Accuracy of confirmation of relativistic velocity addition Can you tell me to what accuracy the relativistic velocity addition formula has been confirmed? And in what experiment? This is just between matter and matter.
I've asked many people, they don't answer the question, which is how accurately the formula has been confirmed - directly, not inferring the result via another area of SR. Perhaps D McKee can help. I know particle physics is the area where the experiments have been done, with pions, kaons, muons.
I'm often told that it has been confirmed, for instance, by one pion experiment, where it was shown that c + 0.99975 c = c. Many say that c + v has been refuted, but that's well known and doesn't answer the question.
I've also looked through pages that list the tests of SR, and read papers on it, but no answer appears. Perhaps you can help, thank you very much.
 A: The pion experiment you hint at is most surely 
T. Alväger, F.J.M. Farley, J. Kjellman, L. Wallin, Test of the second postulate of special relativity in the GeV region, 
Physics Letters, Volume 12, Issue 3, 1964, Pages 260-262
http://dx.doi.org/10.1016/0031-9163(64)91095-9
The process $\pi^- \to \mu^- + \bar{\nu}_\mu$ where both the pion and the muon are relativistic is a perfect case to study "v(matter/other matter) + v(other matter/lab)". This is actually used to produce beams of neutrinos but this has also been routinely observed in particle physics experiments for 50 years. I don't think a dedicated study has been done to confirm the relativistic addition of speeds though. The issue is that the detectors used in particle physics measure the particle energy (calorimeters) or momentum (using a track detector in a magnetic field). So at best the data would show that special relativistic energy and momentum are conserved, and therefore that they are the correct ones, not the Newtonian ones.
As for neutrino beamlines, e.g. T2K, using $\pi^- \to \mu^- + \bar{\nu}_\mu$ as I wrote above, or the conjugate process $\pi^+ \to \mu^+ + \nu_\mu$, I bet their monitoring data would make it possible to test that addition law but nobody bothered as far as I can tell. I could try to see whether I can use some old contacts…
I feel it is important to add a theoretical note here. It has been known for more than one century that a velocity addition law can only have the form
$$ V' = \dfrac{v + V}{1 + k v V}\ \ \ \ (I)$$
where $k$ is an unknown constant. Most importantly this does not rely on any postulate about light speed, or any hypothesis about any physical phenomenon. This is just a straightforward consequence of the mathematics of inertial frames. Indeed, first the velocity addition law shall come from the transformation of coordinates from one frame $R$ to another frame $R'$, and then such transformations can only take the form
$$\begin{aligned} 
t' &= g(v)(t - k v x)\\
x' &= g(v)(x - vt)
\end{aligned}$$
where $v$ is the speed of $R'$ with respect to $R$, and $g(v)$ is a function of that speed, arbitrary at this stage. The only hypotheses required to get his result are the very definition of the speed $v$, and then the fact that performing the transformation from $R$ to $R'$ and then from $R'$ to a third frame $R''$ shall result in the transformation from $R$ to $R''$. Basically, this results come for free because there would be no way to do any kinematics without these assumptions. H.Rothe and P.Frank[1] were the first to develop such an idea as far as I know (beware the article is in German!). Lee and Kalotas[2] wrote a particularly elegant and modern variant of this kind of demonstration. It is so simple that I could easily reproduce the crux of it here if you are interested.
Now the point I am driving at is that such an addition law as (I) only needs to be tested for $V$ equal to the speed of light to prove that $k$ must be equal to $\frac{1}{c^2}$. Rephrased differently, it would be impossible for the addition law to work for one range of speeds and/or particles but not for another range and/or particles. For that to happen, we would need to throw away the very meaning of inertial frame of reference.
[1] H. Rothe and P. Frank, Über die Transformation der Raumzeitkoordinaten von ruhenden auf bewegte Systeme, Annalen der Physik (Leipzig) 34 (1911), 825--853
[2] A. R. Lee and T. M. Kalotas, Lorentz transformations from the first postulate, American Journal of Physics 43 (1975), 434--437
