The relativistic Doppler effect is the change in frequency, wavelength and amplitude Extract of page 22 of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect, first proposed by Christian Doppler in 1842), when taking into account effects described by the special theory of relativity.
The relativistic Doppler effect is different from the non-relativistic Doppler effect as the equations include the time dilation effect of special relativity and do not involve the medium of propagation as a reference point. They describe the total difference in observed frequencies and possess the required Lorentz symmetry.
Astronomers know of three sources of redshift/blueshift: Doppler shifts; gravitational redshifts (due to light exiting a gravitational field); and cosmological expansion (where space itself stretches). This article concerns itself only with Doppler shifts.
Summary of major results In the following table, it is assumed that for the receiver and the source are moving away from each other, being the relative velocity and the speed of light, and . Scenario Formula Notes Relativistic longitudinalDoppler effect Transverse Doppler effect,geometric closest approach Blueshift Transverse Doppler effect,visual closest approach Redshift TDE, receiver in circularmotion around source Blueshift TDE, source in circularmotion around receiver RedshiftTDE, source and receiverin circular motion aroundcommon centerNo Doppler shiftwhen Motion in arbitrary directionmeasured in receiver frame Motion in arbitrary directionmeasured in source frame
Derivation Relativistic longitudinal Doppler effect
Relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, is often derived as if it were the classical phenomenon, but modified by the addition of a time dilation term. This is the approach employed in first-year physics or mechanics textbooks such as those by Feynman or Morin.
Following this approach towards deriving the relativistic longitudinal Doppler effect, assume the receiver and the source are moving away from each other with a relative speed as measured by an observer on the receiver or the source (The sign convention adopted here is that is negative if the receiver and the source are moving towards each other).
Consider the problem in the reference frame of the source.
Suppose one wavefront arrives at the receiver. The next wavefront is then at a distance away from the receiver (where is the wavelength, is the frequency of the waves that the source emits, and is the speed of light).
The wavefront moves with speed , but at the same time the receiver moves away with speed during a time , which is the period of light waves impinging on the receiver, as observed in the frame of the source. So, where is the speed of the receiver in terms of the speed of light. The corresponding , the frequency at which wavefronts impinge on the receiver in the source's frame, is:
Thus far, the equations have been identical to those of the classical Doppler effect with a stationary source and a moving receiver.
However, due to relativistic effects, clocks on the receiver are time dilated relative to clocks at the source: , where is the Lorentz factor. In order to know which time is dilated, we recall that is the time in the frame in which the source is at rest. The receiver will measure the received frequency to be
The ratio
is called the Doppler factor of the source relative to the receiver. (This terminology is particularly prevalent in the subject of astrophysics: see relativistic beaming.)
The corresponding wavelengths are related by
Identical expressions for relativistic Doppler shift are obtained when performing the analysis in the reference frame of the receiver with a moving source. This matches up with the expectations of the principle of relativity, which dictates that the result can not depend on which object is considered to be the one at rest. In contrast, the classic nonrelativistic Doppler effect is dependent on whether it is the source or the receiver that is stationary with respect to the medium.
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. The transverse Doppler effect is one of the main novel predictions of the special theory of relativity.
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.
On the other hand, Kündig (1963) described an experiment where a Mössbauer absorber was spun in a rapid circular path around a central Mössbauer emitter. As explained below, this experimental arrangement resulted in Kündig's measurement of a blueshift.
Source and receiver are at their points of closest approach
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 2 demonstrates that the ease of analyzing this scenario depends on the frame in which it is analyzed. Figure 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. Figure 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 them. 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,
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 3 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 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
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
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 Figure 2, the signal travels perpendicularly to the receiver path and is blueshifted. In Figure 3, the signal travels perpendicularly to the source path and is redshifted.
As seen in Figure 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.
One object in circular motion around the other
Figure 5 illustrates two variants of this scenario. Both variants can be analyzed using simple time dilation arguments. Figure 5a is essentially equivalent to the scenario described in Figure 2b, and the receiver observes light from the source as being blueshifted by a factor of . Figure 5b is essentially equivalent to the scenario described in Figure 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 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.
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
Suppose source and receiver are located on opposite ends of a spinning rotor, as illustrated in Figure 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 Moon conducted a Mössbauer rotor experiment testing exactly this scenario, and found that the Mössbauer absorption process was unaffected by rotation. They concluded that their findings supported special relativity.
This conclusion generated some controversy. A certain persistent critic of relativity 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, that it is simply not true that a Doppler shift must always be observed between two frames in uniform relative motion. Furthermore, as demonstrated in section 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.