  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.
