Classical analysis of Bianchi types I and II in Ashtekar variables

TL;DR

This paper analytically solves Einstein's equations for Bianchi I and II models using Ashtekar variables, revealing a potential-free dynamical framework and exploring the nature of cosmological bounces.

Contribution

It provides a classical analysis of Bianchi models in Ashtekar variables, including solutions and the characterization of cosmological bounces without potential terms.

Findings

Solutions to Einstein equations in Ashtekar variables for Bianchi I and II.

Revealed a potential-free supermetric governing dynamics.

Explored the nature of the cosmological bounce in this framework.

Abstract

We solve the complex Einstein equations for Bianchi I and II models formulated in the Ashtekar variables. We then solve the reality conditions to obtain a parametrization of the space of Lorentzian solutions in terms of real canonically conjugate variables. In the Ashtekar variables, the dynamics of the universe point particle is governed by only a curved supermetric -- there is no potential term. In the usual metric formulation the particle bounces off a potential wall in flat superspace. We consider possible characterizations of this ``bounce'' in the potential-free Ashtekar variables.