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12.3 — Black Holes
Chapter 6.9 derived the Schwarzschild solution and computed the event horizon, the photon sphere, the innermost stable orbit and the tidal forces. This chapter is about black holes as objects in the universe — how they form, how many there are, what they do to the matter around them, why they evaporate, and what was in the two photographs.
The zoo
| Class | Mass | Origin | Example |
|---|---|---|---|
| Stellar | 3–100 M_\odot | Core collapse | Cygnus X-1 (21 M_\odot) |
| Intermediate | 10^2–10^5\,M_\odot | Uncertain | Poorly established |
| Supermassive | 10^6–10^{10}M_\odot | Uncertain | Sgr A* (4.3\times10^{6}) |
| Primordial | Any | Early universe | Hypothetical |
Stellar-mass black holes form as Chapter 12.2 described, and the Milky Way is estimated to contain about 10^{8} of them.
Supermassive black holes sit at the centre of essentially every large galaxy. How they got so big so fast is an open problem — quasars with billion-solar-mass black holes are observed at redshift 7, when the universe was under 800 million years old, and the Eddington limit (Chapter 12.1) makes growing that fast by ordinary accretion very difficult.
The intermediate class is the gap. Very few confirmed examples, which is itself informative about how supermassive ones form.
The M–sigma relation
One of the most striking correlations in astrophysics.
M_{\text{BH}} \approx 2\times10^{8}M_\odot\left(\frac{\sigma}{200\ \text{km/s}}\right)^{4}
where \sigma is the velocity dispersion of stars in the galaxy's bulge.
The black hole's mass is about 0.1 % of the bulge mass, across four orders of magnitude.
Why this is remarkable: the black hole's gravitational influence extends over about 10 parsecs, and the bulge is thousands of parsecs across. The black hole cannot be directly controlling the bulge and the bulge cannot be directly controlling the black hole.
The favoured explanation is feedback. As the black hole accretes it releases enormous energy, which heats and expels gas from the galaxy, shutting off both the accretion and the star formation. The two grow together and stop together.
This is why galaxy evolution and black hole growth are now studied as one subject.
Accretion
Matter falling towards a black hole cannot fall straight in. It has angular momentum, so it settles into a disc and spirals inwards, losing angular momentum through viscosity.
And the disc gets extremely hot. Gravitational potential energy is converted to heat, and the temperature at radius r scales roughly as r^{-3/4}.
The efficiency was computed in Chapter 6.9: matter spiralling to the innermost stable circular orbit releases
\eta = 1-\sqrt{\frac{8}{9}} = 5.7\ \%
of its rest mass for a non-rotating hole, and up to 42 % for a maximally rotating one.
Compare with fusion's 0.7 % (Chapter 6.4).
\boxed{\text{Accretion onto a black hole is the most efficient energy source in the universe.}}
Quasars
A quasar is a supermassive black hole accreting rapidly.
Typical luminosity: 10^{40} W — a thousand times an entire galaxy, from a region the size of the solar system.
How much matter does that take?
\dot{m} = \frac{L}{\eta c^2} = \frac{10^{40}}{(0.1)(9\times10^{16})} = 1.1\times10^{24}\ \text{kg/s}
= 3.5\times10^{31}\ \text{kg/year} = 18\ M_\odot\ \text{per year}
Eighteen solar masses a year, which is a substantial fraction of a galaxy's entire star formation rate.
The Eddington limit (Chapter 12.1) caps it. Setting L = L_{\text{Edd}}:
\dot{M}_{\text{Edd}} = \frac{4\pi GMm_p}{\eta\sigma_Tc} \quad\Longrightarrow\quad \frac{dM}{dt} = \frac{M}{t_{\text{Sal}}}
with the Salpeter time:
t_{\text{Sal}} = \frac{\eta\sigma_Tc}{4\pi Gm_p} \approx 4.5\times10^{7}\ \text{years}
So the mass grows exponentially with an e-folding time of 45 million years. Starting from a 100-solar-mass seed:
t = t_{\text{Sal}}\ln\frac{10^{9}}{100} = (4.5\times10^{7})\ln(10^{7}) = (4.5\times10^{7})(16.1) = 7.2\times10^{8}\ \text{years}
720 million years to reach a billion solar masses — which just barely works for the earliest quasars, and only if accretion runs at the Eddington limit continuously from a seed that already exists. This is the tension driving current work on direct-collapse black hole formation.
Jets
Many accreting black holes launch relativistic jets — collimated beams of plasma at over 0.99c, sometimes extending millions of light years.
The Blandford–Znajek mechanism is the leading explanation: magnetic field lines threading a rotating black hole are twisted by frame dragging (Chapter 6.9), and the twisting extracts rotational energy electromagnetically. This is the Penrose process in magnetic clothing, and it can extract up to 29 % of a maximally rotating hole's mass–energy.
M87's jet is 5000 light years long and was photographed in 1918, decades before anyone had a theory for it.
