Page curves for a family of exactly solvable evaporating black holes

TL;DR

The work analyzes entanglement entropy and Page curves in a one-parameter family of exactly solvable 2D dilaton gravities (RST-BPP) coupled to a conformal field theory. Using the quantum extremal surface/island framework, it shows that island configurations are invariant across the parameter, while the geometry-dependent CFT contribution governs the detailed entropy evolution; in evaporating cases the Page time scales with the black-hole lifetime and the maximum fine-grained entropy is bounded, with a third of the lifetime appearing as a characteristic transition. When two geometries are glued along a null surface, the island can jump between exterior and interior locations, producing a stretched Page curve with two distinct transitions and revealing subtle features about purification rates and the state of the radiation. Overall, the results extend the Page-curve analysis beyond JT-like AdS settings to a broader, exactly solvable, asymptotically flat 2D class and elucidate how gluing and backreaction shape information transfer in evaporating black holes.

Abstract

We study the entanglement entropy of a one-parameter family of exactly solvable gravities in the 2-dimensional asymptotically-flat space. The islands and Page curves of eternal, evaporating and bath-removed black holes are investigated. The different theories in this parameter class are identified through field redefinitions which leave the island invariant. The Page transition is found to occur at the first a third of the black hole life time in the evaporating case for this family of solutions. In addition, we consider gluing the equilibrium black hole and the evaporating one along a null trajectory and study the effect of gluing on the islands and Page curves. In the glued space, the island jumps across two different geometries at a certain retarded time. As a result, the Page transition is stretched and split into two separate ones -- the first transition happens when the net entropy generation stops and the second one occurs as the early radiation effectively starts to become purified. Finally, we discuss the issues concerning the inconsistent rates of purification and the paradox related to the state of the radiation.

Figures (7)

• Figure 1: Penrose diagram for an eternal dilaton black hole. The island is between the boundary points $I'$ and $I$. The radiation is collected beyond the cutoff surface at $A'$ and $A$ to the spacial infinity as indicated by $R$ and $R'$.
• Figure 2: The Penrose diagram for the formation of a dilaton black hole by a collapsing shell of an f wave. The vertical red lines are the boundaries of the spacetime. The green line represents the infalling matter wave. EH stands for the event horizon and AH is the apparent horizon. The radiation is collected near the null infinity as indicated by the blue line along $J^+$.
• Figure 3: The Page curve for an evaporating black hole formed by a shell collapse.
• Figure 4: Sketch of the Penrose diagram for an black hole with thermal bath removed at $x_0^+$. To the left of the null line $x_0^+$, the geometry is identical to that of an eternal black hole or the black hole in equilibrium with the infalling radiation emitted from an external source. "AH" represents the apparent horizon, "EH" stands for the event horizon. $J^-$ and $J^+$ represent the past and future null infinity, respectively. After the evaporation, the space returns to the asymptotically flat space with the boundary.
• Figure 5: The Page curve for a bath-removed evaporating black hole.