Carrollian conservation laws and Ricci-flat gravity

TL;DR

This work defines Carrollian momenta as the variation of a Carrollian-action with respect to the geometry data $(\Omega,b_i,a_{ij})$ and derives Carrollian-constrained conservation laws that replace the relativistic energy--momentum conservation in ultra-relativistic limits. By connecting these momenta to Weyl-Carroll covariant structures and studying the flat case, the authors show how Carrollian physics emerges as the $c\to 0$ limit of relativistic theories, and how conserved charges can be built from Carrollian Killing vectors. They establish two families of charges, demonstrate their algebraic representations, and apply the framework to Ricci-flat gravity: in 3D asymptotically flat spacetimes the charges match covariant phase-space surface charges, while in 4D linearized gravity radiation introduces a conformal anomaly that spoils conservation. The analysis is then illustrated on Robinson–Trautman and Kerr–Taub–NUT spacetimes, reinforcing the Carrollian interpretation of boundary dynamics in flat holography and suggesting avenues for a full Carrollian treatment of flat-space gravity.

Abstract

We construct the Carrollian equivalent of the relativistic energy--momentum tensor, based on variation of the action with respect to the elementary fields of the Carrollian geometry. We prove that, exactly like in the relativistic case, it satisfies conservation equations that are imposed by general Carrollian covariance. In the flat case we recover the usual non-symmetric energy--momentum tensor obtained using Nœther procedure. We show how Carrollian conservation equations emerge taking the ultra-relativistic limit of the relativistic ones. We introduce Carrollian Killing vectors and build associated conserved charges. We finally apply our results to asymptotically flat gravity, where we interpret the boundary equations of motion as ultra-relativistic Carrollian conservation laws, and observe that the surface charges obtained through covariant phase-space formalism match the ones we defined earlier.