On the Constraint Algebra of Degenerate Relativity

TL;DR

This paper compares the constraint algebra of an alternative degenerate extension of general relativity with Ashtekar's original theory, classifying differences based on triad degeneracy and surface-forming properties.

Contribution

It explicitly evaluates the constraint algebra of a specific alternative theory and provides a classification of differences from Ashtekar's original degenerate extension.

Findings

When triad vectors are degenerate and surface-forming, secondary constraints are weaker.

If triad vectors are not surface-forming, the theories are equivalent at the secondary constraint level.

The classification depends on degeneracy and surface-forming properties of triad vectors.

Abstract

As shown by Ashtekar in the mid 80's, general relativity can be extended to incorporate degenerate metrics. This extension is not unique, however, as one can change the form of the hamiltonian constraints and obtain an {\it alternative} degenerate extension of general relativity that disagrees with Ashtekar's original theory when the triads vectors are degenerate. In this paper, the constraint algebra of a particular alternative theory is explicitly evaluated and compared with that of Ashtekar's original degenerate extension. A generic classification of the difference between the two theories is given in terms of the degeneracy and surface-forming properties of the triad vectors. (This classification is valid when the degeneracy and surface-forming properties of the triad vectors is the same everywhere in an open set about a point in space.) If the triad vectors are degenerate and surface-forming, then all the secondary constraints of the alternative degenerate extension are satisfied as a consequence of the primary constraints, and the constraints of this theory are weaker than those of Ashtekar's. If the degenerate triad vectors are not surface-forming, then the first secondary constraint of the alternative theory already implies equivalence with Ashtekar's degenerate extension. What happens when the degeneracy and surface-forming properties of the triad vectors change from point to point is an open question.