The black hole information paradox in a brane world

Abstract

Recent progress in our understanding of the black hole information paradox has led to a new prescription for calculating entanglement entropy, which involves special subsystems in regions where gravity is dynamical, called quantum extremal islands. We present a simple holographic framework where the emergence of quantum extremal islands can be understood in terms of the standard Ryu-Takayanagi prescription, used for calculating entanglement entropy under the anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. Our setup describes a d-dimensional boundary CFT coupled to a (d-1)-dimensional defect, which are dual to a (d+1)-dimensional global AdS spacetime containing a codimension-one brane. Through the Randall-Sundrum mechanism, graviton modes become localized at the brane and, in a certain parameter regime, an effective description of the brane is given by Einstein gravity on a d-dimensional AdS background coupled to two copies of the boundary CFT. Within this effective description, the standard Ryu-Takayanagi formula implies the existence of quantum extremal islands in the gravitating region, whenever Ryu-Takayanagi surfaces cross the brane. Considered with Rindler and Poincaré coordinates respectively, our setup may be viewed as a special class of non-extremal and extremal black holes on the brane, in equilibrium with non-gravitational bath systems. For non-extremal black holes in any dimension, the appearance of quantum extremal islands has the right behaviour to avoid the information paradox and we show that the calculation of the full Page curve is possible. In the case of extremal black holes in higher dimensions, we find no quantum extremal islands for a wide range of parameters. The main benefit of our setup is that it allows for a high degree of analytic control as compared to previous work in higher dimensions. In two dimensions, we find agreement with previous work at leading order; however, a finite ultraviolet cutoff introduced by the brane results in subleading corrections.