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Figure
1. A source of light waves moving to the right, relative to observers,
with velocity 0.7c. The frequency is higher for observers on the right,
and lower for observers on the left.
The relativistic Doppler effect is the change in frequency (and wavelength) of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect), 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.
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 , so
where
is
the speed of the receiver in terms of the speed of light, and where
is the period of light waves impinging on the receiver,
as observed in the frame of the source. The corresponding frequency
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
- Eq. 1:
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
- Eq. 2:
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
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 2 demonstrates that the ease of analyzing this scenario
depends on the frame in which it is analyzed.
- Fig. 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. 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,
- Eq. 3:
Receiver sees the source as being at its closest point
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 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
- 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
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. 2, the signal travels perpendicularly to the receiver
path and is blueshifted. In Fig. 3, the signal travels perpendicularly
to the source path and is redshifted.
As seen in Fig. 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. 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. 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
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. 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.
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.
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 and the absorber at radius anywhere on the rotor, the ratio of the emitter frequency, and the absorber frequency, is given by
- Eq. 5:
where
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.
Motion in an arbitrary direction
Figure 7. Doppler shift with source moving at an arbitrary angle with respect to the line between source and receiver.
The analysis used in section Relativistic longitudinal Doppler effect
can be extended in a straightforward fashion to calculate the Doppler
shift for the case where the inertial motions of the source and receiver
are at any specified angle.
Fig. 7 presents the scenario from the frame of the receiver, with the source moving at speed at an angle measured in the frame of the receiver. The radial component of the source's motion along the line of sight is equal to
The equation below can be interpreted as the classical Doppler
shift for a stationary and moving source modified by the Lorentz factor
- Eq. 6:
In the case when , one obtains the transverse Doppler effect:
In his 1905 paper on special relativity,
Einstein obtained a somewhat different looking equation for the Doppler
shift equation. After changing the variable names in Einstein's
equation to be consistent with those used here, his equation reads
- Eq. 7:
The differences stem from the fact that Einstein evaluated the angle with respect to the source rest frame rather than the receiver rest frame. is not equal to because of the effect of relativistic aberration. The relativistic aberration equation is:
- Eq. 8:
Substituting the relativistic aberration equation Equation 8 into Equation 6 yields Equation 7, demonstrating the consistency of these alternate equations for the Doppler shift.
Setting in Equation 6 or in Equation 7 yields Equation 1, the expression for relativistic longitudinal Doppler shift.
A four-vector approach to deriving these results may be found in Landau and Lifshitz (2005).
Visualization
Figure 8. Comparison of the relativistic Doppler effect (top) with the non-relativistic effect (bottom).
Fig. 8 helps us understand, in a rough qualitative sense, how the relativistic Doppler effect and relativistic aberration differ from the non-relativistic Doppler effect and non-relativistic aberration of light.
Assume that the observer is uniformly surrounded in all directions by
yellow stars emitting monochromatic light of 570 nm. The arrows in each
diagram represent the observer's velocity vector relative to its
surroundings, with a magnitude of 0.89 c.
- In the non-relativistic case, the light ahead of the observer is
blueshifted to a wavelength of 300 nm in the medium ultraviolet, while
light behind the observer is redshifted to 5200 nm in the intermediate
infrared. Because of the aberration of light, objects formerly at right
angles to the observer appear shifted forwards by 42°.
- In the relativistic case, the light ahead of the observer is
blueshifted to a wavelength of 137 nm in the far ultraviolet, while
light behind the observer is redshifted to 2400 nm in the short
wavelength infrared. Because of the relativistic aberration of light,
objects formerly at right angles to the observer appear shifted forwards
by 63°.
- In both cases, the monochromatic stars ahead of and behind the
observer are Doppler-shifted towards invisible wavelengths. If, however,
the observer had eyes that could see into the ultraviolet and infrared,
he would see the stars ahead of him as brighter and more closely
clustered together than the stars behind, but the stars would be far
brighter and far more concentrated in the relativistic case.
