Editing Spacetime (section)

==== Transverse Doppler effect ====
[[File:Transverse Doppler effect scenarios 2.svg|thumb|upright=1.4|Figure 3–7. Transverse Doppler effect scenarios]]
Suppose that a source and a receiver, both approaching each other in uniform inertial motion along non-intersecting lines, are at their closest approach to each other. It would appear that the classical analysis predicts that the receiver detects no Doppler shift. Due to subtleties in the analysis, that expectation is not necessarily true. Nevertheless, when appropriately defined, transverse Doppler shift is a relativistic effect that has no classical analog. The subtleties are these:<ref name="Morin2008">{{cite book |last1=Morin |first1=David |title=Introduction to Classical Mechanics: With Problems and Solutions |date=2008 |publisher=Cambridge University Press |isbn=978-0-521-87622-3 |url=https://archive.org/details/introductiontocl00mori }}</ref>{{rp|541–543}}
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* Fig. 3-7a. What is the frequency measurement when the receiver is geometrically at its closest approach to the source? This scenario is most easily analyzed from the frame S′ of the source.<ref group=note>The ease of analyzing a relativistic scenario often depends on the frame in which one chooses to perform the analysis. '''[[:File:Transverse Doppler effect scenarios 3.svg|In this linked image]]''', we present alternative views of the transverse Doppler shift scenario where source and receiver are at their closest approach to each other.

(a) 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.

(b) 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. Since the receiver's clocks are time-dilated relative to the source, the light that the receiver receives is therefore blue-shifted by a factor of ''gamma''.</ref>
* Fig. 3-7b. What is the frequency measurement when the receiver ''sees'' the source as being closest to it? This scenario is most easily analyzed from the frame S of the receiver.

Two other scenarios are commonly examined in discussions of transverse Doppler shift:
* Fig. 3-7c. If the receiver is moving in a circle around the source, what frequency does the receiver measure?
* Fig. 3-7d. If the source is moving in a circle around the receiver, what frequency does the receiver measure?
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In scenario (a), the point of closest approach is frame-independent and represents the moment where there is no change in distance versus time (i.e. dr/dt&nbsp;=&nbsp;0 where ''r'' is the distance between receiver and source) and hence no longitudinal Doppler shift. The source observes the receiver as being illuminated by light of frequency {{′|''f''}}, but also observes the receiver as having a time-dilated clock. In frame&nbsp;S, the receiver is therefore illuminated by [[blueshifted]] light of frequency
: <math>f = f' \gamma = f' / \sqrt { 1 - \beta ^2  }</math>

In scenario (b) the illustration shows the receiver being illuminated by light from when the source was closest to the receiver, even though the source has moved on. Because the source's clocks are time dilated as measured in frame S, and since dr/dt was equal to zero at this point, the light from the source, emitted from this closest point, is [[redshifted]] with frequency
: <math>f = f' / \gamma = f' \sqrt { 1 - \beta ^2  }</math>

Scenarios (c) and (d) can be analyzed by simple time dilation arguments. In (c), the receiver observes light from the source as being blueshifted by a factor of <math>\gamma</math>, and in (d), the light is redshifted. The only seeming complication is that the orbiting objects are in accelerated motion. However, if an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation. (The converse, however, is not true.)<ref name="Morin2008" />{{rp|541–543}} Most reports of transverse Doppler shift refer to the effect as a redshift and analyze the effect in terms of scenarios (b) or (d).<ref group="note">Not all experiments characterize the effect in terms of a redshift. For example, the [[Ives–Stilwell experiment#Relativistic Doppler effect|Kündig experiment]] measures transverse blueshift using a Mössbauer source setup at the center of a centrifuge rotor and an absorber at the rim.</ref>

{{anchor|Energy and momentum}}