However you understand it, SR is fundamenatlly correct.

The lifetime of the muon wiil depend upon its relative speed to the observer.

If fthe observer is going far slower relative to the muon iit wiil appear to have a longer lifespan.

If the observer were going at the same speed as the muon it would appear to have a much shorter lifespan etc.

And a common mistake is to assume there can be a perfectly stationary observer  - not so it all depends on relative motion.

Own life time of muon is defined, but measured lifetime by an observer is relative.  

Andrew,

SR is a theory and it is considered mostly correct in its predictions. Unfortunately the Lorentz Transformations have been pushed to situation where they should not be applieed. In principle with the Tangherlini transformations it is possible to determine all the types of time dilation and in principle the time dilation in circular motion cannot be predicted by LT which is not applicable in that case.

“Own life time of muon is defined, but measured lifetime by an observer is relative” 

-  that is true and is true for any unstable particle. With a small correction, though, relating to the notion “defined” that isn’t defined in the quote – the “own” life time of a particle can be defined only relating to the Matter’s absolute [5]4D Euclidian spacetime, where every, including unstable, particles move uninterruptedly always simultaneously:

-  with 4D speeds  of light c (“c” is c-vector) in the 4D Euclidian sub-spacetime

-  and with the 1D speed  of light in 5-th [“true time”] cttrue dimension.

 At that motion along the 4-th [“coordinate time”] ctcoor dimension is the changing of internal states of the particles [the motion along cttrue axis is every changing of the state internal states and a changing of a spatial position].

 Every particle is some close-loop algorithm that operates with some frequency, ω; if the particle is at absolute 3D spatial rest the frequency is maximal and is ω0=m0c2/ћ what corresponds maximal [=the speed of light] speed along the 4-th axis. If the particle moves in 3D space with a speed V also, its speed Vt in the coordinate time becomes be slowed V’t=(c2-V2)1/2 and the frequency decreases correspondingly in the Lorentz factor. Since a probability of a crack of the algorithm in unstable particles is non-zero in every cycle, the life time increases in the Lorentz factor also.

But, again – all that in the absolute reference frame that is at absolute 3D spatial rest. If some reference frame moves with some speed, the measured particles lifetime depends on relative speed of the frame and the particle and differs from the real value. For example if the frame’s and the particle’s speeds are equal, the measured lifetime will be equal to the “own” one – since clocks’ tick rates in the frame are slowed in the Lorentz factor also. In different frames the measured life times’ values will be different, but seems evident that it is impossible when the some particle simultaneously has a number of the life times and dependently on some sets of clocks and rules that are called “reference frames”, when these clocks and rules have no interactions with the particle; they can be and cannot be …

More see https://www.researchgate.net/publication/273777630_The_Informational_Conception_and_Basic_Physics

Cheers

Article The Informational Conception and Basic Physics

A muon is a type of subatomic particle that is created in particle collisions. More properly called a “mu-meson”, it has a charge equal to that of an electron and a mass 206 times heavier. As such it is often thought of as a heavy electron. Unlike electrons however, muons are short lived and will quickly decay into other particles, typically an electron and some neutrinos. Laboratory experiments show that their average life span (or rather half-life) is 2.2 microseconds. That is, if we start with 1000 muons, after 2 microseconds we would expect around 500 to remain. Then after a further 2 microseconds, 250 will remain, then 125, etc. Muons are also created at high altitudes where the top of the atmosphere gets bombarded by solar and cosmic protons. The generated muons rain down at high speed, some of them decaying partway down and others making it all the way to the ground at sea level. According to our knowledge of decay times, we should expect the average distance they travel before decaying will depend on their initial speed. For example if a muon were going at 50% of light speed, we’d expect it to travel (on average) 300 metres before decaying . Based on this we should expect the number of muons detected at that 300m distance to be half of the original. And at 600m down we'd expect a quarter of the original to remain, at 900m one eighth would remain, etc. We can continue this down to sea level and determine what percentage would survive based on their original velocity. Now it is noted that the number that make it to sea level is somewhat greater than what might be expected. In fact the number that make it the ground is more than should be possible even if muons were assumed to be travelling at light speed (where the average decay distance calculates as 600m). However, if we take into account the predictions of Special Relativity (SR) in which “moving clocks run slow”, we should expect muons moving at high velocity be time-dilated and thus decay more slowly. In this way a muon would move a farther distance before terminating. Most interesting though, is the actual counts of muons do indeed appear to match the amounts predicted by SR. For example, muons going at 0.87c, where the Lorentz factor is 2, travel twice the distance before decaying. And muons going at 0.995c, where the factor is 10, travel 10 times the distance before decaying. This experiment would appear to be very good evidence for SR. Because if we consider a muon to be a clock, it shows that its decay time is directly influenced by its speed relative to Earth and by the degree calculated according to SR. But at the same time this explanation could also not be correct because speed is relative: From the point of view of the atmospheric muon, it is standing still, and muons on the ground are moving. And so the ground muons should be decaying more slowly instead. This point should be immediately obvious but apparently it isn’t. So it’s worth detailing the issue with a ‘thought experiment’ as follows. Choose two locations; one at ground-level and the other at 2000ft. There is an observer at both places, each with a convenient box in hand. We place a flashbulb at the mid-point, i.e. at 1000ft. The flashbulb is lit and gives off a brief pulse. When the observer at the 2000ft altitude sees the flash, he quickly places the box around a group of a hundred muons that just happen to be passing by and in the downward direction at exactly 0.995c. The box is given a quick downward push so that it matches the 0.995c velocity and heads down with the muons, without making contact with any of them. When the observer at the ground-level altitude sees the flash, he quickly places the box around a group of a hundred muons that he finds sitting stationary just above the ground (assume these muons aren’t affected by gravity). The box and muons then stay at that level without moving.

Particle n-body decay width can be written in a Lorentz invariant way,

W = (4π2/h) (|mif|2 / Π 2Ej)  Rn(E)

Where Rn(E) is the Lorentz invariant phase space factor. http://www-pnp.physics.ox.ac.uk/~libby/Teaching/Lecture4.pdf

also see

http://www-pnp.physics.ox.ac.uk/~mjohn/B5.pdf

I propose the following syllogism:

- If the muons in an accelerated system are not affected by the apparent gravity but only by the velocity

- We deduce that the time behaves in the same way

- However much empirical evidence shows that the true gravity affects the time

- It follows that the Equivalence Principle is violated

- And it is that General Relativity is completely based on the principle of equivalence

- We can conclude that in general relativity there is something that does not work

GR works in the case when the EP is not violated infact. It means it works where we are in presence of pure gravitational effects, orbital motion, no exchange of energy and trajectory of EM waves with 0 mass. Wherever there are transitories which imply an infringement of EP GR does not work, or better it is an approximation which rapidly gets bad as the transitory is big.

Claudio,

“…If the muons in an accelerated system are not affected by the apparent gravity but only by the velocity…"

-  that is true;

“…We deduce that the time behaves in the same way…”

-  that isn’t true, time cannot “behave” principally; and

“…However much empirical evidence shows that the true gravity affects the time…”

isn’t true also; again, any material object cannot affect the time.

 Though indeed the rate of internal processes in any object [including objects that show their own such rates, i.e. clocks], which [the rate] in statics is proportional practically to the object’s mass, slows down if the object is in a gravity field – proportionally, correspondingly, to the object’s gravitational mass defect.

At that since total mass defect of a system of, say, two masses is divided equally between masses, this slowing down is twice lesser then the GR predicts.

“…We can conclude that in general relativity there is something that does not work…”

that is again true, but isn’t some unique reason for  such conclusion…

Cheers