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.Comment: 31 page