Kinematic Self-Similarity

TL;DR

This paper reviews self-similarity in general relativity, defines kinematic self-similarity in relativistic fluids, and finds exact solutions for specific cases, revealing properties of these spacetimes and their cosmological implications.

Contribution

It introduces the covariant notion of kinematic self-similarity in relativistic fluid mechanics and derives integrability conditions for perfect fluid models, providing new exact solutions and insights.

Findings

In geodesic case, orthogonal 3-spaces are Ricci-flat.

In dust models, the expansion differential equation is integrable.

Solutions include shear-free static and stiff fluid models.

Abstract

Self-similarity in general relativity is briefly reviewed and the differences between self-similarity of the first kind and generalized self-similarity are discussed. The covariant notion of a kinematic self-similarity in the context of relativistic fluid mechanics is defined. 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 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.