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Title: The holographic shape of entanglement and Einstein’s equations

Abstract

We study shape-deformations of the entanglement entropy and the modular Hamiltonian for an arbitrary subregion and state (with a smooth dual geometry) in a holographic conformal field theory. More precisely, we study a double-deformation comprising of a shape deformation together with a state deformation, where the latter corresponds to a small change in the bulk geometry. Using a purely gravitational identity from the Hollands-Iyer-Wald formalism together with the assumption of equality between bulk and boundary modular flows for the original, undeformed state and subregion, we rewrite a purely CFT expression for this double deformation of the entropy in terms of bulk gravitational variables and show that it precisely agrees with the Ryu-Takayanagi formula including quantum corrections. As a corollary, this gives a novel, CFT derivation of the JLMS formula for arbitrary subregions in the vacuum, without using the replica trick. Finally, we use our results to give an argument that if a general, asymptotically AdS spacetime satisfies the Ryu-Takayanagi formula for arbitrary subregions, then it must necessarily satisfy the non-linear Einstein equation.

Authors:
 [1];  [2]
  1. Stanford Univ., CA (United States). Stanford Inst. for Theoretical Physics, Dept. of Physics
  2. Univ. of Pennsylvania, Philadelphia, PA (United States). David Rittenhouse Lab.
Publication Date:
Research Org.:
Duke Univ., Durham, NC (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1507703
Grant/Contract Number:  
FG02-05ER41367
Resource Type:
Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2018; Journal Issue: 5; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AdS-CFT Correspondence; Conformal Field Theory

Citation Formats

Lewkowycz, Aitor, and Parrikar, Onkar. The holographic shape of entanglement and Einstein’s equations. United States: N. p., 2018. Web. doi:10.1007/jhep05(2018)147.
Lewkowycz, Aitor, & Parrikar, Onkar. The holographic shape of entanglement and Einstein’s equations. United States. doi:10.1007/jhep05(2018)147.
Lewkowycz, Aitor, and Parrikar, Onkar. Wed . "The holographic shape of entanglement and Einstein’s equations". United States. doi:10.1007/jhep05(2018)147. https://www.osti.gov/servlets/purl/1507703.
@article{osti_1507703,
title = {The holographic shape of entanglement and Einstein’s equations},
author = {Lewkowycz, Aitor and Parrikar, Onkar},
abstractNote = {We study shape-deformations of the entanglement entropy and the modular Hamiltonian for an arbitrary subregion and state (with a smooth dual geometry) in a holographic conformal field theory. More precisely, we study a double-deformation comprising of a shape deformation together with a state deformation, where the latter corresponds to a small change in the bulk geometry. Using a purely gravitational identity from the Hollands-Iyer-Wald formalism together with the assumption of equality between bulk and boundary modular flows for the original, undeformed state and subregion, we rewrite a purely CFT expression for this double deformation of the entropy in terms of bulk gravitational variables and show that it precisely agrees with the Ryu-Takayanagi formula including quantum corrections. As a corollary, this gives a novel, CFT derivation of the JLMS formula for arbitrary subregions in the vacuum, without using the replica trick. Finally, we use our results to give an argument that if a general, asymptotically AdS spacetime satisfies the Ryu-Takayanagi formula for arbitrary subregions, then it must necessarily satisfy the non-linear Einstein equation.},
doi = {10.1007/jhep05(2018)147},
journal = {Journal of High Energy Physics (Online)},
number = 5,
volume = 2018,
place = {United States},
year = {2018},
month = {5}
}

Journal Article:
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Cited by: 8 works
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Figures / Tables:

Figure 1 Figure 1: An illustration of the Euclidean path integral representation of the reduced density matrix. The solid blue line denotes the region R, the dashed blue line is Rc, and the solid black dot is $\partial$R. Also shown are the cylindrical tube surrounding the entangling surface $\partial$RB, and the cutmore » at θ = 0.« less

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