Quantum Extremal Islands Made Easy, Part IV: Massive Black Holes on the Brane

TL;DR

This paper analyzes the entanglement entropy evolution of massive two-dimensional black holes in braneworld models, revealing phase transitions in the Page curve influenced by bath size and central charge ratios.

Contribution

It extends the study of quantum extremal islands to massive black holes on branes, including the effects of intrinsic JT gravity and bath size on the Page curve.

Findings

Late-time entropy depends on bath size, matching thermal or horizon entropy.

Page time scales with the ratio of central charges.

Transition from linear growth to constant entropy at late times.

Abstract

We study two-dimensional eternal black holes with non-zero mass, where each asymptotic boundary is in contact with a CFT on a circle, following the doubly holographic braneworld models constructed in [arXiv:2006.04851, arXiv:2010.00018, arXiv:2010.16398]. We compute the Page curve of the black hole (or the bath CFTs), which amounts to finding different geodesics in the bulk BTZ geometry with a Randall-Sundrum brane falling into the black hole. We also explore the possibility of including an intrinsic JT gravity action on the brane. As expected, the generalized entropy rises linearly at early times. However, there is a transition to a late-time phase in which the entropy remains constant. The value of the late-time entropy depends on the size of the thermal baths. For a small size, it corresponds to the thermal entropy of the baths, while for large size, it corresponds to twice the horizon entropy of the black hole. The critical size and the Page time are proportional to ratio of the central charges of the conformal defect and the bath CFT.