Physics · Gravitation and the universe · Year 4

Black holes

Which rays fall into a black hole, and why is the shadow bigger than the horizon?

The situation

In April 2019 the Event Horizon Telescope (EHT) team published, for the first time in history, an image of the surroundings of a black hole: a bright ring and, inside it, a dark “shadow”, around a black hole in the centre of the galaxy M87 that is about 6.5 billion times heavier than the Sun. The black hole itself emits nothing — what we see is hot gas and light rays bent by the curvature of space-time. Which rays fall into the black hole, which escape, and why is the shadow so much bigger than the horizon?

Topic

General relativity, black holes. Bending of light around a black hole

Key ideas

Schwarzschild radius
Bending of light
Photon sphere and ISCO

Formulas

r_{s} = \frac{2 G M}{c^{2}} \approx 2.95 km \cdot \frac{M}{M_{⊙}}

The Schwarzschild radius is the radius of the event horizon.

\frac{d^{2} u}{d φ^{2}} + u = \frac{3}{2} r_{s} u^{2}, u = \frac{1}{r}

The equation of a light ray (null geodesic). If the right-hand side is zero, the path is a straight line.

r_{p h} = \frac{3}{2} r_{s}, b_{c} = \frac{3 \sqrt{3}}{2} r_{s} \approx 2.598 r_{s}, α \approx \frac{2 r_{s}}{b} (b ≫ r_{s})

The photon sphere, the critical impact parameter and the deflection angle in a weak field.

\frac{d τ}{d t} = \sqrt{1 - \frac{r_{s}}{r}}

A stationary clock at distance $r$ runs slower than a distant one; at the horizon it $\to 0$ .

\frac{d^{2} u}{d φ^{2}} + u = \frac{r_{s} c^{2}}{2 h^{2}} + \frac{3}{2} r_{s} u^{2}, r_{I S C O} = 3 r_{s}, Δ φ \approx \frac{3 π r_{s}}{a (1 - e^{2})}

Orbit of a massive particle: the innermost stable orbit and the precession of the perihelion.

Theory

The Schwarzschild black hole

If a mass $M$ is squeezed into a volume smaller than the radius $r_{s} = 2 G M / c^{2}$ , an event horizon appears: no signal can get out from inside it. For the Sun $r_{s} \approx 2.95$ km, for the Earth ≈ 9 mm. Near the horizon time slows down: a stationary clock runs at the rate $\sqrt{1 - r_{s} / r}$ . A distant observer sees the clock of an object falling towards the horizon almost stop, and its light redden and fade.

Light rays and the shadow

The term $\frac{3}{2} r_{s} u^{2}$ in the ray equation is the correction of general relativity. A ray passing at a distance is deflected through an angle $α \approx 2 r_{s} / b$ (twice the Newtonian estimate — confirmed during the solar eclipse of 1919). At $r = 1.5 r_{s}$ is the photon sphere: a ray can travel round a circle, but this equilibrium is unstable. A ray with impact parameter $b < b_{c} \approx 2.598 r_{s}$ is certainly absorbed — so the radius of the black shadow is ≈ 2.6 times larger than the horizon. If you bring $b$ close to $b_{c}$ , the ray goes round the photon sphere several times; the stars behind appear in the form of rings and arcs (the Einstein ring).

Particle orbits, ISCO

The equation for a massive particle contains the same term $\frac{3}{2} r_{s} u^{2}$ . It turns the orbit forward on every revolution (perihelion precession: for Mercury this is 43″ per century). In a strong field the shift is very large — the orbit takes the shape of a “flower”. At $r < 3 r_{s}$ there is no stable circular orbit (ISCO): the particle falls in along a spiral. That is why the inner edge of an accretion disc lies at about $3 r_{s}$ .

