(a) The Einstein velocity-addition law, Eq. (1.13), has a simpler form if we introduce the concept of the velocity parameter u, defined by the equation
v = tanh u.
Notice that for −∞v = tanh u
And
w = tanh U,
then Eq. (1.13) implies
w' = tanh(u + U).
This means that velocity parameters add linearly.
(b) Use this to solve the following problem. A star measures a second star to be moving away at speed v = 0.9 c. The second star measures a third to be receding in the same direction at 0.9 c. Similarly, the third measures a fourth, and so on, up to some large number N of stars. What is the velocity of the Nth star relative to the first? Give an exact answer and an approximation useful for large N.