Scenario: We have a particle moving through the universe
at some uniform velocity. We have an observer, Moe, comoving
with the particle.
Question A: What is the particle's 4-velocity, relative to
Moe's coordinate system?
Correct answer A: u(P) = [1, 0, 0, 0]@M
Question B: What is Moe's 4-velocity, relative to Moe's
coordinate system?
Correct answer B: u(M) = [1, 0, 0, 0]@M
Question C: What is the particle's 4-velocity relative to
Moe?
Correct answer C: the relative velocity is
u(P) - u(M) = zero = [0, 0, 0, 0]
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My point is, if you want answer (A), don't ask question (C).
In typical relativity situations, question (A) is AFAICT
what people usually mean, even when it sounds like they
are asking question (C).
As you know, I don't like talking about misconceptions unless
they are unusually prevalent. I think this one qualifies.
I get more than 5000 hits from http://www.google.com/search?q=+%224-velocity+relative+to%22+observer
and an alarmingly high number of them fall into the booby
trap.
People like to anthropomorphize things. It's cute. I do
it myself ... but this is an object lesson as to why you
don't want to take it too far.
I hope Moe doesn't take it personally, but for most physics
purposes we don't care about Moe. What we care about is the
coordinate system. Moe himself is moving toward the future
at the rate of 60 minutes per hour, in accordance with
answer (B), but his coordinate system is not. In particular
the contours of constant time in his coordinate system are
not moving, and those are what we normally use to measure
spacetime positions, velocities, et cetera.
Moe himself is an artifice ... and he's the wrong artifice.
============
Continuing that thought: Another thing I don't like is
teaching that depends on subtle nuances in the terminology.
In particular, even if we could get every teacher and every
textbook to oh-so-carefully say "relative to Moe's coordinate
system" rather than "relative to Moe", I'm not so sure that
students would appreciate the distinction.
Therefore I am seriously considering changing my own
terminology, so as to write Moe out of the script entirely,
and instead talk about "the redwich coordinate system"
and "the bluewich coordinate system" or something like
that.