1. The problem statement, all variables and given/known data In Minkowski spacetime, two observers, A and B, are moving at uniform speeds u and v, respectively, along different trajectories, each parallel to the y-axis of some inertial frame S. Observer A emits a photon with frequency nu_A that travels in the x-direction in S and is received by observer B with frequency nu_B. Show that the Doppler shift nu_B/nu_A in the photon frequency is independent of whether A and B are travelling in the same direction or opposite directions. 2. Relevant equations $$\lambda/\lambda' = \nu_B/\nu_A = \gamma(1-\beta\cos\theta)$$ Aberration formula: $$\cos\theta' = (\cos\theta - \beta)/(1-\beta\cos\theta) = -\beta$$ (for transverse case) 3. The attempt at a solution The answer is apparently that the Doppler shift is independent of the relative direction of motion. I have tried to transform to the frame S' where B is stationary, finding the velocity of A using the addition of velocities formula - to then get gamma. I have used the abberation formula to insert cos(theta') into the Doppler shift formula above to get $$\nu_B/\nu_A = \gamma(1+\beta^2)$$Plugging in the velocity of the emitter A in frame S' doesn't seem to get the required result.