Global aspects of shear-free perfect fluids in general relativity

Abstract

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 purpose of the present article is to provide a pictorial description of the solutions of these equations, to depict qualitatively similar and distinct subclasses of solution, and hence to discover in some detail all possible global behaviors of the associated space-times. In all cases, it is found that, when the space-time is sufficiently extended, the fluid exhibits unphysical properties. The conclusion is that shear-free expanding (or contracting) relativistic perfect fluids that obey an equation of state p=p( μ) must be FRW, or else must be restricted to ‘‘local’’ regions, by means of a suitable extension in which at least one of the conditions defining the entire family is relaxed.