Proper conformal symmetries in self-dual Einstein spaces

Abstract

Proper conformal symmetries in self-dual Einstein spaces are considered. It is shown that such symmetries are admitted only by the Einstein spaces of the type \documentclass[12pt]{minimal}\begin{document}$[\textrm {N},-] \otimes [\textrm {N},-]$\end{document}[N,−]⊗[N,−]. Spaces of the type \documentclass[12pt]{minimal}\begin{document}$[\textrm {N}] \otimes [-]$\end{document}[N]⊗[−] are considered in details. Existence of the proper conformal Killing vector implies existence of the isometric, covariantly constant, and null Killing vector. It is shown that there are two classes of \documentclass[12pt]{minimal}\begin{document}$[\textrm {N}] \otimes [-]$\end{document}[N]⊗[−]-metrics admitting proper conformal symmetry. They can be distinguished by analysis of the associated anti-self-dual (ASD) null strings. Both classes are analyzed in details. The problem is reduced to single linear partial differential equation (PDE). Some general and special solutions of this PDE are presented.