The images
M87*, released 10 April 2019. Mass 6.5\times10^{9}M_\odot, distance 55 million light years, shadow diameter 42 microarcseconds.
Sagittarius A*, released 12 May 2022. Mass 4.3\times10^{6}M_\odot, distance 27,000 light years, shadow diameter 52 microarcseconds.
Chapter 11.9 explained why they are similar in apparent size despite a factor of 1500 in mass and 2000 in distance, and why Sgr A* was harder — it varies on minute timescales, so the image smears during the observation.
What the image shows:
The dark centre is the shadow, not the horizon. It is \sqrt{27}/2 \approx 2.6 Schwarzschild radii in radius — about five times the horizon's area — because light that would have missed a Newtonian object is bent in (Chapter 6.9).
The bright ring is the photon sphere region, where light orbits before escaping.
The asymmetry is Doppler beaming. The side of the disc rotating towards us is brighter by a large factor, because relativistic beaming concentrates the emission forwards. From the asymmetry, the rotation direction is determined.
And the shadow size measures the mass, independently of any orbital dynamics, giving a check on the value from stellar orbits. They agree.
Hawking radiation
Black holes are not black.
Hawking's 1974 result came from applying quantum field theory in curved spacetime — not a full quantum gravity theory, but quantum fields on a fixed black hole background.
The argument
The popular version: virtual pairs form near the horizon, one falls in and the other escapes, and the escaping one is real radiation.
This picture is a useful shorthand and it is not the derivation, and Hawking himself was uneasy with it. Three problems with it: it does not explain why the escaping particle carries positive energy and the infalling one negative, it gets the emission spectrum wrong if taken literally, and most of the radiation is actually produced well outside the horizon.
The real argument rests on a fact from Chapter 7.9: what counts as "no particles" depends on the observer.
The vacuum is defined by which modes count as positive frequency, and that depends on the observer's time coordinate. An accelerating observer in flat space sees a thermal bath — the Unruh effect. A black hole horizon does the same thing.
More precisely: a field mode that is purely positive-frequency in the far past becomes a mixture of positive and negative frequency after propagating through the collapsing star's spacetime. That mixing is exactly what particle creation means, and the resulting spectrum is thermal.
The temperature:
\boxed{T_H = \frac{\hbar c^3}{8\pi GMk_B}}
Note what is in it: \hbar (quantum), c (relativity), G (gravity), k_B (thermodynamics). All four fundamental constants in one formula — the only equation in physics where they all appear.
Numbers
For a solar-mass black hole:
T_H = \frac{(1.055\times10^{-34})(2.7\times10^{25})}{8\pi(6.674\times10^{-11})(1.989\times10^{30})(1.381\times10^{-23})}
= \frac{2.849\times10^{-9}}{4.605\times10^{-2}} = 6.2\times10^{-8}\ \text{K}
Sixty nanokelvin.
The cosmic microwave background is at 2.725 K (Chapter 12.6), so a stellar black hole absorbs about 10^{8} times more than it emits. It is growing, not evaporating, and will be until the universe has expanded enough for the CMB to cool below 60 nK — in about 10^{19} years.
The temperature is inversely proportional to mass, so smaller means hotter.
| Mass | Temperature | Lifetime |
|---|---|---|
| 10^{9}M_\odot | 6\times10^{-17} K | 10^{96} y |
| 1M_\odot | 6\times10^{-8} K | 10^{67} y |
| 10^{12} kg | 10^{11} K | 10^{10} y |
| 10^{5} kg | 10^{18} K | 1 s |
A black hole of 10^{12} kg — the mass of a mountain, in a space smaller than a proton — would be evaporating now if any formed in the early universe.
The evaporation timescale:
t \approx 2\times10^{67}\left(\frac{M}{M_\odot}\right)^3\ \text{years}
The M^3 makes it run away at the end. The final second releases about 10^{22} J — a large nuclear bomb — as a burst of increasingly energetic gamma rays.
Searches for such bursts have found nothing, which constrains the abundance of primordial black holes in that mass range.
Black hole thermodynamics
Bekenstein and Hawking together established a complete thermodynamic description, and it is one of the deepest results in theoretical physics.
\boxed{S = \frac{k_Bc^3A}{4G\hbar} = \frac{k_BA}{4\ell_P^2}}
Entropy proportional to the horizon area, in units of the Planck area \ell_P^2 = 2.6\times10^{-70} m².
Not volume — area. Every other entropy in physics is extensive, scaling with volume.
How big is it? For a solar-mass black hole, A = 4\pi r_s^2 = 4\pi(2950)^2 = 1.09\times10^{8} m²:
S = \frac{1.09\times10^{8}}{4\times2.6\times10^{-70}}k_B = 1.05\times10^{77}k_B
Compare with the Sun's entropy as a star: about 10^{58}k_B.
