The Initial Value Problem in Light of Ashtekar's Variables

Summary

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.

Introduction

It is with great pleasure that we dedicate this paper to Dieter Brill, our teacher, advisor, and colleague, on the occasion of his 60th birthday. Our contribution concerns the initial value problem for general relativity, which is amongst Dieter's many areas of expertise. As is the case with almost all research activity developed around the general relativity group at the University of Maryland, the ideas we will present have benefitted from Dieter's always kind and sometimes maddening insightful questioning. Of course it is our wish that this paper will prompt some more such questioning.