Carroll versus Galilei Gravity

TL;DR

This paper identifies two consistent action‑level limits of General Relativity—the ultra‑relativistic Carroll and the non‑relativistic Galilei limits—where certain spin‑connection components remain undetermined and act as Lagrange multipliers enforcing geometric constraints. It develops both first‑order and second‑order formulations for Carroll and Galilei gravity, derives the relevant geometric constraints (such as $K_{ab}=0$ in Carroll and $R_{ab}(H)=0$ in Galilei), and analyzes matter couplings with explicit examples for spin 0, spin 1/2, and electromagnetism to illustrate how dynamics reduce to time or space derivatives in these limits. The work highlights key distinctions between Carroll and Galilei structures, including special features in three dimensions and the role of constraints in shaping the resulting geometries. Potential connections to flat space holography and future directions, such as Hamiltonian quantization and higher‑spin extensions, are discussed as avenues for further exploration.

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.