Claes Uggla

Integrability of irrotational silent cosmological models

We revisit the issue of integrability conditions for the irrotational silent cosmological models. We formulate the problem both in 1 + 3 covariant and 1 + 3 orthonormal frame notation and show that there exists a series of constraint equations that need to be satisfied. These conditions hold identically for FLRW-linearized silent models, but not in the general exact nonlinear case. Thus there is a linearization instability and it is highly unlikely that there is a large class of silent models. We conjecture that there are no spatially inhomogeneous solutions with Weyl curvature of Petrov type I and indicate further issues that await clarification.

Exact Hypersurface-Homogeneous Solutions in Cosmology and Astrophysics

A framework is introduced which explains the existence and similarities of most exact solutions of the Einstein equations with a wide range of sources for the class of hypersurface-homogeneous spacetimes which admit a Hamiltonian formulation. This class includes the spatially homogeneous cosmological models and the astrophysically interesting static spherically symmetric models as well as the stationary cylindrically symmetric models. The framework involves methods for finding and exploiting hidden symmetries and invariant submanifolds of the Hamiltonian formulation of the field equations. It unifies, simplifies and extends most known work on hypersurface-homogeneous exact solutions. It is shown that the same framework is also relevant to gravitational theories with a similar structure, like Brans-Dicke or higher-dimensional theories.

Compactified and reduced dynamics for locally rotationally symmetric Bianchi type IX perfect fluid models

The field equations for the locally rotationally symmetric Bianchi type IX perfect fluid models are reduced to a regularised first-order system of differential equations for three new bounded gravitational variables. The entire dynamics is described in a single coordinate patch covering the evolution all the way from the initial big bang singularity to the final big crunch singularity. This is made possible by exploitation of the algebraic properties of the Hamiltonian describing these models and use of the scale invariance symmetry. By investigating the compactified reduced phase space with qualitative and numerical techniques the authors are able to follow all solutions from the initial singularity to the final singularity, thus obtaining a complete picture of the solution space.

Stationary Bianchi type II perfect fluid models

Einstein’s field equations for stationary Bianchi type II models with a perfect fluid source are investigated. The field equations are rewritten as a system of autonomous first-order differential equations. Dimensionless variables are subsequently introduced for which the reduced phase space is compact. The system is then studied qualitatively using the theory of dynamical systems. It is shown that the locally rotationally symmetric models are not asymptotically self-similar for small values of the independent variable. A new exact solution is also given.

Hypersurface homogeneous and hypersurface self-similar models

Dynamical systems for hypersurface homogeneous and hypersurface self-similar models with non-null symmetry surfaces are derived. The equations are cast in a geometric form based on properties of the symmetry surfaces that emphasize the close connection between the various models. Perfect fluid models are discussed in particular. It is shown how the models form a hierarchical structure where simpler models act as building blocks for more complicated ones. Expressions for the fluids kinematic properties are given and discussed in the context of various special cases.

Extended dynamics and symmetries in perfect fluid Bianchi cosmologies

Symmetry compatible time reparametrisations are calculated for orthogonal perfect fluid Hamiltonian Bianchi cosmologies by using an extended phase space containing the time variable itself. As in the vacuum case treated in an earlier article, the scale symmetry and the non-unimodular automorphism can be coupled to yield a constant of the motion which in the perfect fluid case is necessarily explicitly time dependent except for the stiff matter equation of state. To obtain the symmetry compatible lapse functions it is necessary to start from one of three source-adapted time gauges, two of which are the Bogoyavlensky-Novikov and Jacobi time gauges. The latter reduces the field equations to geodesic flows on certain conformally flat Lorentzian geometries. In these Jacobi geometries, the symmetry giving rise to the time-dependent constant of the motion is just a homothetic motion.

Extended dynamics and symmetries in vacuum Bianchi cosmologies

Symmetries and time reparametrisations are used in a series of articles to study the Hamiltonian equations for certain vacuum and orthogonal perfect fluid Bianchi cosmologies. Using the framework of an extended phase space containing the time variable itself, one may couple the scale invariance symmetry and the non-unimodular automorphism admitted by all the non-semisimple Bianchi types to obtain a variational symmetry whose associated constant of the motion in general depends explicitly on the time. In the vacuum case this is time independent for certain preferred choices of time gauge, corresponding to preferred choices of lapse functions which may be constructed from symmetry invariants. One such preferred choice of lapse is the Jacobi lapse, for which the field equations are reduced to a geodesic flow on a conformally flat Lorentzian geometry. This article studies such preferred choices of lapse functions and the associated constants of the motion for the vacuum case.

Asymptotic cosmological solutions: orthogonal Bianchi type-II models

The time evolution of the orthogonal spatially homogeneous Bianchi type-II models is analysed by applying qualitative techniques to the reduced and regularised differential system of Rosquist and Jantzen (1988). This enables the author to obtain the asymptotic behaviour of solutions near the initial singularity and in the distant future. A global picture emerges in which the general orthogonal type-II solution is shown to be spiralling about the non-LRS solution found by Collins (1971). The author also gives an example how one can obtain exact solutions with help of the series method developed by Picard (1928).

Classifying Einstein's field equations with applications to cosmology and astrophysics

The field equations for spacetimes with finite-dimensional Hamiltonian dynamics are discussed. Examples of models belonging to this class are the cosmological spatially homogeneous models, the astrophysically interesting static spherically symmetric models, static cylindrically symmetric models, and certain cosmological self-similar models. A number of different sources are considered. Although these models arise from quite different physical contexts, their field equations all share a common mathematical structure. This motivates a classification of Einsteins field equations. Several classification schemes, based on properties under various variable transformations, are presented. It is shown how these schemes can be used to classify dynamical properties of the models and how one can thereby obtain qualitative information. It is also shown how one scheme can be used in order to find symmetries and exact solutions.

Bianchi type V perfect fluid cosmologies

Bianchi type V solutions of the Einstein equations are studied using the Hamiltonian approach. Explicit expressions depending on a single quadrature are given for the metric components in the general orthogonal perfect fluid case. It is shown that the quadrature can be evaluated in terms of elementary or elliptic integrals when the parameter γ in the equation of statep=(γ−1)ρ takes the values 1, 10/9, 4/3, 14/9, 5/3, 2.