Observables and Hamilton-Jacobi approaches to general relativity. I. The Earlier History

TL;DR

This paper reviews the historical development of Hamilton-Jacobi methods in classical general relativity, highlighting key contributions and contrasting different approaches to diffeomorphism invariance and canonical transformations.

Contribution

It provides a comprehensive historical analysis of Hamilton-Jacobi techniques and their relation to various formulations of general relativity from the 1960s to the 1970s.

Findings

Historical connections between different approaches to general relativity

Comparison of diffeomorphism invariance methods

Tracing origins to Cartan's invariant dynamics

Abstract

The main focus is on the Hamilton-Jacobi techniques in classical general relativity that were pursued by Peter Bergmann and Arthur Komar in the 1960's and 1970's. They placed special emphasis on the ability to construct the factor group of canonical transformations, where the four-dimensional diffeomorphism phase space transformations were factored out. Equivalence classes were identified by a set of phase space functions that were invariant under the action of the four-dimensional diffeomorphism group. This is contrasted and compared with approaches of Paul Weiss, Julian Schwinger, Richard Arnowitt, Stanley Deser, Charles Misner, Karel Kuchar - and especially the geometrodynamical program of John Wheeler and Bryce DeWitt where diffeomorphism symmetry is replaced by a notion of multifingered time. The origins of all of these approaches are traced to Elie Cartan's invariant integral formulation of classical dynamics. A related correspondence concerning the thin sandwich dispute is also documented.