Relativistic Doppler effect

Diagram 1. A source of light waves moving to the right with velocity 0.7c. The frequency is higher on the right, and lower 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.

Visualization

Diagram 2. Demonstration of aberration of light and relativistic Doppler effect.

In Diagram 2, the blue point represents the observer, and the arrow represents the observer's velocity vector. 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. Note, the grid itself has not changed, but its appearance for the observer has. The main point in the Doppler Effect is to individualize each wave out sourcing the central point and expanding the wave given due to circulatory vibrations at a certain focal point.

Analogy

Understanding relativistic Doppler effect requires understanding the Doppler effect, time dilation, and the aberration of light. As a simple analogy, consider two people playing catch. Imagine that a stationary pitcher tosses one ball each second (1 Hz) at one meter per second to a catcher who is standing one meter away. The stationary catcher will receive one ball per second (1 Hz). Then the catcher walks away from the pitcher at 0.5 meters per second and catches a ball every 2 seconds (0.5 Hz). Finally, the catcher walks towards the pitcher at 0.5 meters per second and catches three balls every two seconds (1.5 Hz). The same would be true if the pitcher moved toward or away from the catcher. By analogy, the relativistic Doppler effect shifts the frequency of light as the emitter or observer moves toward or away from the other.

Diagram 1 shows an emitter traveling to the right, whereas Diagram 2 shows the observer traveling right. While the color shift appears similar, the aberration of light is opposite. To understand this effect, again imagine two people playing catch. If the pitcher is moving to the right and the catcher is standing still, then the pitcher must aim behind the catcher. Otherwise the ball will pass the catcher on the right. Also, the catcher must turn in front of the pitcher, or the ball will hit on the catcher's left. Conversely, if the pitcher is stationary and the catcher is moving to the right, then the pitcher must aim in front of the catcher. Otherwise, the ball will pass the catcher on the left. Also, the catcher must turn to the back of the pitcher, or the ball will hit on the catcher's right. The degree to which the pitcher and catcher must turn to the right or left depends on two things: 1) the instantaneous angle between the pitcher-catcher line and the runner's velocity vector, and 2) the pitcher-catcher velocity relative to the speed of the ball. By analogy, the aberration of light depends on: 1) the instantaneous angle between the emitter-observer line and the relative velocity vector, and 2) the emitter-observer velocity relative to the speed of light.

Motion along the line of sight

Assume the observer and the source are moving away from each other with a relative velocity $v\,$ ($v\,$ is negative if the observer and the source are moving toward each other). Considering the problem in the reference frame of the source, suppose one wavefront arrives at the observer. The next wavefront is then at a distance $\lambda =c/f_{s}\,$ away from him (where $\lambda \,$ is the wavelength, $f_{s}\,$ is the frequency of the wave the source emitted, and $c\,$ is the speed of light). Since the wavefront moves with velocity $c\,$ and the observer escapes with velocity $v$, the time (as measured in the reference frame of the source) between crest arrivals at the observer is

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

where $\beta =v/c\,$ is the velocity of the observer in terms of the speed of light (see beta (velocity)).

Due to the relativistic time dilation, the observer will measure this time to be

$t_{o}={\frac {t}{\gamma }},$

where

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

is the Lorentz factor. The corresponding observed frequency is

$f_{o}={\frac {1}{t_{o}}}=\gamma (1-\beta )f_{s}={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{s}.$

The ratio

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

is called the Doppler factor of the source relative to the observer. (This terminology is particularly prevalent in the subject of astrophysics: see relativistic beaming.) The corresponding wavelengths are related by

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

and the resulting redshift

$z={\frac {\lambda _{o}-\lambda _{s}}{\lambda _{s}}}={\frac {f_{s}-f_{o}}{f_{o}}}$

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.

Transverse Doppler effect

The transverse Doppler effect is the nominal redshift or blueshift predicted by special relativity that occurs when the emitter and receiver are at the point of closest approach. Light emitted at this instant will be redshifted. Light received at this instant will be blueshifted.

Assuming the objects are not accelerated, light emitted when the objects are closest together will be received some time later, at reception the amount of redshift will be

${\frac {1}{\gamma }}={\sqrt {1-v^{2}/c^{2}\,}}.$

Light received when the objects are closest together was emitted some time earlier, at reception the amount of blueshift is

$\gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}\,}}}.$

Classical theory does not make a specific prediction for either of these two cases, as the shift depends on the motions relative to the medium.

The transverse Doppler effect is a consequence of the relativistic Doppler effect.

$f_{o}={\frac {f_{s}}{\gamma \left(1+{\frac {v\cos \theta _{o}}{c}}\right)}}$

In the frame of the receiver, θ₀ represents the angle between the direction of the emitter at emission, and the observed direction of the light at reception. In the case when θ₀ = π/2, the light was emitted at the moment of closest approach, and one obtains the transverse redshift

$f_{o}={\frac {f_{s}}{\gamma }}\,$

The transverse Doppler effect is one of the main novel predictions of the special theory. As Einstein put it in 1907: according to special relativity the moving object's emitted frequency is reduced by the Lorentz factor, so that – in addition to the classical Doppler effect – the received frequency is reduced by the same factor.

Reciprocity

Sometimes the question arises as to how the transverse Doppler effect can lead to a redshift as seen by the "observer" whilst another observer moving with the emitter would also see a redshift of light sent (perhaps accidentally) from the receiver.

It is essential to understand that the concept "transverse" is not reciprocal. Each participant understands that when the light reaches her/him transversely as measured in terms of that person's rest frame, the other had emitted the light afterward as measured in the other person's rest frame. In addition, each participant measures the other's frequency as reduced ("time dilation"). These effects combined make the observations fully reciprocal, thus obeying the principle of relativity.

