Relativistic Doppler effect

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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.
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.

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[edit] 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{\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 of astrophysics: see relativistic beaming.)

[edit] General results

[edit] 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 wavelengths 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}.

[edit] 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 \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)

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

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

f_o=\frac {f_s} {\gamma} \,. (2)

However, if the angle \theta_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 = \gamma\left(1-\frac{v\cos\theta_s}{c}\right)f_s. (3)

\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} \,. (4)
The relativistic aberration formula explains why, for \cos \theta_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 \theta_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{v\cos\theta}{c}.


[edit] Visualization

Diagram 1. Demonstration of aberration of light and relativistic Doppler effect.
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 purple, 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.

[edit] See also

[edit] External links