  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.
