Introduction

One of Charles Misner's most influential contributions to the mathematical development of general relativity was his proposal, along with R. Arnowitt and S. Deser, of the ADM program of (Hamiltonian) reduction of Einstein's equations [1, 2]. By ADM reduction we mean not simply the elegant but essentially elementary reexpression of Hilbert's action in terms of canonical variables but rather the complete reduction of Einstein's equations to an unconstrained Hamiltonian system defined over an appropriate phase space of “true degrees of freedom.” This program, though developed over the years by many people, beginning of course with Arnowitt, Deser and Misner themselves, remains largely unfinished even today. Nevertheless many model problems have been worked out, to a large extent by Misner and his former students and other collaborators, and the reduction program has proven itself of value, within these models, for the study of both classical and quantum dynamics. In this article we shall describe some recent developments of the ADM program and discuss some of the problems which remain to be solved.

Some of the work described below in Sect. II is based in part on a Yale senior physics project by Juan Lin, whereas much of that described in Sect. Ill is based upon the Yale Mathematics Ph.D. thesis of John Cameron (1991). The work described in Sect. IV represents research in progress in collaboration with Arthur Fischer (University of California, Santa Cruz). Section V mentions some recent joint work with Yvonne Choquet-Bruhat (Universite Paris VI) and James Isenberg (University of Oregon).