Page curve for an evaporating black hole
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Springer
Received: April 22, 2020 Accepted: April 29, 2020 Published: May 20, 2020
Friðrik Freyr Gautason,a,b Lukas Schneiderbauer,b Watse Sybesmab and Lárus Thorlaciusb a
Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium b Science Institute University of Iceland, Dunhaga 3, 107 Reykjavík, Iceland
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: A Page curve for an evaporating black hole in asymptotically flat spacetime is computed by adapting the Quantum Ryu-Takayanagi (QRT) proposal to an analytically solvable semi-classical two-dimensional dilaton gravity theory. The Page time is found to be one third of the black hole lifetime, at leading order in semi-classical corrections. A Page curve is also obtained for a semi-classical eternal black hole, where energy loss due to Hawking evaporation is balanced by an incoming energy flux. Keywords: 2D Gravity, Black Holes, AdS-CFT Correspondence ArXiv ePrint: 2004.00598
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)091
JHEP05(2020)091
Page curve for an evaporating black hole
Contents 1
2 Page curve from QRT
3
3 The model 3.1 Coupling to matter 3.2 Semi-classical black holes
5 6 8
4 Generalized entropy
11
5 Page curves 5.1 Eternal black hole 5.2 Dynamical black hole 5.2.1 Island configuration 5.2.2 No-island configuration
12 12 15 16 18
6 Discussion
19
1
Introduction
If black hole evaporation is a unitary process, the entanglement entropy between the outgoing radiation and the quantum state associated to the remaining black hole is expected to follow the so-called Page curve as a function of time [1, 2]. Early on, the entanglement entropy is then a monotonically increasing function of time which closely tracks the coarse grained thermal entropy of the radiation that has been emitted up to that point. This changes when the coarse grained entropy of the radiation exceeds the coarse grained entropy associated to the remaining black hole, at which point the entanglement entropy is limited by the black hole entropy and becomes a decreasing function of time. The time when the entanglement entropy transitions from increasing to decreasing is referred to as the Page time. Reproducing the Page curve without explicitly assuming unitarity is an important step towards resolving Hawking’s black hole information paradox [3]. In a recent breakthrough, a Page curve was computed using semi-classical methods by studying black holes in asymptotically anti-de Sitter (AdS) spacetimes coupled to a conformal field theory (CFT) reservoir [4, 5].1 The result hinges on the use of the Quantum Ryu-Takayanagi (QRT) formula [7–10] and the existence of extremal hypersurfaces terminating on so-called islands behind the event horizon [11]. A version of the Page curve can also be obtained for eternal AdS black holes, but in this case the islands extend outside 1
See e.g. [6] for further details on such a setup.
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JHEP05(2020)091
1 Introduction
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