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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

therefore

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.