Hamiltonian Dynamics of Spatially-Homogeneous Vlasov-Einstein Systems

Abstract

We introduce a new matter action principle, with a wide range of applicability, for the Vlasov equation in terms of a conjugate pair of functions. Here we apply this action principle to the study of matter in Bianchi cosmological models in general relativity. The Bianchi models are spatially-homogeneous solutions to the Einstein field equations, classified by the three-dimensional Lie algebra that describes the symmetry group of the model. The Einstein equations for these models reduce to a set of coupled ordinary differential equations. The class A Bianchi models admit a Hamiltonian formulation in which the components of the metric tensor and their time derivatives yield the canonical coordinates. The evolution of anisotropy in the vacuum Bianchi models is determined by a potential due to the curvature of the model, according to its symmetry. For illustrative purposes, we examine the evolution of anisotropy in models with Vlasov matter. The Vlasov content is further simplified by the assumption of cold, counter-streaming matter, a kind of matter that is far from thermal equilibrium and is not describable by an ordinary fluid model nor other more simplistic matter models. Qualitative differences and similarities are found in the dynamics of certain vacuum class A Bianchi models and Bianchi Type I models with cold, counter-streaming Vlasov matter potentials analogous to the curvature potentials of corresponding vacuum models.

23 pages, 2 figures, to be published in Physical Review D (expected to be published in the July 2011 issue)

References in corpus (8)

• Homoclinic chaos in the dynamics of a general Bianchi IX model
• The initial singularity in solutions of the Einstein-Vlasov system of Bianchi type I
• Dynamics of the spatially homogeneous Bianchi type I Einstein-Vlasov equations
• Matter and dynamics in closed cosmologies
• Chaos in Anisotropic Pre-Inflationary Universes
• Dimensional reduction and gauge group reduction in Bianchi-Type cosmology
• Negative Energy Modes and Gravitational Instability of Interpenetrating Fluids
• Homoclinic chaos and energy condition violation

Cited by in corpus (5)

• Synchronous Lagrangian variational principles in General Relativity
• The properties of Vlasov-Maxwell-Einstein equations and its applications to cosmological models
• Evolution of the phase-space density and the Jeans scale for dark matter derived from the Vlasov-Einstein equation
• Variational principle for the Einstein-Vlasov equations
• Relativistic Thermodynamics, a Lagrangian Field Theory for general flows including rotation