Black holes

Builds on: Sections 10.1, 3.6, 8.2 and S4
Reading time: 45 min
Exercises: 3
Olympiad problems: 2

In the constellation Sagittarius, just above the spout of the Teapot, lies the centre of our Galaxy. Dust hides it from ordinary telescopes, but in infrared light a cluster of stars can be seen there, and since the 1990s astronomers have followed their motions year after year. One of them, called S2, goes round an empty-looking point once every 16 years on a long ellipse, and at its closest it passes at 2.6% of the speed of light. Whatever it orbits has more than four million solar masses packed into a space smaller than the Solar System. No cluster of stars or dead stars could stay packed so tightly for long, and the only known object that fits is a black hole. In this section we see what a black hole is, how big it is, and what happens to light that passes close to one.

10.2.1The Schwarzschild radius

Before you readMake a guess first

If the Sun could be squeezed into a black hole, keeping all its mass, about how big would it be?

In Section 10.1 we found that the escape speed from a neutron star is more than half the speed of light. A body of the same mass squeezed only a few times smaller would have an escape speed above the speed of light, and then not even light could leave it. John Michell saw this in 1783 with Newton's theory alone. Setting the escape speed of (S.21) equal to $c$ gives $2 G M / R = c^{2}$ . Newton's theory cannot be trusted for light, and the right answer comes from Einstein's general relativity, which by a happy chance gives the same radius. Karl Schwarzschild found it in 1916.

ResultThe Schwarzschild radius(10.4)

R_{s} = \frac{2 G M}{c^{2}} = 2.95 km \times \frac{M}{M_{⊙}}

$R_{s}$	the Schwarzschild radius, the radius of the event horizon
$M$	the mass of the black hole
$G, c$	the gravitational constant and the speed of light

This is the radius of a black hole that does not spin and has no electric charge. The radius grows in proportion to the mass.

A body squeezed inside its Schwarzschild radius becomes a black hole. The sphere of radius $R_{s}$ round it is the event horizon. The horizon has no material surface, and something falling through it notices nothing special there, apart from the tides, which are gentle at the horizon of a large black hole and tear a body apart before the horizon of a small one. The horizon is a boundary of no return, since inside it every path, even the path of light, leads further in. A dead core heavier than the greatest mass of a neutron star, about 2 to 3 solar masses, has nothing left to hold it up and becomes a black hole.

Quick checkA stellar black hole

What is the Schwarzschild radius of a black hole of 10 solar masses, in kilometres?

Example 10.3Weighing the black hole at the centre of the Galaxy

The star S2 goes round Sagittarius A*, the black hole at the centre of the Galaxy, with a period of 16.05 years. The semi-major axis of its orbit is 0.1251″ seen from 8250 pc, and the eccentricity is 0.885. Find the mass of the black hole, its Schwarzschild radius, and how close S2 comes to it.

Solution

At 8250 pc an angle of 1″ is 8250 au across, so the semi-major axis is $0.1251 \times 8250 = 1032$ au. The mass of S2 is tiny next to the black hole's, so by Kepler's third law in solar units (8.1),

M = \frac{1032^{3}}{{16.05}^{2}} = 4.27 \times 10^{6} M_{⊙} .

By (10.4), $R_{s} = 2.95 \times 4.27 \times 10^{6} = 1.26 \times 10^{7}$ km, which is 0.084 au. At its closest, S2 is $1032 \times (1 - 0.885) = 119$ au from the black hole.

AnswerSagittarius A* has 4.27 million solar masses and a Schwarzschild radius of 12.6 million km, smaller than the orbit of Mercury. S2 comes within 119 au, about 1410 Schwarzschild radii, where the vis-viva equation (3.5) gives it a speed of 7750 km/s.

