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We study a retarded potential solution of a massless scalar field in curvedspace-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. Thesolution indicates that the phase velocity of the retarded potential of anon-moving scalar charge is position dependent, and may easily be greater thanthe speed of light at a given point. In the case of the Schwarzschildspace-time, at the horizon, the phase velocity becomes infinitely faster thanthe coordinate speed of light at that point. Superluminal phase velocity isrelatively common phenomenon, with the the phase velocity of the massiveKlein-Gordon field as the best known example. We discuss why it is possible tohave modes with superluminal phase velocity even for a massless field.

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