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The Momentum Light Ray Transform

Sombuddha Bhattacharyya, Tuhin Mondal, Suman Kumar Sahoo

TL;DR

This work develops the Momentum Light Ray Transform (MLRT) for symmetric tensor fields in space-time, defining $L^{m,k}$ and its restricted variants to study injectivity and reconstruction under full and restricted angular data. It establishes a global injectivity result when all directions are allowed via the highest moment $L^{m,m}$, and characterizes the kernel when directions are restricted to the sphere or to neighborhoods of a fixed direction, showing the MLRT cannot in general recover lower-rank components in restricted settings. The paper then provides explicit inversion formulas and algorithms: (i) for vector fields, via a Fourier-slice framework and explicit reconstruction formulas on an open set $H_n$, and (ii) for higher-rank symmetric tensor fields through a trace-divergence decomposition $f^{(m)}=A^{(m)}+i_{g_{1/c}}f^{(m-2)}$ with $JA^{(m)}=0$, enabling reconstruction from restricted data $\tilde{L}^{m,k}$ with $k=0,1,...,m$ on $B_n(\omega_0,\delta)$. The methods combine tensor tomography, Lorentzian geometry, and microlocal tools to connect MLRT with the X-ray transform and MRT, offering practical inversion schemes for restricted data and advancing inverse problems for higher-order wave phenomena.

Abstract

In this article, we study Momentum Light Ray Transform (MLRT) on symmetric tensor fields. MLRT is an integral transform in time-space domain ($(t,x)\in \mathbb{R}^{1+n}$), which integrates a scalar function or a tensor field along the light rays with a polynomial type weight. We explore necessary and sufficient conditions for injectivity of MLRT, over general order tensors on space dimension $\geq 2$, from full and restricted measurements. Furthermore, we develop an inversion algorithm for MLRTs in the restricted measurement setting. To prove the results, we use tools from tensor tomography, geometry, and analysis.

The Momentum Light Ray Transform

TL;DR

This work develops the Momentum Light Ray Transform (MLRT) for symmetric tensor fields in space-time, defining and its restricted variants to study injectivity and reconstruction under full and restricted angular data. It establishes a global injectivity result when all directions are allowed via the highest moment , and characterizes the kernel when directions are restricted to the sphere or to neighborhoods of a fixed direction, showing the MLRT cannot in general recover lower-rank components in restricted settings. The paper then provides explicit inversion formulas and algorithms: (i) for vector fields, via a Fourier-slice framework and explicit reconstruction formulas on an open set , and (ii) for higher-rank symmetric tensor fields through a trace-divergence decomposition with , enabling reconstruction from restricted data with on . The methods combine tensor tomography, Lorentzian geometry, and microlocal tools to connect MLRT with the X-ray transform and MRT, offering practical inversion schemes for restricted data and advancing inverse problems for higher-order wave phenomena.

Abstract

In this article, we study Momentum Light Ray Transform (MLRT) on symmetric tensor fields. MLRT is an integral transform in time-space domain (), which integrates a scalar function or a tensor field along the light rays with a polynomial type weight. We explore necessary and sufficient conditions for injectivity of MLRT, over general order tensors on space dimension , from full and restricted measurements. Furthermore, we develop an inversion algorithm for MLRTs in the restricted measurement setting. To prove the results, we use tools from tensor tomography, geometry, and analysis.
Paper Structure (9 sections, 16 theorems, 121 equations)

This paper contains 9 sections, 16 theorems, 121 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminary results
  3. Injectivity of MLRT
  4. Direction vector in $(\mathbb{R}^n\setminus\{0\})$
  5. Direction vector restricted on the sphere:
  6. Direction vector near a fixed vector on $\mathbb{S}^{n-1}$:
  7. Inversion formula for MLRT of symmetric tensor fields
  8. Reconstruction of vector fields:
  9. Reconstruction of symmetric $m$-tensor fields:

Key Result

Theorem 1

Assume that, $f^{(m)} \in C_{c}^{\infty}(\mathbb{R}^{1+n};S^{m})$ for $m\geq 0$ and $n\ge 2$. If $L^{m,m}f^{(m)}((t,x),\omega)=0$, for all $(t,x)\in \mathbb{R}^{1+n}$, $\omega \in \mathbb{R}^{n}\backslash \{0\}$, then $f^{(m)}=0$.

Theorems & Definitions (41)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1.3
  • Theorem 4
  • Remark 1.4
  • Theorem 5
  • Lemma 2.1: Fourier slice theorem, ramm1991inverse
  • ...and 31 more