Affine connections for Galilean and Carrollian structures: a unified perspective

TL;DR

This paper unifies the geometric understanding of Galilean and Carrollian structures through affine connections, demonstrating their emergence from Lorentzian structures and constructing an ultra-relativistic gravitational theory.

Contribution

It introduces a unified classification of affine connections for Galilean and Carrollian structures, linking them to Lorentzian limits and developing an ultra-relativistic gravitational framework.

Findings

Unified classification of Galilean and Carrollian affine connections

Demonstration of structures emerging from Lorentzian limits

Construction of an ultra-relativistic gravitational theory

Abstract

We develop a classification of general Carrollian structures, permitting affine connections with both torsion and non-metricity. We compare with a recent classification of general Galilean structures in order to present a unified perspective on both. Moreover, we demonstrate how both sets of structures emerge from the most general possible Lorentzian structures in their respective limits, and we highlight the role of global hyperbolicity in constraining both structures. We then leverage this work in order to construct for the first time an ultra-relativistic geometric trinity of gravitational theories, and consider connections which are simultaneously compatible with Galilean and Carrollian structures. We close by outlining a number of open questions and future prospects.