Past exam of the symmetry and relativity course of the University of Oxford / 2016 / 3 / a / 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 ${\beta }_{\mathrm{app}}=\beta \sin \vartheta /(1-\beta \cos \vartheta )$. NRAO's time sequence of Galactic microquasar ejecta gives a concrete example of the third measurement.^[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 isThis 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, ${\omega }^{\prime }/\omega =\sqrt{(1-\beta )/(1+\beta )}$; a head-on frame blueshifts it. Transverse angles defined in different frames must not be interchanged.