# Recovery of time dependent coefficients from boundary data for hyperbolic equations

## TL;DR

This paper proves the uniqueness of recovering time-dependent magnetic and electric potentials in hyperbolic equations on Lorentzian manifolds using boundary measurements, Gaussian beams, and light ray transform inversion.

## Contribution

It establishes the uniqueness of potential recovery from boundary data for hyperbolic equations on specific Lorentzian manifolds, extending inverse problem theory.

## Key findings

- Uniqueness of potential recovery proved under given conditions.
- Method based on Gaussian beams and light ray transform inversion.
- Applicable to manifolds that are products of Riemannian manifolds and time intervals.

## Abstract

We study uniqueness of the recovery of a time-dependent magnetic vector-valued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet to Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold.