Recently the authors have obtained results on the structure of the set of solutions to the constraint equations of general relativity. In the present article we shall explain how these results tie in with the dynamics of general relativity. Specifically, we want to show how to make the space of solutions of the full non-linear vacuum field equations of general relativity into an honest smooth manifold (under certain technical conditions) and to show how this becomes a symplectic manifold when isometric metrics are identifies. This make use of some very general results of Marsden-Weinstein [11]. This symplectic structure is analogous to that obtained in classical field theories; cf. Lichnetowiez [9], Segal [12] Chernoff-Marsden [2] and the articles of P. L. Garcia and I. Segal in this volume. However the present case is complicated by the presence of constraints and the necessity of passing to a quotient space (when isometric metrics are identified). This necessity of passing to a quotient space is already recognized in the formal constructions of Fadeev [5].
We shall also mention how the results on the constraint set can be used to justify linearization of the filed equations. In other words, we establish conditions under which a solutions of the linearized filed equations actually approximates, to first order, an exact solution to the non-linear filed equations.
We begin by supplying some necessary background.
