Generation of muons due to spatial dilation

Under this post title there is a very interesting topic and it is the generation of muons as secondary radiation due to time dilation or spatial contraction.

Known by all, although it has been seen in movies, is that if you travel at high speeds, close ac, there are phenomena such as temporary dilation (1 second lasts “more” although those who go at that speed do not realize ) and a spatial contraction (1 meter becomes more dwarf, although those who go to those speeds do not realize) all thanks to the special relativity of Einstein.

I, assume from here, that the reader not only has knowledge of it, but a little mathematical idea of ​​what is behind although, obviously, I explain it to you in between.

By basic material physics you will know that the muons, the muon, is an elementary particle (a fermion, go) with spin 1/2 and negative charge. Until here or you understand it or it sounds like Chinese for that reason, for the seconds I’m going to knead the term a bit.

The muon is a particle that was observed in the cosmic radiation (of which I have spoken a lot). But it is a transition particle, that is, a Jeames Dean particle that follows its maxim of “live fast, die young and leave a beautiful corpse”. And is that in the disintegration of particles (come on, when a particle is shattered) the muon generates an electron, a neutrino and an antineutrino. And, because it generates an electron, it can even (during its short life) replace an electron in an atom. If you want to know more, you can ask or simply look for literature (that there is enough) where you will see that he lives very little, something of 2 micro seconds.

N(t)=N_{0}e^{-\frac{t}{\tau }}

Where, as you know, N_{0} is the number of particles in t = 0 and N(t) are the particles that exist at time t with \tau the half-life.

Muons can be at rest or go at great speeds, almost the light (c), in fact to 0’997c but, as they live little, little move. A moving muon does not reach 660m because of its short life (remember James Dean).

Now, as it travels at speeds “practically” that of light, for us, in an inertial system of all life, the life time of a muon in motion is not the life time of the muon itself. By Einstein, time dilates for the inertial system and the 2 micro seconds become 33 microseconds.

If you want to see the math is simple. The time is contracted with the following equation:

t_{2}-t_{1}=\gamma (t_{2}^{'}-t_{1}^{'})

\gamma = \frac{1}{\sqrt{1-\frac {v^2}{c^2}}}

Where \gamma it is the correction factor of the Lorentz transformation and that is something basic. Remember that the Lorentz factor can be calculated using the Galileo transformation as a basis and introducing it as a correction factor for it. It is a very fun exercise to perform knowing that Galileo’s transformation is (for an axis, for example y):


Where, we only have to take into account a system with coordinates (x, y, z) and origin O and another with coordinates (x ‘, y’, z ‘) with origin O’ moving at a speed v with respect to the previous one. If you dedicate yourself to calculate the velocities of a point, its accelerations and you put that, that particle travels at the speed of light (c) you will come to the transformation of Lorentz in a plisplas. It is not complicated but a good experiment.

Returning to the theme of our particle, the muon, in that time will travel, for the inertial system, a pound of meters, 10000. That is, for us, the muon lives longer and travels more, as James Dean’s fans think of his death.

All thanks to the contraction of the lengths that is something simple to see knowing how much $ latex \ gamma $ is and applying it to the distance of an object that is in the direction of the movement of the system. That is, this sounds complicated but, like everything in physics, it is not.

Imagine the same systems that we had before, the O and the O ‘. In the system O ‘we have, in the direction of movement (this is important) something that measures a length L_{p} (it is so called because it is called own length, which is the length measured in that system, nothing else ). Obviously this length will be:


In the other system, length will be:


Now, by Lorentz, Galileo and our friend Einstein, we have, like time, the lengths measured in different inertial systems, traveling one to “all milk” (very fast, it’s a common spanish term, c or close to c):

y^{'}=\gamma (y-vt)

So, if we measure the length at the same instant t and calculate the position of those points in one and another system, we will have.

y_{1}^{'}=\gamma (y_{1}-vt)

y_{2}^{'}=\gamma (y_{2}-vt)

With what remaining for the lengths that we have before.

y_{2}^{'}-y_{1}^{'}=\gamma (y_{2} - y_{1})

L_{p}=\gamma (L)

L = \frac{1}{\gamma} L_{p}

And all this for what?. So simple, if we calculate the number of particles in a classical way and relativistically we will see that the number of muons is not the same.

That is, if we assume a large number of particles (because the decay is exponential) and that we measure it at an altitude of 10000 meters, that is, we imagine that a detector at that height detects us $ latex 10 ^ {8} $ particles, at sea level (0 meters) according to the classic model and, knowing that it takes 33 micro seconds to arrive (for the speed they carry), we would obtain a result, using the first formula of the post:

N(t)=N_{0}e^{-\frac{t}{\tau }}=10^{8}e^{-15}=31

Where t=15 \tau = 33 \mu s that is, 33 micro seconds are (about) 15 periods.

But, oh surprise, if we use the relativistic formula, the muon actually travels (for the, remember that it is in an inertial system) 1 only period since our 10000 meters correspond to 660 meters in its system and therefore in the end to 2 micro seconds and therefore to 1 period, so, using the previous formula.

N(t)=N_{0}e^{-\frac{t}{\tau }}=10^{8}e^{-1}=37*10^{6}

A not inconsiderable amount of 37 million.

This fact has been experimentally seen by Carl Anderson (the accompanying image) in 1937, in an experiment where, at first it was thought that the one who traveled at these speeds towards generating “exponentially” muons when, really, it is a consequence of relativity despite the fact that it was known.

That is, it is not that the muon (like other particles) is generated spontaneously when traveling at high speeds, but that it is simply necessary to take into account the relativistic effects in this (or any) particle.