Lorentzian Homogeneous Ricci-Flat Metrics on Almost Abelian Lie Groups

Abstract

Abstract. We study Lorentzian homogeneous metrics on almost abelian Lie groups of dimensions larger than three. An almost abelian Lie group is a Lie group whose Lie algebra has a codimension one abelian ideal. Ricci-flat and non-flat conditions on the Lie algebra are derived. In particular, we generalize the four-dimensional Petrov solution to arbitrarily higher dimensions.

Keywords: Lorentzian geometry, Ricci-flat metric, Almost abelian Lie group