Relativistic Doppler effect

thumb|right|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, wavelength and amplitude 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 <math>\beta = v/c > 0</math> the receiver <math> r </math> and the source <math> s </math> are moving away from each other, <math>v</math> being the relative velocity and <math>c</math> the speed of light, and <math display="inline">\gamma = 1/\sqrt{1 - \beta^2}</math>.

{| class="wikitable" style="text-align:center"

! Scenario !! Formula !! Notes

| Relativistic longitudinal Doppler effect

| <math>\frac{\lambda_r}{\lambda_s} = \frac{f_s}{f_r} = \sqrt{\frac{1 + \beta}{1 - \beta</math>

| Transverse Doppler effect, geometric closest approach

| <math>f_r = \gamma f_s</math>

| Blueshift

| Transverse Doppler effect, visual closest approach

| <math>f_r = \frac{f_s}{\gamma}</math>

| Redshift

| TDE, receiver in circular motion around source

| <math>f_r = \gamma f_s</math>

| Blueshift

| TDE, source in circular motion around receiver

| <math>f_r = \frac{f_s}{\gamma}</math>

| Redshift

|TDE, source and receiver in circular motion around common center

|<math> \frac{f'}{f} = \left( \frac{c^2 - R^2 \omega ^2 }{ c^2 - R' ^2 \omega ^2 } \right) ^{1/2} </math>

|No Doppler shift when <math>R = R'</math>

| Motion in arbitrary direction measured in receiver frame

| <math> f_r = \frac{f_s}{\gamma\left(1 + \beta \cos\theta_r\right)}</math>

| Motion in arbitrary direction measured in source frame

| <math> f_r = \gamma \left( 1 - \beta \cos \theta_s \right) f_s</math>

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. 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

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 2 demonstrates that the ease of analyzing this scenario depends on the frame in which it is analyzed.

One object in circular motion around the other

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.

Figure 5 illustrates two variants of this scenario. Both variants can be analyzed using simple time dilation arguments.

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

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 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.

Motion in an arbitrary direction

thumb|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.

Figure 7 presents the scenario from the frame of the receiver, with the source moving at speed <math>v</math> at an angle <math>\theta_r</math> measured in the frame of the receiver. The radial component of the source's motion along the line of sight is equal to <math>v \cos{\theta_r}.</math>

The equation below can be interpreted as the classical Doppler shift for a stationary and moving source modified by the Lorentz factor <math>\gamma :</math>

In the case when <math>\theta_r = 90^{\circ}</math>, one obtains the transverse Doppler effect:

<math display="block">f_r = \frac {f_s} {\gamma}. </math>

In his 1905 paper on special relativity,

In electromagnetic waves both the electric and the magnetic field amplitudes E and B transform in a similar manner as the frequency:

<math display="block">\begin{align}

E_r &= \gamma \left( 1 - \beta \cos \theta_s \right) E_s \\

B_r &= \gamma \left( 1 - \beta \cos \theta_s \right) B_s.

\end{align} </math>

Visualization

thumb|300px|Figure 8. Comparison of the relativistic Doppler effect (top) with the non-relativistic effect (bottom).

Figure 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 motionless yellow stars emitting monochromatic light of 570 nm. The arrows in each diagram represent the observer's velocity vector relative to its surroundings (and the medium, in non-relativistic case), with a magnitude of 0.89 c.

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 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 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, they would see the stars ahead of them 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 <math>\frac{f^3}{e^\frac{hf}{kT} - 1}</math> (where <math>T</math> is the source temperature and <math>f</math> 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

thumb|300px|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.

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). 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

thumb|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 require 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. The tuning fork and receiver start at O, at which point the tuning fork starts to vibrate, emitting waves and moving along the negative x-axis while the receiver starts to move along the positive x-axis. The tuning fork continues until it reaches A, at which point it stops emitting waves: a wavepacket has therefore been generated, and all the waves in the wavepacket are received by the receiver with the last wave reaching it at B. The proper time for the duration of the packet in the tuning fork's frame of reference is the length of OA while the proper time for the duration of the wavepacket in the receiver's frame of reference is the length of OB. If <math>n</math> waves were emitted, then while the inverse slope of AB represents the speed of signal propagation (i.e. the speed of sound) to event B. We can therefore write the speed of sound as