The transverse Doppler effect is closely related to time dilation.
In special relativity, a source moving sideways relative to an observer can produce a frequency shift even when there is no ordinary motion toward or away from the observer. The key mechanism is time dilation: the moving source’s clock runs at a different rate in the observer’s frame, so the intervals between emitted wave crests are changed.
This was a striking prediction because classical intuition suggests that motion exactly across the line of sight should produce no Doppler shift. Relativity instead predicts a transverse effect involving the Lorentz factor. Depending on whether the geometry is described at the source’s or observer’s closest approach, discussions can involve a blueshift or a redshift; the experimental setup and angle convention matter.
Frame dragging is a general-relativistic effect near rotating mass, mass defect concerns nuclear binding energy, and length contraction changes measured distances. They are all physics concepts, but time dilation is the direct special-relativistic connection here.