= Relativistic Doppler effect
{title2=$\nu=\gamma\nu'(1+\beta\cos\theta')$}
= Relativistic Doppler shift
{synonym}
The <Lorentz transformation> changes the <photon> <four-momentum> and therefore its frequency, producing the displayed formula for a boost along the first axis. Here the primed ray angle is measured in the emitting frame and $\beta=v/c$. The angle and frequency transforms must be used together: <aberration of light> changes the direction, and a photon-number angular fraction need not equal the corresponding radiant-energy fraction.
Here $c$ is the <vacuum> <speed of light>, $\gamma$ is the <Lorentz factor>, and the two <frequencies> are measured in different inertial <reference frames>. Unlike the <classical Doppler effect for sound>, there is no material rest <reference frame> for <light> in <vacuum>. For small radial recession speed $v_r\ll c$, the <wavelength> shift satisfies $\Delta\lambda/\lambda\simeq v_r/c$. This small-speed limit underlies <Doppler spectroscopy>; it is not an exact sound formula.
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