135 What about apparent faster-than-light motions?
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This article is from the Astronomy FAQ, by Joseph Lazio (jlazio@patriot.net) with numerous contributions by others.
135 What about apparent faster-than-light motions?
By Martin Hardcastle <m.hardcastle@bristol.ac.uk>
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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