  The entanglement "first law" in conformal field theories relates the
entanglement entropy for a ball-shaped region to an integral over the same
region involving the expectation value of the CFT stress-energy tensor, for
infinitesimal perturbations to the CFT vacuum state. In recent work, this was
exploited at leading order in $N$ in the context of large N holographic CFTs to
show that any geometry dual to a perturbed CFT state must satisfy Einstein's
equations linearized about pure AdS. In this note, we investigate the
implications of the leading 1/N correction to the exact CFT result. We show
that these corrections give rise to the source term for the gravitational
equations: for semiclassical bulk states, the expectation value of the bulk
stress-energy tensor appears as a source in the linearized equations. In
particular, the CFT first law leads to Newton's Law of gravitation and the fact
that all sources of stress-energy source the gravitational field. In our
derivation, this universality of gravity comes directly from the universality
of entanglement (the fact that all degrees of freedom in a subsystem contribute
to entanglement entropy).
