# Moment map interpretation of Einstein equation

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?

• Einstein' equation takes values in $\Gamma(S^2T^*M)$. So this should be the dual of a Lie algebra in some sense. Prime candidate is $\Gamma(S^2TM)$ with bracket $[X_1\vee X_1,Y_1\vee Y_2] = ?$ The natural bracket would take values in $\Gamma(S^3TM)$ and is part of a graded bracket. So I do not think there is an obvious natural interpretation. Jul 19, 2019 at 19:11
• @PeterMichor Being next to ignorant of the field I might be completely off mark, but some work of Anton Zeitlin (see some links at his home page, e. g. these slides) seems to express Einstein equations through Maurer-Cartan equations, which in turn seem to give rise to certain moment maps (seen this in a paper by Gan and Ginzburg, section 1.5). Does my comment make any sense? Jul 22, 2019 at 8:47

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).
• Thank you very much Tobias! It would be nice if the diffeomorphism group $Diff(M)$ plays a role in the story since the moduli space of Einstein equations is the space of Riemannian metrics satisfying Einstein equation modulo the action of $Diff(M)$. It would be nice to get this moduli space as a symplectic reduction. Jul 24, 2019 at 15:08