Abstract
Late-time dominance of entanglement islands plays a critical role in addressing the information paradox for black holes in AdS coupled to an asymptotic non-gravitational bath. A natural question is how this observation can be extended to gravitational systems. To gain insight into this question, we explore how this story is modified within the context of Karch-Randall braneworlds when we allow the asymptotic bath to couple to dynamical gravity. We find that because of the inability to separate degrees of freedom by spatial location when defining the radiation region, the entanglement entropy of radiation emitted into the bath is a time-independent constant, consistent with recent work on black hole information in asymptotically flat space. If we instead consider an entanglement entropy between two sectors of a specific division of the Hilbert space, we then find non-trivial time-dependence, with the Page time a monotonically decreasing function of the brane angle---provided both branes are below a particular angle. However, the properties of the entropy depend discontinuously on this angle, which is the first example of such discontinuous behavior for an AdS brane in AdS space.
Highlights
We suggest a modest extension of previous dynamical principles to determine R for KR two-brane setups by studying quantum extremal surfaces (QES) [39] via a doubly-holographic model, for which the location and the associated entropy of the QES in the d-dimensional theory is determined by a classical minimal (RT)
We argue that R should be dynamically determined by the extremization of the entangling surface, as with previous quantum extremal surface prescriptions in the gravitating region, 3A similar statement would be true to other types of geometries with event horizons, such as de Sitter space which was studied in [45,46,47]
We have shown how gravity in the bath completely changes the behavior of the Page curve for conventional entanglement entropy
Summary
An important and interesting advance in our understanding of the quantum mechanics of black holes is the set of recent calculations [1, 2] of the time evolution of the entanglement entropy between a holographic system that contains a black hole and an external bath. (See [3,4,5] for recent reviews). Based on these motivations and observations, we will study two distinct types of entanglement entropy for our system with a gravitating bath: one closer to previous work on the entropy of the black hole radiation that we demonstrate has no time-dependent Page curve and another entropy that we define which follows an interesting Page curve even in this system In both cases we consider a zero temperature (vacuum) setup as well as one with a thermal black string configuration [37, 38]. The left/right division alluded to above, refers to a well-defined internal division of the degrees of freedom of this (d − 1)-dimensional CFT into two subsystems These subsystems are coupled but correspond to distinct factors of the Hilbert space, and so one can ask about the entanglement entropy between them and the time-dependence of this entanglement entropy. Appendix A provides details of the area growth of the Hartman-Maldacena surface in the black string geometry
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