Editing Relativistic Doppler effect (section)

===Transverse Doppler effect===
Suppose that a source and a receiver are both approaching each other in uniform inertial motion along paths that do not collide. The ''transverse Doppler effect'' (TDE) may refer to (a) the nominal [[blueshift]] predicted by [[special relativity]] that occurs when the emitter and receiver are at their points of closest approach; or (b) the nominal [[redshift]] predicted by special relativity when the receiver '''sees''' the emitter as being at its closest approach.<ref name=Morin>{{cite book |title=Introduction to Classical Mechanics: With Problems and Solutions |chapter=Chapter 11: Relativity (Kinematics) |chapter-url=http://www.people.fas.harvard.edu/~djmorin/chap11.pdf |first1=David |last1=Morin |publisher=[[Cambridge University Press]] |year=2008 |isbn=978-1-139-46837-4 |pages=539–543 |archive-url=https://web.archive.org/web/20180404002006/http://www.people.fas.harvard.edu/~djmorin/chap11.pdf |archive-date=4 April 2018}}</ref> The transverse Doppler effect is one of the main novel predictions of the special theory of relativity.<ref>{{cite web |last=Nolte |first=David D. |title=Transverse Doppler Effect |url=https://galileo-unbound.blog/2021/06/03/the-transverse-doppler-effect-and-relativistic-time-dilation/ |website=Galileo Unbound|date=3 June 2021 }}</ref>

Whether a scientific report describes TDE as being a redshift or blueshift depends on the particulars of the experimental arrangement being related. For example, Einstein's original description of the TDE in 1907 described an experimenter looking at the center (nearest point) of a beam of "[[canal rays]]" (a beam of positive ions that is created by certain types of gas-discharge tubes). According to special relativity, the moving ions' emitted frequency would be reduced by the Lorentz factor, so that the received frequency would be reduced (redshifted) by the same factor.<ref group=p name=Einstein1907>{{cite journal |last=Einstein |first=Albert |year=1907 |title= On the Possibility of a New Test of the Relativity Principle (Über die Möglichkeit einer neuen Prüfung des Relativitätsprinzips) |journal=[[Annalen der Physik]] |volume=328 |issue=6 |pages=197–198 |bibcode=1907AnP...328..197E|doi=10.1002/andp.19073280613 |url=https://einsteinpapers.press.princeton.edu/vol2-trans/246 }}</ref>{{refn|group=note|In his seminal paper of 1905 introducing special relativity, Einstein had already published an expression for the Doppler shift perceived by an observer moving at an arbitrary angle with respect to an infinitely distant source of light. Einstein's 1907 derivation of the TDE represented a trivial consequence of his earlier published general expression.<ref group=p name=Einstein1905>{{cite journal |last=Einstein |first=Albert |year=1905 |title=Zur Elektrodynamik bewegter Körper  |language=de |journal=[[Annalen der Physik]] |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004 |url= http://sedici.unlp.edu.ar/handle/10915/2786|bibcode = 1905AnP...322..891E |doi-access=free }} [http://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation: ‘On the Electrodynamics of Moving Bodies’]</ref>}}

On the other hand, Kündig (1963) described an experiment where a [[Mössbauer effect|Mössbauer absorber]] was spun in a rapid circular path around a central Mössbauer emitter.<ref group=p name=Kundig>{{cite journal|author=Kündig, Walter|date=1963|title=Measurement of the Transverse Doppler Effect in an Accelerated System|journal=Physical Review|volume=129|issue=6|pages=2371–2375|doi=10.1103/PhysRev.129.2371|bibcode = 1963PhRv..129.2371K }}</ref> As explained below, this experimental arrangement resulted in Kündig's measurement of a blueshift.

==== Source and receiver are at their points of closest approach ====
[[File:Transverse Doppler effect scenarios 3.svg|thumb|300px|Figure 2. Source and receiver are at their points of closest approach. (a) Analysis in the frame of the receiver. (b) Analysis in the frame of the source.]]
In this scenario, the point of closest approach is frame-independent and represents the moment where there is no change in distance versus time. Figure&nbsp;2 demonstrates that the ease of analyzing this scenario depends on the frame in which it is analyzed.<ref name=Morin/> 
* Fig.&nbsp;2a. If we analyze the scenario in the frame of the receiver, we find that the analysis is more complicated than it should be. The apparent position of a celestial object is displaced from its true position (or geometric position) because of the object's motion during the time it takes its light to reach an observer. The source would be time-dilated relative to the receiver, but the redshift implied by this time dilation would be offset by a blueshift due to the longitudinal component of the relative motion between the receiver and the apparent position of the source.
* Fig.&nbsp;2b. It is much easier if, instead, we analyze the scenario from the frame of the source. An observer situated at the source knows, from the problem statement, that the receiver is at its closest point to him. That means that the receiver has no longitudinal component of motion to complicate the analysis. (i.e. dr/dt = 0 where r is the distance between receiver and source) Since the receiver's clocks are time-dilated relative to the source, the light that the receiver receives is blue-shifted by a factor of gamma. In other words,

