Curvature inheritance symmetry in Riemannian spaces with applications to fluid space times

Abstract

Katzin et al. [G. H. Katzin, J. Levine, and W. R. Davis, J. Math. Phys. 10, 617 (1969)] introduced curvature collineations (CC), defined by a vector ξ, satisfying LξRbcda=0, where Rbcda is the Riemann curvature tensor of a Riemannian space Vn and Lξ denotes the Lie derivative. They proved that a CC is related to a special conformal motion which implies the existence of a covariant constant vector field. Unfortunately, recent study indicates that the existence of a covariant constant vector restricts Vn to a very rare special case with limited physical use. In particular, for a fluid space time with special conformal motion, either stiff or unphysical equations of state are singled out. Moreover, perfect fluid space times do not admit special conformal motions. This information was not available, in 1969, when CC symmetry was introduced. In this paper, CC is generalized to another symmetry called ‘‘curvature inheritance’’ (CI) satisfying LξRbcda=2αRbcda, where α is a scalar function. We prove that a proper CI (i.e., α≠0) has direct interplay with the physically significant proper conformal motions. As an application, we show that a proper CI, which is also a conformal Killing vector (CKV), can generate new and physically relevant solutions for a variety of fluid spacetimes. In particular, it is shown, that, for CI with CKV, the known stiff or unphysical equations of state are not singled out.