Spacetime Groups

TL;DR

This paper classifies 4-dimensional Lie groups with Lorentz metrics, providing an algebraic framework, solving isomorphism problems, and exploring applications to Einstein equations and Petrov types.

Contribution

It offers a complete algebraic classification of spacetime groups and an algorithmic solution for their isomorphism problem, with applications to Einstein equations.

Findings

Complete list of inequivalent spacetime Lie algebras

Algorithmic method for isomorphism determination

Characterization of Petrov types and conformally Einstein examples

Abstract

A spacetime group is a connected 4-dimensional Lie group G endowed with a left invariant Lorentz metric h and such that the connected component of the isometry group of h is G itself. The Newman-Penrose formalism is used to give an algebraic classification of spacetime groups, that is, we determine a complete list of inequivalent spacetime Lie algebras, which are pairs (g,{\eta}), with g being a 4-dimensional Lie algebra and {\eta} being a Lorentzian inner product on g. A full analysis of the equivalence problem for spacetime Lie algebras is given which leads to a completely algorithmic solution to the problem of determining when two spacetime Lie algebras are isomorphic. The utility of our classification is demonstrated by a number of applications. The results of a detailed study of the Einstein field equations for various matter fields on spacetime groups are given, which resolve a number of open cases in the literature. The possible Petrov types of spacetime groups that, generically, are algebraically special are completely characterized. Several examples of conformally Einstein spacetime groups are exhibited. Finally, we describe some novel features of a software package created to support the computations and applications of this paper.