The information problem in quantum gravity

Feed a diary into a shredder and the information is not destroyed — in principle the strips can be reassembled; burn it, and the same holds: the microscopic state of flame and smoke carries the entire text, merely beyond practical recovery. Quantum mechanics underwrites this “never lost in principle”: time evolution, from section 3.9, is unitary, $\hat U^\dagger\hat U=1$ — pure states always evolve into pure states, and distinct initial states always evolve into distinct final states. Unitarity = the universe does not destroy information.

Now throw the diary into a black hole.

Before 1974 one could still shrug and say “the information is inside the horizon, just out of reach”. But that year Hawking proved that a black hole is not an eternal tomb — it evaporates. Doing quantum field theory in the curved spacetime near a black hole, he found that the horizon continuously radiates particles outward; the black hole’s mass slowly bleeds away until (in principle) it is gone entirely. And Hawking’s calculation says the radiation is exactly thermal: its temperature depends only on the black hole’s mass, and it carries no memory whatsoever of “what was eaten”.

The diary went in, the black hole is gone, and all that came out is thermal noise. Where did the information go? This is the black hole information paradox — quantum mechanics (unitarity) and general relativity (the horizon) delivering irreconcilable answers to the same question, which is why it is widely regarded as the most important clue on the road to quantum gravity.

First, get the numbers: how hot is a black hole, how big is its entropy

Order of magnitude: the Hawking temperature and the Bekenstein–Hawking entropyadvanced~10 minexpand

Step 1: the black hole’s only scale.

A black hole of mass $M$ has just one characteristic length — the Schwarzschild radius (the radius at which the escape velocity reaches the speed of light; the classical derivation $\frac12mv^2=\frac{GMm}{r}$ with $v=c$ happens to give the right answer):

$$r_s=\frac{2GM}{c^2}\tag{13.8.1}$$

A solar mass corresponds to $r_s\approx3$ km.

Step 2: guess the temperature by dimensions.

The radiation originates in quantum fluctuations near the horizon, and the only wavelength scale available is $r_s$. A typical photon has $\lambda\sim r_s$, hence energy $\hbar c/r_s$, so

$$k_BT\sim\frac{\hbar c}{r_s}\sim\frac{\hbar c^3}{GM}\tag{13.8.2}$$

Hawking’s rigorous calculation adds only the numerical coefficient:

$$T_H=\frac{\hbar c^3}{8\pi GMk_B}\tag{13.8.3}$$

Plug in a solar mass, $M=2\times10^{30}$ kg:

$$T_H\approx\frac{1.05\times10^{-34}\times(3\times10^8)^3}{8\pi\times6.67\times10^{-11}\times2\times10^{30}\times1.38\times10^{-23}} \approx6\times10^{-8}\ \text{K}\tag{13.8.4}$$

60 nanokelvin — eight orders of magnitude colder than the cosmic microwave background (2.7 K). So stellar black holes are currently net absorbers of heat and nowhere near evaporating; the evaporation timescale is $\sim10^{67}$ years. The paradox is thus purely a contradiction internal to theory — and that is exactly what makes it precious: a thought experiment that forces the two great theories to lay their cards on the table without any experiment being run. Note one perverse feature: $T\propto1/M$, so the more it evaporates, the smaller and hotter it gets — a black hole’s heat capacity is negative.

Step 3: the entropy.

Where there is a temperature there is an entropy. Apply the first law $\dd (Mc^2)=T\dd S$ and integrate over $M$ ($T\propto1/M$ gives $S\propto M^2$), obtaining the Bekenstein–Hawking entropy:

$$S_{BH}=\frac{k_Bc^3A}{4G\hbar}=k_B\,\frac{A}{4\ell_P^2}, \qquad \ell_P=\sqrt{\frac{G\hbar}{c^3}}\approx1.6\times10^{-35}\ \text{m}\tag{13.8.5}$$

Read it as: one unit of entropy (about 0.72 bits) for every 4 Planck areas of horizon.

For a solar mass, $A=4\pi r_s^2\approx10^8\ \text{m}^2$, so

$$S_{BH}/k_B\approx1\times10^{77}\tag{13.8.6}$$

Compare: the Sun, as an ordinary star, has entropy of order $10^{58}\,k_B$. Collapsing into a black hole multiplies the entropy by 19 orders of magnitude — black holes are the highest-entropy-density objects known, and the entropy ledger of the universe is kept almost entirely under their name.

The paradox in precise form: the Page curve

“Is the information lost or not” can be translated into a computable curve. With the tools of section 9.3: let the black hole form from a pure state and radiate as it evaporates, and ask how the entanglement entropy $S_{\text{rad}}$ of the radiation that has already escaped changes with time.

Hawking’s answer: each parcel of radiation is thermal, entangled with a partner inside the horizon; new entanglement only ever accumulates — $S_{\text{rad}}$ rises monotonically until the black hole is gone. But then the black hole no longer exists while the radiation remains entangled with something that isn’t there: the total state is mixed. Pure state to mixed state — unitarity falls (in the language of section 10.2: $\operatorname{Tr}\rho^2$ has dropped from 1 to far below 1).

