energy of the gravitational field

The energy-momentum tensor \(T_{\mu\nu}\) of any non-gravitational field is a local quantity and can be computed from the field and its derivatives at every single point. But for gravitational fields, a local freely-falling frame sets the metric \(g_{\mu\nu}\) and its first derivatives \(\Gamma^\alpha_{\mu\nu}\) to zero, so that the energy momentum-tensor would vanish locally. Where’s the energy and the momentum of a gravitational field?

If one thinks about the question, the situation is even more subtle: Clearly, the gravitational field is contained in the curvature which is computed from the second derivatives of the metric (in pseudo-Riemannian geometry with torsion-free metric compatible connections), but there is a part of the curvature, the Ricci-curvature, which is a local quantity, because it’s only sourced by the energy-momentum tensor \(T_{\mu\nu}\) of the fields or fluids at the same point. But surely do vacuum solutions such as the Schwarzschild solution have a nonzero energy-momentum content, so it must depend on the full Riemann curvature.