A simple derivation of the relativistic Doppler effect in accelerated systems

The theory of the relativistic Doppler effect is well established and validated by several experiments. Most derivations, however, assume that both the reference frame of the source and the reference frame of the receiver are inertial, and the validity of the obtained formulas in the case of a non-inertial source and/or receiver is usually not clarified in higher physics education. As shown in this paper, if such formulas are carelessly applied to non-inertial cases, they can lead to contradictory results. This is probably what happened historically in the interpretation of experiments with a receiver in uniform circular motion around an inertial source, where some authors reported a shift towards the red, some others a shift towards the blue. This confusion points out the importance of establishing Doppler effect formulas that are valid in non-inertial cases. In this paper, we first set a rigorous definition of the relativistic Doppler shift, clarifying that it is the ratio between two proper time intervals relating to two different pairs of events. We then apply the definition to derive the relativistic Doppler shift in the case of a non-inertial source and/or receiver. The proposed derivations are very simple, provide important insights on the correct application of the time dilatation formula to accelerated systems, and highlight possible subtle mistakes that can lead to completely wrong results. This makes them a useful tool for teaching purposes, especially for graduate students.