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

TL;DR

This paper develops Hodge theory on Carrollian $R^ imes$-bundles, constructing the Hodge star and Laplacian, and applies these tools to black hole horizons and Carrollian electromagnetism.

Contribution

It introduces a novel framework for Carrollian geometry using principal bundles, enabling Hodge theory constructions previously obstructed by degenerate metrics.

Findings

Constructed Hodge star and Laplacian on Carrollian $R^ imes$-bundles.

Applied the framework to the event horizon of Schwarzschild black holes.

Proposed a Carrollian version of electromagnetism.

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.