Geometry and BMS Lie algebras of spatially isotropic homogeneous spacetimes

TL;DR

This paper explores the local geometric structures of maximally symmetric homogeneous spacetimes, revealing their symmetry actions, invariant structures, and connections, with a focus on BMS-like infinite-dimensional symmetries.

Contribution

It provides a detailed geometric analysis of these spacetimes, including symmetry actions, invariant forms, and affine connections, extending previous classifications.

Findings

Boost actions have generic non-compact orbits.

Galilean and Carrollian symmetries are often infinite-dimensional.

Invariant affine connections are classified with their torsion and curvature.

Abstract

Simply-connected homogeneous spacetimes for kinematical and aristotelian Lie algebras (with space isotropy) have recently been classified in all dimensions. In this paper, we continue the study of these "maximally symmetric" spacetimes by investigating their local geometry. For each such spacetime and relative to exponential coordinates, we calculate the (infinitesimal) action of the kinematical symmetries, paying particular attention to the action of the boosts, showing in almost all cases that they act with generic non-compact orbits. We also calculate the soldering form, the associated vielbein and any invariant aristotelian, galilean or carrollian structures. The (conformal) symmetries of the galilean and carrollian structures we determine are typically infinite-dimensional and reminiscent of BMS Lie algebras. We also determine the space of invariant affine connections on each homogeneous spacetime and work out their torsion and curvature.