What do Einstein and special relativity have to do with cosmic rays?
The behaviour of muons in secondary cosmic rays is one of the best ways to approach special relativity, which is one of the cornerstones of contemporary physics.
Muons are unstable, i.e. they spontaneously decay into other particles. Measuring their mean-life (τ) in the laboratory is relatively straightforward: muons are slowed down in a block of material and then one waits until an electron appears, which is the by-product and signal of their decay. The statistical result of the measurements is τ = 0.0000021969811 s. On the other hand, the velocity of a particle with mass -such as the muon- has as an unreachable limit the speed of light in vacuum c = 299 792 458 m/s -this is one of the fundamental results of special relativity, with an impressive empirical support-. Experimental data further indicate that the average velocity of cosmic muons near their birthplace is more than 99% of the speed of light and that it does not change much as they approach the surface.
This means that, on average, they will not be able to travel (1) more than about: c · τ ≈ 300 000 000 m/s · 0.000002 s ≈ 600 m. By definition of the half-life (a concept closely related to the mean-life, whose value is slightly smaller), every 600 m on average half of the surviving muons should disintegrate: after the initial 600 m, ½ of the muons that were there at the beginning remain, after 1200 m, only ½ × ½ = ¼ and so on. After 25 stages of 600 m, it turns out that very few of the muons created at 15 km should reach the surface! And yet, there is a flux of about 1 muon per cm² per minute.
It is true that we are only using averages, so some muons will be created further down or have longer lifetimes. However, the result is that, due to muon decay, the surface of the Earth should only be reached by a fraction (½)25 of the original number, that is, one muon per one thousand million of those created in the upper atmosphere. On the contrary, the experimental result is that as many as one in twenty survive!
The explanation lies in the time dilation of special relativity, according to which time in moving clocks - and by clock we mean any system that measures the passage of time - passes at a slower rate than that measured in clocks at rest. This effect grows with the particle velocity.
A value of τ ≈ 0.000002 s has been obtained for the mean-life of muons at rest.
Now, when making measurements with cosmic muons, we are looking at particles that have a typical energy of several GeV and move relative to us at about 99.98% of the speed of light in a vacuum. For that value, special relativity says that we will observe that the muons' "internal clock", the one that “tells them when to decay”, is about 50 times slower than the one in the laboratory. That is, we will measure for muons at these typical energies a mean-life of Ƭ ≈ 50 τ (and more at higher energies), which implies that the particles will be able to travel distances about 50 times longer, enough for many muons to survive the tens of kilometres to the Earth's surface.
Of course, the point of view of the muon - which sees itself at rest and would measure τ ≈ 0.000002 s - is equally valid as ours, so for it it will be our clocks that go slow.
A good reference is this classic didactic experiment in which muon fluxes are measured at different altitudes: Time Dilation : An Experiment With Mu - Mesons (1962).
And do not miss:
Impossible Muons by Minute Physics