This paper discusses General Relativistic Aspects of Cosmology. Thus it will make only passing reference to astrophysical aspects of cosmology, and such features as GUTS, quantum gravity, etc. (these are covered in depth in recent reviews, e.g. Balian et al., 1980; Gibbons, Hawking and Siklos, 1983; Royal Society 1983).

Relativistic Cosmology: Its Nature, Aims and Problems

The different kinds of self-similarity in general relativity are discussed, with special emphasis on similarity of the `first' kind, corresponding to spacetimes admitting a homothetic vector. We then survey the various classes of self-similar solutions to Einstein's field equations and the different mathematical approaches used in studying them. We focus mainly on spatially homogenous and spherically symmetric self-similar solutions, emphasizing their possible roles as asymptotic states for more general models. Perfect fluid spherically symmetric similarity solutions have recently been completely classified, and we discuss various astrophysical and cosmological applications of such solutions. Finally, we consider more general types of self-similar models.

https://iopscience.iop.org/article/10.1088/0264-9381/16/7/201/pdf

Self-similarity in general relativity

The role of spatial topology in the Hamiltonian description of Bianchi models is analysed. It turns out that, in general, the number of degrees of freedom of these models is not uniquely determined by the isometry group but depends in addition on the choice of topology. Consequently, the quantum theory is quite sensitive to this choice. Contrary to one's initial expectation, subtleties arise in the spatially open models-say with topology R3-rather than closed. Finally, it is shown that class B models cannot occur with closed topologies.

https://iopscience.iop.org/article/10.1088/0264-9381/8/12/005/pdf

Bianchi cosmologies: the role of spatial topology

It is known that the symptotic states of a large class of spatially homogeneous cosmological models, near the big bang and at late times, are described by exact self-similar solutions. With this as motivation the authors find all spatially homogeneous solutions of the Einstein field equations which have an orthogonal perfect fluid as source, or are vacuum, and admit a similarity group H4 which acts transitively on spacetime.

http://iopscience.iop.org/article/10.1088/0264-9381/3/6/011/pdf

Self-similar spatially homogeneous cosmologies: orthogonal perfect fluid and vacuum solutions

Self-similarity in general relativity is briefly reviewed and the differences between self-similarity of the first kind (which can be obtained from dimensional considerations and is invariantly characterized by the existence of a homothetic vector in perfect-fluid spacetimes) and generalized self-similarity are discussed. The covariant notion of a kinematic self-similarity in the context of relativistic fluid mechanics is defined. It has been argued that kinematic self-similarity is an appropriate generalization of homothety and is the natural relativistic counterpart of self-similarity of the more general second (and zeroth) kind. Various mathematical and physical properties of spacetimes admitting a kinematic self-similarity are discussed. The governing equations for perfect-fluid cosmological models are introduced and a set of integrability conditions for the existence of a proper kinematic self-similarity in these models is derived. Exact solutions of the irrotational perfect-fluid Einstein field equations admitting a kinematic self-similarity are then sought in a number of special cases, and it is found that: (i) in the geodesic case the 3-spaces orthogonal to the fluid velocity vector are necessarily Ricci-flat; (ii) in the further specialization to dust (i.e. zero pressure) the differential equation governing the expansion can be completely integrated and the asymptotic properties of these solutions can be determined; (iii) the solutions in the case of zero expansion consist of a class of shear-free and static models and a class of stiff perfect-fluid (and non-static) models; and (iv) solutions in which the kinematic self-similar vector is parallel to the fluid velocity vector are necessarily Friedmann - Robertson - Walker (FRW) models. Solutions in which the kinematic self-similarity is orthogonal to the velocity vector are also considered. In addition, the existence of kinematic self-similarities in FRW spacetimes is comprehensively studied. It is known that there are a variety of circumstances in general relativity in which self-similar models act as asymptotic states of more general models. Finally, the questions of under what conditions are models which admit a proper kinematic self-similarity asymptotic to an exact homothetic solution and under what conditions are the asymptotic states of cosmological models represented by exact solutions of Einstein's field equations which admit a generalized self-similarity are addressed.

http://iopscience.iop.org/article/10.1088/0264-9381/14/1/012/pdf

Robert T. Jantzen

Papers

Perfect fluid sources for spatially homogeneous spacetimes

Reduction of Partially Self-Similar Space-Times to Self-Similar Ones in General Relativity