Gravitational dynamics from entanglement thermodynamics

In a general conformal field theory, perturbations to the vacuum state obey the relation delta S = delta E, where delta S is the change in entanglement entropy of an arbitrary ball-shaped region, and delta E is the change in hyperbolic energy of this region. In this note, we show that for holographic conformal field theories, this relation, together with the holographic connection between entanglement entropies and areas of extremal surfaces and the standard connection between the field theory stress tensor and the boundary behavior of the metric, implies that geometry dual to the perturbed state satisfies Einstein's equations expanded to linear order about pure AdS.