Relativistic Doppler effect Information & Relativistic Doppler effect Links at HealthHaven.com

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 of 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.

[edit] 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.

[edit] Analogy

Understanding relativistic Doppler effect requires understanding 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 moves to the right and the catcher is stands 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 still and the catcher moves 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 instanteous 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 instanteous angle between the emitter-observer line and the relative velocity vector, and 2) the emitter-observer velocity relative to the speed of light.

[edit] 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 observers are moving toward each other). Let us consider 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 observed between crests is

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

where $β = v / c$ 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.

[edit] Transverse Doppler effect

The transverse Doppler effect is the nominal redshift component associated with transverse (i.e. lateral) observation resulting from special relativity effects, and is important both theoretically and experimentally.

If the predictions of special relativity are compared to those of a simple flat nonrelativistic light medium that is stationary in the observer’s frame (“classical theory”), SR’s physical predictions of what an observer sees are always “redder”, by the Lorentz factor

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

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

$f_o = \frac{f_s}{\gamma\left(1+\frac{v\cos\theta_o}{c}\right)}$

In the particular case when $θ o = π / 2$ , one obtains the transverse Doppler effect

$f_o=\frac {f_s} {\gamma} \,$

For receding or approaching objects, the redshift factor $\frac{1}{\gamma}$ modifies the redshift or blueshift predictions of "classical theory". Where the two effects act against each other, the propagation-based effects are stronger. But for the case of an object passing directly across the observer’s line of sight, special relativity’s predictions are qualitatively different from "classical theory" – a redshift where the “classical theory” reference model would have predicted no shift effect at all for the case that the observer is at rest in the aether.

Because of this, the transverse Doppler effect is sometimes held up as one of the main new 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.

[edit] 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 aftward 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.

[edit] 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.

[edit] 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,^{[citation needed]} many other longitudinal tests have been performed since with much higher precision [http://* 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)

[edit] 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).

[edit] 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:

$f_o=\frac {f_s} {\gamma} \,$ .

It should be noted that, 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

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

[edit] See also

[edit] References

Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper". Annalen der Physik 322 (10): 891–921. doi:10.1002/andp.19053221004. 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).

[edit] 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

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