

Practice: Find the horizon

Find a black hole's horizon radius from its mass, or its mass from the radius, at about 3 km per Sun.

Anatomy of a black hole

A 3D tour of a feeding black hole: the horizon, the photon ring, the accretion disk and the jets.

Just outside the horizon, at 1.5r_s, is the photon sphere. Light there can orbit the black hole, but the slightest nudge sends it in or out.

Light aimed closer than about 2.6r_s from the centre is captured, so a black hole blocks a patch bigger than its horizon. That dark patch is called its shadow.

Many black holes are fed by an accretion disk of gas heated to millions of degrees. Feeding black holes are among the brightest things in the universe.

Practice: Escape or fall in

A ray is aimed past a black hole; decide whether it escapes or falls in, then watch the traced path.

The singularity

General relativity says everything inside the horizon is crushed into the centre, a place of infinite density called the singularity. In a spinning black hole it is predicted to be a ring rather than a point.

Infinite density almost certainly means the theory has broken down. Knowing what really happens needs a theory of quantum gravity, and nobody has one yet.

The horizon hides the centre, so no telescope can ever look at it. What is really at the heart of a black hole is still unknown.

Types of black holes

Black holes come in very different sizes. Astronomers sort them into types by mass.

Real black holes on a mass scale in powers of ten, from Cygnus X-1 to M87*, with the stellar, intermediate and supermassive bands and each horizon radius.

Stellar black holes are left by dead stars and weigh a few to about 100 Suns. Supermassive ones sit at the centres of galaxies and weigh millions to billions of Suns.

Intermediate black holes, from about 100 to 100,000 Suns, are rare and hard to find. A few strong cases are known, such as one in the star cluster Omega Centauri.

Practice: Classify the black hole

Read about a black hole's mass and where it was found, then choose stellar, intermediate or supermassive.

Gravitational lensing

Mass bends the path of light, so a black hole acts like a lens. Light from a galaxy far behind it bends round both sides on its way to us.

A background galaxy drifts behind a black hole; every pixel of the sky is traced back through the lens equation, giving arcs, two images and an Einstein ring. Drag the galaxy.

We see the galaxy moved, stretched into arcs or split in two. When it sits exactly behind the black hole, its light spreads into a circle called an Einstein ring.

Lensing lets astronomers weigh things they cannot see. In 2022 a lone black hole drifting through our galaxy was found this way, by how it brightened and shifted a star behind it.

Practice: Read the lens

Match each lensed picture to how well the galaxy lines up behind the black hole.

Time near a black hole

Gravity slows time. A clock hovering near a black hole ticks slower than a clock far away.

Time near a black hole: \textcolor{#0d9488}{t_{\text{near}}} = t_{\text{far}}\sqrt{1 - \frac{r_s}{\textcolor{#3b82f6}{r}}}

A probe hovers at a distance r from a black hole; its clock and a far clock tick side by side, and a plot of the clock rate against r falls to zero at the horizon.

Here r is the clock's distance from the centre. At r = 2r_s the near clock runs at about 71% of the far clock's rate, and closer in it slows towards zero.

Nobody near the black hole feels time slow down, because their own watch seems normal. The difference only shows when the two clocks are compared.

Practice: Slow clocks

A probe hovers at a given number of horizon radii; type how many minutes tick on its clock while an hour passes far away.

Spaghettification

Gravity gets stronger the closer you are. Fall in feet first and your feet are pulled harder than your head, so you are stretched.

A 2 m astronaut falls feet first into a 10 Sun black hole and into Sagittarius A*; the stretch, 2GML/r^3, is shown in g as they fall.

Near a small black hole the stretch turns deadly long before the horizon. This is nicknamed spaghettification.

At the horizon of a supermassive black hole the stretch is gentle. You would cross it without noticing, and only be pulled apart later, deep inside.

Practice: Myth or fact

True or false statements about stretching, slow clocks and how black holes pull.

Hawking radiation

In 1974 Stephen Hawking showed that quantum effects should make every black hole glow very faintly. This glow is called Hawking radiation.

Temperature of a black hole: \textcolor{#0d9488}{T} \approx 6 \times 10^{-8}\ \text{K} \times \frac{M_{\text{Sun}}}{\textcolor{#8b5cf6}{M}}

The bigger the black hole, the colder it is. A black hole of one Sun is only about 60 billionths of a degree above absolute zero.

Temperature against mass in powers of ten, from a mountain to M87*, with the 2.7 K glow from the Big Bang marked; a marker reads off the temperature and how long it would take to evaporate.

That is far colder than the 2.7 K glow left over from the Big Bang, so real black holes take in more energy than they give out. Hawking radiation has never been detected.

Practice: Rank by temperature

Order four black holes from hottest to coldest, using the rule that a bigger black hole is colder.

Pictures of black holes

The Event Horizon Telescope links radio dishes around the world into one telescope the size of the Earth. It can pick out details as small as a doughnut on the Moon.

The ring sizes of Sagittarius A* and M87* worked out from their mass and distance, side by side on the same sky scale, against what the Event Horizon Telescope measured.

In 2019 it showed the first picture of a black hole, M87*, 55 million light years away. In 2022 it showed Sagittarius A*, the black hole at the centre of our own galaxy.

Both show a bright ring round a dark shadow, the size general relativity predicts. Pictures from later years show the same ring, while its bright patches and magnetic field shift.

Gravitational waves

Two black holes orbiting each other shake space itself, sending out ripples called gravitational waves. The waves carry energy away, so the pair spiral together and merge.

Two black holes of 36 and 29 Suns spiral together while the wave they send out is drawn beneath, rising in pitch to a chirp, then ringing down.

On 14 September 2015 the two LIGO detectors in the USA caught the first one, from black holes of about 36 and 29 Suns. About 3 Suns of mass was turned into the energy of the waves.

LIGO, Virgo in Italy and KAGRA in Japan now listen together. By 2026 their catalogue held about 390 detections, almost all from merging black holes.

Practice: How we know they are there

Sort observations into the four ways we find black holes: X-rays from hot gas, stars in orbit, gravitational waves and pictures of the shadow.

The information paradox

Quantum physics says information is never truly destroyed, only scrambled. Burn a book and its information is still there, in principle, in the smoke and ash.

Hawking radiation seems to depend only on the black hole's mass, spin and charge, not on what fell in. So if a black hole evaporates completely, the information about what fell in seems to vanish.

This clash is the information paradox. Recent work suggests the information does get out in the radiation, but nobody knows how, and it is still unsolved.

Practice: Known or still unknown

Sort statements about black holes into what has been observed, what is predicted but not yet seen, and what is still unknown.