- Relativistic Doppler effect
The

**relativistic Doppler effect**is the change infrequency (andwavelength ) oflight , caused by the relative motion of the source and the observer (as in the classicalDoppler 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 thetime dilation effect ofspecial relativity and do not involve the medium of propagation as a reference point. They describe the total difference in observed frequencies and possess the requiredLorentz 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 thereference 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 thewavelength , $f\_e,$ is thefrequency of the wave the source emitted, and $c,$ is thespeed of light ). Since the wavefront moves with velocity $c,$ and the observer escapes with velocity $v,$, the time observed between crests is:$t\; =\; frac\{lambda\}\{c-v\}\; =\; frac\{1\}\{(1-v/c)f\_e\}.$

However, due to the relativistic

time dilation , the observer will measure this time to be:$t\_o\; =\; frac\{t\}\{gamma\}\; =\; frac\{1\}\{gamma(1-v/c)f\_e\},$

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

:$f\_o\; =\; frac\{1\}\{t\_o\}\; =\; gamma\; (1-v/c)\; f\_e\; =\; sqrt\{frac\{1-v/c\}\{1+v/c,f\_e.$

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 ofastrophysics : seerelativistic 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:$f\_o\; =\; sqrt\{frac\{1-v/c\}\{1+v/c,f\_e,$

where $c,$ is the

speed of light .The corresponding

wavelength s are related by:$lambda\_o\; =\; sqrt\{frac\{1+v/c\}\{1-v/c,lambda\_e,$

and the resulting

redshift $z,$ can be written as:$z\; +\; 1\; =\; frac\{lambda\_o\}\{lambda\_e\}\; =\; sqrt\{frac\{1+v/c\}\{1-v/c.$

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

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

**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:$f\_o\; =\; frac\{f\_s\}\{gammaleft(1+frac\{vcos\; heta\_o\}\{c\}\; ight)\},$ (1)

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

In the particular case when $heta\_o=90,$ and $cos\; heta\_o=0\; ,$ one obtains the

transverse Doppler effect :$f\_o=frac\; \{f\_s\}\; \{gamma\}\; ,$. (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:$f\_o\; =\; gammaleft(1-frac\{vcos\; heta\_s\}\{c\}\; ight)f\_s.$ (3)

$cos\; heta\_o\; ,$ and $cos\; heta\_s\; ,$ are tied to each other via the

relativistic aberration formula ::$cos\; heta\_o=frac\{cos\; heta\_s-frac\{v\}\{c\{1-frac\{v\}\{c\}\; cos\; heta\_s\}\; ,.$ (4)

:The

relativistic aberration formula explains why, for $cos\; heta\_s\; =0\; ,$ one obtains a second formula for thetransverse Doppler effect ::$f\_o=f\_s\; gamma\; ,.$ (5)

:(5) is obtained easily by substituting $cos\; heta\_o\; =-frac\; \{v\}\{c\}\; ,$ 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

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

**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 theaberration 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.**See also***

Doppler effect

*Redshift

*Blueshift

*Special relativity

*Transverse Doppler effect **External links***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

*Wikimedia Foundation.
2010.*

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