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vanhees71 said:robphy said:[snip]
##\gamma \hat u_2=\cosh\theta\ \hat u_2##
and
##\gamma \tilde v=\sinh\theta\ \hat u_{2\bot}##
Thus, the relative velocity (the 3-velocity) is the ratio ##\tilde v=\displaystyle \frac{\sinh\theta \hat u_{2\bot} }{\cosh\theta}=\tanh\theta\hat u_{2\bot}## which is spacelike.
I don't know, what your symbols mean, but it doesn't look right, because ##\vec{v}## is not the spacial part of a four-vector. It's not manifestly covariant but a "three velocity".
All quantities with ##\tilde{\phantom{v}}## (tilde) are 4-vectors.
All quantities with ##^\widehat{\phantom{v}}## (hat) are unit 4-vectors, with square-norm ##1## for timelike and ##-1## for spacelike.
I did not use the arrowhead anywhere. I never wrote ##\vec v##.
There are no explicit 3-vector quantities.
But, in my last post, I am suggesting that any 4-vector constructed with the projection tensor ##h_{ab}## is orthogonal to the ##B_a##~observer and can be identified with a 3-vector for the ##B_a##~observer.
I agree with you that
and I never said that.vanhees71 said:##\vec{v}## is not the spacial part of a four-vector.
From what I wrote above,
$$\gamma \tilde v=\sinh\theta\ \hat u_{2\bot}$$ is the spatial component (spatial part) of a four-vector...
akin to what you wrote
Using your notation, ##\vec v## is ratio ##\frac{\Delta \vec x}{\Delta t}## (like a slope)vanhees71 said:$$u_1=\begin{pmatrix}\gamma \\ \gamma \vec{v} \end{pmatrix}, \quad u_2=\begin{pmatrix} 1 \\ \vec{0} \end{pmatrix}.$$
$$\vec v=\frac{\mbox{spatial part}}{\mbox{temporal part}}=\frac{\gamma \vec v}{\gamma}$$
And, I had formed the same ratio
when I wrote
"Thus, the relative velocity (the 3-velocity) is the ratio $$\tilde v=\displaystyle \frac{\sinh\theta \hat u_{2\bot} }{\cosh\theta}=\tanh\theta\hat u_{2\bot}$$"
So, I think we are saying the same thing... but my notation may be unfamiliar to you.
(I used a similar approach in my PF insight
The Electric Field Seen by an Observer: A Relativistic Calculation with Tensors
which tries to follow the approach in Ch 13 of Geroch's General Relativity lecture notes (draft at http://home.uchicago.edu/~geroch/Course Notes ). The spatial-velocity construction is in Ch 7.
I never liked the 3-vector approach and I never liked the differential approach from old relativity texts to develop relativistic formulas. But when Geroch showed these geometrically-motivated 4-vector methods in class, it was an eye-opener.)