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

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.