  In this paper, we demonstrate the emergence of nonlinear gravitational
equations directly from the physics of a broad class of conformal field
theories. We consider CFT excited states defined by adding sources for scalar
primary or stress tensor operators to the Euclidean path integral defining the
vacuum state. For these states, we show that up to second order in the sources,
the entanglement entropy for all ball-shaped regions can always be represented
geometrically (via the Ryu-Takayanagi formula) by an asymptotically AdS
geometry. We show that such a geometry necessarily satisfies Einstein's
equations perturbatively up to second order, with a stress energy tensor
arising from matter fields associated with the sourced primary operators. We
make no assumptions about AdS/CFT duality, so our work serves as both a
consistency check for the AdS/CFT correspondence and a direct demonstration
that spacetime and gravitational physics can emerge from the description of
entanglement in conformal field theories.
