About the Muons Experiment

There are three events in this case:

A – The moving muons reach the height of 10Km (Earth frame)

B – The stationary muons are created on Earth surface.

D – The moving muons reach Earth surface.

We will use x and t as the position and time coordinates of Earth, and x' and t' as position and time of the moving muons. We choose the origin of our coordinates system at A, so that x_A = x'_A = t_A = t'_A = 0. We will use units of Km for distance and seconds for time.

c = 3x10⁵

β = v/c = 0.98

γ = 5

We have seen in the original experiment explanation that:

x_D = 10,  t_D = 3.4x10^-5

x'_D = 0,  t'_D = t_D/γ = 6.8x10^-6

B is synchronized with A on Earth frame and is on Earth surface so:

x_B = 10,   t_B = 0

x'_B =  x_B/γ = 2

We can calculate:

ct'_B = γ(ct_B – βx_B) = -5 x 0.98 x 10 = -49

t'_B = -49 / 300000 = -1.633x10^-4

The half life is always measured in the rest frame of the muons, so what we need is the time interval between events B and D in earth frame. From Earth frame we can directly measure (we switch to microseconds here):

t_D – t_B = 34

In the moving muons' frame:

t'_D – t'_B = 6.8 + 163.3 = 170

But the muons know that the time interval of Earth is factored by gamma, so the interval between the events in Earth frame must be 170/5 = 34

So, it is agreed by both frames that in earth frame 34μs pass from the creation of the stationary muons to the meeting with the moving muons. This means that the 49000 moving muons meet only 1/3 stationary muon.

Here is a schematic sketch of the three events in time-space coordinates of both frames. I could not draw it in real proportions because then the lines would be too close to each other.