Real stars are not monochromatic, but emit a range of wavelengths approximating a black body
distribution. It is not necessarily true that stars ahead of the
observer would show a bluer color. This is because the whole spectral
energy distribution is shifted. At the same time that visible light is
blueshifted into invisible ultraviolet wavelengths, infrared light is
blueshifted into the visible range. Precisely what changes in the colors
one sees depends on the physiology of the human eye and on the spectral
characteristics of the light sources being observed.
Doppler effect on intensity
The Doppler effect (with arbitrary direction) also modifies the
perceived source intensity: this can be expressed concisely by the fact
that source strength divided by the cube of the frequency is a Lorentz
invariant
This implies that the total radiant intensity (summing over all
frequencies) is multiplied by the fourth power of the Doppler factor for
frequency.
As a consequence, since Planck's law describes the black-body radiation as having a spectral intensity in frequency proportional to (where T is the source temperature and ν the frequency), we can draw the conclusion that a black body spectrum seen through a Doppler shift (with arbitrary direction) is still a black body spectrum with a temperature multiplied by the same Doppler factor as frequency.
This result provides one of the pieces of evidence that serves to distinguish the Big Bang theory from alternative theories proposed to explain the cosmological redshift.
Experimental verification
Since the transverse Doppler effect is one of the main novel
predictions of the special theory of relativity, the detection and
precise quantification of this effect has been an important goal of
experiments attempting to validate special relativity.
Ives and Stilwell-type measurements
Figure 9. Why it is difficult to measure the transverse Doppler effect accurately using a transverse beam.
Einstein (1907) had initially suggested that the TDE might be measured by observing a beam of "canal rays" at right angles to the beam.
Attempts to measure TDE following this scheme proved it to be
impractical, since the maximum speed of particle beam available at the
time was only a few thousandths of the speed of light.
Fig. 9 shows the results of attempting to measure the 4861
Angstrom line emitted by a beam of canal rays (a mixture of H1+, H2+,
and H3+ ions) as they recombine with electrons stripped from the dilute
hydrogen gas used to fill the Canal ray tube. Here, the predicted result
of the TDE is a 4861.06 Angstrom line. On the left, longitudinal
Doppler shift results in broadening the emission line to such an extent
that the TDE cannot be observed. The middle figures illustrate that even
if one narrows one's view to the exact center of the beam, very small
deviations of the beam from an exact right angle introduce shifts
comparable to the predicted effect.
Rather than attempt direct measurement of the TDE, Ives and Stilwell (1938)
used a concave mirror that allowed them to simultaneously observe a
nearly longitudinal direct beam (blue) and its reflected image (red).
Spectroscopically, three lines would be observed: An undisplaced
emission line, and blueshifted and redshifted lines. The average of the
redshifted and blueshifted lines would be compared with the wavelength
of the undisplaced emission line. The difference that Ives and Stilwell
measured corresponded, within experimental limits, to the effect
predicted by special relativity.
Various of the subsequent repetitions of the Ives and Stilwell
experiment have adopted other strategies for measuring the mean of
blueshifted and redshifted particle beam emissions. In some recent
repetitions of the experiment, modern accelerator technology has been
used to arrange for the observation of two counter-rotating particle
beams. In other repetitions, the energies of gamma rays emitted by a
rapidly moving particle beam have been measured at opposite angles
relative to the direction of the particle beam. Since these experiments
do not actually measure the wavelength of the particle beam at right
angles to the beam, some authors have preferred to refer to the effect
they are measuring as the "quadratic Doppler shift" rather than TDE.
Direct measurement of transverse Doppler effect
The advent of particle accelerator
technology has made possible the production of particle beams of
considerably higher energy than was available to Ives and Stilwell. This
has enabled the design of tests of the transverse Doppler effect
directly along the lines of how Einstein originally envisioned them,
i.e. by directly viewing a particle beam at a 90° angle. For example,
Hasselkamp et al. (1979) observed the Hα line emitted by hydrogen atoms moving at speeds ranging from 2.53×108 cm/s to 9.28×108 cm/s,
finding the coefficient of the second order term in the relativistic
approximation to be 0.52±0.03, in excellent agreement with the
theoretical value of 1/2.
