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.
Highlights
While the naive ‘Hawking paradox’ is not robust to small non-perturbative corrections, as argued for example in [3], interest was reinvigorated by the discovery [4,5,6] of a version that is robust to these small corrections
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
To explore the question of whether such a topology transition happens, we study various models in which the bath is a holographic system. (See [61], which previously studied a case with a gravitational bath and found similar results for the Page curve.) As a cautionary remark, we remind the reader that the question of the topology of the spatial slice in the bulk dual of a state is a somewhat meaningless one, since there is no non-perturbatively diffeomorphism-invariant linear operator that can measure the ‘connectedness’ of spacetime [62,63,64]
Summary
The 2d gravity theory we work with throughout this paper is Jackiw-Teitelboim (JT). gravity with end-of-the-world (ETW) branes. Because of the dilaton equation of motion, the bulk is always AdS2 The dynamics of this theory reduces to the dynamics of a boundary particle, whose location is the boundary of the ‘cutout’ of AdS2. Since the bulk is always AdS2, we can use conventional coordinate systems. In real time, they are Kruskal-Szekeres : Poincare : Global : ds. We adopt a consistent notation, which we summarise in table 1, for these quantities
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