Pavón (arXiv:2112.01931, 2021) derived an approximate expression for the entropy of Hawking radiation in stable thermal equilibrium with a Schwarzschild black hole inside a perfectly reflecting spherical cavity, by combining Page's approximate renormalized stress-energy tensor in the Hartle-Hawking state with the local thermodynamic Euler relation. The Bekenstein entropy bound is satisfied, but the computation yields negative radiation entropy for small boxes (1≤R/2M≲4.9121\le R/2M\lesssim 4.912), a negative time derivative of the total entropy in a window of unstable equilibrium, and negative entropy changes in a box-removal thought experiment - apparent violations of the generalized second law (GSL). The author attributes these to possible artifacts of neglected ingredients: the approximate stress tensor, the ignored vacuum polarization of the geometry and of the cavity wall, and the ignored backreaction on the metric. The still-open task is to decide, by an approximation-controlled recomputation, whether the violations are genuine or artifacts; to establish the validity threshold of the local Euler relation for boxes comparable to the dominant Hawking wavelength; to relate the thermodynamic radiation entropy to the Funai-Sugawara entanglement-entropy account; and to delimit the GSL's fate in the late, quantum-gravity-dominated evaporation stage.

Background

A Schwarzschild black hole of mass MM emits thermal Hawking radiation at temperature Tbh=(8πM)−1T_{bh}=(8\pi M)^{-1} and carries the Bekenstein-Hawking entropy Sbh=4πM2S_{bh}=4\pi M^2, one quarter of its horizon area (Hawking 1976). The generalized second law (GSL) conjectures that SbhS_{bh} plus the entropy of everything outside the hole never decreases in time. Because an evaporating hole heats up as it shrinks while its radiation cools, equilibrium is impossible in unbounded space; it becomes possible if the radiation is confined to a perfectly reflecting spherical box ('cavity') of radius R>2MR>2M - the black-hole-in-a-box setting. There Tolman's law fixes the local equilibrium temperature of a static gravitating body, T(r)=f(r)−1/2TbhT(r)=f(r)^{-1/2}T_{bh} with f(r)=1−2M/rf(r)=1-2M/r, so that T−gttT\sqrt{-g_{tt}} is constant across the system. The stability of such equilibrium was analyzed by Pavón and Israel (1984): stable equilibrium requires the total energy M+EradM+E_{rad} to have a minimum, is impossible if Erad>M/4E_{rad}>M/4, and for each RR exists only below a critical ratio xmin=R/2Mx_{min}=R/2M; for R=200R=200 in Planck units, xmin≃38.558x_{min}\simeq 38.558.

Beyond the flat-space thermal estimate for the radiation entropy, Pavón (2021) used the renormalized stress-energy tensor ⟨Tμν⟩\langle T^\mu{}_\nu\rangle (the state-dependent expectation value of the quantum stress tensor after ultraviolet renormalization) of a conformally invariant scalar field in the Hartle-Hawking state - the state describing a hole in thermal equilibrium with its radiation, regular on the horizon - as approximated analytically by Page (1982). Page's tensor has negative energy density near the horizon, a robust feature of the state: it persists in exact numerical computations (Howard and Candelas 1984), and the ansatz is the minimal one compatible with the state's general properties (Visser 1997). Pavón then assigned a local entropy density through the Euler relation of thermodynamics for radiation with zero chemical potential, srad=(ρ+P)/Ts_{rad}=(\rho+P)/T, with ρ\rho and PP the local energy density and pressure, and integrated over the proper volume, obtaining SradS_{rad} as a function of x=R/2Mx=R/2M only. Since ρ+P<0\rho+P<0 near the horizon, srads_{rad} is negative there, and the integrated SradS_{rad} is negative for 1≤x≲4.9121\le x\lesssim 4.912 - an interval extending beyond x=3x=3, the floor below which the paper itself deems the Euler relation unreliable (the box should be at least three times the dominant Hawking wavelength, which is of order 2M2M). The Bekenstein bound S/E≤2πRS/E\le 2\pi R (Bekenstein 1981) is nevertheless satisfied: η=S/(2πRE)≤1\eta=S/(2\pi RE)\le 1.

