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From: "Paul.B.Andersen" <relativity@paulba.no>
Newsgroups: sci.physics.relativity
Subject: Re: Muon paradox
Date: Sat, 5 Apr 2025 20:57:33 +0200
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Den 04.04.2025 23:47, skrev LaurenceClarkCrossen:
> "AI Overview
> While muons, formed at high altitudes from cosmic ray interactions, are
> subject to Earth's gravity, their incredibly high speeds and short
> lifespans, coupled with relativistic effects, mean that gravity's
> influence on their trajectories is negligible." -Google AI

Of course the effect of gravitation is negligible.

Muons are created by collisions between cosmic rays at ~15 km,
and will after the creation have an initial energy ~ 6 GeV
which is equivalent to a speed ~ 0.999845g1 ⋅c.

Since the muon is moving at high speed through air, it will loose
energy and have an energy ~ 4 GeV when it reaches the ground.
This is equivalent to a speed ~ 0.999651⋅c.

The average energy is ~ 5 GeV equivalent to a speed ~ 0.999776⋅c

> 
> Considering that time dilation is allegedly caused by acceleration in
> GR, how can this negligible amount of acceleration cause the amount of
> time dilation of the muons?

Since gravitation is negligible there is no reason to invoke GR.

"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

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 = 14.178 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.


-- 
Paul

https://paulba.no/