Einstein's famous equation relates the geometry of a (4-dimensional) manifold to the matter content in that manifold.

*Is there a way to obtain Einstein's equation as a moment map?*

More precisely, given a Riemannian 4-fold $M$ (maybe Lorentzian), is there an infinite-dimensional symplectic manifold $X$ with a hamiltonian action of a group $G$ such that the moment map is Einstein's equation?

For simplicity take for instance the case of an empty space (no matter). So Einstein's equation just tells that the Ricci tensor vanishes. Since the moment map is a map from $X$ to the dual Lie algebra of $G$, we could ask which group $G$ has a dual Lie algebra formed by Ricci tensors?