Editing Relativistic Doppler effect (section)

==== Source and receiver both in circular motion around a common center ====
[[File:Transverse Doppler effect scenarios 7.svg|thumb|Figure 6. Source and receiver are placed on opposite ends of a rotor, equidistant from the center.]]
Suppose source and receiver are located on opposite ends of a spinning rotor, as illustrated in Fig.&nbsp;6. Kinematic arguments (special relativity) and arguments based on noting that there is no difference in potential between source and receiver in the pseudogravitational field of the rotor (general relativity) both lead to the conclusion that there should be no Doppler shift between source and receiver.

In 1961, Champeney and [[Philip Burton Moon|Moon]] conducted a [[Mössbauer effect|Mössbauer rotor experiment]]  testing exactly this scenario, and found that the Mössbauer absorption process was unaffected by rotation.<ref name="Champeney" group=p>{{cite journal |last1=Champeney |first1=D. C. |last2=Moon |first2=P. B. |title=Absence of Doppler Shift for Gamma Ray Source and Detector on Same Circular Orbit |journal=Proc. Phys. Soc. |date=1961 |volume=77 |issue=2 |pages=350–352|doi=10.1088/0370-1328/77/2/318 |bibcode=1961PPS....77..350C }}</ref> They concluded that their findings supported special relativity.

This conclusion generated some controversy. A certain persistent critic of relativity{{who?|date=April 2023}} maintained that, although the experiment was consistent with general relativity, it refuted special relativity, his point being that since the emitter and absorber were in uniform relative motion, special relativity demanded that a Doppler shift be observed. The fallacy with this critic's argument was, as demonstrated in section [[#Point of null frequency shift|Point of null frequency shift]], that it is simply not true that a Doppler shift must always be observed between two frames in uniform relative motion.<ref name="Sama">{{cite journal |last1=Sama |first1=Nicholas |title=Some Comments on a Relativistic Frequency-Shift Experiment of Champeney and Moon |journal=American Journal of Physics |date=1969 |volume=37 |issue=8 |pages=832–833 |doi=10.1119/1.1975859|bibcode=1969AmJPh..37..832S }}</ref> Furthermore, as demonstrated in section [[#Source and receiver are at their points of closest approach|Source and receiver are at their points of closest approach]], the difficulty of analyzing a relativistic scenario often depends on the choice of reference frame. Attempting to analyze the scenario in the frame of the receiver involves much tedious algebra. It is much easier, almost trivial, to establish the lack of Doppler shift between emitter and absorber in the laboratory frame.<ref name="Sama"/>

As a matter of fact, however, Champeney and Moon's experiment said nothing either pro or con about special relativity. Because of the symmetry of the setup, it turns out that virtually ''any'' conceivable theory of the Doppler shift between frames in uniform inertial motion must yield a null result in this experiment.<ref name="Sama"/>

Rather than being equidistant from the center, suppose the emitter and absorber were at differing distances from the rotor's center. For an emitter at radius <math>R'</math> and the absorber at radius <math>R</math> ''anywhere'' on the rotor, the ratio of the emitter frequency, <math>f',</math> and the absorber frequency, <math>f,</math> is given by

{{NumBlk||<math display="block"> \frac{f'}{f} = \left( \frac{c^2 - R^2 \omega ^2 }{ c^2 - R' ^2 \omega ^2 } \right) ^{1/2} </math>|{{EquationRef|5|Eq. 5}}}}

where <math>\omega</math> is the angular velocity of the rotor. The source and emitter do not have to be 180° apart, but can be at any angle with respect to the center.<ref name="Synge" group=p>{{cite journal |last1=Synge |first1=J. L. |title=Group Motions in Space-time and Doppler Effects |journal=Nature |date=1963 |volume=198 |issue=4881 |page=679|doi=10.1038/198679a0 |bibcode=1963Natur.198..679S |s2cid=42033531 |doi-access=free }}</ref><ref name="Keswani">{{cite book |last1=Keswani |first1=G. H. |title=Origin and Concept of Relativity |date=1965 |publisher=Alekh Prakashan |location=Delhi, India |pages=60–61 |url=https://books.google.com/books?id=_wKbOfv3bpQC&pg=PA60 |access-date=13 October 2018}}</ref>