Wave 4-vector. Doppler Effect

If we divide the momentum-energy 4-vector by the rationalized Planck constant (or Dirac constant) , we get the wave 4-Vector:

          or                                 (25)

where  is the wave vector associated with the wave number . It is related to the momentum  , by . The energy can be expressed as , where  .

            The Lorentz transformation allows us to derive the wave frequency an observer sees according to the wave frequency emitted in the source frame.

               (26)

From the matrices product, we get:

                   (27)

The last one is the most important of them, representing the Doppler effect, i.e., a change in the observer frequency of a wave as a result of relative motion between the source and the observer.

 can be written as ,

Therefore:

            (28)

But, , so

;  Because , we obtain:

  (29)

            If one consider the case in which the direction of the propagation of wave and the relative velocity of the S and S’ frame do coincide, we obtain longitudinal Doppler effect.

            If the source travels strait away from the observer,

and              (30)

If the source travels toward the observer, , so

            (31)

If an observer moving relative to the source’s frame S, looks perpendicularly at the wave source which emits at the frequency n, he will measure a modified frequency.

            Hence:

, so

            (32)

content