Quantum extremal islands made easy. Part II. Black holes on the brane
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Springer
Received: October 19, 2020 Accepted: October 29, 2020 Published: December 3, 2020
Quantum extremal islands made easy. Part II. Black holes on the brane
a
Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada b Department of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada c Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, 14476 Potsdam, Germany
E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: We discuss holographic models of extremal and non-extremal black holes in contact with a bath in d dimensions, based on a brane world model introduced in [1]. 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. We show that the appearance of quantum extremal islands in those models is a consequence of the well-understood phase transition of RT surfaces, and does not make any direct reference to ensemble averaging. For non-extremal black holes the appearance of quantum extremal islands has the right behaviour to avoid the information paradox in any dimension. We further show that for these models the calculation of the full Page curve is possible in any dimension. The calculation reduces to numerically solving two ODEs. In the case of extremal black holes in higher dimensions, we find no quantum extremal islands for a wide range of parameters. In two dimensions, our results agree with [2] at leading order; however a finite UV cutoff introduced by the brane results in subleading corrections. For example, these corrections result in the quantum extremal surfaces moving further outward from the horizon, and shifting the Page transition to a slightly earlier time. Keywords: AdS-CFT Correspondence, Black Holes, Gauge-gravity correspondence ArXiv ePrint: 2010.00018
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2020)025
JHEP12(2020)025
Hong Zhe Chen,a,b Robert C. Myers,a Dominik Neuenfeld,a Ignacio A. Reyesc and Joshua Sandora
Contents 1 Introduction
1
2 Braneworld framework 2.1 Braneworlds in higher dimensions 2.2 Two dimensions and black holes
6 6 9 12 15 16 20 21 24 26 27 28
4 Numerical results 4.1 General behavior of the islands 4.2 The Page curve in d > 2
30 30 34
5 Extremal horizon in equilibrium with T = 0 bath 5.1 Island phase 5.2 No-island phase 5.3 Islands at T = 0 for d > 2
41 44 49 49
6 Two dimensions revisited 6.1 Brane action 6.2 Bulk and brane geometries 6.3 Entropies: island and no-island phases 6.4 Page curve
52 53 55 57 64
7 Discussion
64
1
Introduction
Understanding the quantum description of black holes remains a central question in theoretical physics. One unresolved question is the fate of information during black hole evaporation. In his seminal work, Hawking argued that in a quantum theory black holes evaporate into a mixed state of radiation, independently of how the black hole was formed [3–5].
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JHEP12(2020)025
3 Black hole
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