Observationally homogeneous shear-free perfect fluids

Abstract

It has been conjectured that, in general relativity, shear-free perfect fluids which obey any reasonable barotropic equation of state are necessarily either non-expanding or nonrotating. We prove that this is valid in the restricted case when the fluid's expansion and energy density are assumed to be functionally dependent. In a cosmological context, this condition of functional dependence is of interest, because it is closely related to a recently proposed criterion of observational (spatial) homogeneity, which has been enunciated in the Postulate of Uniform Thermal Histories (indeed, the two are equivalent when the fluid's expansion is nonzero). Our result on shear-free fluids may be readily specialized to the case of hypersurface-homogeneous spacetimes, and in particular to that of spatially homogeneous cosmological models. We briefly examine all subcases in which the fluid's expansion is nonzero and focus attention on the one-parameter family of solutions which are not hypersurface-homogeneous.