Holographic equivalence between the first law of entanglement entropy and the linearized gravitational equations

TL;DR

This paper demonstrates a holographic equivalence between the first law of entanglement entropy and linearized gravitational equations using integral transformations and symmetries in the AdS/CFT framework.

Contribution

It provides a concise proof of the holographic equivalence leveraging integral transformations and vacuum modular Hamiltonian symmetries.

Findings

Established a compact proof of the holographic equivalence

Utilized integral transformations and symmetries in the proof

Built upon the Wald formalism and previous frameworks

Abstract

We use the intertwining properties of integral transformations to provide a compact proof of the holographic equivalence between the first law of entanglement entropy and the linearized gravitational equations, in the context of the AdS/CFT-correspondence. We build upon the framework developed by Faulkner et al. [1] using the the Wald formalism, and exploit the symmetries of the vacuum modular Hamiltonian of ball-shaped boundary regions.