Superluminal Motion: an Illusion

by Dr. G.H. George, Mathematics Section, Gulf Polytechnic.
This article appeared in the Voice of Technology, #7, pp.36-37, (Bahrain), 1987

The “Voice of Technology” was an in-house publication of the then Gulf Polytechnic (later the University of Bahrain) and was founded in the 1980’s.


It is shown from purely classical considerations that apparent superluminal (faster than light) motion, transverse to the line of sight, can persist over several degrees of the field of view, even if the true velocity is subluminal.

Key words:   superluminal motion, quasars.


Nothing can travel faster than light in free space.   This fact is the foundation of Einstein’s first (special) theory of relativity and it is one of the few parts of relativity theory which are widely known outside the physics community.

Therefore it came as quite a surprise when the British scientific journal “Nature” published an article showing radio maps of objects apparently moving away from each other faster than the speed of light, (Pearson et al., Nature 1981 Vol. 290, pages 365-368).   The objects are thought to be collections of hot plasma streaming out from a quasar in a narrow jet.


Quasars have very faint images which look like stars when viewed telescopically.   However, their spectra indicate very high speeds of recession, (between 15% and 97% of the speed of light).   Most astronomers today [1987] believe that galactic speeds of recession are approximately proportional to distance.

This would place the quasars so far away that the light by which we see the farthest of them now set out before our solar system even existed.   At such immense distances, (thousands of millions of light years), almost any measurable movement across the sky over the period of a year would require superluminal (faster than light) motion.

If the method of determining quasar distances is correct, then the apparent relative speed of the quasar components is anything up to ten times the speed of light.   How can this happen, when nothing can move faster than light?

The Paradox Resolved

Let   c   be the speed of light.
In figure 1, a particle moves with uniform velocity   v   along the line NPQ.

From triangle ONP,   r2 = b2 + R 2
From triangle ONQ,   (r + dr)2 = b2 + (R + dR) 2
but   b2 + R 2 = r2, so, to first order,
delta_r = R deltaR / sqrt{b^2 + R^2}

line NPQ, distance b from origin
Figure 1

Any light signal from   P   reaches an observer at   O   after   r/c   time units.
The particle travels from   P   to   Q   in time   (dR / v)   time units.
Any light signal from   Q   will reach observer   O   ((r + dr) / c)   time units later.

Hence the time which elapses between the observer seeing the particle at   P   and seeing it at   Q   is

delta_T = delta_R / v + R deltaR / (c sqrt{b^2 + R^2})

Hence the apparent speed of the particle is
V' = v / (1 + (v/c) sqrt{1 + (b/R)^2}^(-1/2))
sqrt{1 + (b/R)^2}^(-1/2) = cos theta
V' = v / [1 + (v/c) cos theta]

We are interested in the component of apparent velocity transverse to the line of sight, (i.e. the apparent speed across the sky),

V_T = v sin theta / [1 + (v/c) cos theta]

When theta < pi/2 the particle is receding,
cos theta > 0  and  V_T < v   for all positive true speeds   v.
When theta > pi/2 the particle is approaching,
cos theta < 0  and  V_T > v sin theta
In this case it makes sense to replace the obtuse angle   q   by its supplementary acute angle phi = pi - theta.   Then

V_T = v sin phi / [1 - (v/c) cos phi]

For a given true velocity   v , the apparent transverse velocity has a maximum where
d V_T / d phi = 0
cos phi = v/c
maximum apparent transverse velocity
     = v / sqrt{1 - (v/c)^2}
V_T max > c  when  v > c / sqrt(2)

V_T > c  when  v sin(phi + pi/4) > c / sqrt(2)
So for c/sqrt(2) << v < c, apparent transverse superluminal motion can persist over a significant range of angles f.   There is no paradox!

Let us find the range of angles,   Df , over which apparent transverse velocity > U
v sin phi / [1 - (v/c) cos phi] > U
Define a new angle   z   as in Figure 2:

tan zeta = U/c
Figure 2

[dividing by hypotenuse]
v sin(theta + zeta) > sin zeta
sin(theta + zeta) > U / [v sqrt{1 + (U/c)^2}]
Therefore apparent transverse superluminal motion persists over a range of angles
theta1 < theta < theta2  where
    theta1 = Arcsin(U / [v sqrt{1 + (U/c)^2}]) - zeta
    theta2 = pi - theta1 - 2 zeta
Delta theta = pi - 2 Arcsin(U / [v sqrt{1 + (U/c)^2}])

True velocity
v / c
Maximum value
of V_T / c

1.00   infinity
0.99 7.018
0.98 4.925
0.95 3.042
0.90 2.065
0.85 1.614
0.80 1.333
0.75 1.134
0.71 1.000
0.70 0.980
0.60 0.750
0.50 0.577

As can be seen from figure 3, true subluminal velocities can cause apparent superluminal transverse motion at over ten times the speed of light, which persists for over ten degrees of the sky.   In this way, the observations can be explained.

Figure 3:   Range of angles of apparent transverse superluminal motion
versus true velocity as a proportion of the speed of light
Figure 3:  chart of Delta phi against v/c

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Created 2008 06 02 and most recently modified 2008 06 02 by Dr. G.H. George.