Relativistic Doppler effect

The relativistic Doppler effect is the frequency shift of light from a source in inertial motion as measured by an observer in any other inertial frame. For a source emitting at proper frequency f₀ and receding from the observer along the line of sight at velocity v = βc, the observed frequency is f = f₀ √((1 − β)/(1 + β)) = f₀/(γ(1 + β)). For approach (v < 0, β < 0), the formula gives a blueshift f > f₀; for recession (β > 0), a redshift f < f₀. The structure is the product of two factors: the classical Doppler shift (1 − β)/(1 + β)^(1/2) accounting for the changing path length of successive wavefronts, and the time-dilation factor 1/γ accounting for the slower ticking of the source's emission clock as measured in the observer's frame.

Christian Doppler in 1842 derived the classical formula for sound and light by considering only the path-length effect. The relativistic version adds the kinematic time-dilation factor and applies symmetrically to electromagnetic waves in vacuum (no medium needed). The redshift-distance relation for galaxies (Hubble 1929) is dominated by cosmological expansion rather than special-relativistic Doppler, but the formula remains exact for galaxies in our cosmological neighbourhood up to peculiar-velocity contributions. Particle physics uses the relativistic Doppler shift continuously: pion lifetimes in flight, atomic transitions in fast ion beams, and the spectroscopy of relativistic jets from active galactic nuclei all rely on inverting the formula to recover the source frame's rest frequency from the observed frequency. The transverse-Doppler effect, where the source has zero radial velocity at the moment of emission but non-zero velocity perpendicular to the line of sight, is the limiting case f = f₀/γ — pure time dilation, no classical contribution.