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

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.

Take the wave along positive $x$, so $k_x=\omega /c$. A longitudinal boost $\mathbf{V}=(v_x,0,0)$ givesA receiver moving in the propagation direction measures a redshift. The same signed expression gives a blueshift when $v_x<0$. If the original wave propagates along negative $x$ instead, retain $k_x=-\omega /c$ in the general formula.

For $\mathbf{V}=(0,v_y,0)$, the laboratory wavevector is perpendicular to the boost. The dot product vanishes, soThis transverse-observer transformation is a blueshift. It must not be confused with the inverse factor often quoted when transverse refers to a ray perpendicular to the source velocity in the receiver's frame. The angle condition is specified in different frames in those two statements; relativistic aberration relates them.