Shear-free fluids in general relativity

Abstract

For a large class of shear-free general-relativistic perfect fluids that obey a barotropic equation of state, either the expansion or the rotation is zero; well-known examples include the Friedmann–Robertson–Walker (FRW) models, the Gödel solution, and stationary axisymmetric systems in rigid rotation. It has been conjectured that this is necessarily the case. Several results prove that restricted versions of this conjecture are valid, although no proof is known for the general case. A survey of these special results is given, together with physical and mathematical reasons for the study of shear-free fluids.If the conjecture is true, then there are three mutually exclusive subclasses, according to whether or not the expansion and the rotation are zero separately or simultaneously. Of these, the physically most interesting subclass is that in which the expansion is not zero, since this subclass might be thought to contain space-times that are suitable for the description of collapsing stars or expanding cosmologies. All space-times of this particular subclass are given, and their global properties are investigated. It turns out that the FRW models are the only ones in which the matter is physically reasonable on a global scale. This consequently provides a global uniqueness theorem for the FRW models.