Carrollian $\mathbb{R}^\times$-bundles III: The Hodge Star and Hodge--de Rham Laplacians
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918107184889856 |
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| author | Bruce, Andrew James |
| author_facet | Bruce, Andrew James |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21906 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Carrollian $\mathbb{R}^\times$-bundles III: The Hodge Star and Hodge--de Rham Laplacians Bruce, Andrew James Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics 53C50, 58A14, 83C99 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. |
| title | Carrollian $\mathbb{R}^\times$-bundles III: The Hodge Star and Hodge--de Rham Laplacians |
| topic | Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics 53C50, 58A14, 83C99 |
| url | https://arxiv.org/abs/2507.21906 |