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Covariant tomography of fields

Radosław Antoni Kycia

TL;DR

This work addresses the inverse boundary value problem for covariant fields by introducing covariant tomography on star-shaped domains, linking boundary data to interior currents and gauge potentials through a geometric decomposition of differential forms. The method rests on the homotopy-based decomposition $\Lambda^{k}(U)=\mathcal{E}^{k}(U)\oplus\mathcal{A}^{k}(U)$ and the covariant derivative $d^{\nabla}=d+A\wedge$, enabling a two-step Extension/Projection framework with three extension strategies (constant along rays, heat-diffusion, harmonic) and a projection step to solve $d^{\nabla}\phi=J$ or $A\wedge\phi=J-d\phi$. Key contributions include explicit local solutions and convergence criteria, analysis of non-uniqueness due to gauge modes, and the extension of the framework to geometric-based PDEs such as Maxwell equations, supported by low-dimensional and $\mathbb{R}^{3}$ electromagnetic examples. The results provide a systematic, geometry-aware tomographic toolkit for interior field reconstruction from boundary measurements in physics and engineering, with clear implications for gauge-field and current tomography in covariant settings.

Abstract

This paper addresses the Inverse Boundary Value Problem (IBVP) for classical fields, specifically focusing on the recovery of parallelly transformed fields within a region based on known boundary data. We introduce a local solution framework, termed "covariant tomography," that uses geometric decomposition to reconstruct interior fields and currents within star-shaped open subsets. The core of our approach involves decomposing differential forms into exact and antiexact components, enabling the formulation of the parallel transport equation via a homotopy operator. We examine three primary extension techniques - radial, heat equation, and harmonic - to map boundary values into the interior, noting that the choice of extension directly influences the regularity of the resulting currents. The proposed methodology provides a systematic way to identify the realizability of boundary values and offers solutions for both current and gauge field tomography. Finally, we demonstrate the utility of this framework through illustrative examples in low-dimensional spaces and electromagnetic potential reconstruction in $\mathbb{R}^{3}$.

Covariant tomography of fields

TL;DR

This work addresses the inverse boundary value problem for covariant fields by introducing covariant tomography on star-shaped domains, linking boundary data to interior currents and gauge potentials through a geometric decomposition of differential forms. The method rests on the homotopy-based decomposition and the covariant derivative , enabling a two-step Extension/Projection framework with three extension strategies (constant along rays, heat-diffusion, harmonic) and a projection step to solve or . Key contributions include explicit local solutions and convergence criteria, analysis of non-uniqueness due to gauge modes, and the extension of the framework to geometric-based PDEs such as Maxwell equations, supported by low-dimensional and electromagnetic examples. The results provide a systematic, geometry-aware tomographic toolkit for interior field reconstruction from boundary measurements in physics and engineering, with clear implications for gauge-field and current tomography in covariant settings.

Abstract

This paper addresses the Inverse Boundary Value Problem (IBVP) for classical fields, specifically focusing on the recovery of parallelly transformed fields within a region based on known boundary data. We introduce a local solution framework, termed "covariant tomography," that uses geometric decomposition to reconstruct interior fields and currents within star-shaped open subsets. The core of our approach involves decomposing differential forms into exact and antiexact components, enabling the formulation of the parallel transport equation via a homotopy operator. We examine three primary extension techniques - radial, heat equation, and harmonic - to map boundary values into the interior, noting that the choice of extension directly influences the regularity of the resulting currents. The proposed methodology provides a systematic way to identify the realizability of boundary values and offers solutions for both current and gauge field tomography. Finally, we demonstrate the utility of this framework through illustrative examples in low-dimensional spaces and electromagnetic potential reconstruction in .
Paper Structure (9 sections, 5 theorems, 46 equations, 1 figure)

This paper contains 9 sections, 5 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results
  4. Extension problem
  5. Projection problem
  6. Convergence, stability, and non-uniqueness
  7. Tomography for geometric-based equations
  8. Examples
  9. Conclusions

Key Result

Theorem 1

(Theorem 1 of KyciaSilhan) The unique nontrivial solution to the equation with the condition $dH\phi= c\in \mathcal{E}(U,V)\setminus ker(A\wedge\_)$, $c\neq 0$, is given by where $c$ is an arbitrary form, $(H(A\wedge \_))^{0}=Id$, and is the $l$-fold composition of the operator $H\circ A \wedge \_$. The series in (Eq.Solution_homogenous_k_gt_0) is uniformly convergent to the continuous form ($\

Figures (1)

  • Figure 1: Geometry of the problem.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Theorem 3
  • Definition 1
  • Example 1: $\Lambda^{0}$ on $(0,1)$
  • Example 2: $\Lambda^{1}$ on $(0,1)$
  • Corollary 1
  • Example 3: $\Lambda^{0}$ on $(0,1)$ - radial extension
  • Example 4: Electromagnetic potential in $\mathbb{R}^{3}$ knowing $J$.
  • ...and 1 more