Black Holes

Motion along the line of sight in Relativistic Doppler effect

Assume the observer and the source are moving away from each other with a relative velocity $v\,$ ( $v\,$ is negative if the observer and the source are moving toward each other). Considering the problem in the reference frame of the source, suppose one wavefront arrives at the observer. The next wavefront is then at a distance $\lambda=c/f_s\,$ away from him (where $\lambda\,$ is the wavelength, $f_s\,$ 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 (as measured in the reference frame of the source) between crest arrivals at the observer is

$t = \frac{\lambda}{c-v} = \frac{c}{(c-v)f_s} = \frac{1}{(1-\beta)f_s},$

where $\beta = v / c\,$ is the velocity of the observer in terms of the speed of light (see beta (velocity)).

Due to the relativistic time dilation, the observer will measure this time to be

$t_o = \frac{t}{\gamma},$

where

$\gamma = \frac{1}{\sqrt{1-\beta^2}}$

is the Lorentz factor. The corresponding observed frequency is

$f_o = \frac{1}{t_o} = \gamma (1-\beta) f_s = \sqrt{\frac{1-\beta}{1+\beta}}\,f_s.$

The ratio

$\frac{f_s}{f_o} = \sqrt{\frac{1+\beta}{1-\beta}}$

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.) The corresponding wavelengths are related by

$\frac{\lambda_o}{\lambda_s} = \frac{f_s}{f_o} = \sqrt{\frac{1+\beta}{1-\beta}},$