# 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:

- 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:
- 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}

}

*Citation information provided by*

Web of Science

Web of Science

#### Figures / Tables:

*R*, the dashed blue line is

*R*, and the solid black dot is $\partial$

^{c}*R*. Also shown are the cylindrical tube surrounding the entangling surface $\partial$

*R*

_{B}, and the cutmore »

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Figures / Tables found in this record:

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*