1 Kinematic Self-similarity

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; (1) in the geodesic case the 3-spaces orthogonal to the fluid velocity vector are necessarily Ricci-flat and (ii) in the further specialisation 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.