Godel metric as a squashed anti-de Sitter geometry

TL;DR

This paper explores a family of 2+1 dimensional geometries that interpolate between Godel and anti-de Sitter spaces, revealing their causal structures and embeddings in higher-dimensional flat spaces.

Contribution

It introduces a one-parameter family of geometries connecting Godel and anti-de Sitter metrics, providing new insights into their causal structures and embeddings.

Findings

Godel metric is part of a family including anti-de Sitter space.

Lightcone deformations illustrate causal boundary between safe and pathological spaces.

Global embeddings demonstrate the presence of closed timelike curves.

Abstract

We show that the non flat 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 a 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 space-times, 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.