relativistic Doppler effect

The Doppler effect is the change in a wave's frequency when source and observer move relative to one another — the same reason a passing siren drops in pitch. For light, relativity reshapes the classical formula because the messenger always travels at the same speed c, and because moving clocks run slow. A source rushing toward you has its light shifted to higher frequency (blueshift); one receding has it shifted lower (redshift).

For motion straight along the line of sight, the relativistic factor is sqrt((1 + beta)/(1 - beta)), with beta = v/c. This blends the familiar bunching-up or stretching-out of wave crests with an extra slowing of the source's own clock. The result is symmetric and frame-consistent in the way relativity demands, unlike the lopsided classical sound formula that treats the medium as special.

The most striking feature has no classical counterpart at all: the transverse Doppler effect. Even when a source flies past at its closest point, moving exactly sideways so it is neither approaching nor receding, its light still arrives redshifted. The shift is pure time dilation — the moving source's clock ticks slow, so every wave crest is delayed. Measuring this transverse redshift is a direct, laboratory-grade confirmation that moving clocks really do run slow.