On May 9, 6:44*am, Albertito wrote:
Let us consider the simple case of addition of velocities
along a straight line. The incompleteness of SR resides
in the fact that a speed v can't currently be experimentally
distinguished from its rapidity r = arctanh(v/c), for values
of that beta = v/c *below the third-order term approximation.
The power series expansion of r = arctanh(v/c) is

* * * * * r = v/c + v*3/3c^3+ v^5/5c^5 + v^7/7c^7 + ...

Provide references of any experimental test of SR, showing
that the rapidity r can be distinguished from its beta v/c, beyond
its second-order approximation. Prove at least that the third-order
term v*3/3c^3 lives outside the error bars. Since, we can't *still
perform such accurate experimental tests, we must conclude
the addition of velocities still remains within the euclidean sum
of vectors

* * * * * * * *w = u + v.

That sum can't still be experimentally distinguished from the sum
of rapidities

* * * * * * *arctanh(w/c) = arctanh(u/c) + arctanh(v/c).

In addition, we must also conclude that the relativistic Doppler

* * * * * * * *f' = Exp(-r) f ,
* * * * * * * *where r = arctanh(v/c),

can't still be experimentally distinguished from this one

* * * * * * * *f' = Exp(-v/c) f

There is actually an abundance of these tests. I'll mention one.
Muon beamlines are created by allowing charged pions to decay in
flight. The pions have momenta such that their speed is close to that
of light. Since it is an exoenergetic decay, the muon has extra
kinetic energy from the decay. Since the pion's decay mechanism
doesn't give a whit whether the pion is in flight when it decays (and
in fact, the principle of relativity says the physics of the pion
decay has to be the same for pions at rest vs. pions in free flight),
we can guess what that kinetic energy profile is for the moving pions
by using the distribution for decaying pions at rest. Or, put another
way, we can take the velocity distribution of muons in the pion rest
frame and boost them to the frame where the pions are moving close to
the speed of light. Using rapidities to perform that boost results in
a velocity distribution that agrees with direct measurement. The
Galilean transformations, on the other hand, predict that the muon
velocity would have a peak in the forward direction that exceeds c.
Muon time of flight can be measured directly via a triplet of
scintillator paddles, and in fact this is routinely done in muon
beamlines. The muon speed distribution never exceeds c.

PD