= Solution
Three relevant observable effects are the relativistic <Doppler effect> in spectral-line shifts, including the transverse time-dilation contribution; <relativistic aberration>, observable as a change of apparent light direction with observer motion; and <relativistic beaming> of moving emitters, associated with strongly asymmetric jet brightness. <Apparent superluminal motion> supplies a related light-travel-time test: $v_{\mathrm{app}}/c=\beta\sin\vartheta/(1-\beta\cos\vartheta)$ can exceed one without a material speed exceeding $c$. These interpretations require an emission and source-motion model; a cosmological or gravitational redshift by itself is not a direct test of special-relativistic kinematics.
For <Compton scattering> from an initially stationary <electron>, let incoming and outgoing <photon> <energies> be $\varepsilon$ and $\varepsilon'$ and their mutual angle be $\psi$. <Four-momentum conservation> gives $P+Q=P'+Q'$. Squaring $P'=P+Q-Q'$ and using both <electron> mass shells and both <photon> null norms gives
$$
2P\cdot(Q-Q')=2Q\cdot Q',\qquad
m_e(\varepsilon-\varepsilon')=\frac{\varepsilon\varepsilon'}{c^2}(1-\cos\psi).
$$
Consequently the outgoing angular frequency and shift are
$$
\boxed{\omega'=\frac{\omega}{1+\frac{\hbar\omega}{m_ec^2}(1-\cos\psi)},\qquad
\Delta\omega=-\omega\frac{\frac{\hbar\omega}{m_ec^2}(1-\cos\psi)}{1+\frac{\hbar\omega}{m_ec^2}(1-\cos\psi)}.}
$$
Equivalently, the <Compton wavelength shift> is $\lambda'-\lambda=h(1-\cos\psi)/(m_ec)$. The <electron> recoil is essential; treating its <energy> as unchanged would suppress the effect.
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