Gauging the Carroll Algebra and Ultra-Relativistic Gravity

TL;DR

This paper explores the gauging of the Carroll algebra to develop a geometric framework for ultra-relativistic gravity, extending the understanding of space-time structures in the limit where the speed of light approaches zero.

Contribution

It introduces a method to gauge the Carroll algebra and constructs the most general affine connection for Carrollian space-times, expanding the geometric and gravitational models in ultra-relativistic regimes.

Findings

Gauged the Carroll algebra to develop Carrollian geometry.

Constructed theories of ultra-relativistic gravity in 2+1 dimensions.

Identified cases with anisotropic Weyl invariance for z=0.

Abstract

It is well known that the geometrical framework of Riemannian geometry that underlies general relativity and its torsionful extension to Riemann-Cartan geometry can be obtained from a procedure known as gauging the Poincare algebra. Recently it has been shown that gauging the centrally extended Galilei algebra, known as the Bargmann algebra, leads to a geometrical framework that when made dynamical gives rise to Horava-Lifshitz gravity. Here we consider the case where we contract the Poincare algebra by sending the speed of light to zero leading to the Carroll algebra. We show how this algebra can be gauged and we construct the most general affine connection leading to the geometry of so-called Carrollian space-times. Carrollian space-times appear for example as the geometry on null hypersurfaces in a Lorentzian space-time of one dimension higher. We also construct theories of ultra-relativistic (Carrollian) gravity in 2+1 dimensions with dynamical exponent z<1 including cases that have anisotropic Weyl invariance for z=0.