A slice theorem for the space of solutions of Einstein's equations

J. Isenberg and J. E. Marsden

Phys. Rep., 89, (1982), 179-222

Abstract:

A slice for the action of a group G on a manifold X at a point x $\in$ X is, roughly speaking, a submanifold S_x which is transverse to the orbits of G near x . Ebin and Palais proved the existence of a slice for the diffeomorphism group of a compact manifold acting on the space of all Riemannian metrics. We prove a slice theorem for the group $\mathcal {D}$ of diffeomorphisms of spacetime acting on the space $\mathcal {E}$ of spatially compact, globally hyperbolic solutions of Einstein's equations. New difficulties beyond those encountered by Ebin and Palais arise because of the Lorentz signature of the spacetime metrics in $\mathcal {E}$ and because $\mathcal {E}$ is not a smooth manifold-- it is known to have conical singularities at each spacetime metric with symmetries. These difficulties are overcome through the use of the dynamic formulation of general relativity as an infinite dimensional Hamiltonian system (ADM formalism) and through the use of constant mean curvature foliations of the spacetimes in $\mathcal {E}$ . (We devote considerable space to a review and extension of some special properties of constant mean curvature surfaces and foliations that we need.) The conical singularity structure of $\mathcal {E}$ , the sympletic aspects of the ADM formalism, and the uniqueness of constant mean curvature foliations play key roles in the proof of the slice theorem for the action of $\mathcal {D}$ on $\mathcal {E}$ . As a consequence of this slice theorem, we find that the space $\mathcal {G}$ = $\mathcal {E}$/$\mathcal {D}$ of gravitational degrees of freedom is a stratified manifold with each stratum being a sympletic manifold. The spaces for homogeneous cosmologies of particular Bianchi types give rise to special finite dimensional symplectic strata in this space $\mathcal {G}$ . Our results should extend to such coupled field theories as the Einstein-Yang-Mills equations, since the Yang-Mills system in a given background spacetime admits a slice theorem for the action of the gauge transformation group on the space of Yang-Mills solutions, since there is a satisfactory Hamiltonian treatment of the Einstein-Yang-Mills system, and since the singularity structure of the solution set is known.