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

Isenberg, James and Marsden, Jerrold E. (1982) A slice theorem for the space of solutions of Einstein's equations. Physics Reports, 89 (2). pp. 179-222. ISSN 0370-1573. doi:10.1016/0370-1573(82)90066-7.

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A slice for the action of a group G on a manifold X at a point x ε X is, roughly speaking, a submanifold Sx 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 D of diffeomorphisms of spacetime acting on the space 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 E and because 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 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 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 D on E. As a consequence of this slice theorem, we find that the space D = E / 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 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

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Additional Information:© 1982 North-Holland Publishing Company. Available online 18 September 2002. An early attempt at a slice theorem which contained some of the ideas of this paper was done in collaboration with Arthur Fischer around 1975. We also thank Judy Arms, Robert Jantzen, Vincent Moncrief, Dick Palais, Iz Singer, Abe Taub and Tony Tromba for their encouragement and suggestions.
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Chaim Weizmann Postdoctoral FellowshipUNSPECIFIED
Miller InstituteUNSPECIFIED
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ID Code:19508
Deposited By: Ruth Sustaita
Deposited On:19 Aug 2010 22:35
Last Modified:08 Nov 2021 23:53

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