File:Transverse Doppler effect scenarios 3.svg

English: The ease of analyzing a relativistic scenario often depends on the frame in which one chooses to perform the analysis. Here we analyze transverse Doppler shift for the case where source and receiver are at their closest approach to each other.

(left) If we analyze the scenario in the frame of the receiver, we find that the analysis is more complicated than it should be. The apparent position of a celestial object is displaced from its true position (or geometric position) because of the object's motion during the time it takes its light to reach an observer. The source would be time-dilated relative to the receiver, but the redshift implied by this time dilation would be offset by a blueshift due to the longitudinal component of the relative motion between the receiver and the apparent position of the source.

(right) It is much easier if, instead, we analyze the scenario from the frame of the source. An observer situated at the source knows, from the problem statement, that the receiver is at its closest point to him. That means that the receiver has no longitudinal component of motion to complicate the analysis. Since the receiver's clocks are time-dilated relative to the source, the light that the receiver receives is therefore blue-shifted by a factor of gamma.