Apparent speeds greater than the speed of light

References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Chapter 12, Post 6.

Although no object can travel faster than light, it is possible for the apparent speed of an object to be greater than {c}. A common example is that of the apparent motion of a star across the sky. Suppose a star is a distance {a=L} from Earth at time {t_{a}} and that its velocity {\mathbf{v}} is towards Earth at an angle {\theta} to the line of sight. At time {t_{b}} it arrives at a point {b} whose distance from Earth is

\displaystyle  b=L-v\cos\theta\left(t_{b}-t_{a}\right) \ \ \ \ \ (1)

The times of arrival at Earth of the light emitted at distances {a} and {b} are

\displaystyle   T_{a} \displaystyle  = \displaystyle  t_{a}+\frac{L}{c}\ \ \ \ \ (2)
\displaystyle  T_{b} \displaystyle  = \displaystyle  t_{b}+\frac{1}{c}\left[L-v\cos\theta\left(t_{b}-t_{a}\right)\right] \ \ \ \ \ (3)

During this time, the star moves a distance perpendicular to the line of sight of {v\sin\theta\left(t_{b}-t_{a}\right)}, so the apparent speed as seen from Earth is

\displaystyle   u \displaystyle  = \displaystyle  \frac{v\sin\theta\left(t_{b}-t_{a}\right)}{T_{b}-T_{a}}\ \ \ \ \ (4)
\displaystyle  \displaystyle  = \displaystyle  \frac{v\sin\theta\left(t_{b}-t_{a}\right)}{\left(t_{b}-t_{a}\right)\left(1-\frac{v}{c}\cos\theta\right)}\ \ \ \ \ (5)
\displaystyle  \displaystyle  = \displaystyle  \frac{v\sin\theta}{1-\frac{v}{c}\cos\theta} \ \ \ \ \ (6)

This speed has a maximum at an angle {\theta} which can be found by setting the derivative to zero:

\displaystyle   \frac{du}{d\theta} \displaystyle  = \displaystyle  {v\cos\left(\theta\right)\left(1-{\frac{v\cos\left(\theta\right)}{c}}\right)^{-1}}-{\frac{{v}^{2}\left(\sin\left(\theta\right)\right)^{2}}{c}\left(1-{\frac{v\cos\left(\theta\right)}{c}}\right)^{-2}}\ \ \ \ \ (7)
\displaystyle  \displaystyle  = \displaystyle  {\frac{\left(\cos\left(\theta\right)c-v\right)cv}{\left(\cos\left(\theta\right)\right)^{2}{v}^{2}-2\, c\cos\left(\theta\right)v+{c}^{2}}}\ \ \ \ \ (8)
\displaystyle  \displaystyle  = \displaystyle  0\ \ \ \ \ (9)
\displaystyle  \cos\theta \displaystyle  = \displaystyle  \frac{v}{c} \ \ \ \ \ (10)

At this angle, the apparent speed is

\displaystyle   u_{max} \displaystyle  = \displaystyle  \frac{v\sqrt{1-v^{2}/c^{2}}}{1-v^{2}/c^{2}}\ \ \ \ \ (11)
\displaystyle  \displaystyle  = \displaystyle  \gamma v \ \ \ \ \ (12)

Since {\gamma\rightarrow\infty} as {v\rightarrow c}, {u_{max}} can be much larger than {c} even though the actual speed of the star is less than {c}. This again illustrates the importance of correctly interpreting the raw data that we see, and of allowing for the travel time of the light.

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