Integrability and Explicit Solutions in Some Bianchi Cosmological Dynamical Systems

Abstract

The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G ₂ cosmologies. By using Darboux’s theory in order to study ordinary differential equations in the complex projective plane ??² we solve the Bianchi V models totally. Moreover, we carry out a study of Bianchi VI models and first integrals are given in particular cases.     

An investigation of the dynamical evolution of a class of Bianchi VI−1/9 cosmological models

We consider the evolution of a subclass of the orthogonal spatially homogeneous cosmologies of Bianchi type VI_–1/9. Expansion normalized variables are introduced to write the Einstein field equations for these models as a three-dimensional autonomous system of ordinary differential equations. This system is analyzed qualitatively using the techniques of dynamical systems, and a cosmological interpretation of the phase portraits is given.     

Newtonian Cosmology in Lagrangian Formulation: Foundations and Perturbation Theory

The Newtonian theory of spatially unbounded, self-gravitating, pressureless continua in Lagrangian form is reconsidered. Following a review of the pertinent kinematics, we present alternative formulations of the Lagrangian evolution equations and establish conditions for the equivalence of the Lagrangian and Eulerian representations. We then distinguish open models based on Euclidean space R³ from closed models based (without loss of generality) on a flat torus T³. Using a simple averaging method we show that the spatially averaged variables of an inhomogeneous toroidal model form a spatially homogeneous background model and that the averages of open models, if they exist at all, in general do not obey the dynamical laws of homogeneous models. We then specialize to those inhomogeneous toroidal models whose (unique) backgrounds have a Hubble flow, and derive Lagrangian evolution equations which govern the (conformally rescaled) displacement of the inhomogeneous flow with respect to its homogeneous background. Finally, we set up an iteration scheme and prove that the resulting equations have unique solutions at any order for given initial data, while for open models there exist infinitely many different solutions for given data.     

Photoconductivity derived from an exact modelling of the uncompensated one-donor model. III—problem of fitting the exact model with approximate models

Abstract

The paper is essentially devoted to a quantitative comparison of the exact (1D) model developed in Part I, with two approximate models indicated by Kittel (1D_a1) and by Van der Ziel (1D_a2). And this is made for identical values of their common parameters. This allows to show that models 1D and 1D_a1 generally present no suitable intervals for fitting. The model of Van der Ziel is more suited, for, a good fitting with the 1D model can be obtained, for relatively reduced G values, and for temperatures bounded upward by the characteristic temperature T ₀ defined previously. The role played by the sub-models, obtained after simplifying the 1D_ai model equations, is studied also.     

A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules

Tanaka,⁽¹⁸⁾ showed a way to relate the measure solution {P _t } _t of a spatially homogeneous Boltzmann equation of Maxwellian molecules without angular cutoff to a Poisson-driven stochastic differential equation: {P _t } is the flow of time marginals of the solution of this stochastic equation. In the present paper, we extend this probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. Then we derive from this interpretation a numerical method for the concerned Boltzmann equations, by using easily simulable interacting particle systems.     

Homogeneous Bianchi type VI0 perfect fluid space-times

An algorithm is presented for generating new exact solutions of the Einstein equations for spatially homogeneous cosmological models of Bianchi type VI₀. The energy-momentum tensor is of perfect fluid type. Starting from Dunn and Tupper's dust-filled universe, new classes of solutions are obtained. The solutions represent anisotropic universes filled with perfect fluid not satisfying the equation of state. Some of their physical properties are studied.     

Equations with Singular Diffusivity

Recently models of faceted crystal growth and of grain boundaries were proposed based on the gradient system with nondifferentiable energy. In this article, we study their most basic forms given by the equations u _t=(u _x/|u _x|) _x and u _t=(1/a)(au _x/|u _x|) _x , where both of the related energies include a |u _x| term of power one which is nondifferentiable at u _x=0. The first equation is spatially homogeneous, while the second one is spatially inhomogeneous when a depends on x. These equations naturally express nonlocal interactions through their singular diffusivities (infinitely large diffusion constant), which make the profiles of the solutions completely flat. The mathematical basis for justifying and analyzing these equations is explained, and theoretical and numerical approaches show how the solutions of the equations evolve.     

4D Homogeneous Isotropic Cosmological Models Generated by the 5D Vacuum

4-Dimensional homogeneous isotropic cosmological models obtained from solutions of vacuum 5-dimensional Einstein equations are considered. It is assumed, that the G₅₅-component of the 5D metric simulates matter in the comoving frame of reference. The observable 4D metric is defined up to conformal transformations of the metric of the 4D section g, with a conformal factor as a function of the component G₅₅. It is demonstrated, that the form of this function determines the matter equation of state. Possible equations of state are analyzed separately for flat, open and closed models.     

