As the saying goes, experience teaches you to recognize a
mistake when you've made it again.
I recently caught my self mentioning 3-velocity. This is
a mistake that I've made more than once.
Here's the story:
1) The 4-velocity is u = [dt/dτ, dx/dτ, dy/dτ, dz/dτ]
where τ is the proper time. This 4-velocity is often
very useful.
2) The _reduced velocity_ is v = [dx/dt, dy/dt, dz/dt]
which is equal to the
spacelike part of the 4-velocity _divided_ by gamma,
i.e. _reduced_ by a factor of dt/dτ.
This reduced velocity is sometimes useful. It is
sometimes called the "classical" velocity, because
usually a naive measurement of the velocity ends up
measuring the reduced velocity.
MTW call this the "coordinate velocity" ... but I've
never found that name to have much explanatory or
mnemonic value.
3) Alas, the reduced velocity is all-too-often called
the "3-velocity". That is a disaster, because of the
following broken parallelism:
++ the 3-vector position is the spacelike part of the
4-vector position
++ the 3-momentum is the spacelike part of the 4-momentum
-- the so-called "3-velocity" is *not* the spacelike
part of the 4-velocity.
Any student who learns an expression involving the "3-velocity"
is just about guaranteed to make mistakes, because the name is
easier to remember than the definition. This leads to bug
after bug after bug.
Constructive suggestion: Call it the reduced velocity, or
the classical velocity, or the coordinate velocity. I like
calling it the reduced velocity, because that has some mnemonic
value. Don't ever use the term 3-velocity. Just don't do it.
It's taboo.