The Initial Value Problem in Light of Ashtekar's Variables

TL;DR

This paper explores the initial value problem in Ashtekar's formulation of general relativity, presenting algebraic solutions to constraints, extending methods to include matter, and applying them to specific cosmological models.

Contribution

It introduces an algebraic method for solving initial value constraints in Ashtekar's variables, including matter, and applies it to vacuum Bianchi models.

Findings

Algebraic solution for diffeomorphism constraints with non-degenerate curvature

Extension of the method to include matter fields

Application to vacuum Bianchi models

Abstract

The form of the initial value constraints in Ashtekar's hamiltonian formulation of general relativity is recalled, and the problem of solving them is compared with that in the traditional metric variables. It is shown how the general solution of the four diffeomorphism constraints can be obtained algebraically provided the curvature is non-degenerate, and the form of the remaining (Gauss law) constraints is discussed. The method is extended to cover the case when matter is included, using an approach due to Thiemann. The application of the method to vacuum Bianchi models is given. The paper concludes with a brief discussion of alternative approaches to the initial value problem in the Ashtekar formulation.