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Consider a photon with energy moving at an angle with respect to the x- axis. Its three- velocity is , so its four- momentum is
where h is Planck's constant and the frequency.
In a frame with three- velocity (v,0,0) relative to the frequency is . Using the Lorentz transformations we get
so we get the following result:
If , so that the photon moves in the same direction as , we have
For low velocities this reduces to
This is the usual Doppler shift , modified at large v.
If , so the photon moves perpendicular to , we have
This is the transverse Doppler shift and is a consequence of time dilation.