Covariant tomography of fields
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908856620154880 |
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| author | Kycia, Radosław Antoni |
| author_facet | Kycia, Radosław Antoni |
| contents | This paper develops 'covariant tomography', a local framework for solving Inverse Boundary Value Problems (IBVP) for parallel transport equation on star-shaped domains. By integrating geometric decomposition with specific interior extensions - radial, heat equation, or harmonic - the method reconstructs currents and gauge potentials from boundary data. The choice of extension directly dictates the regularity of the recovered interior fields. A primary contribution is the 'tower' algorithm, which reduces higher-order systems, such as Maxwell equations, to a sequence of coupled first-order equations. We establish a formal solvability criterion (Theorem \ref{Th_TowerTheorem}), proving that higher-order IBVPs are solvable if and only if this tower is sequentially solvable. The framework is validated through low-dimensional examples and electromagnetic potential reconstruction in $\mathbb{R}^{3}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13261 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Covariant tomography of fields Kycia, Radosław Antoni Mathematical Physics Analysis of PDEs Differential Geometry 58A10, 53Z05, 51P05, 35R30, 70S15 This paper develops 'covariant tomography', a local framework for solving Inverse Boundary Value Problems (IBVP) for parallel transport equation on star-shaped domains. By integrating geometric decomposition with specific interior extensions - radial, heat equation, or harmonic - the method reconstructs currents and gauge potentials from boundary data. The choice of extension directly dictates the regularity of the recovered interior fields. A primary contribution is the 'tower' algorithm, which reduces higher-order systems, such as Maxwell equations, to a sequence of coupled first-order equations. We establish a formal solvability criterion (Theorem \ref{Th_TowerTheorem}), proving that higher-order IBVPs are solvable if and only if this tower is sequentially solvable. The framework is validated through low-dimensional examples and electromagnetic potential reconstruction in $\mathbb{R}^{3}$. |
| title | Covariant tomography of fields |
| topic | Mathematical Physics Analysis of PDEs Differential Geometry 58A10, 53Z05, 51P05, 35R30, 70S15 |
| url | https://arxiv.org/abs/2601.13261 |