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Green's function of a massless scalar field in curved space-time and superluminal phase velocity of the retarded potential

We study a retarded potential solution of a massless scalar field in curved space-time. In a special ansatz for a particle at rest whose magnitude of the (scalar) charge is changing with time, we found an exact analytic solution. The solution indicates that the phase velocity of the retarded potential of a non-moving scalar charge is position dependent, and may easily be greater than the speed of light at a given point. In the case of the Schwarzschild space-time, at the horizon, the phase velocity becomes infinitely faster than the coordinate speed of light at that point. Superluminal phase velocity is relatively common phenomenon, with the the phase velocity of the massive Klein-Gordon field as the best known example. We discuss why it is possible to have modes with superluminal phase velocity even for a massless field.