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Emergent Gravity requires (kinematic) non-locality

This work refines arguments forbidding non-linear 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 non-linear 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.