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I have a question regarding relativistic Doppler effect. The following page is taken from "Introduction to Special Relativity" by Robert Resnick. Relativistic Transverse Doppler effect

My question is why Eq. (2.25b) needs to be rearranged to analyze the effect? The equation that is finally used has $\nu'$ that belongs to the moving frame whereas $\theta$ belongs to the stationary frame. What is the rationale behind this?

Relativity is not my field of expertise. Excuse me if this question sounds stupid. Thanking in advance.

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  • $\begingroup$ What has your question got to do with the transverse relativistic Doppler effect? $\endgroup$
    – ProfRob
    Jun 6, 2021 at 14:37

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This is the form that is most useful for experiments.

Suppose that we have a particle that emits some radiation with a characteristic frequency. If we send those particles through the lab at high speed then we can use the lab frame as the unprimed frame and the particle’s frame as the primed frame.

We then use the known characteristic frequency in the particle’s frame and the experimentally setup angle in the lab frame to predict the measured frequency in the lab frame.

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  • $\begingroup$ In that case why not Eq. (2.25a)? Thanks for your answer and edit. Because, depending on which form you choose, factor $\gamma$ changes the position. And, the position of this factor determine blueshift/redshift. $\endgroup$
    – Kansara
    Jun 6, 2021 at 13:28
  • $\begingroup$ @Kansara 2.25a uses the angle in the particle’s frame which is not as easy for the experimenter to determine as the angle in the lab frame. $\endgroup$
    – Dale
    Jun 6, 2021 at 14:17
  • $\begingroup$ okay. thanks, I got the point. $\endgroup$
    – Kansara
    Jun 6, 2021 at 18:55

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