Page Curve for an Evaporating Black Hole

TL;DR

This work demonstrates that the quantum Ryu-Takayanagi (QRT) prescription can be applied to evaporating black holes in asymptotically flat spacetimes by leveraging a solvable 2D dilaton gravity model with holographic matter. By computing the generalized entropy with and without island configurations, the authors obtain Page curves for both dynamical and eternal black holes, finding a Page time of $t_{ ext{Page}} = rac{1}{3\, ext{ε}}$ for evaporation and $t_{ ext{Page}} = rac{1}{4\epsilon} = \frac{12M}{c}$ for the eternal case, with islands located inside the horizon for evaporation and outside for the eternal case. The bulk entropy is evaluated via AdS_3/CFT_2 holography, enabling analytic control over backreaction through the RST modification of CGHS gravity, and supporting the robustness of the island paradigm beyond AdS/CFT. Overall, the paper provides concrete, semi-classical evidence that QRT-type entanglement salvage via islands yields a Page curve consistent with unitary evolution in a flat spacetime setting, while clarifying the limitations and the regime of validity of the approach.

Abstract

A Page curve for an evaporating black hole in asymptotically flat spacetime is computed by adapting the Quantum Ryu-Takayanagi (QRT) proposal to an analytically solvable semi-classical two-dimensional dilaton gravity theory. The Page time is found to be one third of the black hole lifetime, at leading order in semi-classical corrections. A Page curve is also obtained for a semi-classical eternal black hole, where energy loss due to Hawking evaporation is balanced by an incoming energy flux.

Figures (5)

• Figure 1: Page curve for an evaporating RST black hole.
• Figure 2: Penrose diagram of an evaporating RST black hole formed from collapsing matter (green). A timelike anchor curve separates the spacetime into interior and exterior regions. As time evolves along this curve, more and more Hawking radiation has passed through it on its way to future null infinity. The island moves with time along the purple curve inside the event horizon.
• Figure 3: A Penrose diagram of an eternal black hole. A pair of timelike anchor curves (blue curves) separates the spacetime into an interior and two exteriors. The two spatial hypersurfaces intersect the anchor curves at different times. On the late time surface the generalized entropy is dominated by the area term associated to the islands denoted by purple dots.
• Figure 4: Page curve for the eternal RST black hole with $t_{\text{Page}}=6S_{\text{BH}}/c$. The graph plots $S_\text{gen}-\frac{c}{3}\sigma_A$ as a function of retarded time on the anchor curve.
• Figure 5: Penrose diagram of a dynamical RST black hole with two spacelike hypersurfaces indicated, one before the Page time and the other after, corresponding to the no-island and island configurations, respectively.