Covariant Carrollian Electric and Magnetic Limits of General Relativity

TL;DR

This work uncovers a covariant Carrollian structure for gravity by performing 1+3 and 1+2+1 decompositions on FLRW backgrounds and systematically contracting to Carrollian limits. It demonstrates that, unlike the Galilean case, GR splits into two Carrollian theories: an Electric Limit with static tidal fields and vanishing gravito-magnetic content, and a Magnetic Limit where the magnetic Weyl tensor $H_{ab}$ remains dynamical and is sourced by spacetime shear. The magnetic Carrollian theory is shown to be the consistent, non-trivial Carrollian gravity framework, with direct relevance to null surfaces, horizons, and holographic considerations, whereas the electric limit yields a static, ultra-local regime for tidal effects. These results provide a robust geometric foundation for Carrollian gravity, informing black hole horizon dynamics, gravitational memory, and potential holographic dualities, and suggest avenues for extending to non-linear regimes and higher-dimensional or modified gravity theories.

Abstract

The Carrollian limit ($c \to 0$) of General Relativity provides the geometric language for describing null hypersurfaces, such as black hole event horizons and null infinity. Motivated by the well-established electric and magnetic limits of Galilean electromagnetism, we perform a systematic analysis of the low-velocity limit of linearized gravity to derive its Carrollian counterparts. Using a 1+3 covariant decomposition, we study the transformation properties of linear tensor perturbations (gravitational waves) on a Friedmann-Lemaitre-Robertson-Walker background under Carrollian boosts. We demonstrate that, analogous to the electromagnetic case, the full set of linearized Einstein's equations is not Carrollian-invariant. Instead, the theory bifurcates into two distinct and consistent frameworks: a Carrollian Electric Limit and a Carrollian Magnetic Limit. In the electric limit, dynamics are frozen, leaving a static theory of tidal forces ($E_{ab}$) constrained by the matter distribution. In contrast, the Magnetic Limit yields a consistent dynamical theory where the magnetic part of the Weyl tensor ($H_{ab}$), which governs gravito-magnetic and radiative effects, remains well-defined and is sourced by the spacetime shear. This framework resolves ambiguities in defining Carrollian gravity and provides a robust theory for gravito-magnetic dynamics in ultra-relativistic regimes. Our results have direct implications for the study of black hole horizons, gravitational memory, and the holographic principle.