The relativistic Doppler effect

The classical Doppler effect, familiar from sound, arises purely from relative motion changing how quickly successive wave crests reach an observer: a source approaching an observer compresses the crests together, raising the observed frequency, while a receding source stretches them out, lowering it. For light, this same relative-motion effect operates, but special relativity adds a second contribution that classical sound waves do not have: the source's own clock runs slow, as measured by the observer, due to ordinary time dilation, and this stretches every wave period by an additional factor of γ regardless of the direction of motion.

The combined relativistic Doppler formula folds both effects together. For a source moving directly along the line of sight, the classical compression-or-stretching effect and the time-dilation effect both point the same way — approach produces blueshift, recession produces redshift — so the relativistic formula looks similar in spirit to the classical one but with an extra γ factor baked in from time dilation. The genuinely new prediction appears for transverse motion, where a source crosses the line of sight exactly perpendicular to the observer at the moment of closest approach. Classically, a source moving purely sideways relative to an observer produces no Doppler shift at all — the distance to the observer is momentarily neither growing nor shrinking. Relativistically, the source's clock is still time-dilated regardless of the direction of its motion, so the observed light is still redshifted even in this purely transverse case — the transverse Doppler effect, a distinctively relativistic prediction with no classical acoustic analogue, and one that has been directly measured in particle-accelerator and satellite experiments as confirmation of time dilation itself.

Unlike the sound Doppler effect, which depends on the source's and observer's motion relative to the medium (air) that carries the sound wave, the relativistic Doppler effect for light depends only on the relative motion between source and observer — light requires no medium, consistent with the postulates of special relativity.