Canonical analysis of the Bianchi models in an Ashtekar formulation

Abstract

The authors discuss the reduction of the physical degrees of freedom in the Ashtekar formulation of general relativity which leads to the Bianchi models. They also perform a canonical analysis of the Bianchi models on compact spacelike surfaces. In the case of the class B Bianchi models the constraint algebra does not close because some of the constraints are second class. This property does not exclude a consistent dynamics, but due to purely geometrical reasons, the class B models cannot exist on compact spacelike surfaces. In the case of the class A models the constraints are all first class. However, the number of independent components of the vector constraint varies from 0 (type I) to 3 (type IX). This is an agreement with the results of a similar analysis recently performed by Ashtekar and Samuel (1991) in the metric formulation. The reduced phase space is analysed through a gauge fixing procedure in the case of the type I, II and IX Biachi models.