\boxed{\text{Collapsing the Sun into a black hole increases its entropy by a factor of } 10^{19}.}
A black hole is the maximum-entropy state of any region. This gives the Bekenstein bound: the information that can be stored in a region is limited by its surface area, not its volume — about 10^{69} bits per square metre.
And the four laws of black hole mechanics map exactly onto thermodynamics:
| Thermodynamics | Black holes |
|---|---|
| T constant at equilibrium | Surface gravity constant on horizon |
| dU = TdS-PdV | dM = \frac{\kappa}{8\pi}dA+\Omega dJ |
| dS \geq 0 | dA \geq 0 |
| T = 0 unreachable | Extremal (T=0) unreachable |
The correspondence was noticed before Hawking radiation was derived and was thought to be a formal analogy. Hawking's result showed it is not an analogy — it is thermodynamics.
The information paradox
The sharpest unsolved problem in theoretical physics.
The setup:
Quantum mechanics is unitary (Chapter 7.3). Information is never destroyed; the Schrödinger equation is reversible.
Hawking radiation is exactly thermal. A thermal spectrum depends only on temperature, and therefore only on the black hole's mass. It carries no information about what fell in.
So when the black hole evaporates completely, the information about everything that fell in is gone.
\boxed{\text{Unitarity and Hawking radiation cannot both be right as stated.}}
Hawking argued for decades that information is genuinely lost and that quantum mechanics must be modified. He conceded a bet in 2004, accepting that information probably escapes — though without a mechanism.
Proposed resolutions
Information escapes in subtle correlations. The radiation is not exactly thermal; correlations between early and late photons carry the information. The problem: showing this requires understanding the evaporation's final stages, which needs quantum gravity.
Remnants. Evaporation stops at the Planck mass, leaving an object holding all the information. The problem: an object of 10^{-8} kg would need to store unbounded information, which causes trouble elsewhere.
Firewalls. Almheiri, Marolf, Polchinski and Sully argued in 2012 that unitarity, locality and the equivalence principle cannot all hold, and proposed that an infalling observer hits a wall of high-energy radiation at the horizon. The problem: this violates the equivalence principle (Chapter 6.6), which says nothing special happens at a horizon.
Holography. The AdS/CFT correspondence (Chapter 8.7) says a gravitational theory in a volume is equivalent to a non-gravitational theory on its boundary. In the boundary theory, evolution is manifestly unitary, so information cannot be lost. This is the strongest argument that the paradox has an answer, and it does not say what the mechanism is in the bulk.
Islands and the Page curve. Since 2019, calculations using the "quantum extremal surface" prescription have reproduced the Page curve — the entanglement entropy of the radiation rising and then falling, as unitarity requires — from gravitational path integrals. This is genuine progress and the interpretation is still argued about.
The problem is not solved. It is, by wide agreement, the clearest signpost towards quantum gravity.
Falling in
Chapter 6.9 covered the two accounts and they are worth restating with the numbers.
The distant observer sees the infalling astronaut slow, redden, dim and freeze at the horizon, fading exponentially with a timescale of r_s/c — 20 microseconds for a solar-mass hole. They never see the crossing.
The astronaut crosses in finite proper time, feeling nothing locally, because the equivalence principle says a freely falling observer feels no gravity and the horizon is not a place.
Tidal forces were computed in Chapter 6.P: lethal 124 Schwarzschild radii out for a stellar hole, and negligible even inside the horizon for a supermassive one.
Inside the horizon, the roles of r and t swap. In the Schwarzschild metric, g_{00} and g_{rr} change sign at r = r_s. The radial coordinate becomes timelike, which means moving towards r = 0 becomes as unavoidable as moving forwards in time.
\boxed{\text{Inside a black hole, the singularity is not a place. It is a moment in your future.}}
Firing rockets outward does not help; it only shortens the proper time to the singularity. The longest possible survival is achieved by free fall, and for a solar-mass hole it is about 10 microseconds.
Where this shows up in your life
Quasars are the most distant objects routinely observed, and their absorption spectra probe the intergalactic medium along the whole line of sight.
The galactic centre black hole was weighed by tracking stellar orbits for 25 years, winning the 2020 Nobel Prize.
Gravitational wave astronomy (Chapter 6.10) is now dominated by black hole mergers, with over 200 catalogued.
Black hole thermodynamics produced the holographic principle, which is one of the most influential ideas in theoretical physics even outside gravity.
And Hawking radiation is the only place where quantum mechanics, relativity, gravity and thermodynamics all meet in one equation — which is why the information paradox is taken so seriously as a guide to what comes next.
What the next chapter fixes
Black holes are the endpoint of stellar evolution and the anchor of galaxies. The galaxies themselves are the next scale up, and measuring how they rotate produced the single most persistent unexplained observation in modern astronomy. Chapter 12.4 covers the structure of the Milky Way from inside it, the classification of galaxies, the rotation curves that do not match the visible matter, and the large-scale structure that the universe has organised itself into.