Carrollian conservation laws and Ricci-flat gravity

TL;DR

This paper develops a Carrollian framework for conservation laws in flat spacetime gravity, deriving a Carrollian energy-momentum tensor, and connecting it to asymptotic flat gravity boundary conditions and conserved charges.

Contribution

It introduces a Carrollian energy--momentum tensor, derives its conservation laws, and applies these to asymptotically flat gravity, linking ultra-relativistic limits to boundary dynamics.

Findings

Carrollian energy--momentum tensor satisfies conservation equations.

Conservation laws emerge from ultra-relativistic limit of relativistic equations.

Surface charges in flat gravity match Carrollian conserved charges.

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 N\oe ther 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.