The physics

This level gives the formulas behind the sizes and energies in the level before. The constants are from JPL, and the results are worked out here unless a source is named.

The size of the horizon. For a mass $M$ that does not spin, the horizon is a sphere whose radius is

$$r_s = \frac{2 G M}{c^2}$$

This is the Schwarzschild radius, after Karl Schwarzschild, who found the solution of Einstein’s equations that it belongs to. Everything in it is a constant except the mass, so the radius grows in step with the mass. For the Sun it is 2.953 km.

Mass	$r_s$	Width, $2 r_s$
The Earth	8.87 mm	17.7 mm
The Sun	2.953 km	5.9 km
10 Suns	29.5 km	59 km
4.3 million Suns	12.7 million km	25.4 million km

The last is 18 times the width of the Sun, and its radius is 0.085 au, well inside the orbit of Mercury.

Newton’s escape speed, $\sqrt{2GM/r}$, reaches the speed of light at exactly this radius. That is a neat way to remember the formula, but it is not where the formula comes from. In Newton’s picture something thrown upwards more slowly than escape speed still rises for a while before it falls back. At a real horizon nothing moves outwards at all.

Not always dense. Divide a black hole’s mass by the volume of a sphere of radius $r_s$ and you get a mean density of

$$\bar\rho = \frac{3 c^6}{32 \pi G^3 M^2}$$

It falls with the square of the mass. This is bookkeeping, since the mass is not spread evenly inside, but it shows that a large black hole need not be dense at all.

Mass	Mean density inside the horizon
The Sun	$1.8 \times 10^{19}$ kg/m³
10 Suns	$1.8 \times 10^{17}$ kg/m³
4.3 million Suns	1.0 million kg/m³, a thousand times water
136 million Suns	1,000 kg/m³, the same as water
A billion Suns	18 kg/m³, less than a fiftieth of water

To make a black hole from one star, matter has to be squeezed to more than a hundred thousand million million kilograms in every cubic metre. To make one from a hundred million stars’ worth of gas, it needs only to reach the density of water.

Weighing the one in our galaxy. Kepler’s third law gives the mass at the centre of an orbit. With the period $P$ in years, the semi-major axis $a$ in astronomical units and the mass in Suns it is simply

$$M = \frac{a^3}{P^2}$$

The star S2 goes round the centre of the galaxy every 16 years. The GRAVITY team’s fit to its orbit and those of three other stars gives $4.30 \times 10^6$ Suns, to about a quarter of one per cent. A 16 year orbit round that mass has a semi-major axis of about 1,030 au.

The orbit is a long ellipse. At its closest S2 is 120 au from the centre, moving at 7,650 km/s, which is 2.6 per cent of the speed of light. That distance is about 1,400 times $r_s$. So 4.3 million Suns sit inside a space only four times the width of Neptune’s orbit, and nothing known but a black hole can be that heavy and that small. At this distance the team measured two effects that Newton’s laws do not allow: the star’s light is stretched as it climbs away, and the ellipse itself turns slowly, as Mercury’s does.

Mass turned into ripples. When two black holes merge, the final hole has less mass than the two that made it. The difference leaves as gravitational waves, with energy

$$E = \Delta m\, c^2$$

In the first merger detected, holes of 36 and 29 Suns made one of 62, and 3.0 Suns of mass were radiated. That is $5.4 \times 10^{47}$ joules, 4.6 per cent of the mass the two began with. The Sun, at its present brightness, would take 44 trillion years to give out as much.

Why falling gas shines so brightly. Gas in a circular orbit of radius $r$ has given up an energy of $GMm/2r$ for each mass $m$, compared with gas at rest far away. As a share of the gas’s own rest energy $mc^2$ that is

$$\frac{E}{m c^2} = \frac{G M}{2 r c^2} = \frac{r_s}{4 r}$$

Round a black hole that does not spin, circular orbits are stable only outside $3 r_s$. Gas that has spiralled in that far has given up about $1/12$ of its rest energy by this Newtonian reckoning, or 8 per cent. Observations of quasars point to about a tenth. For comparison, fusing hydrogen into helium releases 0.7 per cent. Gas falling towards a black hole turns far more of its mass into light than a star ever does, which is why the brightest steady sources of light known are black holes being fed.