Lie Algebra Expansions and Actions for Non-Relativistic Gravity

TL;DR

The paper shows that Lie algebra expansions provide a systematic route to construct non-relativistic gravity algebras and their invariant actions from the relativistic Poincaré algebra. By expanding Maurer-Cartan forms and consistently truncating, it reproduces NR algebras such as Galilei, Bargmann, extended Bargmann, extended Newtonian, and extended string Newton-Cartan, along with corresponding actions. It also derives invariance conditions for the actions and demonstrates several explicit examples across dimensions, clarifying when invariant actions exist and how higher-order truncations are needed. The approach generalizes to ultra-relativistic Carroll gravity and non-relativistic supergravity, offering a unifying framework with potential connections to non-relativistic limits and holography.

Abstract

We show that the general method of Lie algebra expansions can be applied to re-construct several algebras and related actions for non-relativistic gravity that have occurred in the recent literature. We explain the method and illustrate its applications by giving several explicit examples. The method can be generalized to include the construction of actions for ultra-relativistic gravity, i.e. Carroll gravity, and non-relativistic supergravity as well.

Figures (1)

• Figure 1: Existence conditions of the terms $B_{1},\ B_{2}$ and $B_{3}$ in the action that could be read from Table \ref{['table:table1']}.