  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
($\partial_{0}g_{\mu\nu}=0$ and $g_{0j}=0$). The equality depends on on the
Gauss's law constraint but not on the dynamical Euler-Lagrange equations.
