2+1 Sector of 3+1 Gravity

TL;DR

This paper investigates the rank-2 sector of 3+1 dimensional Ashtekar gravity, revealing conditions for consistent evolution and the behavior of degenerate triads resembling 2+1 dimensional spacetimes with a propagating field.

Contribution

It demonstrates that the degeneracy of triads is preserved during evolution and links the 2+1 dimensional behavior to the degenerate sector of Ashtekar gravity.

Findings

Degeneracy of triads is preserved during evolution.

2-surfaces behave like 2+1 dimensional spacetimes.

Evidence for evolving non-degenerate into degenerate sectors.

Abstract

The rank-2 sector of classical $3+1$ dimensional Ashtekar gravity is considered. It is found that the consistency of the evolution equations with the reality of the volume requires that the 3-surface of initial data is foliated by 2-surfaces tangent to degenerate triads. In turn, the degeneracy is preserved by the evolution. The 2-surfaces behave like $2+1$ dimensional empty spacetimes with a massless complex field propagating along each of them. The results provide some evidence for the issue of evolving a non-degenerate gravitational field into a degenerate sector of Ashtekar's phase space.