# Relativistic Doppler effect

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Relativistic Doppler effect

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.

The mechanism (a simple case)

Assume the observer and the source are moving away from each other with a relative velocity $v,$ (the sign of $v,$ is simply reversed in the case where the observers are moving toward each other). Let us consider the problem from the reference frame of the source.

Suppose one wavefront arrives at the observer. The next wavefront is then at a distance $lambda=c/f_e,$ away from him (where $lambda,$ is the wavelength, $f_e,$ 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

:$t = frac\left\{lambda\right\}\left\{c-v\right\} = frac\left\{1\right\}\left\{\left(1-v/c\right)f_e\right\}.$

However, due to the relativistic time dilation, the observer will measure this time to be

:$t_o = frac\left\{t\right\}\left\{gamma\right\} = frac\left\{1\right\}\left\{gamma\left(1-v/c\right)f_e\right\},$

where $gamma = 1/sqrt\left\{1-v^2/c^2\right\} = 1/sqrt\left\{\left(1-v/c\right)\left(1+v/c\right)\right\}$, so the corresponding observed frequency is

:$f_o = frac\left\{1\right\}\left\{t_o\right\} = gamma \left(1-v/c\right) f_e = sqrt\left\{frac\left\{1-v/c\right\}\left\{1+v/c,f_e.$

The ratio $f_e / f_o,$ 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.)

General results

For motion along the line of sight

If the observer and the source are moving directly away from each other with velocity $v,$, the observed frequency $f_o,$ is different from the frequency of the source $f_e,$ as

:$f_o = sqrt\left\{frac\left\{1-v/c\right\}\left\{1+v/c,f_e,$

where $c,$ is the speed of light.

The corresponding wavelengths are related by

:$lambda_o = sqrt\left\{frac\left\{1+v/c\right\}\left\{1-v/c,lambda_e,$

and the resulting redshift $z,$ can be written as

:$z + 1 = frac\left\{lambda_o\right\}\left\{lambda_e\right\} = sqrt\left\{frac\left\{1+v/c\right\}\left\{1-v/c.$

In the non-relativistic limit—i.e. when $v ll c,$—the approximate expressions are

:$frac\left\{Delta f\right\}\left\{f\right\} simeq -frac\left\{v\right\}\left\{c\right\}; qquad frac\left\{Delta lambda\right\}\left\{lambda\right\} simeq frac\left\{v\right\}\left\{c\right\}; qquad z simeq frac\left\{v\right\}\left\{c\right\}.$

For motion in an arbitrary direction

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

:$f_o = frac\left\{f_s\right\}\left\{gammaleft\left(1+frac\left\{vcos heta_o\right\}\left\{c\right\} ight\right)\right\},$ (1)

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

In the particular case when $heta_o=90,$ and $cos heta_o=0 ,$ one obtains the transverse Doppler effect

:$f_o=frac \left\{f_s\right\} \left\{gamma\right\} ,$. (2)

However, if the angle $heta_s,$ is measured in the reference frame of the source (at the time when the light is received by the observer), the expression is

:$f_o = gammaleft\left(1-frac\left\{vcos heta_s\right\}\left\{c\right\} ight\right)f_s.$ (3)

$cos heta_o ,$ and $cos heta_s ,$ are tied to each other via the relativistic aberration formula:

:$cos heta_o=frac\left\{cos heta_s-frac\left\{v\right\}\left\{c\left\{1-frac\left\{v\right\}\left\{c\right\} cos heta_s\right\} ,.$ (4)

:The relativistic aberration formula explains why, for $cos heta_s =0 ,$ one obtains a second formula for the transverse Doppler effect:

:$f_o=f_s gamma ,.$ (5)

:(5) is obtained easily by substituting $cos heta_o =-frac \left\{v\right\}\left\{c\right\} ,$ into (1). Turns out that (5) is more useful than (2) being the form used routinely in the Ives-Stilwell experiment.

In the non-relativistic limit, both formulæ become

:$frac\left\{Delta f\right\}\left\{f\right\} simeq -frac\left\{vcos heta\right\}\left\{c\right\}.$

Visualization

[
frame|right|Diagram_1._Demonstration_of_aberration of light and relativistic Doppler effect.]
In diagram 1, the blue point represents the observer. The "x","y"-plane is represented by yellow graph paper. As the observer accelerates, he sees the graph paper change colors. Also he sees the distortion of the "x","y"-grid due to the aberration of light. The black vertical line is the "y"-axis. The observer accelerates along the "x"-axis. If the observer looks to the left, (behind him) the lines look closer to him, and since he is accelerating away from the left side, the left side looks red to him (redshift). When he looks to the right (in front of him) because he is moving towards the right side, he sees the right side as green, blue, and violet, respectively as he accelerates (blueshift). Note that the distorted grid is just the observer's perspective, it is all still a consistent yellow graph, but looks more colored and distorted as the observer changes speed.

*Doppler effect
*Redshift
*Blueshift
*Special relativity
*Transverse Doppler effect

*M Moriconi, 2006, [http://www.iop.org/EJ/abstract/0143-0807/27/6/015 Special theory of relativity through the Doppler effect]
* [http://adamauton.com/warp/ Warp Special Relativity Simulator] Computer program demonstrating the relativistic doppler effect.
* [http://www.mpi-hd.mpg.de/ato/rel/doppler-symposium.tgif.pdf] Presentation of the Guido Saathoff modern reenactment of the Ives-Stilwell experiment
* [http://mathpages.com/home/kmath587/kmath587.htm The Doppler Effect] at MathPages

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