Emergent gravity requires kinematic nonlocality

This Letter refines arguments forbidding nonlinear dynamical gravity from appearing in the low energy effective description of field theories with local kinematics, even for those with instantaneous long-range interactions. Specifically, we note that gravitational theories with universal coupling to energy-an intrinsically nonlinear phenomenon-are characterized by Hamiltonians that are pure boundary terms on shell. In order for this to be the low energy effective description of a field theory with local kinematics, all bulk dynamics must be frozen and, thus, irrelevant to the construction. The result applies to theories defined either on a lattice or in the continuum, and requires neither Lorentz invariance nor translation invariance.