Entanglement islands and black hole decay in regular dilaton gravity

TL;DR

This work examines the black hole information paradox in two-dimensional dilaton gravity with linear dilaton vacua, focusing on CGHS and regular sinh--CGHS models. It applies the island formula to compute entanglement entropy of Hawking radiation, obtaining a unitary Page-curve behavior for CGHS, while near-extremal regular black holes in the sinh--CGHS model yield divergences that challenge unitarity. To address endpoint issues, the authors develop a semiclassical regularization of shock-wave tunneling in a Vaidya-like spacetime, deriving a tunneling probability ${P}_{fi}\sim\exp(-\Delta S_ ext{BH})$ and showing extremal remnants are unstable, decaying into horizonless spacetimes on a calculable timescale ${\tau}$. This suggests that horizonless endpoints could restore unitary evolution, albeit with caveats about information transfer across dissolving horizons and the role of nonperturbative saddles. The results point to a potential path to unitary black-hole evaporation within a controlled toy-model setting and motivate extensions to higher dimensions and more complete quantum-gravity frameworks.

Abstract

We consider a class of two-dimensional dilaton gravity models with linear dilaton vacuum including Callan-Giddings-Harvey-Strominger (CGHS) model as the special case. General thermodynamic properties of black holes in such models are evaluated. We focus on the CGHS model and its modification with regular black holes as empty-space solutions characterized by ever-present finite curvature. We find generalized entanglement entropy blows-up for near-extremal regular black holes considered as remnants. That signalling a possible breakdown of the semiclassical approximation near the endpoint of evaporation. We conjecture that remnants are unstable and decay by quantum fluctuations into horizonless spacetimes. We give an estimate for the decay amplitude by using a semiclassical regularization method and propose a path to mitigate the unitarity loss problem.

Figures (7)

• Figure 1: Penrose diagram depicting setup for entanglement island. Legend: A, A$^*$ are anchor points, R, R$^*$ are parts of Cauchy surface containing old Hawking radiation, Q, Q$^*$ are QEPs, and I is island. Wavy line simbolizes geodesic incompleteness of spacetime either with singularities or Cauchy horizons.
• Figure 2: Plots of generalized entropy of the Hawking radiation for the CGHS model \ref{['eq:ent-cghs']} (dashed) and the sinh-CGHS model \ref{['eq:exact-sh-cghs-gen']} (thick). The asymptotic behaviour is given by Eq. \ref{['eq:large-m']} at $M\to\infty$ (thin) and Eq. \ref{['eq:sh-ent-mext']} at $M\to M_\mathrm{ext}$ (dashed thin).
• Figure 3: Page curve for sinh--CGHS black hole during evaporation. Monotonic dashed curve is Hawking answer for radiation $S_\mathrm{rad}(t)=2(S_\mathrm{BH}(0)-S_\mathrm{BH}(t))$. Contribution from island falls before it approaches $M_\mathrm{qb}$ and then starts to grow again exceeding Hawking answer. Remnants should decay to protect unitary evolution on timescale $\tau$.
• Figure 4: Spacetime diagram for solution of Eq. \ref{['eq:vaidya']}. Thin lines are null rays satisfying equation $M_\mathrm{ext}\cosh(2\lambda r(v))(1-2r'(v))={\cal M}(v)$. Vertical strip represents wave packet of matter forming initial black hole with closed apparent horizon (dashed curve). Black hole evaporates linearly producing a bulk of Hawking radiation. On the final stage it cools down forming a remnant which eventually decays into last burst of quantum radiation leaving empty space.
• Figure 5: Quantum xerox. Vertical strips are wave packets falling into black hole. After scrambling time the quantum information returns with Hawking radiation quanta (thin line). One can make a slicing $\Sigma$ so that quantum state $|\psi\rangle$ is represented simultaneously inside the black hole interior and in the Hawking radiation.