Belinfante Tensors Induced by Matter-Gravity Couplings

We show that any generally covariant coupling of matter fields to gravity gives rise to a conserved, on-shell symmetric energy-momentum tensor equivalent to the canonical energy-momentum tensor of the flat-space theory. For matter fields minimally coupled to gravity our algorithm gives the conventional Belinfante tensor. We establish that different matter-gravity couplings give metric energy-momentum tensors differing by identically conserved tensors. We prove that the metric energy-momentum tensor obtained from an arbitrary gravity theory is on-shell equivalent to the canonical energy-momentum tensor of the flat-space theory.