Muon confusion

[psmitty]

Hi guys, just wanted to ask a question related to muon
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 distance Earth needs to travel in Earth'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)?

[psmitty]

Ok, nevermind, I found the answer elsewhere. I made a mistake in my calculations:

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.

But distance to Earth at t'=0 (2km) is not the total distance between Earth and muon, because (which gets clearer from the Minkowski diagram) Earth in muon's frame starts to travel before t'=0.

To get the distance to Earth in muon's frame, I should have used delta_x=0 (because Earth is not moving in its frame), and then delta_x' would be 106km (which is the total distance to Earth in muons frame).

2km is the distance at t'=0, which Earth travels for the last 7µs of its trip.

Thanks anyway! Cheers!