Can information be transferred faster than light? II. The relativistic Doppler effect on electromagnetic wave packets with suboptic and superoptic group velocities

Abstract

It is shown that (a) both the dispersion relations between the mean frequency θ₀ and the mean wave number k ₀ are invariant under the Lorentz transformation; and (b) the relativistic Doppler effects on θ ₀ and k ₀ differ. In the suboptic packet there is anomalous red shift in the mean wave number k' ₀ received from a source receding with speed v: k′ ₀ changes sign through zero as v goes through the value v_g, the mean group velocity in the packet. In the superoptic packet the anomalous red shift is in the mean frequency θ′ ₀ which reverses sign through zero as v goes through the value v_p, the mean phase velocity in the packet. This last finding indicates a blackout of the superoptic signal when propagated toward a receiver moving away from the source at a speed greater than v_p. There is no violation of causality involved, and the final conclusion of the paper is that it is not a fundamental axiom of “special relativity,” as usually believed, to deny that information can be transmitted at speeds greater than c.