Re: Muon paradox

Paul.B.Andersen:

Den 31.03.2025 22:40, skrev LaurenceClarkCrossen:

Do muons move at a different velocity in the laboratory than in the
atmosphere?
>
"No, muons generally do not move at a different velocity in a laboratory
setting compared to their velocity in the atmosphere; they both travel
at speeds very close to the speed of light, typically around 99.8% of
the speed of light" - Google search AI.
>
Then why would they "time dilate" in the atmosphere?

 
The speed of muons is v = ~ 0.999668⋅c through the atmosphere
which also is within the laboratory with open roof.
γ = 38.8.
 
The mean proper lifetime of a muon is t₀ = 2.2 μs.
But measured in the Earth's rest frame the lifetime of the muon
is  tₑ = 2.2e-6⋅γ s = 85.36 μs (time dilation!).
 
Since muons are created at a height ~15 km, and the time for
a muon to reach the earth is t = 15e3/v = 5.005 s,
then the part of the muon flux that will reach the Earth is
 N/N₀ = exp(-t/tₑ) = 0.556, so 55.6% of the muons would reach the Earth.
 
If the lifetime of the muons had been 2.2 μs, then the part of
the muon flux that will reach the Earth would be:
 N/N₀ = exp(-t/t₀) = 1.32e-10.
So only 0.0000000132% of the muons would reach the Earth.
 
Can toy guess which of them is closest to what is observed?
 

@PaulBAndersen

There is one flaw I find in the SR explanation, can you confirm if it is
true:

What is really measured are these (the facts):

1. The mean proper lifetime of a muon is t₀ = 2.2 μs.
2. muons are created at a height ~15 km
3. The speed of the muons is ~c, so travel time is ~50.05 μs
4. muon flux measured on the Earth's surface is about 55.6% of what it
is at 15km.

From 1, 2 and 3, the expected muon flux on the Earth's surface is:

N/N₀ = exp(-t/t₀) = exp(-50.05/2.2) = 1.32e-10 = 0.0000000132%

The important point (the flaw) is that the speed of the muon has not
actually been measured to be 0.999668⋅c, but instead is computed.

N/N₀ = exp(-t/γt₀) = .556 => γ = 38.8 => v = 0.999668⋅c

The SR explanation would have been more convincing, if the speed had
actually been measured to that many significant figures.