# Carroll versus Galilei Gravity

## TL;DR

This paper explores two novel limits of General Relativity, called Carroll and Galilei gravity, formulated at the action level, revealing unique geometric constraints and differences in matter coupling.

## Contribution

It introduces first- and second-order formalisms for Carroll and Galilei gravity, highlighting their constraints and differences from standard GR.

## Key findings

- Both theories impose geometric constraints on spacetime.
- Independent connection components act as Lagrange multipliers.
- Distinct differences between Carroll and Galilei gravity are identified.

## Abstract

We consider two distinct limits of General Relativity that in contrast to the standard non-relativistic limit can be taken at the level of the Einstein-Hilbert action instead of the equations of motion. One is a non-relativistic limit and leads to a so-called Galilei gravity theory, the other is an ultra-relativistic limit yielding a so-called Carroll gravity theory. We present both gravity theories in a first-order formalism and show that in both cases the equations of motion (i) lead to constraints on the geometry and (ii) are not sufficient to solve for all of the components of the connection fields in terms of the other fields. Using a second-order formalism we show that these independent components serve as Lagrange multipliers for the geometric constraints we found earlier. We point out a few noteworthy differences between Carroll and Galilei gravity and give some examples of matter couplings.