= Solution
A boost into a frame moving with <velocity> $\mathbf V$ transforms the temporal component of the <four-wavevector> as $K'^0=\Gamma(K^0-\mathbf V\cdot\mathbf k/c)$. Therefore the general <relativistic Doppler effect> is
$$
\boxed{\omega'=\Gamma(\omega-\mathbf V\cdot\mathbf k)
=\Gamma\omega(1-\beta\cos\vartheta),}
$$
where $\vartheta$ is the laboratory angle between the boost and propagation directions. This is also the frequency measured from the phase rate along the observer's <world line>, $\omega'=K\cdot U_{\mathrm{obs}}$. The frequency is an observer-dependent contraction of the invariant wave phase.
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