Gradient conformal killing vectors and exact solutions

Abstract

We establish a connection between conformally related Einstein spaces and conformai killing vectors (CKV). We begin with the conformal map and prove that (a) under the conformal mapping¯g ik=Ω−2gik, the necessary and sufficient condition for the tracefree part of the Ricci tensor (S ik=Rik−(R/4)g ik) to remain invariant is thatΩ i is a CKV ofg ik, and (b) the most general form forΩ for conformally flat Einstein space, which is the de Sitter space, is composed of three terms each of which alone represents a flat space. The existence of gradient CKV (GCKV) is examined in relation to vacuum and perfect fluid spacetimes.