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

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 $\mathrm{AdS}/\mathrm{CFT}$ correspondence. We build upon the framework developed by Faulkner et al. using the Wald formalism and exploit the symmetries of the vacuum modular Hamiltonian of ball-shaped boundary regions.

Figure 1

Summary.

Figure 2

The Killing vector field $\xi$. The dashed line represents the boundary ball $B$ and the dotted line represents the RT-surface $\tilde{B}$. The shaded area is a Cauchy surface $\mathrm{\Sigma }(\tilde{B})$ that lies on a constant-time slice.