Re: Muon paradox

Den 06.04.2025 15:15, skrev Paul.B.Andersen:

----------------------------
 I will post it again. Please read it this time.
And make one reply, where you quote what you respond to.
If you think something is wrong specify exactly what, and quote
what you think is wrong.
 | "Time dilation" is predicted by SR.
|
| Let's see how:
|
| As you know, a muon may decay at any time. (it's radioactive decay)
| The proper mean lifetime is 2.2 μs. This means that if we know
| the muon exists, then the probability that it still will exist
| after 1 μs is exp(-1e-6/2.2e-6) = 0.63. That means that the probability
| that the muon will decay after less than 1 μs  is 0.27.
|
| We will define that the muon we are going to analyse decays exactly
| τ₀ = 1 μs after its creation, measured in the muon's rest frame.
| The speed of the muon in the Earth frame is v = 0.999776⋅c, γ = 47.3
 A comment:
A vast number of muons are created every second,
so there will always be some muons which have
a proper lifetime close to 1 μs.
 |
| We will use two frames of references:
| The muon frame K(x,t) and the Earth-frame K'(x',c').
| The muon frame is moving at the speed v relative to the Earth-frame.
| The muon is created 15 km above ground, we will set this event to
| have the coordinates in K': x₀'= 0, t₀'= 0
| and the coordinates in  K:  x₀ = 0, t₀ = 0
| The muon is stationary at x₀ = 0
|
| event creation:  t₀'= 0, t₀ = 0
|
| K':------|--------------------------------|----> x'
|     x₀'= 0                                15 km (ground)
|
| K: ------M------------------------------------> x  v->
|     x₀ = 0
|
| ================================
|
| event decay:
|
| K':------|-------------------------|------|----> x'
|     x₀'= 0                    x₁'= L      15 km (ground)
|
| K: --------------------------------M-----------> x  v->
|                               x₁ = 0
|
| When the muon decays, the coordinates  are
| In K:  x₁ = 0,  t₁ = τ₀
| In K':
|  L = x₁'= γ(x₁ + v⋅t₁) = γ⋅v⋅τ₀
|  t₁'= γ(t₁ + v⋅x₁/c²) =  γ⋅τ₀
|
| Putting in numbers:
| L = γ⋅v⋅τ₀ = 14.177 km (the muon will decay before it hits the ground)
| t₁'= γ⋅τ₀ = 47.3 μs
| t₁ = τ₀ = 1 μs
|
| Conclusion:
| In this scenario, there is but one single muon with one single life.
| The muon lives from its creation to its decay.
| This single life is measured to last 1 μs in the muon frame K,
| and is measured to last 47.3 μs in the Earth-frame K'.
|
| Time dilation is the phenomenon that the measured time
| between two events on an object's world-line depend
| on the frame of reference in which it is measured.
|
| So this is "time dilation" by definition.
|
| And as you know, the phenomenon "time dilation"
| is thoroughly proven to exist in the real world.
|
| It simply is how Mother Nature works.
 

I have now realised that Laurence Clark Crossen is too
ignorant of elementary physics and mathematics to understand
anything of the above.
So he will keep claiming that what he doesn't understand must
be nonsense even if it is experimentally confirmed.
Case closed.
--
Paul
https://paulba.no/