Islands with Gravitating Baths: Towards ER = EPR

TL;DR

The paper investigates how entanglement between two JT gravity systems, each coupled to a gravitating bath, can trigger ER=EPR-like transitions between disconnected black holes and wormhole-connected geometries. By studying three models—a doubled PSSY setup, dynamical coupling via a 2d CFT bath, and real-time coupling of ends of a two-sided black hole—the authors compute Page curves and entanglement entropies using the island rule and quantum extremal surfaces, uncovering entanglement-driven Hawking-Page-like transitions in some cases and more subtle, operator-dependent behavior in dynamical scenarios. A key result is that, in the doubled PSSY model, sufficiently large entanglement rank induces a ket-ket wormhole, realizing ER=EPR at the bulk level, while in dynamical settings the dominance of connected saddles can depend on conformal factors, insertions, and ground-state projections. Overall, the work links entanglement structure to bulk topology in two-dimensional gravity, offering concrete realizations and raising questions about generalization to higher dimensions and the precise role of averaging and replica wormholes in ER=EPR-like physics.

Abstract

We study the Page curve and the island rule for black holes evaporating into gravitating baths, with an eye towards establishing a connection with the ER=EPR proposal. We consider several models of two entangled 2d black holes in Jackiw-Teitelboim (JT) gravity with negative cosmological constant. The first, "doubled PSSY," model is one in which the black holes have end-of-the-world (ETW) branes with a flavour degree of freedom. We study highly entangled states of this flavour degree of freedom and find an entanglement-induced Hawking-Page-like transition from a geometry with two disconnected black holes to one with a pair of black holes connected by a wormhole, thus realising the ER = EPR proposal. The second model is a dynamical one in which the ETW branes do not have internal degrees of freedom but the JT gravity is coupled to a 2d CFT, and we entangle the black holes by coupling the two CFTs at the $AdS$ boundary and evolving for a long time. We study the entanglement entropy between the two black holes and find that the story is substantially similar to that with a non-gravitating thermal bath. In the third model, we couple the two ends of a two-sided eternal black hole and evolve for a long time. Finally, we discuss the possibility of a Hawking-Page-like transition induced by real-time evolution that realises the ER = EPR proposal in this dynamical setting.

Figures (19)

• Figure 1: Schematic representation of the entanglement wedge transition in a model of two coupled black holes. The purple line is the HRT surface whose generalised entropy calculates the EE between the two boundary CFTs. At early times, each black hole interior is contained in the entanglement wedge of the corresponding boundary CFT. At late times, based on the island rule, we expect that the build-up of entanglement between the interiors causes one of the interiors to become part of the entanglement wedge of the other CFT. In this paper, we ask if this is the same as the entanglement lines condensing into a geometric wormhole.
• Figure 2: The geometries contributing to \ref{['eq:dpssy-norm']}. Here $\ell_1$ and $\ell_2$ are the lengths of one half (i.e., the "ket" part) of the Euclidean boundaries.
• Figure 3: The naive bulk dual of $\ket{\psi}$.
• Figure 4: The disconnected and connected histories, without the splitting quench. The two boundaries are coupled at $u = 0$, which produces a symmetric pair of shocks (orange), kicking the boundary particle outward and moving the future horizon towards the boundary; all causal horizons are green lines. We have also indicated the rough position of the late time QESs.
• Figure 5: On the left, the flat strip is a rectangle in $y$ coordinates. An infinite rectangle given by $u > u_{0}$ is squeezed into a small finite volume in Poincare $x$ coordinates. This is the source of the large conformal factor in \ref{['eqn:logZ-disc']}. A similar squeezing occurs in the connected history, except that we consider global rather than Poincare coordinates.