Generalized transform of the I relativistic phase velocity

The purpose of this article is to derive the generalized formula of relativistic phase velocity transformation, by utilizing the Lorentz transforms on the wavevector k and frequency w for the moving inertial frame relative to the static system moving uniformly in an arbitrary direction. Furthermore, some significant cases are discussed. R 6 w & Cet essai est pour dCriver le gCniralisateur formule de transformation de vitesse de phase relativiste, via utiliser le Lorentz transformation sur la vecteur de ondulatoue k et la friquence w pour le r.5fCrentiel inertiel de mouvoir relativement au referentiel immobile se mouvoir uniformbment vers la direction arbitrairement. Ensuite, quelques cas signifiant sont a discutes. We consider the inertial frame S' moving at a velocity v with respect to another inertial frame S uniformly in an arbitrary direction, and let their corresponding coordinate axes coincide respectively when I = 0. From the transforms of the four-vector, we find formulae of relativistic transformation between (k,w/c) and (k',w'/c) as follows [ I ] k = ( y l)(k'.v)u/t?+k'tyw'v/? (la) w = y(w' + k'.v) ( Ib) where labelled and unlabelled primed quantities are observed from the inertial frames S and S' respectively and y = 1/(1 t?/c2)'". The inverse formulae, expressing if, w ' in terms of k, w, are most easily obtained by changing v to -U (since the frame S moves with velocity -U relative to the frame SI), namely k '=(yI ) (k .v )v /~+k-ywv/ ' c2 (2a) U' = y ( w k.v) . (26) The generalized formula of relativistic phase velocity transformation can be obtained by using formulae (1) and (2), and the definition of the phase velocity which is given by [Z]