The equivalence principle and experimental confirmations

The equivalence principle: inside a freely falling laboratory no experiment can detect gravity, and acceleration in a closed rocket is indistinguishable from gravity. Its basis is the equality of inertial and gravitational mass (all bodies fall with the same acceleration; see the “Law of universal gravitation” experiment for details). From this Einstein concluded that gravity is the curvature of space-time; this is the main idea of the theory of general relativity. It leads to the gravitational redshift: the frequency of a photon raised to a height $H$ decreases by $Δ f / f = g H / c^{2}$ (and the frequency of a photon that has climbed away from a black hole is multiplied by $\sqrt{1 - r_{s} / r}$ ). In 1960 R. Pound and G. Rebka, in a tower at Harvard University about 22.5 m high, measured this shift of $2.5 \cdot 10^{- 15}$ to an accuracy of ≈ 10 % (in 1964 Pound and Snider reached ≈ 1 %). This was made possible by the Mössbauer effect (R. Mössbauer, 1958): a $^{57} Fe$ nucleus in a crystal emits and absorbs a 14.4 keV γ-quantum without recoil, and the relative width of the line is only $\approx 3 \cdot 10^{- 13}$ . The shift was compensated, by the Doppler effect, by moving the source at a speed $v = g H / c$ . Mössbauer spectroscopy is still used today to determine the chemical state of iron in matter (for example, on the Mars rovers). Other confirmations: the bending of light (1919), the perihelion precession of Mercury, GPS clocks (the “Special relativity” experiment), gravitational waves (2015) and the EHT images.

Compact objects and gravitational lensing

When the fuel in a star's core runs out, the core contracts. If the remnant's mass is small, it becomes a white dwarf the size of the Earth (held up by electron pressure; the limit is ≈ 1.4 $M_{⊙}$ — the Chandrasekhar limit); if it is heavier, a neutron star with a radius of about 10–12 km (fast-spinning ones are pulsars). No pressure can stop a remnant heavier than about 2–3 $M_{⊙}$ — a black hole forms. Stellar evolution and the formation of heavy elements are covered in the “Physics of stars” experiment. Merging pairs of black holes and neutron stars are the source of gravitational waves: on 14 September 2015 LIGO recorded the merger of black holes of about 36 and 29 solar masses (“Interferometers and LIGO”). Gravitational lensing: if the source, the lens and the observer are in a line, the source appears as an Einstein ring with angular radius $θ_{E} = \sqrt{\frac{4 G M}{c^{2}} \frac{D_{L S}}{D_{L} D_{S}}}$ . For a stellar lens $θ_{E} \sim 1$ milliarcsecond — the images cannot be resolved, but when the lens passes in front of the source the source temporarily brightens: $A = \frac{u^{2} + 2}{u \sqrt{u^{2} + 4}}$ , where $u$ is the angle between the source and the lens (in units of $θ_{E}$ ). This is microlensing; a short extra peak on the curve reveals a planet around the lens star. Clusters of galaxies lens distant galaxies into arcs (dark matter and the expanding universe — see the “The expanding universe” experiment).

Model

A non-rotating, uncharged (Schwarzschild) black hole; the camera is a stationary observer. For every pixel the ray path is taken from an exact table of null geodesics (512 rays in $b$ , RK4), so the lensing, the shadow and the warped image of the disc are the result of calculation. The disc is thin; its colour is drawn approximately from a temperature profile, with the Doppler and gravitational redshift $g = \sqrt{1 - r_{s} / r} / γ (1 - β \cos α)$ . The diagram lines (rays, circles) are drawn in $r, φ$ coordinates and are not themselves lensed. The unit of time is $r_{s} / c$ ; at 1× 1 s = 10 $r_{s} / c$ .

Did you know?

The EHT image of M87* was published on 10 April 2019; the angular diameter of the bright ring is about 42 microarcseconds. The image of Sgr A* at the centre of our Galaxy (mass about 4 million solar masses) came out in 2022.
The 2020 Nobel Prize in Physics was awarded to Roger Penrose (for showing that black holes are an inevitable consequence of general relativity) and to Reinhard Genzel and Andrea Ghez (for the discovery of a supermassive object at the centre of the Galaxy).

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.