Dynamics of emission from a semiconductor laser with an external resonator

Abstract

We develop a Hamiltonian formulation of the Bianchi type I spacetime in conformal gravity-the theory described by the Lagrangian L ∝ √ −g C abcd C abcd , which involves the quadratic curvature invariant constructed from the Weyl tensor, in a four-dimensional spacetime. 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 Painlevé 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. We adopt Wald's conventions of sign, definitions of curvature tensors (but the extrinsic curvature) and abstract index notation unless otherwise stated [4]. 0264-9381/99/030749+20$19.50 © 1999 IOP Publishing Ltd † Latin indices i, j, k, . . . refer to spacelike components in t whereas Greek indices α, β, γ, . . . refer to components with respect to a specific spacetime basis, such as the ADM basis with vectors e n := n = (∂ t − N i ∂ i )/N and e i = ∂ i .