Hamiltonian Formulation and Exact Solutions of Bianchi type-I Model in Conformal Gravity

TL;DR

This paper formulates the Hamiltonian approach for Bianchi type-I models in conformal gravity, derives explicit equations, and finds all possible exact solutions using a global involution algorithm, revealing their relation to Einstein spaces.

Contribution

It introduces a Hamiltonian formulation for Bianchi type-I models in conformal gravity and develops a global involution algorithm to systematically find all exact solutions.

Findings

Derived explicit super-Hamiltonian and constraints for the model

Developed a global involution algorithm to find all particular solutions

Showed the system is not integrable via Painleve analysis

Abstract

We develop a Hamiltonian formulation of Bianchi type-I cosmological model in conformal gravity, i.e. the theory described by a Lagrangian which involves the quadratic curvature invariant constructed from the Weyl tensor, in four dimensions. We derive the explicit forms of the super-Hamiltonian and the constraint expressing the conformal invariance of the theory, and we write down the system of canonical equations. To seek out exact solutions to this system we add extra constraints on the canonical variables and we go through a global involution algorithm that possibly leads to the closure of the constraint algebra. This enables us to extract all possible particular solutions that may be written in closed analytical form. On the other hand, probing the local analytical structure we show that the system does not possess the Painleve property (presence of movable logarithms) and that it is therefore not integrable. We stress that there is a very fruitful interplay of local integrability-related methods such as the Painleve test and global techniques such as the involution algorithm. Strictly speaking, we demonstrate that the global involution algorithm has proven to be exhaustive in the search for exact solutions. The conformal relationship of the solutions, or absence thereof, with Einstein spaces is highlighted.