  Entanglement entropy obeys a 'first law', an exact quantum generalization of
the ordinary first law of thermodynamics. In any CFT with a semiclassical
holographic dual, this first law has an interpretation in the dual
gravitational theory as a constraint on the spacetimes dual to CFT states. For
small perturbations around the CFT vacuum state, we show that the set of such
constraints for all ball-shaped spatial regions in the CFT is exactly
equivalent to the requirement that the dual geometry satisfy the gravitational
equations of motion, linearized about pure AdS. For theories with entanglement
entropy computed by the Ryu-Takayanagi formula $S=A/(4G_N)$, we obtain the
linearized Einstein equations. For theories in which the vacuum entanglement
entropy for a ball is computed by more general Wald functionals, we obtain the
linearized equations for the associated higher-curvature theories. Using the
first law, we also derive the holographic dictionary for the stress tensor,
given the holographic formula for entanglement entropy. This method provides a
simple alternative to holographic renormalization for computing the stress
tensor expectation value in arbitrary higher derivative gravitational theories.
