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If entanglement entropy in a small geodesic ball is maximized at fixed volume in the vacuum, then it should be stationary under variation to a nearby state. I will show that this stationarity condition is equivalent to the semiclassical Einstein equation. If the matter QFT is not conformal, then the derivation requires a further assumption about QFT, whose validity is currently under investigation. [Based on http://arxiv.org/abs/1505.04753]

We discuss area terms in entanglement entropy and show that a recent formula by Rosenhaus and Smolkin is equivalent to the Adler-Zee sum rule for the renormalization of the Newton constant in terms of correlator of traces of the stress tensor. We elaborate on how to fix the ambiguities in these formulas: Improving terms for the stress tensor of free fields, boundary terms in the modular Hamiltonian, and contact terms in the Euclidean correlation functions. We make computations for free fields and show how to apply these calculations to understand some results for interacting theories which have been studied in the literature. We check the sum rule holographicaly. We also discuss an application to the F-theorem.