Relativistic Aberration of Light

Figure 3.10: Relativistic aberration of light.

Consider a light ray that travels from a distant source to an observer located at the origin of some inertial frame, $S$. Let ${\bf c}$ be the phase velocity of the light ray. Of course, $\vert{\bf c}\vert=c$, where $c$ is the speed of light in vacuum. Suppose that ${\bf c}$ lies in the $x$-$y$ plane, such that its direction subtends an angle $\theta$ with the $-x$-direction, as shown in Figure 3.10. It is clear from the figure that $c_x = - c\,\cos\theta$ and $c_y=-c\,\sin\theta$. Suppose that a second observer, moving with velocity ${\bf v} = v\,{\bf e}_x$ with respect to the first, observes the light ray. Let ${\bf c}'$ be the phase velocity of the light ray in the second observer's frame, $S'$, which is in a standard configuration with respect to frame $S$. Of course, $\vert{\bf c}'\vert=c$. Suppose that ${\bf c}'$ lies in the $x'$-$y'$ plane, such that its direction subtends an angle $\theta'$ with the $-x'$-direction, as shown in Figure 3.10. It is clear from the figure that $c_x' = - c\,\cos\theta'$ and $c_y=-c\,\sin\theta'$. The transformation of velocity, (3.122)–(3.124), yields

$$\tan\theta'=\frac{-u_y'}{-u_x'}=\frac{-u_y}{-\gamma\,(u_x-v)}=\frac{c\,\sin\theta}{-\gamma\,(-c\,\cos\theta-v)},$$ (3.129)

or

$$\tan\theta' = \frac{\sin\theta}{\gamma\,(\cos\theta+v/c)}.$$ (3.130)

Thus, the direction of the light ray, and, hence the angular position of the source, appears different to the two observers.

In particular, suppose that the first observer is located in the rest frame of the Sun, and the second is located on the Earth, whose instantaneous orbital velocity about the Sun is ${\bf v} = v_e\,{\bf e}_x$, where $v_e = 2.977\times 10^4\,{\rm m\,s^{-1}}$. In this case, the previous equation yields

$$\tan\theta' = \frac{\sin\theta}{\gamma\,(\cos\theta+\kappa)},$$ (3.131)

where

$$\kappa = \frac{v_e}{c}= 9.930\times 10^{-5},$$ (3.132)

$$\gamma = \frac{1}{\sqrt{1-v_e^{\,2}/c^2}}= 1.000000005.$$ (3.133)

It can be seen that formula (3.131) is almost indistinguishable from the classical aberration formula, (3.23). Thus, it is clear that special relativity is capable of accounting for Bradley's observation of the aberration of starlight. (See Section 3.1.3.) Furthermore, it is obvious that the relativistic aberration of light is associated with the properties of the Lorentz transformation between two frames of reference, moving with respect to one another, rather than the velocity of light with respect to any particular medium. It follows that special relativity is also capable of accounting for Airy's observation of the aberration of starlight. (See Section 3.1.4.)