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 action with respect to space-time metric, is a under general coordinate transformations, symmetric in its indices, and BRST invariant. canonical has none these properties but Hamiltonian does correctly generate time dependence fields. It shown that $\int d^{3}x\,\sqrt{g}\;T^{0}_{\;\; 0}$ equal theory coupled spin 1/2 0 matter fields (including an $R\phi^{2}$ term) static background metric ($\partial_{0}g_{\mu\nu}=0$ $g_{0j}=0$). equality depends on Gauss's law constraint not dynamical Euler-Lagrange equations.