Other direct tests of the TDE on rotating platforms were made possible by the discovery of the Mössbauer effect, which enables the production of exceedingly narrow resonance lines for nuclear gamma ray emission and absorption.
Mössbauer effect experiments have proven themselves easily capable of
detecting TDE using emitter-absorber relative velocities on the order of
2×104 cm/s. These experiments include ones performed by Hay et al. (1960), Champeney et al. (1965), and Kündig (1963).
Time dilation measurements
The
transverse Doppler effect and the kinematic time dilation of special
relativity are closely related. All validations of TDE represent
validations of kinematic time dilation, and most validations of
kinematic time dilation have also represented validations of TDE. An
online resource, "What is the experimental basis of Special Relativity?"
has documented, with brief commentary, many of the tests that, over the
years, have been used to validate various aspects of special
relativity. Kaivola et al. (1985) and McGowan et al. (1993)
are examples of experiments classified in this resource as time
dilation experiments. These two also represent tests of TDE. These
experiments compared the frequency of two lasers, one locked to the
frequency of a neon atom transition in a fast beam, the other locked to
the same transition in thermal neon. The 1993 version of the experiment
verified time dilation, and hence TDE, to an accuracy of 2.3×10−6.
Relativistic Doppler effect for sound and light
Figure 10. The relativistic Doppler shift formula is applicable to both sound and light.
First-year physics textbooks almost invariably analyze Doppler shift
for sound in terms of Newtonian kinematics, while analyzing Doppler
shift for light and electromagnetic phenomena in terms of relativistic
kinematics. This gives the false impression that acoustic phenomena
requires a different analysis than light and radio waves.
The traditional analysis of the Doppler effect for sound
represents a low speed approximation to the exact, relativistic
analysis. The fully relativistic analysis for sound is, in fact, equally
applicable to both sound and electromagnetic phenomena.
Consider the spacetime diagram in Fig. 10. Worldlines for a
tuning fork (the source) and a receiver are both illustrated on this
diagram. Events O and A represent two vibrations of the tuning fork. The period of the fork is the magnitude of OA, and the inverse slope of AB represents the speed of signal propagation (i.e. the speed of sound) to event B. We can therefore write:
- (speed of sound)
- (speeds of source and receiver)
and are assumed to be less than
since otherwise their passage through the medium will set up shock
waves, invalidating the calculation. Some routine algebra gives the
ratio of frequencies:
- Eq. 9:
If and are small compared with , the above equation reduces to the classical Doppler formula for sound.
If the speed of signal propagation approaches , it can be shown that the absolute speeds and
of the source and receiver merge into a single relative speed
independent of any reference to a fixed medium. Indeed, we obtain Equation 1, the formula for relativistic longitudinal Doppler shift.
Analysis of the spacetime diagram in Fig. 10 gave a general
formula for source and receiver moving directly along their line of
sight, i.e. in collinear motion.
Figure
11. A source and receiver are moving in different directions and speeds
in a frame where the speed of sound is independent of direction.
Fig. 11 illustrates a scenario in two dimensions. The source moves with velocity (at the time of emission). It emits a signal which travels at velocity towards the receiver, which is traveling at velocity at the time of reception. The analysis is performed in a coordinate system in which the signal's speed is independent of direction.
The ratio between the proper frequencies for the source and receiver is
- Eq. 10:
The leading ratio has the form of the classical Doppler effect, while
the square root term represents the relativistic correction. If we
consider the angles relative to the frame of the source, then and the equation reduces to Equation 7, Einstein's 1905 formula for the Doppler effect. If we consider the angles relative to the frame of the receiver, then and the equation reduces to Equation 6, the alternative form of the Doppler shift equation discussed previously.