Quantum-extremal-surface derivation of the fine-grained Page curve of nonequilibrium Schwarzschild–de Sitter evaporation

Easson (Phys. Rev. D 113, 084014 (2026)) constructed the first fully analytic, backreacted solution of Schwarzschild–de Sitter (SdS) black-hole evaporation: a spherically reduced two-dimensional dilaton-gravity model with Polyakov trace-anomaly backreaction in the Unruh–de Sitter state, carrying a conserved Killing flux from the black-hole to the cosmological horizon, with provably monotonic mass loss and satisfaction of the generalized second law. Its information-theoretic content, however, is only a thermo-controlled proxy: the Page time is estimated by branch-crossing between a coarse-grained entropy-production proxy and the black-hole Bekenstein–Hawking entropy. The open problem is to replace this estimate by a microscopic computation: extremize the generalized entropy $S_{\mathrm{gen}}=X/(4G_2)+S_{\mathrm{out}}$ on the fully backreacted, time-dependent SdS geometry, with an explicitly specified radiation region, quantum state, and ultraviolet subtraction convention, in order to determine the precise island location, which saddle (inner black-hole-side or cosmological-side island) dominates for static-patch observers in the presence of the steady flux, and the Page time. This core remains open.

Background

Schwarzschild–de Sitter (SdS) spacetime is the unique spherically symmetric solution of Einstein's equations with a positive cosmological constant $\Lambda$; it describes a black hole embedded in an expanding universe. A static observer in it is confined to a finite static patch bounded by two Killing horizons (null surfaces on which the patch's time-translation symmetry degenerates): the black-hole horizon at $r_b$ and the cosmological horizon at $r_c$. Each horizon has its own surface gravity ($\kappa_b$, $\kappa_c$) and hence its own Hawking temperature $T_h=\kappa_h/2\pi$; generically $\kappa_b\neq\kappa_c$, so the two horizons radiate at different temperatures and the patch is intrinsically out of thermal equilibrium — a steady heat current flows from the hotter black hole toward the cooler cosmological horizon. The degenerate Nariai configuration, in which the two horizons coincide, is the only zero-flux equilibrium.

Because a full four-dimensional backreacted calculation is not analytically tractable, a standard laboratory is the spherical reduction of Einstein gravity to a two-dimensional dilaton-gravity theory, in which the areal radius is encoded in a dilaton field $X=r^2$ and the dominant s-wave sector (the spherically symmetric, zero-angular-momentum mode) of $N$ conformal matter fields is retained. In two dimensions the quantum backreaction of conformal matter is captured by the anomaly-induced Polyakov effective action (the effective action encoding the two-dimensional trace anomaly), often written using an auxiliary scalar field $\psi$ obeying $\Box\psi=R$. The quantum state relevant to evaporation is the Unruh–de Sitter (UdS) state — regular on both future horizons, like the Unruh state of an evaporating black hole but adapted to the two-horizon patch — which carries a conserved Killing-energy flux $J\propto N(\kappa_b^2-\kappa_c^2)$. In the Polyakov framework the quantum state is fixed by two functions of the null coordinates, the state functions $t_u=t(u)$ and $t_v=t(v)$ entering the auxiliary field, whose choice determines on which horizons the state is regular and what flux it carries; the Unruh–de Sitter state is the choice regular on both future horizons. A strictly static geometry cannot support this flux (the momentum constraint forces the mixed stress-tensor component to vanish), so the backreacted solution is necessarily time dependent, treated adiabatically with the mass-loss law $\dot M=-J$.

In this framework D. A. Easson ('The fate of Schwarzschild–de Sitter black holes: nonequilibrium evaporation', Phys. Rev. D 113, 084014 (2026)) constructed the first globally analytic, backreacted solution of SdS evaporation: he proved $\kappa_b>\kappa_c$ throughout the physical static patch (so the black hole loses mass monotonically and evaporates toward empty de Sitter), verified the generalized second law (GSL — the statement that the total horizon entropy plus the entropy outside never decreases; here the sum of the black-hole and cosmological horizon entropies grows monotonically along the evolution), and computed locally measured temperatures and fluxes for static observers.

