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.

Figures (2)

• Figure 1: This picture is drawn using the sausage coordinates from the Appendix. It shows anti-de Sitter space as a cylinder (with a conformal boundary). The timelike congruence consists of timelike spirals ruling a set of helicoids. To the right we show that the flow becomes null on the boundary.
• Figure 2: Again using sausage coordinates we show the null surface $X = V$, and how it is ruled by the spacelike congruence. To the right we show that the flow becomes null on the boundary, and where it has fixed points. There is a special point acting as a sink for all those members of the congruence that belong to the null surface shown.