On May 13, 4:53*am, Albertito wrote:
On May 13, 10:04 am, PD wrote:



On May 13, 3:31 am, Albertito wrote:

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?

Well, for one thing, it doesn't happen. As I mentioned to you, the
time of flight of muons is *measured*. We know what the speed is. It's
not greater than 1.

That's only a problem
if you use the relativistic kinetic energy.

No, sir. It's a problem with measurement.

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).

You can't keep track of single muons to measure their
'time of flight'. All you can do is to perform statistics and
extrapolate to single muons. You can't say a muon called
A was emitted by a pion called P at time t=0, and then A
was detected at distance R at time t'.

Of course you can. You didn't read what I wrote. You time a muon in
flight the same way you could a car on the highway: you time its
crossings on successive, spaced gates. (Note you don't have to follow
the car all the way from the garage to do so.) You make a scintillator
paddle triplet, with each paddle separated by some appreciable
difference (say, 20m), and then the signals from each of the
scintillators to an o'scope or a TDC by an equal length cable. 20 m
would be about 60 ns if the speed were as high as c, trivial to
measure by either of the devices mentioned. The presence of the MIP
signal in all three paddles assures that you are seeing the passage of
the muon. You measure the distance, and you measure the time it takes
the muon to cover that distance. You have a direct measurement of the
velocity. It does not exceed c. *Measured*.

It might do you good to go into the laboratory to see for yourself,
rather than just jacking around with piddly little ideas.

PD