Cut loci of Berger type Lorentzian structures

Consider the deformation of the standard Lorentzian metric on the anti de-Sitter space along the fibers of the Hopf fibration. We study the universal covering of this Lorentzian manifold to exclude a priori presence of time-like cycles. We describe the sets attainable by admissible curves and study the question of the existence of the longest arcs. Next, we investigate Lorentzian geodesics for optimality: we find the cut time and the cut locus. As a geometric application we compute the injectivity radius of the corresponding Lorentzian manifold.