Relativistic aberration and detector synchronization
Astronomers have been aware of a phenomenon called
aberration for a long time[1].
It occurs when viewing the angular orientation of a distant star and is due to
the earth’s motion on it orbit. It is largest
when the velocity of the earth (ve is
about 30 km/sec) is perpendicular to the line between the earth and the
star. Thus when an astronomer views a
star at
where c is the speed of light. The ratio ve/c is about 0.0001 so the effect is readily measurable and the doubled angle is about 41 seconds of arc. It is due to the fact that the wave fronts appear tilted when viewed in the moving frame of reference.
This effect might be explained with the following simple example:
Suppose a very narrow detector is approaching along the x axis (at speed v) a collimated light beam of width w. The detector reaches the left edge of the light beam at time t=0 and the right edge, dx, at time t=w/v. While the detector is traversing the width of the beam, the wave fronts have moved along the y direction a distance dy= c (w/v). Therefore the apparent tilt of the wave front is dy/dx=dy/w=c/v which, for small v/c, is the inverse of the value given above. Since aberration is a first order effect in v/c, I am at a loss to explain why this simple calculation fails even for small v/c. Obviously, for small v/c and reasonably wide beams, any wavefront moves by many, many wavelengths during the beam traversal. And therefore the tilt of the wave front is ambiguous. In fact, since the narrowest beam width possible is about w=1 wavelength, even for v/c of the order of ½, the tilt dy would be 2 wavelengths which is again fairly ambiguous.
To animate aberration, I chose to use a tilted moving array of detectors angled at
where v is the linear speed of the detectors along a direction perpendicular to the direction to the transmitter. I also tilted the wave fronts at this same angle.
Effectively this is analogous to someone moving with velocity v perpendicular to the light source and tilting a single receiver plane by the angle q which is the same angle as the tilt of the wave fronts. In both cases, the detector receives the entire wave front simultaneously rather than having multiple phases spread across the receiver plane,