C. B. Collins

More qualitative cosmology

Standard geometric techniques of differential equation theory are employed to determine the qualitative behaviour of a set of non-rotating perfect-fluid cosmologies, whose spatially homogeneous hypersurfaces admit a 3-parameter group of isometries of Bianchi types I, II, III, V, or VI. In this way we are led to some new exact solutions of the field equations.The field equations for a broad class of cosmological models are presented in a regularised form, limitations on the use of this procedure are examined, and some suggestions are made of ways of avoiding the difficulties that arise.

Global structure of the ’’Kantowski–Sachs’’ cosmological models

A discussion is given of the ’’Kantowski–Sachs’’ cosmological models; these are defined locally as admitting a four‐parameter continuous isometry group which acts on spacelike hypersurfaces, and which possesses a three‐parameter subgroup whose orbits are 2‐surfaces of constant curvature (i.e., the models possess spherical symmetry, combined with a translational symmetry, and can thus be regarded as nonempty analogs of part of the extended Schwarzschild manifold). It is shown that all general relativistic models in which the matter content is a perfect fluid satisfying reasonable energy conditions are geodesically incomplete, both to the past and to the future, and that at each resulting singularity the fluid energy density is infinite. In the case where the fluid obeys a barotropic equation of state (which includes all known exact perfect fluid solutions) the field equations are shown to decouple to form a plane autonomous subsystem. This subsystem is examined using qualitative (Poincare–Bendixson) theory, and phase–plane diagrams are drawn depicting the behavior of the fluid’s energy density and shear anisotropy in the course of the models’ evolution. Further diagrams depict the conformal structure of these universes, and a table summarizes the asymptotic properties of all physically relevant variables.

Singularities in Bianchi cosmologies

Abstract We discuss how infinite density singularities may be shown to occur in Friedmann-Robertson-Walker universes and orthogonal spatially homogeneous universes, but how very different behaviours are possible in tilted homogeneous cosmologies. After considering various possibilities that arise in this case, we illustrate them by examining the behaviour of exact solutions of Einsteins equations for a homogeneous cosmology which is a locally rotationally symmetric tilted Bianchi type V universe. These universes - which can be arbitrarily similar to a Robertson-Walker universe at late times - show a variety of singular behaviours quite different from those in the ‘orthogonal’ case. In particular, there exist such universes in which two singularities occur at the early stages of the universe, but in which the density of matter is finite at all times.

Tilting at cosmological singularities

A detailed investigation is made of the simplest type of general relativistic perfect fluid cosmological models that possess a singularity at which all physical quantities are well-behaved. These models are spatially homogeneous, axisymmetric generalisations of the open (k/it=−1) Robertson-Walker universes. A pictorial description of the evolution of the models is obtained by using the qualitative theory of differential equations.The most surprising feature that emerges is that for some (non-empty) models the matter density may become zero, within a finite time, on a null hypersurface which acts as a Cauchy horizon for the models. This result is generalized to most other types of spatially homogeneous models.It is also discovered that the behaviour of the models varies dramatically with the type of matter content. This casts some doubts on the validity of assuming definite equations of state in general relativity, and suggests an investigation of the structural stability of Einsteins field equations.

Qualitative magnetic cosmology

The technique of phase plane analysis, which was used in a previous paper [4] to study the behaviour of a class of perfect-fluid anisotropic cosmological models, is applied to some simple anisotropic models that contain a uniform magnetic field. A formal correspondence is established between these magnetic models (of Bianchi type I) and certain perfect fluid models (of Bianchi type II), and new exact solutions are consequently discovered.

Shear‐free perfect fluids with zero magnetic Weyl tensor

Rotating, shear‐free general‐relativistic perfect fluids are investigated. It is first shown that, if the fluid pressure, p, and energy density, μ, are related by a barotropic equation of state p=p( μ) satifying μ+p≠0, and if the magnetic part of the Weyl tensor (with respect to the fluid flow) vanishes, then the fluid’s volume expansion is zero. The class of all such fluids is subsequently characterized. Further analysis of the solutions shows that, in general, the space‐times may be regarded as being locally stationary and axisymmetric (they admit a two‐dimensional Abelian isometry group with timelike orbits, which is in fact orthogonally transistive), although various specializations can occur, with the ‘‘most special’’ case being the well‐known Godel model, which is space‐time homogeneous (it admits a five‐dimensional isometry group acting multiply transitively on the space‐time). all solutions are of Petrov type D. The fact that there are any solutions in the class at all means that a theorem appeari...

A class of shear‐free perfect fluids in general relativity. II

We continue our previous investigation of shear‐free perfect fluids in general relativity, under the assumptions that the fluid satisfies an equation of state p=p(μ) with μ+p≠0, and that the vorticity and acceleration of the fluid are parallel (and possibly zero). We classify algebraically the set of such solutions into thirteen invariant nonempty cases. In each case, we investigate the allowed isometry groups and Petrov types, and invariantly characterize the special subcases that arise. We also show how the various subcases are related to each other and to the works of previous authors.

Global aspects of shear‐free perfect fluids in general relativity

In an earlier investigation of the class of shear‐free expanding (or contracting) irrotational perfect fluids obeying a barotropic equation of state p=p( μ), and satisfying the field equations of general relativity, it was shown that the space‐times form three distinct classes. In one class, the fluid acceleration is zero (i.e., the flow is geodesic), and the space‐times are the well‐known spatially homogeneous and isotropic Friedmann–Robertson–Walker (FRW) models. In the other two classes, the acceleration is nonzero, and the space‐times are spatially anisotropic and are less familiar. One of these classes consists of the spherically symmetric Wyman solutions, whereas models that are plane symmetric, and either spatially or temporally homogeneous, constitute the final class. Analytic forms for these anisotropic space‐time metrics were given, although in each case their exact determination would depend upon the solution of a single nonlinear ordinary differential equation, which has not been achieved. The...

Static relativistic perfect fluids with spherical, plane, or hyperbolic symmetry

This article examines the Einstein field of equations of general relativity, when the source of the gravitational field is a perfect fluid, and the geometry is static and possesses spherical, plane, or hyperbolic symmetry. This examination unifies, extends, and amends some earlier works. It is shown that a previous qualitative treatment of static spherically symmetric perfect fluids that obey a γ‐law equation of state can be extended to include the cases of plane and hyperbolic symmetry. In the case of plane symmetry, the exact solution is provided for general values of γ. This indicates defects in an earlier prescription that was given for a general equation of state.

All solutions to a nonlinear system of complex potential equations

This work employs the powerful geometric methods previously developed in order to determine all solutions of the nonlinear system (∇γ)2≡∇γ⋅∇γ=f (γ), (∇2γ)2−(1/2)(N+2) f′ (γ) ∇2γ+(1/4)(N+1)[f′ (γ)]2 −(1/2) Nf (γ) f\ (γ) =−N∇γ⋅∇ (∇2γ), where f (γ) is an arbitrarily assigned function and N is an arbitrary constant. The circumstances are determined under which compatible solutions exist, not only when γ is real, but also when γ is complex, and all of the corresponding solutions are found. This is done by referring the system of equations to a set of coordinates based on the (real or complex) equipotential surfaces of constant γ. The relationship between the solutions and the geometry of the equipotential surfaces is examined, and a close association is discovered between the set of allowable equipotential surfaces and the class of surfaces of constant total (Gaussian) curvature. These results are analogous to those in Collins, Math. Proc. Camb. Phil. Soc. 80, 165–87 (1976), where the system under study was as...