Stability estimate and the Aharonov-Bohm effect for the Klein-Gordon equation with time-dependent vector potentials

We consider the relativistic Schr\"odinger (Gordon-Klein) equation with a time dependent vector and a scalar potential on a bounded cylindrical domain. Using a standard Geometric Optics Ansatz, we establish a logarithmic stability estimate for the recovery of the vector potentials. We also study the recovery of the potentials in the presence of obstacles. It is shown that under some geometric restrictions, the recovery of the potentials is possible up to a gauge transform.