TITLE: Moment map interpretation of Einstein equation
QUESTION [5 upvotes]: 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?

REPLY [3 votes]: I'm not aware of a momentum map interpretation of the Einsteins's equation, but you can bring Einstein's equations in a Hamiltonian form with momentum map constraints (this is due to Fischer & Marsden).
In more detail, you write spacetime $M$ (locally) as the product $\mathbb{R} \times \Sigma$, where $\Sigma$ is a spatial Cauchy surface.
Then Einstein's equation split into dynamical equations of a time-dependent Riemannian metric on $\Sigma$ and two constraint equations (called the momentum constraint and the Hamiltonian constraint). The momentum constraints turns out to be of the form $J = 0$ for the momentum map $J$ of the natural $Diff(\Sigma)$ action on the cotangent bundle $T^* Metr(\Sigma)$. The Hamiltonian constraint is also related to the $Diff(M)$-symmetry of Einstein's equation but its interpretation as a momentum map constraint is more complicated (for this have a look at recent work by Blohmann & Weinstein).