  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.
