In another thread, Bob Sciamanda wrote something that could be interpreted
as equating the 3-velocity to the spatial projection of the 4-velocity.
This is at most only tangentially related to the point he was trying
to make ... but since is is a notorious source of confusion, it is worth
discussing in its own right.
According to the usual definitions, the 4-velocity is defined as
d X / d tau [1]
where X is the 4-vector position. The spatial projection of this
is just
d x / d tau [2]
where small x is the 3-vector position, i.e. the spatial projection
of X.
However, it is conventional to define the actual 3-velocity as
d x / d t [3]
where x is the spatial part of X, and t is the temporal component
of X. Expression [3] is definitely not equivalent to expression
[2]. For example, in the case of acceleration in a straight
line,
d x / d tau goes like sinh(rho)
where rho is the rapidity, and grows without bound. Meanwhile
d x / d t goes like tanh(rho)
and never exceeds the speed of light.
It would be "nice" if the 3-velocity were just the spatial part of
the 4-velocity, but this is not the case, according to well-established
conventions. This drives students nuts, but there are good reasons
for the conventional definitions, and they are not likely to change
anytime soon.
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