Relativistic phase velocity transformation

Abstract

By utilizing the appropriate Doppler formulas, it is shown that phase velocities in both Galilean and relativistic kinematics transform according to rules different from the ordinary velocity addition laws. As corollaries, it is demonstrated that (i) phase velocities parallel to the inertial frames’ relative velocity transform according to the Lorentz velocity addition rule; (ii) regardless of its direction, the velocity of light in vacuum transforms as a relativistic particle velocity; and (iii) the relativistic phase velocity transformation rule reduces to the Galilean rule in the nonrelativistic limit.