On May 12, 11:21 pm, PD wrote:
On May 12, 6:56 am, Albertito wrote:



On May 11, 10:07 pm, PD wrote:

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

Interpretation under my model and that provided
by SR can't still be experimentally distinguished.

Consider this sum of betas,

v/c = v_p/c + v_mu'/c ,
with
v_p, pion speed at the moment of its decay wrt muon
detector.
v_mu', the relative speed of a muon with respect
to the point where the pion has decayed

Consider this sum of rapidities,

arctanh(w/c) = arctanh(v_p/c) + arctanh(v_mu'/c)

In both sums, neither v_p nor v_m_u' exceed c.

SR assumes the final speed of the muon in the detector
must be

w = c tanh(arctanh(v_p/c) + arctanh(v_mu'/c)),

but it needs also the assumption there is time dilation
of the muon lifetime, as t = t_0 / sqrt(1 - w^2/c^2),
in order to fit the prediction to the observation.

In my model, the final speed of the muon in the detector
is
v = v_p + v_mu',
without the assumption of any time dilation at all.

If it still is not clear to you, I can repeat,

v/c =(v_p + v_mu')/c

And, if you do the math, you find out the left-hand-side of this
formula should be greater than 1, given the amount of kinetic energy
released in pions decaying to muons. Go ahead, plug in the numbers.


What's the problem with finding out the left-hand-side
of this formula is greater than 1? That's only a problem
if you use the relativistic kinetic energy. Of course,
v/c = (v_p + v_mu')/c is greater than 1, because v_p/c
is very close to 1, and v_mu'/c can be greater than
(1 - v_p/c).