Equality of the Hilbert Hamiltonian and the canonical Hamiltonian for gauge theories in a static spacetime

The Hilbert energy-momentum tensor for gauge-fixed non-Abelian gauge theories, defined by the variational derivative of the action with respect to the space-time metric, is a tensor under general coordinate transformations, symmetric in its indices, and BRST invariant. The canonical energy-momentum tensor has none of these properties but the canonical Hamiltonian does correctly generate the time dependence of the fields. It is shown that the Hilbert Hamiltonian $\int d^{3}x\sqrt{g}{T}_0^0$ is equal to the canonical Hamiltonian for a general gauge theory coupled to spin 1/2 and spin 0 matter fields (including an $R{\phi }^2$ term) in a static background metric (${\mathrm{\partial }}_0{g}_{\mu \nu }=0$ and $g_{0j}=0$). The equality depends on on the Gauss’s law constraint but not on the dynamical Euler-Lagrange equations.