Islands in cosmology
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Springer
Received: September 19, 2020 Accepted: October 11, 2020 Published: November 20, 2020
Thomas Hartman, Yikun Jiang and Edgar Shaghoulian Department of Physics, Cornell University, Ithaca, New York, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: A quantum extremal island suggests that a region of spacetime is encoded in the quantum state of another system, like the encoding of the black hole interior in Hawking radiation. We study conditions for islands to appear in general spacetimes, with or without black holes. They must violate Bekenstein’s area bound in a precise sense, and the boundary of an island must satisfy several other information-theoretic inequalities. These conditions combine to impose very strong restrictions, which we apply to cosmological models. We find several examples of islands in crunching universes. In particular, in the four-dimensional FRW cosmology with radiation and a negative cosmological constant, there is an island near the turning point when the geometry begins to recollapse. In a twodimensional model of JT gravity in de Sitter spacetime, there are islands inside crunches that are encoded at future infinity or inside bubbles of Minkowski spacetime. Finally, we discuss simple tensor network toy models for islands in cosmology and black holes. Keywords: AdS-CFT Correspondence, Black Holes ArXiv ePrint: 2008.01022
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)111
JHEP11(2020)111
Islands in cosmology
Contents 1 Introduction 1.1 Summary of examples
1 4
2 Review of the island rule
6 8 8 11 12 13 14 14 16
4 Matter entropy in FRW 4.1 Thermofield double setup 4.2 Two copies of Minkowski 4.3 Turning on the scale factor
17 17 19 21
5 No 5.1 5.2 5.3
22 22 24 25
islands in radiation-dominated FRW Application of the general constraints Explicit check Positive cosmological constant
6 Islands in recollapsing FRW 6.1 Bekenstein area bound and the quantum normal regions 6.2 Islands for |tR | . β 6.2.1 Subleading analysis and the QNEC bound 6.3 Islands for |tR | & β
25 26 28 29 29
7 dS2 7.1 Islands in dS2 7.2 Bekenstein area bound and the quantum normal regions 7.3 Pure de Sitter in higher dimensions?
30 33 35 37
8 Minkowski bubble in dS2
38
9 Tensor network picture 9.1 Evaporating black hole 9.2 Tensor network for FRW
40 41 42
A Entanglement across the thermofield double in 2d CFT
43
–i–
JHEP11(2020)111
3 General conditions on islands 3.1 Condition 1: the area bound is violated 3.2 Condition 2: I is quantum normal 3.3 Condition 3: G is quantum normal 3.4 Comments on the three conditions 3.5 The island is a maximum in the time direction 3.6 Examples 3.7 Comments on the Bousso bound
B Derivation of the timelike-maximum requirement
45
C Details of the CGHS/RST example
46
D Derivation of the bound on |∂Smat |
49
Introduction
The holographic principle suggests that the entropy of a region in quantum gravity is bounded by its area in Planck units, S≤
Area . 4
(1.1)
For a st
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