For the GSL, differentiating along slow evaporation (M˙≃−αM−2\dot M\simeq-\alpha M^{-2}, valid for M≫1M\gg 1) gives S˙=λ(R,x)∣M˙∣\dot S=\lambda(R,x)|\dot M|, with λ<0\lambda<0 in a window (xmin,x0)(x_{min},x_0) (for R=200R=200: x0≃40.24x_0\simeq 40.24) where the system sits in unstable equilibrium, so the total entropy decreases. A separate thought experiment - gently removing an inner box so the radiation re-equilibrates in a larger one at conserved total energy - yields ΔS≃−4.02\Delta S\simeq-4.02 (for R1=100R_1=100, R2=300R_2=300, total energy E=4.2E=4.2) and ΔS≃−4.536\Delta S\simeq-4.536 (E=6E=6), again negative, at box-to-hole ratios far above the Euler-relation floor. Pavón is explicit that these apparent violations may be artifacts of four neglected ingredients: the approximate nature of Page's tensor, the vacuum polarization (quantum stress-tensor expectation value) of the background geometry, the vacuum polarization of the wall, and the backreaction of the radiation on the metric; a further entropy source may also have been overlooked. He further leaves open the relation between this thermodynamic entropy and the entanglement entropy of the radiation computed by Funai and Sugawara in local quantum field theory - the von Neumann entropy of the radiation reduced state, quantifying its quantum correlations with the black hole interior - a computation in which unitarity requires replacing the Schwarzschild singularity by a solid core. In the modern fine-grained literature, radiation entropy is instead tracked by von Neumann entropy with entanglement islands - interior regions that must be joined to the radiation region when computing its fine-grained entropy, their boundary area entering through the replica-wormhole ('island') formula (Almheiri et al. 2021) - and reflecting cavities have been used to regulate the infrared behavior of eternal Schwarzschild (Ageev et al. 2025). The fine-grained GSL, for generalized entropy (horizon area/4/4 plus exterior von Neumann entropy), has been proven under semiclassical assumptions (Wall 2010). None of these frameworks answers the coarse-grained question the source paper raises.

Problem Statement

Determine whether the generalized second law is respected, in its coarse-grained thermodynamic form, by a Schwarzschild black hole in equilibrium with its Hawking radiation inside a perfectly reflecting spherical cavity - and specify which entropy definition and which cavity regimes make the statement secure. The source paper (Pavón, arXiv:2112.01931) leaves four linked issues open.

(1) Primary semiclassical strand. Recompute the radiation entropy SradS_{rad} and its evolution for the cavity system with (a) the backreaction of the radiation on the Schwarzschild metric included, (b) a renormalized stress-energy tensor more exact than Page's approximation, and (c) the vacuum polarization of both the background geometry and the cavity wall included - and decide whether the negative near-horizon entropy density, the integrated Srad<0S_{rad}<0 for 1≤R/2M≲4.9121\le R/2M\lesssim 4.912, the window with S˙<0\dot S<0 during evaporation (e.g. xmin<x<x0x_{min}<x<x_0 for R=200R=200, with xmin≃38.558x_{min}\simeq 38.558 and x0≃40.24x_0\simeq 40.24), and the negative entropy changes ΔS\Delta S of the box-removal thought experiment persist or disappear. If a further entropy source is required for consistency, identify it and its contribution.

(2) Validity of the local entropy assignment. The paper assumes the local Euler relation s=(ρ+P)/Ts=(\rho+P)/T is reliable only when the box is at least about three times the wavelength of the dominant Hawking mode (of order 2M2M), and it lists RR being of the order of MM among the possible unphysical origins of the negative SradS_{rad}. Determine whether this local-equilibrium entropy assignment remains adequate in the regimes where the apparent violations arise: the negative-SradS_{rad} interval extends to R/2M≃4.9R/2M\simeq 4.9, near the stated floor, and the thought-experiment violations occur at R/2M≃8.3R/2M\simeq 8.3 and 11.911.9.

(3) Entropy bookkeeping. Establish the relation between the thermodynamic radiation entropy so defined and the entanglement entropy of the radiation derived by Funai and Sugawara in local quantum field theory - keeping in mind the source's noted caveat that, in that approach, unitarity of the evaporation process requires replacing the Schwarzschild singularity by a solid core.

