# LOG#041. Muons and relativity.

QUESTION: Is the time dilation real or is it an artifact of our current theories?

There are solid arguments why time dilation is not an apparent effect but a macroscopic measurable effect. Today, we are going to discuss the “reality” of time dilation with a well known result:

Muon detection experiments!

Muons are enigmatic elementary particles from the second generation of the Standard Model with the following properties:

1st. They are created in upper atmosphere at altitudes of about 9000 m, when cosmic rays hit the Earth and they are a common secondary product in the showers created by those mysterious yet cosmic rays.
2nd. The average life span is $2\times 10^{-6}s\approx 2ms$
3rd. Typical speed is 0.998c or very close to the speed of light.
So we would expect that they could only travel at most $d=0.998c\times 2 \times 10^{-6}\approx 600m$
However, surprisingly at first sight, they can be observed at ground level! SR provides a beautiful explanation of this fact. In the rest frame S of the Earth, the lifespan of a traveling muon experiences time dilation. Let us define

A) t= half-life of muon with respect to Earth.

B) t’=half-life of muon of the moving muon (in his rest frame S’ in motion with respect to Earth).

C) According to SR, the time dilation means that $t=\gamma t'$, since the S’ frame is moving with respect to the ground, so its ticks are “longer” than those on Earth.

A typical dilation factor $\gamma$ for the muon is about 15-100, although the value it is quite variable from the observed muons. For instance, if the muon has $v=0.998c$ then $\gamma \approx 15$. Thus, in the Earth’s reference frame, a typical muon lives about 2×15=30ms, and it travels respect to Earth a distance

$d'=0.998c\times 30ms\approx 9000m$.

If the gamma factor is bigger, the distance d’ grows and so, we can detect muons on the ground, as we do observe indeed!

Remark:  In the traveling muon’s reference frame, it is at rest and the Earth is rushing up to meet it at 0.998c. The distance between it and the Earth thus is shorter than 9000m by length contraction. With respect to the muon, this distance is therefore 9000m/15 = 600m.

An alternative calculation, with approximate numbers:

Suppose muons decay into other particles with half-life of about 0.000001sec. Cosmic ray muons have speed now about v = 0.99995 c.
Without special relativity, muon would travel

$d= 0.99995 \times 300000 km/s\times 0.00000156s=0.47 km$ only!

Few would reach earth’s surface in that case. It we use special relativity, then plugging the corresponding gamma for $v=0.99995c$, i.e.,  $\gamma =100$, then muons’ “tics” run slower and muons live 100 times longer. Then, the traveled distance becomes

$d'=100\times 0.9995\times 300000000 m/s\times 0.000001s= 30000m$

Conclusion: a lot of muons reach the earth’s surface. And we can detect them! For instance, with the detectors on colliders, the cosmic rays detectors, and some other simpler tools.

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