Replica wormholes and the entropy of Hawking radiation
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Springer
Received: February 17, 2020 Accepted: April 19, 2020 Published: May 5, 2020
Ahmed Almheiri,a Thomas Hartman,b Juan Maldacena,a Edgar Shaghoulianb and Amirhossein Tajdinib a
Institute for Advanced Study, Princeton, New Jersey, U.S.A. b Department of Physics, Cornell University, Ithaca, New York, U.S.A.
E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: The information paradox can be realized in anti-de Sitter spacetime joined to a Minkowski region. In this setting, we show that the large discrepancy between the von Neumann entropy as calculated by Hawking and the requirements of unitarity is fixed by including new saddles in the gravitational path integral. These saddles arise in the replica method as complexified wormholes connecting different copies of the black hole. As the replica number n → 1, the presence of these wormholes leads to the island rule for the computation of the fine-grained gravitational entropy. We discuss these replica wormholes explicitly in two-dimensional Jackiw-Teitelboim gravity coupled to matter. Keywords: 2D Gravity, Black Holes, Models of Quantum Gravity ArXiv ePrint: 1911.12333
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)013
JHEP05(2020)013
Replica wormholes and the entropy of Hawking radiation
Contents 1 Introduction 1.1 The island rule for computing gravitational von Neumann entropies 1.2 Two dimensional eternal black holes and the information paradox 1.3 Replica wormholes to the rescue
1 3 5 6 9 9 11
3 Single interval at finite temperature 3.1 Geometry of the black hole 3.2 Quantum extremal surface 3.3 Setting up the replica geometries 3.4 Replica solution as n → 1 3.5 Entropy 3.6 High-temperature limit
15 17 17 18 20 21 22
4 Single interval at zero temperature 4.1 Quantum extremal surface 4.2 Replica wormholes at zero temperature
23 23 24
5 Two intervals in the eternal black hole 5.1 Review of the QES 5.2 Replica wormholes 5.3 Purity of the total state
25 25 26 28
6 Comments on reconstructing the interior
28
7 Discussion
30
A Derivation of the gravitational action
33
B Linearized solution to the welding problem
35
C The equation of motion in Lorentzian signature
36
1
Introduction
Hawking famously noted that the process of black hole formation and evaporation seems to create entropy [1]. We can form a black hole from a pure state. The formation of the black hole horizon leaves an inaccessible region behind, and the entanglement of quantum fields across the horizon is responsible for the thermal nature of the Hawking radiation as well as its growing entropy.
–1–
JHEP05(2020)013
2 The replica trick for the von Neumann entropy 2.1 The replicated action for n ∼ 1 becomes the generalized entropy 2.2 The two dimensional JT gravity theory plus a CFT
A useful diagnostic for information loss is the fine-grained (von Neumann) entropy of the Hawking radiation, SR = −Tr ρR log ρR , where ρR is the density matrix of the radiation. This entrop
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