In this entry, we are going to study a relativistic effect known as “stellar aberration”.
From the known Lorentz transformations of velocities (inverse case), we get:
The classical result (galilean addition of velocities) is recovered in the limit of low velocities or sending the light speed get the value “infinite” . Then,
Let us define
Thus, we get the component decomposition into the xy and x’y’ planes:
From this equations, we get
From the last equation, we get
From this equation, if , i.e., if and with , we obtain the result
By these formalae, the angle of a light beam propagating in space depends on the velocity of the source respect to the observer. We can observe this relativistic effect every night (supposing a good approximation that Earth’s velocity is non-relativistic, as it shows). The physical interpretation of the above aberration formulae (for the stars we watch during a sky night) is as follows: due to the Earth’s motion, a star in the zenith is seen under an angle .
Other important consequence from the stellar aberration is when we track ultra-relativistic particles (). Then, and then, the observer moves close to the source of light. In this case, almost every star (excepting those behind with ) are seen “in front of” the observer. If the source moves with almost the speed of light, then the light is “observed” as it were concentrated in a little cone with an aperture .