Relativistic Doppler Effect of Light and Matter Waves
... −ωt + k · r represents the scalar product of a four component vector (iω/c, k) with the space-time four-vector
xµ = (ict, x, y, z). This implies that the four component
vector (iω/c, k) is also a four-vector.
This is a rather long procedure, but once this fact is
established, ( ω, k ) transformation ...