Scalar radiation from a radially infalling source into a Schwarzschild black hole in the framework of quantum field theory

We investigate the radiation to infinity of massless scalar field from a source falling radially towards a Schwarzschild black hole using the framework of quantum field theory at tree level. In the case where the source falls from infinity, the monopole radiation is dominant for low initial velocities but higher multipoles become dominant at high initial velocities. It is found that, as in the electromagnetic and gravitational cases, at high initial velocities the energy spectrum for each multipole with $l \geq 1$ is approximately constant up to the fundamental quasinormal frequency and then drops to zero. We also investigate the case where the source falls from rest at a finite distance from the black hole. We find that the monopole and dipole contributions are dominant in this case. We point out that this case needs to be distinguished carefully from the unphysical process where the source abruptly appears at rest and starts falling, which would result in radiation of an infinite amount of energy. We also investigate the radiation of massless scalar field to the horizon of the black hole, finding some features similar to the gravitational case.