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 (like in the regular Doppler effect), when taking into account effects of the special theory of relativity.

The relativistic Doppler effect is different from the true (non-relativistic) Doppler effect as the equations include the time dilation effect of special relativity. 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 $v\,$. 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 her (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 $v\,$, they will meet after a time

$t={\frac {\lambda }{c-v}}={\frac {1}{(1-v/c)f_{e}}}$

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 $\gamma =1/{\sqrt {1-v^{2}/c^{2}}}$, so the corresponding frequency is

$f_{o}={\frac {1}{t_{o}}}=\gamma (1-v/c)f_{e}={\sqrt {\frac {1-v/c}{1+v/c}}}\,f_{e}$

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 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:

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

Note: In all the expressions in this section it is assumed that the observer and the source are moving away from each other. If they are moving towards each other, $v\,$ should be taken negative.

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 $\theta \,$ relative to the direction from the observer to the source (at the time when the light is emitted), the frequency changes as

$f_{o}={\frac {f_{s}}{\gamma \left(1+{\frac {v\cos \theta }{c}}\right)}}$

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

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

$f_{o}=\gamma \left(1-{\frac {v\cos \theta }{c}}\right)f_{s}$

In the non-relativistic limit:

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

Visualization

Diagram 1. Demonstration of Aberration of light and Relativistic Doppler effect.

On diagram 1 observer is represented by blue point. We have xy-plane in the space, which is represented by yellow graph paper. If observer moves then he sees the graph paper in different colors. Also he will see the distortion of xy grid due to aberration of light .

See also

• Doppler effect
• Redshift
• Blueshift
• Special relativity
• Transverse Doppler effect

External links

• Warp Special Relativity Simulator Computer program demonstrating the relativistic doppler effect.