Islands in asymptotically flat 2D gravity
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Springer
Received: May 28, 2020 Accepted: June 15, 2020 Published: July 3, 2020
Thomas Hartman,a Edgar Shaghouliana and Andrew Stromingerb a
Department of Physics, Cornell University, Ithaca, New York, U.S.A. b Department of Physics, Harvard University, Cambridge, MA, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: The large-N limit of asymptotically flat two-dimensional dilaton gravity coupled to N free matter fields provides a useful toy model for semiclassical black holes and the information paradox. Analyses of the asymptotic information flux as given by the entanglement entropy show that it follows the Hawking curve, indicating that information is destroyed in these models. Recently, motivated by developments in AdS/CFT, a semiclassical island rule for entropy has been proposed. We define and compute the island rule entropy for black hole formation and evaporation in the large-N RST model of dilaton gravity and show that, in contrast, it follows the unitary Page curve. The relation of these two observations, and interesting properties of the dilaton gravity island rule, are discussed. Keywords: 2D Gravity, Black Holes, Models of Quantum Gravity ArXiv ePrint: 2004.13857
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)022
JHEP07(2020)022
Islands in asymptotically flat 2D gravity
Contents 1
2 Review of the RST model 2.1 Large-N action and equation of motion 2.2 Solutions 2.2.1 Linear dilaton vacuum 2.2.2 Eternal black hole 2.3 Evaporating black hole in RST 2.3.1 Entanglement entropy
4 4 6 6 7 8 9
3 The island rule for an evaporating black hole 3.1 Quantum extremal surface 3.2 Evaporating black hole 3.2.1 When does the QES hit the singularity? 3.3 Page curve 3.4 Scrambling time
11 11 13 14 15 16
4 The eternal black hole
18
A Local analysis of replica wormholes A.1 Replica geometries A.2 Derivation of the QES
21 21 23
1
Introduction
Many years ago, Gibbons and Hawking [1] showed that the entropy of a black hole can be formally but simply calculated in the semiclassical limit from the Euclidean path integral. That calculation is now understood, at least in some cases, to correctly enumerate the quantum microstate degeneracy of the black hole. It is a wonderful and deep surprise that semiclassical gravity is smart enough to reproduce this degeneracy. In the intervening decades, the unreasonable efficacy of semiclassical gravity has been repeatedly demonstrated in disparate contexts. Most cases however involve symmetries of one kind or another — such as time translations. Recently [2–6] a remarkable proposal has been made invoking semiclassical gravity to analyze the information flow out of black holes. The starting point is the famous and well-understood Ryu-Takayanagi formula [7] for antideSitter space (AdS), which semiclassically computes microscopic quantum entanglement entropies in terms of areas of extremal surfaces. Recent proposed generalizations of this formula define an ‘island rule entropy’
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