From: ggeorge@kean.ucs.mun.ca (G.H. George)
Subject: FTL images (was Re: help needed with a thought problem)
Organization: Memorial University. St.John's Nfld, Canada
Date: Wed, 29 Jul 1992 12:50:58 GMT

In article <1992Jul29.025114.15848@organpipe.uug.arizona.edu>, 
kachun@astro.as.arizona.edu (Ka Chun Yu) writes:
> 
> A more interesting question of FTL communication posits an immense solid
> spherical shell surrounding a very bright light source, e.g. a star.  (I
> don't remember where I first read about this--anyone have a clue?)  An
> object is made to orbit about the light source at an extremely high angular
> velocity.  If the spherical shell is large enough and if the orbital speed
> was high enough, one could envision that at one point, the shadow of the
> orbiting object on the inside of the shell will be moving faster than the
> speed of light with respect to an observer standing on the inside surface
> of the shell.  The question then is, can this be considered some kind of
> signal that is 'faster than light'?
                                     - Not really, as no observers on the
shell could use the shadow to transmit any information between them faster
than light.

Another interesting thought experiment concerns a powerful laser lighthouse
rotating at a modest angular velocity on Earth.   If its tightly collimated
beam is allowed to sweep across the lunar surface, then the spot can easily be
made to travel faster than light.   Now imagine what an observer with a very
high speed camera on the Moon's surface would see as the spot passed by ... 

The spot is not a physical object and is not subject to any Lorentz
contraction.   However, the finite speed of light from the image of the spot
to the observer will create severe distortions.   For a superluminal spot
nothing will be seen of it until after it has passed by the observer (or until
the instant of its arrival if the observer is placed on the path of the spot).
Thereafter the observer will see *two* images of the spot, one travelling
forwards and the other (faster) image backwards along the path.  

If an image of some object is projected with the spot, then any lines parallel
to the line of motion remain parallel to the line of motion, although their
lengths may change.   Any line perpendicular to the line of motion becomes two 
arcs of the same ellipse, one arc being in each image.

If the observer starts a clock at the instant when the centre of the forward
image of the spot is at its closest apparent approach (at impact parameter  d )
to the observer and if a coordinate system is set up with the x-axis pointing 
in the direction of true motion (speed  v = b c) and the observer at the
origin, then the apparent position of any other point in the spot (true 
coordinates (x,y) relative to the centre of the spot) can be shown to be

x'(x,y,t) = b/(b^2 - 1) * {-[ct + d + x/b] plus or minus 
            sqrt[ b^2 * (ct + d + x/b)^2 - (b^2 - 1)*(d + y)^2 ] ;

                    y'(x,y,t) = d + y

The positive root corresponds to the forward image and the negative root to the
backward image.   The centre of the spot does not become visible until time
t = - d/c , at which instant the image is travelling infinitely fast in both
directions!   The angle between the velocity vector and the line of sight at
the instant of first sight for any part of the image of the spot is
theta = Arctan(sqrt[b^2 - 1]).

It is quite an interesting exercise to verify these conclusions!

G.H. George                                       ggeorge@bigvax.engr.mun.ca
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
St. John's,   Newfoundland
CANADA     A1B 3X5
Xref: sparky sci.skeptic:13415 alt.paranormal:1983