Anti-de Sitter space, squashed and stretched

TL;DR

This paper explores the geometric and causal properties of 2+1 dimensional anti-de Sitter spaces that are squashed or stretched, revealing impacts on their boundary structure and connections to black hole horizons.

Contribution

It introduces Lorentzian analogues of squashed 3-spheres in anti-de Sitter space and analyzes how squashing or stretching affects their causal and conformal boundary properties.

Findings

Squashing or stretching alters the causal structure of AdS space.

Such deformations destroy the conformal boundary of the original spacetime.

Near horizon geometry of extremal Kerr black holes relates to squashed AdS space.

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.