Schr\"odinger equation in a general curved space-time geometry

We consider relativistic quantum field theory in the presence of an external electric potential in a general curved space-time geometry. We utilise Fermi coordinates adapted to the time-like geodesic to describe the low-energy physics in the laboratory and calculate the leading correction due to the curvature of the space-time geometry to the Schr\"odinger equation. We then compute the non-vanishing probability of excitation for a hydrogen atom that falls in or is scattered by a general Schwarzschild black hole. The photon that is emitted from the excited state by spontaneous emission extracts energy from the black hole, increases the decay rate of the black hole and adds to the information paradox.