Editing Spacetime (section)

=== Doppler effect ===
{{Main|Doppler effect|Relativistic Doppler effect}}

The [[Doppler effect]] is the change in frequency or wavelength of a wave for a receiver and source in relative motion. For simplicity, we consider here two basic scenarios: (1) The motions of the source and/or receiver are exactly along the line connecting them (longitudinal Doppler effect), and (2) the motions are at right angles to the said line ([[transverse Doppler effect]]). We are ignoring scenarios where they move along intermediate angles.

==== Longitudinal Doppler effect ====
The classical Doppler analysis deals with waves that are propagating in a medium, such as sound waves or water ripples, and which are transmitted between sources and receivers that are moving towards or away from each other. The analysis of such waves depends on whether the source, the receiver, or both are moving relative to the medium. Given the scenario where the receiver is stationary with respect to the medium, and the source is moving directly away from the receiver at a speed of ''v<sub>s</sub>'' for a velocity parameter of ''β<sub>s</sub>'', the wavelength is increased, and the observed frequency ''f'' is given by
: <math>f = \frac{1}{1+\beta _s}f_0</math>

On the other hand, given the scenario where source is stationary, and the receiver is moving directly away from the source at a speed of ''v<sub>r</sub>'' for a velocity parameter of ''β<sub>r</sub>'', the wavelength is ''not'' changed, but the transmission velocity of the waves relative to the receiver is decreased, and the observed frequency ''f'' is given by
: <math>f = (1-\beta _r)f_0</math>

[[File:Spacetime Diagram of Relativistic Doppler Effect.svg|thumb|Figure 3–6. Spacetime diagram of relativistic Doppler effect]]
Light, unlike sound or water ripples, does not propagate through a medium, and there is no distinction between a source moving away from the receiver or a receiver moving away from the source. Fig.&nbsp;3-6 illustrates a relativistic spacetime diagram showing a source separating from the receiver with a velocity parameter <math>\beta ,</math> so that the separation between source and receiver at time <math>w</math> is <math>\beta w </math>. Because of time dilation, <math>w = \gamma w' .</math> Since the slope of the green light ray is −1, <math>T =  w + \beta w = \gamma w' (1 + \beta) .</math> Hence, the [[relativistic Doppler effect]] is given by<ref name="Bais" />{{rp|58–59}}
: <math>f = \sqrt{\frac{1 - \beta}{1 + \beta}}\,f_0.</math>

==== 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}}
{{plainlist|
* 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?
}}<!—end plainlist—>
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}}