Relativistic Doppler effect

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.

Derivation

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.^[1]^[2] This is the approach employed in first-year physics or mechanics textbooks such as those by Feynman^[3] or Morin^[4].

Following this approach towards deriving the longitudinal Doppler effect, assume the receiver and the source are moving away from each other with a relative velocity $v\,$ (The sign convention adopted here is that $v\,$ 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 $\lambda =c/f_{s}\,$ away from the receiver (where $\lambda \,$ is the wavelength, $f_s\,$ is the frequency of the waves that the source emits, and $c\,$ is the speed of light).

The wavefront moves with velocity $c\,$ , but at the same time the receiver moves away with velocity $v$ , so $\lambda +vt=ct$ .

The period $t_{r,s}$ and frequency $f_{r,s}$ of light waves impinging on the receiver, as observed in the frame of the source, are

t_{r,s}={\frac {\lambda }{c-v}}

={\frac {c}{(c-v)f_{s}}}={\frac {1}{(1-\beta )f_{s}}},

f_{r,s}=1/t_{r,s}=(1-\beta )f_{s}

where $\beta = v / c\,$ is the velocity of the receiver in terms of the speed of light.

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. The receiver will measure the received frequency to be

Eq. 1:

f_{r}=\gamma f_{r,s}

=\gamma (1-\beta )f_{s}

={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{s}.

where

\gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}

is the Lorentz factor.

The ratio

{\frac {f_{s}}{f_{r}}}={\sqrt {\frac {1+\beta }{1-\beta }}}

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:

{\frac {\lambda _{r}}{\lambda _{s}}}={\frac {f_{s}}{f_{r}}}={\sqrt {\frac {1+\beta }{1-\beta }}},

and the resulting redshift

z={\frac {\lambda _{r}-\lambda _{s}}{\lambda _{s}}}={\frac {f_{s}-f_{r}}{f_{r}}}

can be written as

z={\sqrt {\frac {1+\beta }{1-\beta }}}-1.

In the non-relativistic limit (when $v \ll c$ ) this redshift can be approximated by

z \simeq \beta = \frac{v}{c},

corresponding to the classical Doppler effect.

Identical expressions for relativistic Doppler shift are obtained when performing the analysis in the reference frame of the receiver with a moving source.^[3]^[4]

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.^[4] 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.^{[p 1]}^{[note 1]}

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.^{[p 3]} 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.^[4]

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,

f_{r}=\gamma f_{s}

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.^[4] 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

f_{r}={\frac {f_{s}}{\gamma }}

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.^[5]

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.^[4] 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 $\gamma .$ . 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.^[6]

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 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.^{[p 4]} 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.^[7] 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.^[7]

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.^[7]

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 $R'$ and the absorber at radius $R$ anywhere on the rotor, the ratio of the emitter frequency, $\nu ',$ and the absorber frequency, $\nu ,$ is given by

Eq. 3:

{\frac {\nu '}{\nu }}=\left({\frac {1-R^{2}\omega ^{2}}{1-R'^{2}\omega ^{2}}}\right)^{1/2}

where $\omega$ 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.^{[p 5]}^[8]

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 Longitudinal Doppler effect can be extended in a straightforward fashion to calculate the Doppler shift for the case were the inertial motions of the source and receiver are at any specified angle.^[2]^[9]

Fig. 7 presents the scenario from the frame of the receiver, with the source moving at speed $v$ at an angle $\theta _{r}$ measured in the frame of the receiver. The radial component of the source's motion along the line of sight is equal to $v\cos {\theta _{r}}.$

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

Eq. 4:

f_{r}={\frac {f_{s}}{\gamma \left(1+\beta \cos \theta _{r}\right)}}.

In the case when $\theta _{r}=90^{\circ }$ and $\cos \theta _{r}=0,$ one obtains the transverse Doppler effect:

f_{r}={\frac {f_{s}}{\gamma }}.\,

In his 1905 paper on special relativity,^{[p 2]} 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. 5:

f_{r}=\gamma \left(1-\beta \cos \theta _{s}\right)f_{s}.

The differences stem from the fact that Einstein evaluated the angle $\theta _{s}$ with respect to the source rest frame rather than the receiver rest frame. $\theta _{s}$ is not equal to $\theta _{r}$ because of the effect of relativistic aberration. The relativistic aberration equation is:

Eq. 6:

\cos \theta _{r}={\frac {\cos \theta _{s}-\beta }{1-\beta \cos \theta _{s}}}\,

Substituting the relativistic aberration equation Eq. 6 into Eq. 4 yields Eq. 5, demonstrating the consistency of these alternate equations for the Doppler shift.^[9]

Substituting $\theta _{r}=0$ in Eq. 4 or $\theta _{s}=0$ in Eq. 5 yields Eq. 1, the expression for longitudinal Doppler shift.