A black hole has a mass and a radius, so it has a mean density, its mass divided by $\frac{4}{3} π R_{s}^{3}$ . Since $R_{s}$ grows as $M$ , this density falls as $1 / M^{2}$ . A black hole of one solar mass has a mean density of $1.84 \times 10^{19}$ kg/m³, higher than a neutron star's. Sagittarius A* has a mean density of $1.0 \times 10^{6}$ kg/m³, and one of $10^{8}$ solar masses only 1840 kg/m³, less than twice the density of water. What makes a black hole is a mass squeezed inside its own Schwarzschild radius, and for a large mass that can happen at a modest density.

10.2.2Light near a black hole

Light passing a black hole is pulled towards it and bent. A ray that passes far from the hole, at a distance $b$ from its centre, is bent by $2 R_{s} / b = 4 G M / (c^{2} b)$ radians, which is the subject of Section 10.6. Close in the bending becomes much stronger. At 1.5 Schwarzschild radii the pull is just enough to bend light into a circle, so light could go round the hole there. This sphere is the photon sphere. The orbit is unstable, and light on it soon either falls in or escapes.

A ray coming from far away is described by its impact parameter $b$ , the distance by which it would miss the centre if it went straight on. Rays with $b$ less than $\frac{3 \sqrt{3}}{2} R_{s} = 2.60 R_{s}$ come inside the photon sphere and fall in. Rays with larger $b$ swing round and leave. Seen against a bright background, a black hole is therefore a dark disc of radius $2.60 R_{s}$ , called its shadow, more than twice the size of its horizon.

Figure 10.2Light passing a black hole

Impact parameter

b

The ray
Closest approach
Bending

The figure shows rays of light coming in from the left past a black hole H, drawn as the hatched disc of its horizon, with lengths in Schwarzschild radii. The dashed circle P is the photon sphere at

1.5 R_{s}

. The heavy ray L has the chosen impact parameter

b

, and the thin rays show a fan of others. Rays between the dotted edges of the shadow, at

b = 2.60 R_{s}

, fall in. Running the paths backwards shows why this makes a shadow. A distant observer on the right who looks towards the hole within

2.60 R_{s}

of its centre is looking along a path that ends on the horizon, so no light reaches the observer from there. Just outside the edges a ray can wind once or more round the hole before it leaves, and far out the bending approaches

2 R_{s} / b

radians. The figure can be set to , , or .

The bending grows quickly as a ray passes closer. A ray with $b = 10 R_{s}$ is bent by 13.5°, where the rule $2 R_{s} / b$ for distant rays gives 11.5°, and in the figure the heavy ray at $b = 4 R_{s}$ is bent by about 49°.

In 2019 the Event Horizon Telescope, a network of radio dishes spread over Earth, made the first image of a black hole's shadow, for the black hole of 6.5 billion solar masses at the centre of the galaxy M87. It shows a dark disc inside a ring of light from hot gas that swirls round it. Close to a black hole circular orbits become unstable, and for a black hole that does not spin the innermost stable circular orbit is at 3 Schwarzschild radii, and this sets the inner edge of the discs of gas that we meet in Section 10.5.

10.2.3Gravitational redshift

Light that climbs away from a mass loses energy, just as a ball thrown upwards does. Its speed cannot change, so its frequency falls and its wavelength grows. This is the gravitational redshift. In the same way a clock close to a mass runs slow when it is compared with one far away, because the light that carries each tick arrives stretched.

ResultGravitational redshift(10.5)

\begin{aligned} 1 + z & = \frac{λ_{obs}}{λ_{emit}} = \frac{1}{\sqrt{1 - R_{s} / r}} \\ z & \approx \frac{G M}{r c^{2}} for r ≫ R_{s} \end{aligned}

$z$	the redshift of light that leaves a source at rest at radius $r$ and is seen far away
$λ_{emit}, λ_{obs}$	the wavelengths where the light leaves and where it is seen
$r$	the distance from the centre where the light leaves, outside the body
$R_{s}$	the Schwarzschild radius of the body

The source must be at rest, since a moving source adds a Doppler shift. The same factor

\sqrt{1 - R_{s} / r}

gives the rate of a clock at rest at radius

r

compared with one far away. At

3 R_{s}

such a clock runs at 0.82 of the rate of a distant one and light is redshifted by

z = 0.22

, and at the horizon the redshift becomes infinite.

