Characterizing the curvature and its first derivative for imperfect fluids

Abstract

The curvature tensor and its derivatives up to any order can be covariantly characterized by a minimal set of spinor quantities. On the other hand it might be useful, particularly in cosmology, to describe the geometry of a spacetime in a (1+3) formalism, based on an invariantly defined fluid velocity. In this work, we consider an imperfect fluid possessing both isotropic and anisotropic pressure. For these fluids, we determine the (1+3) matter terms of the curvature as well as the parts of the first order covariant derivative of the curvature (∇R) determined pointwise by the matter via the Bianchi identities. Explicit relations between the set of such terms obtained from the (1+3) and spinor decomposition of ∇R are given. We show that in both sets there are 36 independent terms.