In Misner, Thorne and Wheeler's Gravitation exercise 21.3, the authors suggest
For a more ambitious project, show that any stress-energy tensor derived from a field Lagrangian by the prescription of equation (21.33) will automatically satisfy the conservation law $T_{\alpha\beta}^{\hphantom{\alpha\beta};\beta}=0$.
Equation 21.33: $$T_{\alpha\beta}=-2\frac{\delta L_{field}}{\delta g^{\alpha\beta}}+g_{\alpha\beta}L_{field}.\tag{21.33}$$
I was able to show that for a scalar field Lagrangian of the form $L(\phi,g^{\mu\nu}\phi_{,\mu}\phi_{,\nu})$ the Euler-Lagrange equation together with the definition of the stress-energy tensor in equation (21.33) imply that $T_{\alpha\beta}^{\hphantom{\alpha\beta};\beta}=0$. The key point seemed to be the connection between derivatives with respect to $g^{\alpha\beta}$ and those with respect to the field and its derivatives given by the Euler-Lagrange equation. In the case of a vector field, however, the possible forms of terms in the Lagrangian seem to be infinite (it seems to always be possible to contract a higher tensor power of $\nabla A$ along combinations of "slots" that do not come from combinations of lower powers, unlike the scalar case).
Is this "ambitious project" well-defined and doable? If so, what is the right way to approach the vector field (or more general) case?
