The holographic shape of entanglement and Einstein’s equations
Abstract
We study shapedeformations 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 doubledeformation 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 HollandsIyerWald 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 RyuTakayanagi 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 RyuTakayanagi formula for arbitrary subregions, then it must necessarily satisfy the nonlinear Einstein equation.
 Authors:

 Stanford Univ., CA (United States). Stanford Inst. for Theoretical Physics, Dept. of Physics
 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:
 FG0205ER41367
 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 10298479
 Publisher:
 Springer Berlin
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AdSCFT 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 shapedeformations 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 doubledeformation 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 HollandsIyerWald 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 RyuTakayanagi 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 RyuTakayanagi formula for arbitrary subregions, then it must necessarily satisfy the nonlinear 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}
}
Web of Science
Figures / Tables:
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