Linearized Constraints in the Connection Representation: Hamilton-Jacobi Solution

TL;DR

This paper presents an explicit Hamilton-Jacobi solution to linearized constraints in the connection representation of general relativity, clarifying the origin of singularities in previous solutions and advancing understanding of Ashtekar variables.

Contribution

It provides the first explicit Hamilton-Jacobi solution for all linearized constraints, highlighting differences from prior singular solutions and their limitations.

Findings

Explicit solution to linearized constraints

Clarification of singularity origins in previous work

Insights into the structure of the connection representation

Abstract

Newman and Rovelli have used singular Hamilton-Jacobi transformations to reduce the phase space of general relativity in terms of the Ashtekar variables. Their solution of the gauge constraint cannot be inverted and indeed has no Minkowski space limit. Nonetheless, we exhibit an explicit Hamilton-Jacobi solution of all the linearized constraints. The result does not encourage an iterative solution, but it does indicate the origin of the singularity of the Newman-Rovelli result.