Godel metric as a squashed anti-de Sitter geometry

We show that the non-hat factor of the Godel metric belongs to a one-parameter family of (2+1)-dimensional geometries that also includes the anti-de Sitter metric. The elements of this family allow a generalization ci la Kaluza-Klein of the usual (3 + 1)-dimensional Godel metric. Their lightcones can be viewed as deformations of the anti-de Sitter ones, involving tilting and squashing. This provides a simple geometric picture of the causal structure of these spacetimes, anti-de Sitter geometry appearing as the boundary between causally safe and causally pathological spaces. Furthermore, we construct a global algebraic isometric embedding of these metrics in (4 + 3)- or (3 + 4)-dimensional flat spaces, thereby illustrating in another way the occurrence of the closed timelike curves.