Page’s answer (1993): if unitarity holds and evaporation is a “thoroughly scrambling but law-abiding” unitary process, then $S_{\text{rad}}$ should first rise — early radiation really is entangled with the black hole — peak around the midpoint of evaporation (the Page time), and then turn over and fall, returning to zero as the black hole disappears: the late radiation gradually “takes back” the entanglement of the early radiation, and the information flows out with the radiation as a whole, in thoroughly scrambled form — as ashes are to a diary.

This rise-then-fall Page curve is the verdict criterion: whoever can derive the descending branch from a theory of gravity has solved the core of the paradox. The difficulty: every step of Hawking’s semiclassical calculation looks sound, yet it can only ever yield the monotonic rise — and for over forty years nobody could point to the step that fails.

The 2019 turn: islands, and a gravitational derivation of the Page curve

The breakthrough came from taking the gravitational path integral (the gravity version of the ideas of chapter 11) seriously all the way to the end. In 2019, two groups (Penington; Almheiri–Engelhardt–Marolf–Maxfield) discovered that the gravitational path integral for the radiation entropy contains previously overlooked saddle points — replica wormholes — which take over as dominant after the Page time and produce a new entropy formula (the island formula): the radiation’s entropy is the minimum over all ways of “assigning a region inside the horizon (an island) to the radiation’s side of the ledger”. Late in the evaporation, the island-including assignment wins and the entropy comes down — for the first time, the Page curve was derived from a (semiclassical) gravity calculation, with no complete theory of quantum gravity required.

What this means, and what it does not, are worth separating:

• It means: semiclassical gravity is smarter than Hawking gave it credit for — the unitary Page curve was hiding in the gravitational path integral all along, tucked into nontrivial saddles. The “information is conserved” side can all but declare victory.
• It does not mean: that we know how the information is written into the radiation, or what any individual Hawking photon carries; the physical interpretation of the wormhole saddles, and the formula’s standing for real four-dimensional evaporating black holes, remain hotly debated. The verdict is in; the stolen goods have not all been recovered.

∎Key formulas

• Hawking temperature

  $$T_H=\frac{\hbar c^3}{8\pi GMk_B}\approx6\times10^{-8}\ \text{K}\ (M=M_\odot)$$

  T ∝ 1/M: negative heat capacity, hotter as it shrinks; stellar-mass evaporation takes ~10⁶⁷ years

• Bekenstein–Hawking entropy

  $$S_{BH}=k_B\,\frac{A}{4\ell_P^2}\approx10^{77}k_B\ (M=M_\odot)$$

  Entropy ∝ area, not volume — the starting point of holography

• Unitarity

  $$\hat U^\dagger\hat U=1\ \Rightarrow\ \text{pure}\to\text{pure}$$

  One side of the paradox; the other is the exact thermality of Hawking radiation

• Page curve

  $$S_{\text{rad}}(t)\ \text{rises, then falls to zero as the black hole vanishes}$$

  The criterion for unitary evaporation; derived in 2019 from the island formula and the gravitational path integral

?Self-check3 questions

1. 1.

   The core conflict of the black hole information paradox is:

   • Quantum mechanics demands unitary evolution (pure states stay pure), while Hawking's calculation gives evaporation into thermal radiation carrying no information about the initial state
   • General relativity forbids black holes from evaporating
   • Information falls into the black hole faster than light
   • The singularity inside makes the wavefunction diverge

   Submit
2. 2.

   What is the Hawking temperature of a solar-mass black hole (M = 2×10³⁰ kg), in nanokelvin? (T_H = ħc³/(8πGMk_B))

   nK800% relative tolerance

   Submit
3. 3.

   Which statements about the Page curve and the post-2019 developments are correct? (Select all that apply.)

   Select all that apply

   • If evaporation is unitary, the radiation entanglement entropy should rise then fall, reaching zero when the black hole disappears
   • Hawking's original calculation gives a monotonically rising entropy, corresponding to information loss
   • The island formula and replica wormholes yielded the first derivation of the Page curve from a semiclassical gravity calculation
   • These developments fully explain how the information is written into each Hawking photon

   Submit

Where to dig deeper

At the popular level: Susskind’s The Black Hole War chronicles the paradox’s first three decades of combat — partisan, but physically sound. At the lecture-note level: Harlow’s black hole information notes (arXiv:1409.1231) are the standard graduate text; for the post-2019 developments, see the review by Almheiri et al., The entropy of Hawking radiation (arXiv:2006.06872). The tools these references use — entanglement entropy, density matrices, path integrals — you have already met, all of them, in chapters 9, 10 and 11; the only missing layer is curved spacetime, and the reading will feel closer than you expect.

The next section is the book’s last: from cosmic scales back to the keyboard in front of you — when quantum computing meets machine learning, what is substance and what is froth.