The apparently faster-than-light motions observed in the jets of some
radio-loud quasars have misled a number of people into believing that
the speed of light is not really a limit on velocity and that
astrophysics has provided a disproof of the theory of relativity.  In
fact, these motions can be easily understood without any new physics;
you just need trigonometry and the idea of the constancy of the speed of
light.

Consider the situation shown in the diagram below.  A blob B of
radio-emitting plasma starts at O and moves with velocity v at some
angle a to our line of sight.  At a time t, B has moved across the sky
a distance vt sin a.  The light from when it was at O has travelled a
distance ct towards us (c is the speed of light).  But the light from
its position at time t only has to travel an additional distance 
(ct - vt cos a) to reach us. Thus we measure the time between the two
events as (distance / speed of light) = t(1 - (v/c) cos a).  If we
derive an apparent velocity by dividing the (measurable) transverse
motion of the source by the measured time difference, we get

                 vt sin a               v sin a
v(apparent) = ------------------  =  ---------------
              t(1 - (v/c) cos a)     1 - (v/c) cos a


                       ^     O          ^
                       |     |\         |
                       |     | \        |
                       |     |  \       vt cos a
                       |     | a \      |
                       ct    |    \     |
                       |     |     \    |
                       |     |      B   v
                       |     |          ^
                       |     |          ct - vt cos a
                       v     |          v



                            \_____I_____/
                             (Earth, radio telescope)

This apparent velocity can clearly be greater than c if a is small and
v is close to c.  There are other independent reasons for believing
that the jets in radio-loud quasars have velocities close to c and are
aligned close to the line of sight, so that this explanation is a
plausible one.

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