Visualization

In Fig. 8, the blue point represents the observer, and the arrow represents the observer's velocity vector relative to its surroundings. When the observer is stationary, the x,y-grid appears yellow and the y-axis appears as a black vertical line. Increasing the observer's velocity to the right shifts the colors and the aberration of light distorts the grid. When the observer looks forward (right on the grid), points appear green, blue, and violet (blueshift) and grid lines appear farther apart. If the observer looks backward (left on the grid), then points appear red (redshift) and lines appear closer together. The grid has not changed, but its appearance for the observer has.

Fig. 9 illustrates that the grid distortion is a relativistic optical effect, separate from the underlying Lorentz contraction which is the same for an object moving toward an observer or away.

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^{[p 6]} (here, "source strength" refers to spectral intensity in frequency, i.e., power per unit solid angle and per unit frequency, expressed in watts per steradian per hertz; for spectral intensity in wavelength, the cube should be replaced by a fifth power). 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 $\nu ^{3}/\left(e^{h\nu /kT}-1\right)$ (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.

Experimental verification and observational support

Experiments supporting transverse Doppler effect

In practice, experimental verification of the transverse effect usually involves looking at the longitudinal changes in frequency or wavelength due to motion for approach and recession: by comparing these two ratios together we can rule out the relationships of "classical theory" and prove that the real relationships are "redder" than those predictions. The transverse Doppler shift is central to the interpretation of the peculiar astrophysical object SS 433.

The first longitudinal experiments were carried out by Herbert E. Ives and Stilwell in (1938), and many other longitudinal tests have been performed since with much higher precision.^{[p 7]} Also a direct transverse experiment has verified the redshift effect for a detector actually aimed at 90 degrees to the object.^{[p 8]}

Notes

↑ In his seminal paper of 1905 introducing special relativity, Einstein had already published an expression for the Doppler shift perceived by an observer moving at an arbitrary angle with respect to an infinitely distant source of light. Einstein's 1907 derivation of the TDE represented a trivial consequence of his earlier published general expression.^{[p 2]}

Primary sources

↑ Einstein, Albert (1907). "On the Possibility of a New Test of the Relativity Principle (Über die Möglichkeit einer neuen Prüfung des Relativitätsprinzips)". Annalen der Physik. 328 (6): 197–198. Bibcode:1907AnP...328..197E. doi:10.1002/andp.19073280613.
1 2 Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper". Annalen der Physik (in German). 322 (10): 891–921. Bibcode:1905AnP...322..891E. doi:10.1002/andp.19053221004. English translation: ‘On the Electrodynamics of Moving Bodies’
↑ Kündig, Walter (1963). "Measurement of the Transverse Doppler Effect in an Accelerated System". Physical Review. 129 (6): 2371–2375. Bibcode:1963PhRv..129.2371K. doi:10.1103/PhysRev.129.2371.
↑ Champeney, D. C.; Moon, P. B. (1961). "Absence of Doppler Shift for Gamma Ray Source and Detector on Same Circular Orbit". Proc. Phys. Soc. (London). 77: 350–352.
↑ Synge, J. L. (1963). "Group Motions in Space-time and Doppler Effects". Nature. 198: 679.
↑ Johnson, Montgomery H.; Teller, Edward (February 1982). "Intensity changes in the Doppler effect". Proc. Natl. Acad. Sci. USA. 79 (4): 1340. Bibcode:1982PNAS...79.1340J. doi:10.1073/pnas.79.4.1340. PMC 345964. PMID 16593162.
↑ Ives, H. E.; Stilwell, G. R. (1938). "An experimental study of the rate of a moving atomic clock". Journal of the Optical Society of America. 28 (7): 215. Bibcode:1938JOSA...28..215I. doi:10.1364/JOSA.28.000215.
↑ Hasselkamp, D.; E. Mondry; A. Scharmann (1979-06-01). "Direct observation of the transversal Doppler-shift". Zeitschrift für Physik A. 289 (2): 151–155. Bibcode:1979ZPhyA.289..151H. doi:10.1007/BF01435932.

References

↑ Sher, D. (1968). "The Relativistic Doppler Effect". Journal of the Royal Astronomical Society of Canada. 62: 105–111. Retrieved 11 October 2018.
1 2 Gill, T. P. (1965). The Doppler Effect. London: Logos Press Limited. pp. 6–9. Retrieved 12 October 2018.
1 2 Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (February 1977). "Relativistic Effects in Radiation". The Feynman Lectures on Physics: Volume 1. Reading, Massachusetts: Addison-Wesley. pp. 34–7 f. ISBN 9780201021165. LCCN 2010938208.
1 2 3 4 5 6 Morin, David (2008). "Chapter 11: Relativity (Kinematics)". Introduction to Classical Mechanics: With Problems and Solutions (PDF). Cambridge University Press. pp. 539–543. ISBN 978-1-139-46837-4. Archived from the original (PDF) on 4 Apr 2018.
↑ Brown, Kevin S. "The Doppler Effect". Mathpages. Retrieved 12 October 2018.
↑ Misner, C. W., Thorne, K. S., and Wheeler, J. A, (1973). Gravitation. Freeman. p. 163. ISBN 0716703440.
1 2 3 Sama, Nicholas (1969). "Some Comments on a Relativistic Frequency-Shift Experiment of Champeney and Moon". American Journal of Physics. 37: 832–833. doi:10.1119/1.1975859.
↑ Keswani, G. H. (1965). Origin and Concept of Relativity. Delhi, India: Alekh Prakashan. pp. 60–61. Retrieved 13 October 2018.
1 2 Brown, Kevin S. "Doppler Shift for Sound and Light". Mathpages. Retrieved 6 August 2015.

