Ricci collineation vectors in fluid space-times

Abstract

The properties of fluid space-times that admit a Ricci collineation vector (RCV) parallel to the fluid unit four-velocity vector ua are briefly reviewed. These properties are expressed in terms of the kinematic quantities of the timelike congruence generated by ua. The cubic equation derived by Oliver and Davis [Ann. Inst. Henri Poincaré 30, 339 (1979)] for the equation of state p=p(μ) of a perfect fluid space-time that admits an RCV, which does not degenerate to a Killing vector, is solved for physically realistic fluids. Necessary and sufficient conditions for a fluid space-time to admit a spacelike RCV parallel to a unit vector na orthogonal to ua are derived in terms of the expansion, shear, and rotation of the spacelike congruence generated by na. Perfect fluid space-times are studied in detail and analogues of the results for timelike RCVs parallel to ua are obtained. Properties of imperfect fluid space-times for which the energy flux vector qa vanishes and na is a spacelike eigenvector of the anisotropic stress tensor πab are derived. Fluid space-times with anisotropic pressure are discussed as a special case of imperfect fluid space-times for which na is an eigenvector of πab.