The holographic shape of entanglement and Einstein’s equations

Lewkowycz, Aitor; Parrikar, Onkar

doi:10.1007/jhep05(2018)147

The holographic shape of entanglement and Einstein’s equations

Journal Article · Wed May 23 00:00:00 EDT 2018 · Journal of High Energy Physics (Online)

DOI:https://doi.org/10.1007/jhep05(2018)147· OSTI ID:1507703

Lewkowycz, Aitor ^[1]; Parrikar, Onkar ^[2]

Stanford Univ., CA (United States). Stanford Inst. for Theoretical Physics, Dept. of Physics; Stanford Univ., CA (United States)
Univ. of Pennsylvania, Philadelphia, PA (United States). David Rittenhouse Lab.

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.

View Accepted Manuscript (DOE)

Research Organization:: Duke Univ., Durham, NC (United States)

Sponsoring Organization:: USDOE

Grant/Contract Number:: FG02-05ER41367

OSTI ID:: 1507703

Alternate ID(s):: OSTI ID: 22743085

Journal Information:: Journal of High Energy Physics (Online), Journal Name: Journal of High Energy Physics (Online) Journal Issue: 5 Vol. 2018; ISSN 1029-8479

Publisher:: Springer BerlinCopyright Statement

Country of Publication:: United States

Language:: English

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