The G\"odel Universe as a Lie group with left-invariant Lorentz metric and\newline the Iwasawa decomposition

TL;DR

This paper explores the structure of the G"odel Universe modeled as Lie groups with left-invariant Lorentz metrics, utilizing Iwasawa decomposition to analyze geometric and isometric properties of related Lie groups.

Contribution

It introduces a novel application of Iwasawa decomposition to model the G"odel Universe as Lie groups with Lorentz metrics and examines isometries between sub-Riemannian Lie groups.

Findings

Models G"odel Universe as Lie groups with Lorentz metrics

Shows isometry induced by Iwasawa decomposition of SL(2,R)

Analyzes geometric structures of specific Lie groups

Abstract

We discuss models of the G\"odel Universe as Lie groups with left-invariant Lorentz metric for two simply connected four-dimensional Lie groups, the Iwasawa decomposition for semisimple Lie groups, and left-invariant Lorentz metric on ${\rm SL}(2,\mathbb{R})$, following K.-H.~Neeb. Also we show that the isometry between two non-isomorphic sub-Riemannian Lie group, constructed by A.~Agrachev and D.~Barilari, is induced by some Iwasawa decomposition of ${\rm SL}(2,\mathbb{R})$.