Towards a general solution of the Hamiltonian constraints of General Relativity

TL;DR

This paper explores the connection between Ashtekar formalism and Poincaré Gauge Theory in gravity, and investigates solutions to Einstein's constraints involving metric-dependent and independent components, especially for isotropic metrics.

Contribution

It establishes relationships between different gauge-theoretical approaches to gravity and proposes new solutions to the Hamiltonian constraints of General Relativity.

Findings

Identifies the link between Ashtekar formalism and Poincaré Gauge Theory.

Proposes solutions involving Cotton-York tensor contributions.

Analyzes metric-independent contributions in isotropic cases.

Abstract

The present work has a double aim. On the one hand we call attention on the relationship existing between the Ashtekar formalism and other gauge-theoretical approaches to gravity, in particular the Poincar\'e Gauge Theory. On the other hand we study two kinds of solutions for the constraints of General Relativity, consisting of two mutually independent parts, namely a general three-metric-dependent contribution to the extrinsic curvature $K_{ab}$ in terms of the Cotton-York tensor, and besides it further metric independent contributions, which we analyze in particular in the presence of isotropic three-metrics.