(4) Late-stage strand, acknowledged to be quantum-gravity-dependent. The fate of the GSL during the late, possibly nonperturbative stages of evaporation, when MM is no longer large - which the paper itself expects to require a reliable theory of quantum gravity or a consistent unitary account of evaporation.

Strands (1)-(3) share one objective: an approximation-controlled, interpretable statement of the GSL for the finite-cavity black-hole-radiation system; strand (4) delimits what any such statement can cover. A resolution must address strands (1)-(3); strand (4) is out of scope and need only be delimited. The source itself cautions that a resolution "may not be round the corner" and may await a reliable theory of quantum gravity.

References

  1. Diego Pavón, 'The entropy of Hawking radiation and the generalized second law', arXiv:2112.01931 (2021), DOI 10.48550/arXiv.2112.01931, https://arxiv.org/abs/2112.01931 (primary source)
  2. Don N. Page, 'Thermal stress tensors in static Einstein spaces', Physical Review D 25, 1499 (1982), DOI 10.1103/PhysRevD.25.1499, https://doi.org/10.1103/PhysRevD.25.1499
  3. Jacob D. Bekenstein, 'Universal upper bound on the entropy-to-energy ratio for bounded systems', Physical Review D 23, 287 (1981), DOI 10.1103/PhysRevD.23.287, https://doi.org/10.1103/PhysRevD.23.287
  4. S. W. Hawking, 'Black holes and thermodynamics', Physical Review D 13, 191 (1976), DOI 10.1103/PhysRevD.13.191, https://doi.org/10.1103/PhysRevD.13.191
  5. D. Pavón and W. Israel, 'Stability of thermal equilibrium for a radiating black hole in a box', General Relativity and Gravitation 16, 563 (1984), DOI 10.1007/BF00769629, https://doi.org/10.1007/BF00769629
  6. K. W. Howard and P. Candelas, 'Quantum Stress Tensor in Schwarzschild Space-Time', Physical Review Letters 53, 403 (1984), DOI 10.1103/PhysRevLett.53.403, https://doi.org/10.1103/PhysRevLett.53.403
  7. Matt Visser, 'Gravitational vacuum polarization IV: Energy conditions in the Unruh vacuum', arXiv:gr-qc/9703001, https://arxiv.org/abs/gr-qc/9703001
  8. Shotaro Shiba Funai and Hirotaka Sugawara, 'Calculation of Hawking Radiation in Local Field Theory', arXiv:2102.04344 (2021, v3), https://arxiv.org/abs/2102.04344
  9. Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian and Amirhossein Tajdini, 'The entropy of Hawking radiation', Reviews of Modern Physics 93, 035002 (2021), DOI 10.1103/RevModPhys.93.035002, https://doi.org/10.1103/RevModPhys.93.035002
  10. Aron C. Wall, 'Proof of the generalized second law for rapidly evolving Rindler horizons', Physical Review D 82, 124019 (2010), DOI 10.1103/PhysRevD.82.124019, https://doi.org/10.1103/PhysRevD.82.124019
  11. Dmitry S. Ageev, Irina Ya. Aref'eva and Timofei A. Rusalev, 'Black holes, cavities, and blinking islands', Physical Review D 111, 026002 (2025), DOI 10.1103/PhysRevD.111.026002, arXiv:2311.16244, https://arxiv.org/abs/2311.16244
  12. G. G. L. Nashed, Alnadhief H. A. Alfedeel and Tiberiu Harko, 'Analytic semiclassical backreaction of a Schwarzschild black hole in a finite cavity: horizon shift, temperature renormalization, and canonical stability in the Hartle-Hawking State', arXiv:2604.10346 (2026, accepted in EPJC), https://arxiv.org/abs/2604.10346
  13. D. S. Ageev, I. Ya. Aref'eva, A. I. Belokon, A. V. Ermakov, V. V. Pushkarev and T. A. Rusalev, 'Infrared Regularization and Finite Size Dynamics of Entanglement Entropy in Schwarzschild Black Hole', arXiv:2209.00036 (2022), https://arxiv.org/abs/2209.00036