Degenerate Sectors of the Ashtekar Gravity

TL;DR

This paper fully solves the Einstein-Ashtekar equations for two remaining degenerate sectors in 3+1 dimensions, revealing their structure and gauge fixing methods, with implications for classical gravity solutions.

Contribution

It provides a complete solution to the degenerate sectors of Einstein-Ashtekar equations using gauge fixing and coordinate choices, advancing understanding of degenerate gravitational configurations.

Findings

Complete solutions for degenerate sectors are obtained.

Hamiltonian weakly Poisson commutes with sector-defining conditions.

Summary of solutions is provided in the appendix.

Abstract

This work completes the task of solving locally the Einstein-Ashtekar equations for degenerate data. The two remaining degenerate sectors of the classical 3+1 dimensional theory are considered. First, with all densitized triad vectors linearly dependent and second, with only two independent ones. It is shown how to solve the Einstein-Ashtekar equations completely by suitable gauge fixing and choice of coordinates. Remarkably, the Hamiltonian weakly Poisson commutes with the conditions defining the sectors. The summary of degenerate solutions is given in the Appendix.