- 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 effectas the equations include the time dilationeffect of special relativityand 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 (the sign of is simply reversed in the case where the observers are moving toward each other). Let us consider the problem from the
reference frameof the source.
wavefrontarrives at the observer. The next wavefront is then at a distance away from him (where is the wavelength, is the frequencyof the wave the source emitted, and is the speed of light). Since the wavefront moves with velocity and the observer escapes with velocity , the time observed between crests is
However, due to the relativistic
time dilation, the observer will measure this time to be
where , so the corresponding observed frequency is
The ratio 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.)
For motion along the line of sight
If the observer and the source are moving directly away from each other with velocity , the observed
frequencyis different from the frequency of the source as
where is the
speed of light.
wavelengths are related by
and the resulting
redshiftcan be written as
In the non-relativistic limit—i.e. when —the approximate expressions are
For motion in an arbitrary direction
If, in the
reference frameof the observer, the source is moving away with velocity at an angle relative to the direction from the observer to the source (at the time when the light is emitted), the frequency changes as
In the particular case when and one obtains the
transverse Doppler effect
However, if the angle is measured in the
reference frameof the source (at the time when the light is received by the observer), the expression is
and are tied to each other via the
relativistic aberration formula:
relativistic aberration formulaexplains why, for one obtains a second formula for the transverse Doppler effect:
:(5) is obtained easily by substituting into (1). Turns out that (5) is more useful than (2) being the form used routinely in the
In the non-relativistic limit, both formulæ become
aberration of lightand 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.
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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