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

TL;DR

The work classifies and constructs three-dimensional Chern–Simons gravity theories for all spatially isotropic homogeneous spacetimes by pairing kinematical Klein pairs with invariant metrics. Central to the approach is the Medina–Revoy double-extension bootstrap, which endows nonsemisimple algebras with nondegenerate invariant bilinear forms, enabling well-defined CS actions and limits from (A)dS. The authors organize the theories into a tesseract of limits linking AdS/dS, Galilei/Carroll, and aristotelian variants, and extend the framework to higher dimensions using double extensions and coadjoint extensions to produce metric Lie algebras such as Maxwell and AdS-Galilei. The paper provides a unified action principle S_CS with a decomposed structure, analyzes central extensions, non-contracting cases, and geometrical underpinnings via Cartan geometry, and discusses potential boundary dynamics and holographic interpretations. Overall, it furnishes a systematic methodology to realize metric, gauge-theoretic gravity for a broad spectrum of non-Lorentzian spacetimes and clarifies how invariant structures survive various limits.

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.

Figures (2)

• Figure 1: This tesseract describes the limits starting with the (anti-)de Sitter theories extended by two (trivial) central extensions. They are added such that they are nontrivial in the nonrelativistic ($c_{i} \to 0$) limit and render all theories well-defined (✓) in the flat ($\Lambda \to 0$) and ultrarelativistic ($\tau \to 0$) limit. This spans the outer cube which is described in Table \ref{['tab:mostgen']}. The $\alpha,\beta \to 0$ limit trivializes the central extensions and leads to the inner cube. The nonrelativistic theories are, unlike the carrollian, not well-defined anymore. This is due to degeneracy of their invariant metric as indicated by a cross (✗) and can be seen in Table \ref{['tab:kin']}. See also Table \ref{['tab:CSlimit']} for a summary of the properties of the algebras appearing in the tesseract.
• Figure 2: Cartan geometry is a generalization of both Riemannian geometry, by allowing the tangent spaces to be different from $\mathbb{R}^n$, and of Klein geometry, by nontrivially gluing together Klein geometries.

Theorems & Definitions (1)

• Theorem 1