Carrollian ${\mathbb{R}}^{\times }$-bundles III: The Hodge Star and Hodge--de Rham Laplacians

Abstract

Carrollian ${\mathbb{R}}^{\times }$-bundles (${\mathbb{R}}^{\times }:=\mathbb{R}\setminus \{0\}$) offer a novel perspective on intrinsic Carrollian geometry using the powerful tools of principal bundles. Given a choice of principal connection, a canonical Lorentzian metric exists on the total space. This metric enables the development of Hodge theory on a Carrollian ${\mathbb{R}}^{\times }$-bundle; specifically, the Hodge star operator and Hodge--de Rham Laplacian are constructed. These constructions are obstructed on a Carrollian manifold due to the degenerate metric. The framework of Carrollian ${\mathbb{R}}^{\times }$-bundles bridges the gap between Carrollian geometry and (pseudo)-Riemannian geometry. As an example, the question of the Hodge--de Rham Laplacian on the event horizon of a Schwarzschild black hole is addressed. A Carrollian version of electromagnetism is also proposed.

9 pages. Comments welcomed