Black Holes

Sunday, 9 June 2013

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)

In the particular case when $\theta_o=90^{\circ}\,$ and $\cos\theta_o=0 \,$ one obtains the transverse Doppler effect:

$f_o = \frac {f_s} {\gamma}. \,$

Due to the finite speed of light, the light ray (or photon, if you like) perceived by the observer as coming at angle $\theta_o \,$ , was, in the reference frame of the source, emitted at a different angle $\theta_s \,$ . $\cos \theta_o \,$ and $\cos \theta_s \,$ are tied to each other via the relativistic aberration formula:

$\cos \theta_o=\frac{\cos \theta_s-\frac{v}{c}}{1-\frac{v}{c} \cos \theta_s} \,.$

Therefore, Eq. (1) can be rewritten as

$f_o = \gamma\left(1-\frac{v\cos\theta_s}{c}\right)f_s.$ (2)

For example, a photon emitted at the right angle in the reference frame of the emitter ( $\cos \theta_s = 0 \,$ ) would be seen blue-shifted by the observer: