Riemannian and Lorentzian Calder\'on problem under Magnetic Perturbation

We study both the Riemannian and Lorentzian Calder\'on problem when a family of Dirichlet-to-Neumann maps are given for an open set of magnetic/electromagnetic potentials. For the Riemannian version, by allowing small perturbations of the magnetic potential, we use the Runge Approximation Theorem to show that the metric can be uniquely determined. There is no gauge equivalence in this case. For the Lorentzian version, we use microlocal analysis to construct the trajectory of null-geodesics via generic perturbations of the electromagnetic potential, hence the conformal class of the metric can be constructed. Moreover, we also show, in the Lorentzian case, the same result can be obtained using generic perturbations of the metric itself.