19. Aberration
In this section aberration is derived in a medium. This is an important result as it shows that the stimulated emission process used in the previous section is also successful in deriving the aberration. Though actually there are two aberrations, one by absolute motion of the observer and one by absolute motion of the source.
When a component of a light ray is perpendicular to the observer's absolute velocity v, there is an apparent change in direction of the light ray. In the following diagram, a ray of light is coming in off-perpendicular by an angle f. Due to this aberration, the light ray appears to come in at an angle differing by q. We are also considering the possibility that the light ray is travelling through a substance with index of refraction n.
Fig.7: Aberration of a ray of light approaching at an angle f.
The diagram shows the various vector contributions (utilizing Eq. (18-4)). We thus have
(1)
This ultimately yields
(2)
It is important to note that the aberration is independent of the index of refraction1.
If an observer who is moving at an unknown absolute velocity sees a distant object (with no physical contact), the true direction of the object is not known, since there will be an aberration dependent on this unknown absolute velocity. If the velocity of the observer changes than there will be a noticeable change in aberration, but the actual absolute velocity of the observer is never revealed.
Now one might think, however, that there could be a physically connected system (observer and light source), where the system could be rotated, then observations of aberration would reveal the absolute velocity of the system. The problem is that there is also an aberration based on the absolute velocity of the source and this prevents any such deflection from being observed.
Imagine a rail-car moving along a track at some velocity, a small projectile is fired from the rail car in the forward direction. When the projectile was fired it was given a certain momentum, but it also had a momentum based on the velocity of the rail-car already by virtue of the projectile's mass. Now for an observer on the ground, the projectile has a total momentum which includes what it had when it was at rest on the rail-car. In order to show this extra momentum the projectiles velocity is proportionately larger, in other words, there is a ballastic addition of velocities. If we replace the projectile with a photon, it turns out that the photon will also have the additional momentum related to the velocity of the rail-car corresponding to the energy of the photon. Of course, it is not shown in the velocity since that is limited to c. However, it will show it in an increased frequency caused by the Doppler effect (see next Section). In other words the momentum related to the absolute velocity of the source must always be maintained.
Now imagine that the projectile is fired perpendicular to the direction of the rail-car's velocity. In order to maintain the proper momentum in relation to an observer on the ground the velocities can be added as vectors. In respect to the ground, the projectile will still have its original momentum in the direction of the moving rail-car. Now the photon must also be deflected forward in a similar fashion. Its net speed relative to the ground (the absolute frame in this analogy) can never be greater than c, but any amount of ballastic addition that would exceed c is essentially converted into a frequency shift to compensate and thereby conserve momentum. But note that the photon is still deflected.
Though this effect is not usually discussed in relation to photons, it is actually well known in the case of electromagnetic radiation where it is often referred to as the headlighting effect2. And, in fact, the headlighting effect performs the same momentum conserving role.
1.This was shown by Airy and Hoek (see Ditchburn10 439).
2. Griffiths1 430 or Jackson2 663.
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