Hi guys, just wanted to ask a question related to muon(adsbygoogle = window.adsbygoogle || []).push({});

experiments (and all other which can be presented this

way) to get some things clearer.

Ok, here goes:

We have lots of muons travelling towards Earth. Their

mean lifetime, measured in lab conditions (at rest) is

2.2µs. Their concentrations (flux) have been measured

at different altitudes, and their speed (0.99c) has

been measured near the Earth surface. By comparing

their concentrations at an altitude of 15km and at sea

level, it has been shown that many more survive than

expected, considering their speed and their mean

lifetime.

SR calculation follows:

Note: I took delta_x (change of height) to be negative,

because it's decreasing, but this is a matter of choice.

Speed of light is c=299792458m/s

Speed of muon is -0.99c = -296794533,4 m/s

Lorentz factor is then: gamma = 7.08881205

EARTH's frame

delta_x: -15km = -15000m (height decreased by 15km)

delta_t: delta_x/v = 50.54µs

It takes 50µs for the muon to travel 15km. v=0.99c.

MUON's frame

delta_t': gamma*(delta_t-(v*delta_x)/(c*c)) = 7.13µs

delta_x': gamma*(delta_x-v*delta_t) = 0m (in this frame, muon is stationary)

distance to Earth at t'=0: delta_x/gamma = -2116m

It takes 7µs for the muon to travel 2km. v=0.99c.

Ok, so far everyhing is as Relativity predicts.

Now the strange part.

What if we started with the fact that it takes 7µs for

the muon to travel 2km at that speed and want to find

out delta_t in Earth's frame? Let's say that muon is

stationary and Earth is travelling towards the muon.

MUON's frame

delta_x: -2.116km = -2116m

delta_t: delta_x/v = 7.13µs

Now we are in muon's frame, and want to find out

the time and distanceEarthneeds to travel inEarth's

frame. We should get 50µs, distance of 0m, but

we should be able to calculate muon's distance also.

Using exactly the same reasoning as when we started,

we get:

EARTH's frame

delta_t': gamma*(delta_t-(v*delta_x)/(c*c)) = 1.01µs

delta_x': gamma*(delta_x-v*delta_t) = 0m (in this frame, Earth is stationary)

distance to muon at t'=0: delta_x/gamma = -298.5m

Shouldn't we be able to get our starting results (50µs, 0m, -15km)?

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# Muon confusion

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