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From: "Paul.B.Andersen" <relativity@paulba.no>
Newsgroups: sci.physics.relativity
Subject: Re: Muon paradox
Date: Tue, 1 Apr 2025 19:56:05 +0200
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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?

-- 
Paul

https://paulba.no/