A Conformal Approach to Carroll Gravity

TL;DR

This paper develops a conformal framework for coupling matter to Carroll gravity, revealing unique features in the magnetic case and providing a foundation for further matter-gravity coupling studies.

Contribution

It introduces a conformal approach to Carroll gravity with matter coupling, detailing transformation rules and the role of intrinsic torsion tensors.

Findings

Coupling a scalar to conformal Carroll gravity is achieved.

The magnetic case shows the Lagrange multiplier is absorbed into spin-connections.

The conformal approach differs from the Galilei case in matter-gravity relations.

Abstract

We show how to take the first step in the conformal program for constructing general matter couplings to Carroll gravity. In particular, we couple a single massless electric/magnetic scalar to conformal Carroll gravity with isotropic dilatations and show how, upon gauge-fixing, we obtain a (non-conformal version of) electric/magnetic Carroll gravity. We determine the full Carroll transformation rules paying special attention to the way the so-called intrinsic torsion tensors occur in these transformation rules. A noteworthy feature in the magnetic case is that the Lagrange multiplier present in the Lagrangian gets absorbed, after coupling to conformal Carroll gravity and gauge-fixing, into one of the independent spin-connections of magnetic Carroll gravity. Our results form a convenient starting point for constructing general matter couplings to Carroll gravity. Surprisingly, we find that the same relation between dynamical matter and gravity, which forms the basis of the conformal program, does not work in the usual way in the Galilei case.