Spatially isotropic homogeneous spacetimes

TL;DR

This paper classifies all simply-connected homogeneous spacetimes with spatial isotropy across all dimensions, revealing new geometries and analyzing their symmetries, invariant structures, and limits between different spacetime types.

Contribution

It provides a comprehensive classification of homogeneous spacetimes with spatial isotropy, including new geometries and their symmetry properties across all dimensions.

Findings

Identified new classes of geometries in low dimensions.

Mapped the limits and contractions between different spacetimes.

Analyzed geometric properties like invariance, torsion, and curvature.

Abstract

We classify simply-connected homogeneous ($D+1$)-dimensional spacetimes for kinematical and aristotelian Lie groups with $D$-dimensional space isotropy for all $D\geq 0$. Besides well-known spacetimes like Minkowski and (anti) de Sitter we find several new classes of geometries, some of which exist only for $D=1,2$. These geometries share the same amount of symmetry (spatial rotations, boosts and spatio-temporal translations) as the maximally symmetric spacetimes, but unlike them they do not necessarily admit an invariant metric. We determine the possible limits between the spacetimes and interpret them in terms of contractions of the corresponding transitive Lie algebras. We investigate geometrical properties of the spacetimes such as whether they are reductive or symmetric as well as the existence of invariant structures (riemannian, lorentzian, galilean, carrollian, aristotelian) and, when appropriate, discuss the torsion and curvature of the canonical invariant connection as a means of characterising the different spacetimes.