References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Chapter 12, Post 27.

[Griffiths’s approach to the relativistic four-velocity is similar to that of Moore, although rather confusingly, he uses different notation (as well as keeping factors of in the equations rather than setting ). To keep the notation consistent with Griffiths, I’ll use his notation here, but anyone attempting to follow both books should beware…]

We can now return to a particle travelling on a hyperbolic trajectory, so its position (one dimensional, on the axis) is

Here, and are the position and time as measured by an observer at rest (so they are not proper time for the particle). We can find the particle’s proper time as a function of by using the relation between time intervals:

We have

Integrating both sides, we get, taking when (the integral can be done by software or looked up as it is a standard integral):

We can write the position as a function of by starting with 1 and using this last result:

For the ordinary velocity, we have

The four velocity is defined as

so we have (I’m assuming that in part (c) of Griffiths’s problem, he meant to ask for in terms of , not , as the latter doesn’t give anything particularly informative):

As a check, we note that