Anti-de Sitter space, squashed and stretched

Abstract

We study the Lorentzian analogues of the squashed 3-sphere, namely 2+1 dimensional anti-de Sitter space, squashed or stretched along fibres that are either spacelike or timelike. The causal structure, and the property of being an Einstein--Weyl space, depend critically on whether we squash or stretch. We argue that squashing, and stretching, completely destroys the conformal boundary of the unsquashed spacetime. As a physical application we observe that the near horizon geometry of the extremal Kerr black hole, at constant Boyer--Lindquist latitude, is anti-de Sitter space squashed along compactified spacelike fibres.