Photoconductivity derived from an exact modelling of the uncompensated one-donor model. II—computer simulation of the steady-state exact model

Abstract

Stationary photoconductivity is treated with the model of Part I, free from any of the usual simplifying approximations of the related equations. This simulation leads to set forth a new concept of characteristic temperature T ₀, at which the donor population is independent of the illumination intensity G. T ₀ separates two intervals of temperature over which G either partially empties (T< T ₀) or fills (T > T ₀) the level. Also T ₀ has, in particular, an influence on the ratio n/p of free electrons and holes. The effective isothermal behaviour of n(G) shows that n(G) ∞ G ^1/2 on the lower side of the G range, and n(G) ∞ G at higher G. Variations of n(T) at constant G also display original, T ₀ dependent, characteristics. Finally, a qualitative comparison is made of the 1D model with the two 1D_ai approximate models described in I, in order to distinguish their most prominent behaviour differences.     

Spherically symmetric equations in gauge theories for an arbitrary semisimple compact lie group

An explicit construction of spherically symmetric equations (not only static and/or self-dual) in gauge theories for the minimal embedding of SU(2) in an arbitrary semisimple compact Lie group G is given. The final equations are written in a form containing only gauge invariant quantities in R₂. The whole group structure is concentrated in the only matrix, which is directly related to the Cartan matrix of G. In particular, the developed technique allows to generalize the Witten duality equation [1] and to obtain the spectrum of pointlike solutions in G.     

Non-static electromagnetic field in general relativity VI

The equations of Rainich's already unified field theory are investigated for the typeG ₃ II acting on the spatially homogeneous hypersurfacesx ⁰=constant. Two solutions of a non-static electromagnetic field for the above case are presented here by using the exterior differential calculus. The space-time admits a three-parameter continuous group of motions, the minimum invariant varieties being the geodesically parallel parametric surfacesx ¹=constant,x ²=constant andx ³=constant orthogonal tox ⁰=constant at points of the geodesics.     

Massive String Cloud Cosmologies in Saez-Ballester Theory of Gravitation

String cloud cosmological models are studied using spatially homogeneous and anisotropic Bianchi type VI₀ metric in Saez-Ballester Scalar-Tensor theory of gravitation. The field equations are solved for massive string cloud with particles attached to them. A more general linear equation of state of the cosmic string tension density with the proper energy density of the universe is considered instead of taking any particular relationships like pure geometric string or the case of the p-string. The pure geometric string and p-string solutions can be easily inferred from the models. For all viable models the possible limiting values of the linear connection between the proper energy density and string tension density have been calculated. The physical and kinematical properties of the models have been discussed in detail.     

Exact rotating and expanding radiation-filled universe

An exact rotating and expanding solution of the Einstein field equations without cosmological constant is given. The solution represents a Bianchi type VI₀ spatially homogeneous universe filled with radiation having the equation of state p = ?/3.     

Integrable Euler equations on SO(4) and their physical applications

For the Lie algebra SO(4) (and other six dimensional Lie algebras) we find some Euler's equations which have an additional fourth order integral and are algebraically integrable (in terms of elliptic functions) in a one parameter set of orbits. Integrable Euler's equations having an additional second order integral and generalizing Steklov's case are presented. Equations for rotation of a rigid body havingn ellipsoid cavities filled with the ideal incompressible fluid being in a state of homogeneous vortex motion are derived. It is shown that the obtained equations are Euler's equations for the Lie algebra of the groupG _n+1=SO(3) × ... × SO(3). New physical applications of Euler's equations on SO(4) are discussed.     

Spatially homogeneous and inhomogeneous cosmologies with equation of statep=μ

This paper gives an algorithm for generating solutions of the Einstein field equations which have an irrotational perfect fluid, with equation of statep=, as source, and which admit a two-parameter Abelian group of local isometries. The algorithm is used to derive a variety of new and known spatially homogeneous cosmological models, both tilted and nontilted. However, since the solutions in general only admit two Killing vectors, spatially inhomogeneous models are also obtained. Finally, it is pointed out that the solution generation technique used in this paper is closely related to solution generation techniques that have been used to generate solutions of the source-free Brans-Dicke field equations, and of the Einstein field equations with a massless scalar field as source.     

Kähler Reduction of Metrics with Holonomy G<Subscript>2</Subscript>

A torsion-free G₂ structure admitting an infinitesimal isometry such that the quotient is a Kähler manifold is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations. Reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to G₂.     

Gauge-invariant perturbations in anisotropie homogeneous cosmological models

In a recent paper K. Tomita and M. Den found a set of coupled differential equations for spatially flat, anisotropic homogeneous,N- dimensional cosmological models. Some particular exact solutions of those differential equations for a few specific equations of state were obtained by D. Lorentz-Petzold. In the present work we solve those differential equations completely.     

Plane Symmetric Vacuum Bianchi Type-III Cosmological Model in Brans-Dicke Theory

We have obtained an exact solution of the vacuum Brans-Dicke (Phys. Rev. 124:925, 1961) field equations for the metric tensor of a spatially homogeneous and anisotropic model. Some physical properties of the model are also studied.