Kinematic self-similarity

Abstract

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.