{{NumBlk||<math display="block">f_r = \gamma f_s</math>|{{EquationRef|3|Eq. 3}}}}

==== Receiver ''sees'' the source as being at its closest point ====
[[File:Transverse Doppler effect scenarios 4.svg|thumb|Figure 3. Transverse Doppler shift for the scenario where the receiver ''sees'' the source as being at its closest point.]]
This scenario is equivalent to the receiver looking at a direct right angle to the path of the source. The analysis of this scenario is best conducted from the frame of the receiver. Figure&nbsp;3 shows the receiver being illuminated by light from when the source was closest to the receiver, even though the source has moved on.<ref name=Morin/> Because the source's clock is time dilated as measured in the frame of the receiver, and because there is no longitudinal component of its motion, the light from the source, emitted from this closest point, is redshifted with frequency

{{NumBlk||<math display="block">f_r = \frac{f_s}{\gamma}</math>|{{EquationRef|4|Eq. 4}}}}

In the literature, most reports of transverse Doppler shift analyze the effect in terms of the receiver pointed at direct right angles to the path of the source, thus ''seeing'' the source as being at its closest point and observing a redshift.

==== Point of null frequency shift ====
[[File:Transverse Doppler effect scenarios 6.svg|thumb|300px|Figure 4. Null frequency shift occurs for a pulse that travels the shortest distance from source to receiver.]]
Given that, in the case where the inertially moving source and receiver are geometrically at their nearest approach to each other, the receiver observes a blueshift, whereas in the case where the receiver ''sees'' the source as being at its closest point, the receiver observes a redshift, there obviously must exist a point where blueshift changes to a redshift. In Fig.&nbsp;2, the signal travels perpendicularly to the receiver path and is blueshifted. In Fig.&nbsp;3, the signal travels perpendicularly to the source path and is redshifted.

As seen in Fig.&nbsp;4, null frequency shift occurs for a pulse that travels the shortest distance from source to receiver. When viewed in the frame where source and receiver have the same speed, this pulse is emitted perpendicularly to the source's path and is received perpendicularly to the receiver's path. The pulse is emitted slightly before the point of closest approach, and it is received slightly after.<ref name="Brown_b">{{cite web |last1=Brown |first1=Kevin S. |title=The Doppler Effect |url=https://www.mathpages.com/home/kmath587/kmath587.htm |publisher=Mathpages |access-date=12 October 2018}}</ref>

==== One object in circular motion around the other ====
[[File:Transverse Doppler effect scenarios 5.svg|thumb|300px|Figure 5. Transverse Doppler effect for two scenarios: (a) receiver moving in a circle around the source; (b) source moving in a circle around the receiver.]]

Fig.&nbsp;5 illustrates two variants of this scenario. Both variants can be analyzed using simple time dilation arguments.<ref name=Morin/> Figure&nbsp;5a is essentially equivalent to the scenario described in Figure&nbsp;2b, and the receiver observes light from the source as being blueshifted by a factor of <math>\gamma</math>. Figure&nbsp;5b is essentially equivalent to the scenario described in Figure&nbsp;3, and the light is redshifted.

The only seeming complication is that the orbiting objects are in accelerated motion. An accelerated particle does not have an inertial frame in which it is always at rest. However, an inertial frame can always be found which is momentarily comoving with the particle. This frame, the [[Proper reference frame (flat spacetime)|''momentarily comoving reference frame'' (MCRF)]], enables application of special relativity to the analysis of accelerated particles. If an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation.<ref name=Misner>{{Cite book|last1=Misner | first1 = C. W. | last2 = Thorne | first2 = K. S. | last3 = Wheeler | first3 = J. A|title=Gravitation|year=1973|page=163|publisher=Freeman|isbn=978-0716703440}}</ref>

The converse, however, is not true. The analysis of scenarios where ''both'' objects are in accelerated motion requires a somewhat more sophisticated analysis. Not understanding this point has led to confusion and misunderstanding.

==== 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>