Experimental verification

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.

Longitudinal tests

The first of these experiments was carried out by Ives and Stilwell in (1938) and although the accuracy of this experiment has since been questioned, many other longitudinal tests have been performed since with much higher precision [1],[2]. These usually claim greater certainty than Ives-Stilwell, but also tend to be more complicated.

• Herbert E. Ives and G.R. Stilwell, “An experimental study of the rate of a moving clock”

J. Opt. Soc. Am 28 215–226 (1938) and part II. J. Opt. Soc. Am. 31, 369–374 (1941)

Transverse tests

To date, only one inertial experiment seems to have verified the redshift effect for a detector actually aimed at 90 degrees to the object.

• D. Hasselkamp, E. Mondry, and A. Scharmann, "Direct Observation of the Transversal Doppler-Shift"

Z. Physik A 289, 151–155 (1979).

Motion in an arbitrary direction

If, in the reference frame of the observer, the source is moving away with velocity $v\,$ at an angle $\theta _{o}\,$ relative to the direction from the observer to the source (at the time when the light is emitted), the frequency changes as

$f_{o}={\frac {f_{s}}{\gamma \left(1+{\frac {v\cos \theta _{o}}{c}}\right)}}.$            (1)

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

$f_{o}={\frac {f_{s}}{\gamma }}.\,$

Due to the finite speed of light, the light ray (or photon, if you like) perceived by the observer as coming at angle $\theta _{o}\,$, was, in the reference frame of the source, emitted at a different angle $\theta _{s}\,$. $\cos \theta _{o}\,$ and $\cos \theta _{s}\,$ are tied to each other via the relativistic aberration formula:

$\cos \theta _{o}={\frac {\cos \theta _{s}-{\frac {v}{c}}}{1-{\frac {v}{c}}\cos \theta _{s}}}\,.$

Therefore, Eq. (1) can be rewritten as

$f_{o}=\gamma \left(1-{\frac {v\cos \theta _{s}}{c}}\right)f_{s}.$            (2)

For example, a photon emitted at the right angle in the reference frame of the emitter ($\cos \theta _{s}=0\,$) would be seen blue-shifted by the observer:

$f_{o}=\gamma f_{s}.\,$

In the non-relativistic limit, both formulæ (1) and (2) give

${\frac {\Delta f}{f}}\simeq -{\frac {v\cos \theta }{c}}.$

Accelerated motion

For general accelerated motion, or when the motions of the source and receiver are analyzed in an arbitrary inertial frame, the distinction between source and emitter motion must again be taken into account.

The Doppler shift when observed from an arbitrary inertial frame:

${\frac {f_{o}}{f_{s}}}={\frac {1-{\frac {\|{\vec {v_{o}}}\|}{\|{\vec {c}}\|}}cos(\theta _{co})}{1-{\frac {\|{\vec {v_{s}}}\|}{\|{\vec {c}}\|}}cos(\theta _{cs})}}{\sqrt {\frac {1-(v_{s}/c)^{2}}{1-(v_{o}/c)^{2}}}}$

where:

${\vec {v_{s}}}$ is the velocity of the source at the time of emission

${\vec {v_{o}}}$ is the velocity of the receiver at the time of reception

${\vec {c}}$ is the light velocity vector

$\theta _{cs}$ is the angle between the source velocity and the light velocity at the time of emission

$\theta _{co}$ is the angle between the receiver velocity and the light velocity at the time of reception

If ${\vec {c}}$ is parallel to ${\vec {v_{s}}}$, then $\theta _{cs}=0^{\circ }$, which causes the frequency measured by the receiver $f_{o}$ to increase relative to the frequency emitted at the source $f_{s}$. Similarly, if ${\vec {c}}$ is anti-parallel to ${\vec {v_{s}}}$, $\theta _{cs}=180^{\circ }$, which causes the frequency measured by the receiver $f_{o}$ to decrease relative to the frequency emitted at the source $f_{s}$.

This is the classical Doppler effect multiplied by the ratio of the receiver and source Lorentz factors.

Due to the possibility of refraction, the light's direction at emission is generally not the same as its direction at reception. In refractive media, the light's path generally deviates from the straight distance between the points of emission and reception. The Doppler effect depends on the component of the emitter's velocity parallel to the light's direction at emission, and the component of the receiver's velocity parallel to the light's direction at absorption. This does not contradict Special Relativity.

The transverse Doppler effect can be analyzed from a reference frame where the source and receiver have equal and opposite velocities. In such a frame the ratio of the Lorentz factors is always 1, and all Doppler shifts appear to be classical in origin. In general, the observed frequency shift is an invariant, but the relative contributions of time dilation and the Doppler effect are frame dependent.

See also

• Doppler effect
• Doppler beaming
• Redshift
• Blueshift
• Time Dilation
• Gravitational Time Dilation
• Special relativity

References

• Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper". Annalen der Physik. 322 (10): 891–921. Bibcode:1905AnP...322..891E. doi:10.1002/andp.19053221004. {{cite journal}}: Cite has empty unknown parameters: |month= and |coauthors= (help) English translation: ‘On the Electrodynamics of Moving Bodies’

• A. Einstein (1907), "Über die Möglichkeit einer neuen Prüfung des Relativitätsprinzips", Annalen der Physik SER.4, no.23
• J. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).

External links

• M Moriconi, 2006, Special theory of relativity through the Doppler effect
• Warp Special Relativity Simulator Computer program demonstrating the relativistic Doppler effect.
• The Doppler Effect at MathPages