Variational principles for natural divergence-free tensors in metric field theories

Let Tab=Tba=0 be a system of differential equations for the components of a metric tensor on RmRm. Suppose that Tab transforms tensorially under the action of the diffeomorphism group on metrics and that the covariant divergence of Tab vanishes. We then prove that Tab=Eab(L) is the Euler–Lagrange expression of some Lagrangian density LL provided that Tab is of third order. Our result extends the classical works of Cartan, Weyl, Vermeil, Lovelock, and Takens on identifying field equations for the metric tensor with the symmetries and conservation laws of the Einstein equations.