= Solution
Three useful observational phenomena are <relativistic Doppler effect> in spectral lines from approaching and receding ejecta, <relativistic beaming> through jet brightness asymmetry, and <apparent superluminal motion> inferred from successive radio images of moving knots. The first compares spectral frequencies with rest-frame lines; the second compares the two sides after accounting for intrinsic emission and absorption; the third tests the light-travel-time relation $\beta_{\rm app}=\beta\sin\vartheta/(1-\beta\cos\vartheta)$. NRAO's time sequence of Galactic microquasar ejecta gives a concrete example of the third measurement.<Microquasars in Our Own Galaxy by National Radio Astronomy Observatory>[VLA time-series observations of GRS 1915+105 and explanation of apparent superluminal separation by subluminal jets directed toward the observer]{ref}
These are tests of relativistic radiation kinematics when source geometry and environmental effects are controlled, rather than claims that any one redshift or brightness contrast by itself uniquely proves <special relativity>.
For the frequency transformation, a photon has <wave four-vector> $K=(\omega/c,\mathbf k)$ with $|\mathbf k|=\omega/c$ in vacuum. The temporal component of a <Lorentz boost> is
$$
\frac{\omega'}c=\gamma_v\left(\frac\omega c-\frac{\mathbf v\cdot\mathbf k}{c}\right),
\qquad
\boxed{\omega'=\gamma_v\omega\left(1-\frac vc\cos\theta\right).}
$$
This is the <relativistic Doppler effect>: frequency changes both because of relative motion along the ray and because clocks in relative motion are time dilated. The angle here is measured in S. A frame chasing the photon redshifts it, $\omega'/\omega=\sqrt{(1-\beta)/(1+\beta)}$; a head-on frame blueshifts it. Transverse angles defined in different frames must not be interchanged.
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