Islands in Asymptotically Flat 2D Gravity

TL;DR

Hartman, Shaghoulian, and Strominger analyze islands in the 2D RST model to test semiclassical island rule entropy against the traditional Hawking flux. They derive the quantum extremal surface and compute the island-rule generalized entropy $S_{ m gen}$ for evaporating and eternal black holes, showing that, at leading order, the island rule reproduces the Page curve and Hayden–Preskill scrambling time, while the actual Hawking flux remains governed by information loss in this model. The results expose a tension between the island-rule predictions and Hawking-flux behavior in 2D dilaton gravity, and they illustrate how island surfaces inside the interior can map to retarded times at ${ m I}^+$, providing a self-consistent check of the island proposal. Overall, the work demonstrates how the island rule can yield unitary information retrieval in a semiclassical setting, at least in these toy models, and highlights the need for a deeper understanding of the gravitational path integral and boundary conditions in asymptotically flat spacetimes.

Abstract

The large-N limit of asymptotically flat two-dimensional dilaton gravity coupled to N free matter fields provides a useful toy model for semiclassical black holes and the information paradox. Analyses of the asymptotic information flux as given by the entanglement entropy show that it follows the Hawking curve, indicating that information is destroyed in these models. Recently, motivated by developments in AdS/CFT, a semiclassical island rule for entropy has been proposed. We define and compute the island rule entropy for black hole formation and evaporation in the large-N RST model of dilaton gravity and show that, in contrast, it follows the unitary Page curve. The relation of these two observations, and interesting properties of the dilaton gravity island rule, are discussed.

Figures (7)

• Figure 1: Linear dilaton vacuum. The physical spacetime is the unshaded region, with a boundary at $\Omega = \frac{1}{4}$.
• Figure 2: Evaporating black hole in Kruskal coordinates. The apparent horizon $AH$ is on the dashed line and the evaporation endpoint is marked $EP$. The dotted line prior to $EP$ is the event horizon.
• Figure 3: Points are labeled by $(\sigma^+_P, \sigma^+_{\overline{P}})$, where $\sigma^+_{\overline{P}}$ is the value of $\sigma^+$ for the image point obtained by reflecting across the timelike boundary.
• Figure 4: Island in the evaporating black hole. The Observer at $P_O$ collects the radiation in region $R$. The island is $I$, and $I \cup B \cup R$ is a Cauchy slice. The endpoint of the island is the quantum extremal surface $P_Q$. The dashed lines are the apparent horizon and the curve of extremal surfaces.
• Figure 5: Page curve (assuming a large initial mass ${M} \gg 1$).