Past exam of the symmetry and relativity course of the University of Oxford / 2017 / 3 / b / Solution

A boost into a frame moving with velocity $\mathbf{V}$ transforms the temporal component of the four-wavevector as $K^{\prime 0}=\Gamma (K^0-\mathbf{V}\cdot \mathbf{k}/c)$. Therefore the general relativistic Doppler effect iswhere $\vartheta$ 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, ${\omega }^{\prime }=K\cdot U_{\mathrm{obs}}$. The frequency is an observer-dependent contraction of the invariant wave phase.