Example 10.4The redshift of Sirius Bfinish it

The spectral lines of a white dwarf are shifted by its gravity. Sirius B has 1.02 solar masses and a radius of 0.0081 solar radii. Find its gravitational redshift and the speed that a Doppler shift of the same size would need.

The start

The radius is far larger than the Schwarzschild radius, so $z \approx G M / (r c^{2})$ from (10.5). With $M = 1.02 \times 1.988 \times 10^{30}$ kg and $r = 0.0081 \times 6.957 \times 10^{8}$ m, this gives $z = 2.67 \times 10^{- 4}$ .

Your stepNow turn the redshift into the speed

v = c z

(S.30).

km/s

Common errorsSection 10.2

A black hole is not a cosmic vacuum cleaner. Far outside its horizon it pulls exactly like any other body of the same mass, so if the Sun became a black hole, Earth would keep its orbit.
The event horizon is the radius inside which nothing can get out, and there is no solid surface there.
The shadow of a black hole is larger than its horizon, with a radius of $2.60 R_{s}$ , because light passing inside the photon sphere is captured too.
Gravitational redshift happens even when the source does not move, because the light is stretched by climbing out of the gravity of the body, so it should not be read as a Doppler shift.

ExercisesSection 10.23 questions

10.2.1
A neutron star has 1.4 solar masses and a radius of 12 km. What is the gravitational redshift $z$ of light from its surface? Use the full form of the result.
10.2.2
The Sun is replaced by a black hole of the same mass. What happens to Earth's orbit?
10.2.3
Using the mass found in Example 10.3, a Schwarzschild radius of $1.26 \times 10^{10}$ m at a distance of 8250 pc, find the angular diameter of the shadow of Sagittarius A*, in microarcseconds.
microarcseconds

Olympiad problemsPast IOAA problems, restated2 problems

2009The density of a supermassive black hole10 points

Adapted from IOAA 2009 (Tehran), theory problem 1. We have restated it in our own words and kept the original data. The official solution uses rounder constants and gives 1850 kg/m³. The solution was last checked on 23 October 2026.

Find the mean density, inside its Schwarzschild radius, of a supermassive black hole of $10^{8}$ solar masses.

I solved it without the solution

AnswerThe mean density is 1840 kg/m³, less than twice that of water. It falls as $1 / M^{2}$ .

2019A telescope to see a black hole10 points

Adapted from IOAA 2019 (Keszthely), theory problem 3. We have restated it in our own words and kept the original data. The problem takes the radius of the shadow to be 3 Schwarzschild radii, a little more than the true 2.60, and the mass of Sagittarius A* to be 3.6 million solar masses, less than the 4.3 million measured since. We keep its values. The solution was last checked on 23 October 2026.

The Event Horizon Telescope imaged the shadow of the black hole at the centre of the galaxy M87 at a wavelength of 1.3 mm. Take the radius of the shadow to be 3 Schwarzschild radii.

(a) How large must a telescope be to resolve the shadow of a black hole of mass $M$ at distance $d$ ? (6 points)

(b) Give this size in Earth radii for (i) the black hole of M87, of $6.5 \times 10^{9}$ solar masses at $5.5 \times 10^{7}$ light-years, and (ii) Sagittarius A*, of $3.6 \times 10^{6}$ solar masses at 8.3 kpc. (1 point each)

(c) Which technique makes such a telescope possible: (A) gravitational lensing by dark matter, (B) interferometry with an array of radio telescopes, (C) slowing photons in a dense medium, (D) correcting the distortion of the incoming wavefront, or (E) focusing neutrinos with strong electromagnetic fields? (2 points)

I solved it without the solution

Answer(a) $D \geq 1.22 λ c^{2} d / (12 G M)$ . (b) (i) 1.12 Earth radii, (ii) 1.0 Earth radius. (c) (B), interferometry with an array of radio telescopes.