External links

Warp Special Relativity Simulator Computer program demonstrating the relativistic Doppler effect.

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[7] In his seminal paper of 1905 introducing special relativity, Einstein had already published an expression for the Doppler shift perceived by an observer moving at an arbitrary angle with respect to an infinitely distant source of light. Einstein's 1907 derivation of the TDE represented a trivial consequence of his earlier published general expression.^{[p 2]}

[5] Einstein, Albert (1907). "On the Possibility of a New Test of the Relativity Principle (Über die Möglichkeit einer neuen Prüfung des Relativitätsprinzips)". Annalen der Physik. 328 (6): 197–198. Bibcode:1907AnP...328..197E. doi:10.1002/andp.19073280613.

[Einstein1905-6] 1 2 Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper". Annalen der Physik (in German). 322 (10): 891–921. Bibcode:1905AnP...322..891E. doi:10.1002/andp.19053221004. English translation: ‘On the Electrodynamics of Moving Bodies’

[8] Kündig, Walter (1963). "Measurement of the Transverse Doppler Effect in an Accelerated System". Physical Review. 129 (6): 2371–2375. Bibcode:1963PhRv..129.2371K. doi:10.1103/PhysRev.129.2371.

[Champeney-11] Champeney, D. C.; Moon, P. B. (1961). "Absence of Doppler Shift for Gamma Ray Source and Detector on Same Circular Orbit". Proc. Phys. Soc. (London). 77: 350–352.

[Synge-13] Synge, J. L. (1963). "Group Motions in Space-time and Doppler Effects". Nature. 198: 679.

[16] Johnson, Montgomery H.; Teller, Edward (February 1982). "Intensity changes in the Doppler effect". Proc. Natl. Acad. Sci. USA. 79 (4): 1340. Bibcode:1982PNAS...79.1340J. doi:10.1073/pnas.79.4.1340. PMC 345964. PMID 16593162.

[17] Ives, H. E.; Stilwell, G. R. (1938). "An experimental study of the rate of a moving atomic clock". Journal of the Optical Society of America. 28 (7): 215. Bibcode:1938JOSA...28..215I. doi:10.1364/JOSA.28.000215.

[hass-18] Hasselkamp, D.; E. Mondry; A. Scharmann (1979-06-01). "Direct observation of the transversal Doppler-shift". Zeitschrift für Physik A. 289 (2): 151–155. Bibcode:1979ZPhyA.289..151H. doi:10.1007/BF01435932.

[1] Sher, D. (1968). "The Relativistic Doppler Effect". Journal of the Royal Astronomical Society of Canada. 62: 105–111. Retrieved 11 October 2018.

[Gill-2] 1 2 Gill, T. P. (1965). The Doppler Effect. London: Logos Press Limited. pp. 6–9. Retrieved 12 October 2018.

[Feynman-3] 1 2 Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (February 1977). "Relativistic Effects in Radiation". The Feynman Lectures on Physics: Volume 1. Reading, Massachusetts: Addison-Wesley. pp. 34–7 f. ISBN 9780201021165. LCCN 2010938208.

[Morin-4] 1 2 3 4 5 6 Morin, David (2008). "Chapter 11: Relativity (Kinematics)". Introduction to Classical Mechanics: With Problems and Solutions (PDF). Cambridge University Press. pp. 539–543. ISBN 978-1-139-46837-4. Archived from the original (PDF) on 4 Apr 2018.

[Brown_b-9] Brown, Kevin S. "The Doppler Effect". Mathpages. Retrieved 12 October 2018.

[Misner-10] Misner, C. W., Thorne, K. S., and Wheeler, J. A, (1973). Gravitation. Freeman. p. 163. ISBN 0716703440.

[Sama-12] 1 2 3 Sama, Nicholas (1969). "Some Comments on a Relativistic Frequency-Shift Experiment of Champeney and Moon". American Journal of Physics. 37: 832–833. doi:10.1119/1.1975859.

[Keswani-14] Keswani, G. H. (1965). Origin and Concept of Relativity. Delhi, India: Alekh Prakashan. pp. 60–61. Retrieved 13 October 2018.

[Brown_a-15] 1 2 Brown, Kevin S. "Doppler Shift for Sound and Light". Mathpages. Retrieved 6 August 2015.