Thermodynamic Entropy versus Fine-Grained Recovery

“The entropy of a black hole” can denote several inequivalent quantities. One must specify the state, region or operator algebra, coarse graining, renormalization, and gravitational approximation before asking whether an entropy increases or follows a Page curve. Horizon thermodynamics and the generalized second law constrain a generalized entropy associated with causal horizon cuts. They do not by themselves establish unitary recovery of the fine-grained quantum state in Hawking radiation.

Required background. Quantum Information supplies reduced states, von Neumann entropy, purification, and the difference between fine- and coarse-grained entropy.

Helpful background. Quantum Fields in Curved Spacetime supplies exterior field algebras; Einstein Equations supplies the geometric response; Black-Hole Geometry and Mechanics supplies horizon cuts; and Semiclassical Gravity supplies the renormalized generalized entropy.

Five different entropy questions

For a horizon cut $\Sigma$, the semiclassical generalized entropy is

$$S_{\mathrm{gen}}[\Sigma] =S_{\mathrm{grav}}^{\mathrm{ren}}[\Sigma] +S_{\mathrm{out}}^{\mathrm{ren}}[\Sigma].$$

Its terms and several neighboring notions answer different questions.

Entropy	State and system	What it diagnoses
Bekenstein–Hawking or Wald entropy	A stationary black-hole solution in a specified gravitational theory and ensemble	Thermodynamic response and the first law
Exterior von Neumann entropy	A quantum state restricted to the algebra outside a chosen cut	Fine-grained correlations across that cut, after renormalization
Generalized entropy	Renormalized gravitational entropy plus exterior entropy for the same cut	The semiclassical GSL and quantum-corrected extremality
Radiation entropy	A state reduced to a specified collected-radiation region or algebra	How much that subsystem is entangled with its complement
Coarse-grained thermodynamic entropy	Macroscopic data with inaccessible microstates averaged over	Irreversibility at the chosen descriptive level

The gravitational term need not be an ordinary von Neumann entropy of a known microscopic tensor factor. Conversely, the fine-grained radiation entropy is not obtained merely by inserting the shrinking horizon area into $A/(4G)$. Wald’s review develops the stationary thermodynamic and GSL notions and stresses the remaining statistical questions Wald 2001, §§4–6, PDF.

Monotonicity and unitary recovery are different claims

Under the causal-horizon hypotheses of the relative-entropy proof, later cuts obey

$$S_{\mathrm{gen}}[\Sigma_2] \geq S_{\mathrm{gen}}[\Sigma_1].$$

This compares two outside algebras associated with nested horizon cuts. A Page curve instead asks for the fine-grained entropy $S(R_u)=-\operatorname{tr}\rho_{R_u}\log\rho_{R_u}$ of radiation collected up to retarded time $u$. In a globally pure, unitarily evolving system, that entropy can first rise and later fall because the complement purifying $R_u$ changes. Unitarity alone does not guarantee the familiar smooth Page curve: one also needs assumptions about the initial purity, which degrees of freedom escape, and their correlations. There is no contradiction: the two statements refer to different algebras and include different geometric terms.

The distinction is already visible without gravity. A pure Bell pair has zero entropy as a whole, while either qubit has entropy $\log 2$. Asking “what is the entropy?” without naming the subsystem admits incompatible correct answers. In gravity, the ambiguity is amplified because the definition of a quantum-corrected region can itself involve extremizing generalized entropy.

Wall’s GSL proof specifies the causal-horizon outside algebra and the complete escaping system in Wall 2012, §§2.6–2.8, printed pp. 18–21, PDF. It proves generalized-entropy monotonicity within its semiclassical domain; it does not reconstruct a unitary evaporation map.

Where quantum extremal surfaces enter

A quantum extremal surface $X$ is stationary under shape variations of generalized entropy,

$$\delta_X S_{\mathrm{gen}}[X]=0.$$

In holographic settings, choosing an appropriate extremum can change which bulk region is associated with a boundary radiation region. Engelhardt and Wall formulate the perturbative quantum extremal surface prescription in Engelhardt and Wall 2015, §§2.1–3.1, PDF. That framework is related to generalized entropy but adds holographic, state, and extremization assumptions not contained in the ordinary horizon GSL.

Page curves, islands, replica saddles, reconstruction, and the evidence for fine-grained information recovery are developed in Black-Hole Information. This is the scientific boundary: the present chapter supplies the thermodynamic and semiclassical entropy inputs, while that volume assesses whether and how the radiation encodes the initial state.

Regime and independent checks

Here $S_{\mathrm{gen}}$ is defined in semiclassical effective field theory through the retained loop order, using one covariant renormalization scheme for the action and exterior entropy. The cut must be smooth relative to the cutoff scale, and any GSL theorem also needs its stated causal-horizon and algebraic hypotheses. Fine-grained radiation entropy requires a specified state and radiation algebra; a thermal spectrum alone determines neither its purification nor its late-time evolution.

Two limiting checks are useful. In a stationary equilibrium state, the horizon area and exterior entropy can exchange under small perturbations while their generalized sum obeys the first law. In a nongravitating bipartite pure state, the geometric term is absent and the two reduced entropies are equal, but the entropy of the full system is zero. The second check shows directly why a subsystem entropy cannot be identified with global information loss.

Exercise

A pure system consists of collected radiation $R$ and its complement $B$. Initially their common reduced entropy is $3.0$. At a later time unitarity and the new bipartition give $S(R)=S(B)=2.2$. Over the same interval, a causal horizon calculation gives $\Delta S_{\mathrm{grav}}=-0.5$ and $\Delta S_{\mathrm{out}}=+0.8$. Compute both relevant entropy changes and state what each establishes.

Solution

The radiation’s fine-grained entropy changes by $\Delta S(R)=2.2-3.0=-0.8$: it lies on a decreasing segment for this chosen bipartition. Purity requires $S(R)=S(B)$ at each time, but does not require either reduced entropy to be monotone.

The generalized-horizon entropy changes by

$$\Delta S_{\mathrm{gen}} =\Delta S_{\mathrm{grav}}+\Delta S_{\mathrm{out}} =-0.5+0.8=+0.3.$$

Thus the GSL is satisfied even while radiation entropy decreases. The first number diagnoses a decrease in fine-grained entropy for $R$, which by itself does not establish complete recovery of an unknown initial state; the second concerns a causal horizon and includes its geometric term. Neither can be substituted for the other without an additional relation between their regions and algebras.

References

• Engelhardt, N., and A. C. Wall (2015), “Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime,” Journal of High Energy Physics 2015(1), 73. Open PDF.
• Wald, R. M. (2001), “The Thermodynamics of Black Holes,” Living Reviews in Relativity 4, 6. Open PDF.
• Wall, A. C. (2012), “A Proof of the Generalized Second Law for Rapidly Changing Fields and Arbitrary Horizon Slices,” Physical Review D 85, 104049; revised 2018. Open PDF.