# Limits of three-dimensional gravity and metric kinematical Lie algebras in any dimension

## TL;DR

This paper classifies and constructs three-dimensional and higher-dimensional gravity theories based on various kinematical Lie algebras, including Carrollian, Galilean, and Aristotelian types, using systematic algebraic extensions.

## Contribution

It introduces a systematic method to define nondegenerate bilinear forms for kinematical algebras, enabling the construction of corresponding gravity theories in any dimension.

## Key findings

- Extended classification of homogeneous spacetimes to Chern--Simons theories.
- Systematic construction of nondegenerate bilinear forms via algebraic extensions.
- Ability to take limits of (A)dS spacetimes at the action level.

## Abstract

We extend a recent classification of three-dimensional spatially isotropic homogeneous spacetimes to Chern--Simons theories as three-dimensional gravity theories on these spacetimes. By this we find gravitational theories for all carrollian, galilean, and aristotelian counterparts of the lorentzian theories. In order to define a nondegenerate bilinear form for each of the theories, we introduce (not necessarily central) extensions of the original kinematical algebras. Using the structure of so-called double extensions, this can be done systematically. For homogeneous spaces that arise as a limit of (anti-)de Sitter spacetime, we show that it is possible to take the limit on the level of the action, after an appropriate extension. We extend our systematic construction of nondegenerate bilinear forms also to all higher-dimensional kinematical algebras.