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Inverse problem for connections in semi-linear wave equations on Lorentzian manifolds

Lauri Oksanen, Ruochong Zhang

TL;DR

The paper addresses recovering a time-dependent Hermitian connection $A$ on a globally hyperbolic Lorentzian manifold from a semilinear wave equation with cubic nonlinearity via the source-to-solution map. It develops a microlocal framework for nonlinear wave interactions, reducing the inverse problem to the inversion of a non-abelian broken light-ray transform along broken null geodesics, and proves recovery of $A$ on the causal diamond up to a gauge, leveraging a gauge-fixing argument and the parallel transport along lightlike geodesics. The work extends prior Minkowski and Riemannian results to curved spacetimes, handles cut and conjugate points, and provides a complete microlocal analysis (including three-fold linearization and Greenleaf–Uhlmann calculus) to establish uniqueness of the broken-ray data and the reconstruction of the connection. This advances inverse problems on Lorentzian manifolds by proving that time-dependent, lower-order coefficients (connections) can be recovered from nonlinear wave measurements, with potential applications to geometric imaging in relativity and gauge-field problems.

Abstract

This paper recovers Hermitian connections of semi-linear wave equations with cubic nonlinearity. The main novelty is in the geometric generality: we treat the case of an arbitrary globally hyperbolic Lorentzian manifold. Our approach is based on microlocal analysis of nonlinear wave interactions, which recovers a non-abelian broken light-ray transform, and the inversion of broken light-ray transforms on globally hyperbolic Lorentzian manifolds.

Inverse problem for connections in semi-linear wave equations on Lorentzian manifolds

TL;DR

The paper addresses recovering a time-dependent Hermitian connection on a globally hyperbolic Lorentzian manifold from a semilinear wave equation with cubic nonlinearity via the source-to-solution map. It develops a microlocal framework for nonlinear wave interactions, reducing the inverse problem to the inversion of a non-abelian broken light-ray transform along broken null geodesics, and proves recovery of on the causal diamond up to a gauge, leveraging a gauge-fixing argument and the parallel transport along lightlike geodesics. The work extends prior Minkowski and Riemannian results to curved spacetimes, handles cut and conjugate points, and provides a complete microlocal analysis (including three-fold linearization and Greenleaf–Uhlmann calculus) to establish uniqueness of the broken-ray data and the reconstruction of the connection. This advances inverse problems on Lorentzian manifolds by proving that time-dependent, lower-order coefficients (connections) can be recovered from nonlinear wave measurements, with potential applications to geometric imaging in relativity and gauge-field problems.

Abstract

This paper recovers Hermitian connections of semi-linear wave equations with cubic nonlinearity. The main novelty is in the geometric generality: we treat the case of an arbitrary globally hyperbolic Lorentzian manifold. Our approach is based on microlocal analysis of nonlinear wave interactions, which recovers a non-abelian broken light-ray transform, and the inversion of broken light-ray transforms on globally hyperbolic Lorentzian manifolds.

Paper Structure

This paper contains 17 sections, 14 theorems, 239 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Forward problem
  3. Main theorem
  4. Previous literature
  5. Broken light-ray transform
  6. Lorentzian geometry
  7. Parallel transport
  8. Recovery of the connection
  9. Microlocal analysis of linear waves
  10. Connection wave operator
  11. Lagrangian distributions
  12. Paired Lagrangian distributions
  13. Distorted plane waves
  14. Nonlinear interactions
  15. Three-fold linearization
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $A$ and $B$ be two Hermitian connections such that $L_{g,A} = L_{g,B}$ and $A=B$ on $\mho$. Then, there exists a gauge transformation $\boldsymbol{\varphi}$ satisfying eqn: natural gauge such that

Figures (1)

  • Figure 1: The blue cylinder is $\mho$ and the black revolution is $\mathbb D$. The parallel transport is from $x$ to $y$ and then $y$ to $z$ through the black broken light-like geodesic. The dashed lines are world lines $\mu_{a_1}$ and $\mu_{a_2}$ through $x$ and $z$, respectively.

Theorems & Definitions (28)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Definition 2.1
  • Theorem 2.1
  • Lemma 2.5
  • proof
  • ...and 18 more