Fluid space–times and conharmonic symmetries

Abstract

The conharmonic curvature tensor is considered as an invariant of the conharmonic transformation defined by Ishii and the necessary and sufficient conditions for the conharmonic curvature tensor in a perfect fluid space–time to be divergence free has been obtained. Conharmonic motion, conharmonic collineation, and conharmonic curvature collineation are introduced as subcases of conformal motion, conformal collineation, and Weyl conformal collineation, respectively, and relations of conharmonic symmetries with inheriting symmetries are investigated. In the case of an existing conharmonic Killing vector along the flow vector, along the anisotropy vector, and perpendicular to both in an anisotropic fluid space–time it is found that no equation of state is singled out unless the conharmonic Killing vector is also a curvature inheritance vector. Conditions are obtained for the symmetries of the anisotropic fluid space–time admitting a conharmonic Killing vector to be inherited. In the case of a conharmonic symmetric space–time and also in the case of a space–time with a divergence-free conharmonic curvature tensor it is found that if the space–time admits an infinitesimal conharmonic Killing vector then the scalar curvature of the space–time vanishes and the space–time is either conharmonically flat or has four distinct principal null directions.