The information-theoretic layer concerns the entanglement entropy of the Hawking radiation. According to the quantum extremal surface (QES) prescription, the fine-grained entropy of a region is computed from the generalized entropy $S_{\mathrm{gen}}=X(\partial\mathcal R)/(4G_2)+S_{\mathrm{out}}[\mathcal R]$ (a boundary 'area' term built from the dilaton, plus the von Neumann entropy of quantum fields in the region), evaluated at its stationary points, where $\partial_\pm S_{\mathrm{gen}}=0$ denote vanishing quantum expansions along the two null directions. If the dominant saddle includes an island — a region beyond the horizon that counts as part of the entanglement wedge (the subregion reconstructible from the radiation's quantum state) of the outside radiation — the radiation entropy stops growing and turns over, producing the Page curve, with the turnover time called the Page time. In de Sitter settings this program has so far been realized only in restricted regimes: two-dimensional de Sitter dilaton models of Jackiw–Teitelboim type (Jackiw–Teitelboim gravity, the minimal two-dimensional dilaton gravity with a linear dilaton potential, the standard solvable model of two-dimensional de Sitter holography) entangled with a non-gravitating bath; small SdS black holes with the cosmological background frozen and radiation collected by distant or future-boundary observers; and eternal (nonevaporating) multi-horizon backgrounds. Easson's model instead provides a fully backreacted, evaporating two-horizon geometry, but its Page statement is only a thermo-controlled estimate: a coarse-grained entropy-production proxy whose slope is set by the two horizon temperatures and the steady flux, crossed against the black-hole Bekenstein–Hawking entropy (the universal horizon-area entropy law) to estimate a Page time. The author states explicitly that a microscopic Page curve would require a full QES extremization in the fully backreacted geometry, and that the role of the conserved nonequilibrium flux in selecting the physically relevant island saddle has not been analyzed. Three further correction channels lie outside this two-dimensional conformal-anomaly framework: greybody factors (the frequency-dependent barrier transmission of four-dimensional propagation, absent in the two-dimensional s-wave reduction), massive or nonconformal fields (whose stress tensors are not fixed by the two-dimensional trace anomaly), and the dimensional-reduction anomaly (state-dependent terms generated when the angular modes of the four-dimensional theory are integrated out). The source paper argues these rescale the flux magnitude without changing the sign structure of the flux or the generalized second law, and lists them as directions for extending the model toward more realistic four-dimensional physics; their quantitative effect on the Page time and island structure is undetermined.

Problem Statement

In the two-dimensional dilaton-gravity framework of D. A. Easson, 'The fate of Schwarzschild–de Sitter black holes: nonequilibrium evaporation' (Phys. Rev. D 113, 084014 (2026)) — the spherical reduction of four-dimensional Einstein gravity with $\Lambda>0$, with Polyakov anomaly backreaction of $N$ conformal fields in the Unruh–de Sitter state, carrying the conserved Killing flux $J$ and the adiabatic mass loss $\dot M=-J$ — derive the microscopic fine-grained Page curve of the radiation emitted by the evaporating Schwarzschild–de Sitter black hole. Specifically:

(1) Specify the radiation region $\mathcal R$ for a static-patch observer (the natural choice being the region accessible to a static observer, bounded by the cosmological horizon), the global quantum state (the Unruh–de Sitter state with its state functions $t_u,t_v$), and an explicit ultraviolet subtraction convention for $S_{\mathrm{out}}$.

(2) Construct the generalized entropy $S_{\mathrm{gen}}[\partial\mathcal R]=X(\partial\mathcal R)/(4G_2)+S_{\mathrm{out}}[\mathcal R]$ on the fully backreacted, time-dependent geometry, and solve the quantum extremality conditions $\partial_\pm S_{\mathrm{gen}}=0$ explicitly (analytically or numerically) to locate the quantum extremal surface / island endpoint(s).

(3) Establish which of the saddle candidates located in item (2) — the inner (black-hole-side) island near $r_b$, a cosmological-side island near $r_c$, if present for the region $\mathcal R$ of item (1), or no island — dominates the fine-grained entropy for the static-patch observer specified in item (1).

(4) Compute the fine-grained Page curve $S_{\mathrm{fine}}(v)=\min\{S_{\mathrm{no\,island}}(v),\ S_{\mathrm{island}}(v)\}$, where $S_{\mathrm{island}}(v)$ is the minimum of the generalized entropy $S_{\mathrm{gen}}[\partial\mathcal I]$ over all island endpoints $\partial\mathcal I$ located in item (2), together with the Page time, and compare with the thermo-controlled estimate of the source paper (its Eqs. (51)–(58)).

The problem is solved when the fine-grained entropy of the model is obtained from an explicit quantum-extremal-surface calculation on the backreacted geometry (items 1–4), including a rigorous determination that no island saddle dominates (and hence no Page turnover occurs) should the extremization yield that outcome; delimiting the regime of adiabatic validity (away from the Nariai limit and the Planck-scale endpoint) is part of the answer.

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