Aberration of light

Relativistic aberration

Suppose that we're in a railway carriage that has a large mirror attached to one of the carriage's side-walls. We stand alongside this mirror, face-on, and fire a light-beam directly at it. If the laws of physics appear to us to be those of a "stationary" observer, we'll expect the beam to strike the mirror at 90 degrees, leave the mirror at 90 degrees, and end up being reflected straight back at the original light-source.

We'll assume that this is what actually happens. 

How does this situation appear from the point of view of a second observer, in a train moving along a second set of tracks parallel to ours? 

The other observer is forced to agree with us that the light leaves our source, bounces off the glass and ends up back at the source again ... but from their perspective, the source has moved during this time. They see the lightbeam to be emitted at position A, moving at an angle that takes it forwards to reach the moving mirror and strike it a glancing blow, and then being reflected forwards to hit the source again at its new position, B.

The second observer can argue that in order for the light to be able to keep up with us, it can't just be aimed sideways, it also has to have a velocity component parallel to the tracks, that exactly matches our velocity ... The ray that we think is aimed at exactly 90 degrees appears to the other observer to be aimed (and reflected) in a slightly different direction. It seems to them that in order to hit the source, our beam must have been aimed at the mirror at an angle other than 90 degrees, so that the reflected ray could point towards the source's future position. They argue that for the beam to hit its target, the lightsource must have been tilted to point more towards the direction that we were moving in.

The condition that the second observer needs the "transverse-aimed" ray to advance along the track at precisely the same rate as its source is enough for us to be able to work out the angular deflection as a function of velocity.
For the "transverse-aimed" ray (aimed at an angle "A = 90 degrees"), this gives us:

(this relationship can be written in various different ways, depending on the scheme that we choose for labeling our angles).

This "relativistic aberration" effect shows up in special relativity [notes] , and also in historical Newtonian ballistic emission theory, which assumed that light was thrown off at a set speed with respect to its emitter.

The Relativistic Ellipse

If we want to generalise this result, and apply the same argument to any rays that we emitted in other directions, we arrive at the Relativistic Ellipse. This gives us the apparent angle-change for light emitted in any direction. It can also be used to calculate a relativistic theory's Doppler shift predictions, for any angle. 

Effects

Relativistic aberration generates two major results:

1. The Searchlight Effect: since light from a "moving" body has to appear to be tilted to point more towards the direction that the body moves in, if the body emits a burst of light that it believes to be equally concentrated in all directions, outsiders for whom that body is seen to be "moving" will see a greater quantity of light to be emitted in the forward direction, and a reduced quantity to be aimed rearwards. The body supplied more intense illumination to objects placed ahead of it than behind it. 

2. Geometrical distortions: The moving object's angles appear more "squashed together" at the front of the object and more "splayed apart" at its rear. If we're looking at the side of the object, the tilt of the rays, which makes it easier for us to see the object's rear and more difficult for us to see its front, might be interpreted as showing an image in which the object appears to be partly rotated, with its nose tilted away from us (Terrell-Penrose rotation [note]). However, the "rotation" interpretation doesn't work so well for the view seen by observers placed in front of (or behind) the object: instead they see more (or less) of the surface than they would have if there had been no relative motion.

The same arguments apply for how the moving observer sees their surroundings. The surrounding universe can be taken as an "object"  that's been turned inside out to face us, and aberration effects tell us that we should see our surrounding starfield to distorted when we move through it at high speed: The starfield should appear to us to be more concentrated in the direction that we're moving in, and less concentrated in the region that we're moving away from. Stars whose positions we believe to be at exactly 90 degrees to our position and direction of motion should be seen by use to  slightly ahead of us.

See also:

• The Relativistic Ellipse
• Gravitational aberration

Not to be confused with:

• Chromatic aberration of light, which is about the annoying tendency of glass lenses to focus different colours at different places. 

References:

• James Bradley "… an Account of a New Discovered Motion of the Fix'd Stars" Philosophical Transactions of the Royal Society 637-660 (1728)
• A. Einstein, " On the Electrodynamics of Moving Bodies " (" Zur Elektrodynamik bewegter Korper ") Annalen der Physik 17 891-921 (1905)
• Terrell -Penrose rotation:
• James Terrell "Invisibility of the Lorentz Contraction" Physical Review 116 1041-1045 (1959)
• Roger Penrose, "The Apparent Shape of a Relativistically Moving Sphere" Proc. Cambridge Phil. Soc. 55 137-139 "Research Notes", letter (1959)
• Aberrated starfields are discussed and illustrated in:
• G.D. Scott and H.J. van Driel "Geometrical Appearances at Relativistic Speeds" Am.J.Phys. 38 971-977 (1970)