Motion in An Arbitrary Direction
If, in the reference frame of the observer, the source is moving away with velocity at an angle relative to the direction from the observer to the source (at the time when the light is emitted), the frequency changes as
- (1)
In the particular case when and one obtains the transverse Doppler effect:
Due to the finite speed of light, the light ray (or photon, if you like) perceived by the observer as coming at angle, was, in the reference frame of the source, emitted at a different angle . and are tied to each other via the relativistic aberration formula:
Therefore, Eq. (1) can be rewritten as
- (2)
For example, a photon emitted at the right angle in the reference frame of the emitter would be seen blue-shifted by the observer:
In the non-relativistic limit, both formulæ (1) and (2) give
Read more about this topic: Relativistic Doppler Effect
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—Ralph Waldo Emerson (1803–1882)