Hamiltonian formulation and exact solutions of the Bianchi type I space-time in conformal gravity

Abstract

We develop a Hamiltonian formulation of the Bianchi type I space-time in conformal gravity, i.e. the theory described by a Lagrangian that is defined by the contracted quadratic product of the Weyl tensor, in a four-dimensional space-time. We derive the explicit forms of the super-Hamiltonian and of the constraint expressing the conformal invariance of the theory and we write down the system of canonical equations. To seek out exact solutions of this system we add extra constraints on the canonical variables and we go through a global involution algorithm which eventually leads to the closure of the constraint algebra. The Painleve approach provides us with a proof of non-integrability, as a consequence of the presence of movable logarithms in the general solution of the problem. We extract all possible particular solutions that may be written in closed analytical form. This enables us to demonstrate that the global involution algorithm has brought forth the complete list of exact solutions that may be written in closed analytical form. We discuss the conformal relationship, or absence thereof, of our solutions with Einstein spaces.