In relativity, confusion can arise because of
multiple concepts of velocity.
We start with the four-vector position x.
The derivative of this with respect to the
proper time (tau) is called the proper
velocity and denoted u:
u := d x / d tau
This has lots of nice properties. Notably
the four-vector momentum is
p = m u
In turn, the four-vector momentum has some
nice properties, including the fact that
its projection onto the spacelike dimensions
gives us the ordinary three-vector momentum.
But we should not congratulate ourselves too
soon. Alas, if we project the proper velocity
u onto the spacelike dimensions, we do _not_
get the ordinary three-vector velocity.
It is easy to see that this must be so; the
momentum p and the proper velocity u grow
without bound (e.g. when a particle is subjected
to a steady force for a long time) ... whereas
v, the ordinary three-vector velocity, is bounded
by c.
We can calculate something called the coordinate
velocity v as
v := d x / d t
Even though x is a four-vector, this v is not
a well-behaved four-vector, even though it has
four components. It does not have the
transformation-invariance we would like our
four-vectors to have. But this v does have the
desired property that its spacelike projection
is the ordinary three-vector velocity.
Also, for a simple boost in a given direction:
u = sinh rho
v = tanh rho
where rho is the rapidity. They are the same
when they're small, but in general they differ
by a factor of cosh rho, i.e. gamma.
Some texts refer to u as "the" velocity. Others
refer to v as "the" velocity. I've been guilty
of inconsistency myself; I just now cleaned up
some of the documents on my web site.
We need to be careful not to inflict this sort
of ambiguity and inconsistency on poor innocent
students.