06 Muon Paradox

   Today, I would like to talk about the muon paradox. There are several subatomic particles that are smaller than atoms, such as a μ-meson. It is called a μ-meson, but it is abbreviated to muon. The lifetime of this muon is fixed. The proper lifetime of this particle is 2.2 μsec. It only survives for a very short moment, and then  it collapses into other particles.

Cosmic-rays are constantly coming out of space due to the explosion of stars in the universe or the nuclear fusion reaction of the sun. At this time, the cosmic-ray and the earth's atmosphere react to create a muon. Muon's speed is near the speed of light when it falls to Earth. It drops at a rate of about 0.998c. However, even if it falls so fast, it cannot reach the ground because its life is short.

If the muon has a lifetime of 2.2 μ sec and a speed of 0.998 c, the muon can run about 650 m from the point where it was created. The point where the usual muon occurs is about 10km above sea level. If so, the muon should never be found theoretically at sea level. However, muon particles are actually found at sea level. What we cannot find is actually discovered, so we call this a ‘muon paradox’.

Figure 1. Overview of muon paradox

In fact, the muon paradox has already been released halfway. It can be explained nicely by interpreting it as a time dilation. Since the muon runs at 0.998c, the Lorentz factor γ is about 30. If so, the lifetime of muon is lengthened by about 30 times. If it does, it will be able to reach sea level easily enough because it runs more than 30 times longer. It's a great success.

The problem appears when interpreting this phenomenon as a length contraction. The effects of relativity always occur together. (time dilation and length contraction) Let's interpret this phenomenon once in length contraction. If interpreted as a length contraction, the muon will have to run a shorter distance than before, and can never reach sea level. So, it is not explained. So, no one could explain the phenomenon of sea level reaching the muon using length contraction. At that time, Banesh Hoffmann courageously explains this phenomenon in his book "Relativity and Its Roots": 1)

Figure 2. The solution of muon paradox claimed by Banesh Hoffmann

He explains that when you look at the Earth from the muon's perspective, the muon is relatively stationary and the Earth rushes toward the muon at 0.998c. So, the length of the earth is about 30 times flattened, and the distance is shorter. He explains that muon can reach sea level easily enough because of the short distance. These explanations also appear in high school students' textbooks these days.
 

At this point, I cannot help myself. In the above explanation, we observed from the earth when describing this using time dilation, and observed from muon when explaining using length contraction. Why did he do that? What is the reason for you to change the position of the observer during the description? What is the scientific basis?
 

Suppose that observing the earth from the perspective of the muon, solves the problem. If so, present a photograph of the flattened earth when observed in the muon. It is not necessarily a photograph. Present any evidence that the earth is shrinking like a pen cake and rushing toward the muon. When I asked for it, someone who was a physics expert asked me:

Do you believe that you must see it with your own eyes?

How do you recognize physical phenomena without seeing them? How do you accept it without proof? It is not a piece of literature or a novel. It is common knowledge to be recognized as a legitimate theory or fact only if it is logically correct and experimentally verified. To claim 'from the muon's point of view, the earth will shrink', you must first present the evidence. You should not claim this before. Physics is science, not faith.

If length contraction theory cannot solve this paradox, you might think that length expansion theory is the same. No. The theory of length expansion has solid evidence. Muon's sea level reach was observed by David and James on Earth. And they published it in paper. The evidence is the following paper.

Frisch, David H. and Smith, James H., “Measurement of the Relativistic Time Dilation Using μ-Mesons”, Am. J. Phys., 32. 342.

Did David and James observe the muon on Earth? Or did they ride the muon and observe the earth? They naturally stood on the earth on both feet and observed muons running at 0.998c. If so, the problem has already been solved. Originally it has to run 650m, but Muon's flight has expanded to 10km. Theoretically, the muon cannot be reached, but it is obviously expanding because it has actually arrived. Thus, the sea surface arrival of muon particles is evidence of length expansion, not evidence of length contraction.

Figure 3. Three theories about muon paradox. Because David and James are people who live on Earth, this paradox must be explained from the perspective of the earth. 

(1) Analysis of length contraction theory 

(2) Analysis of classical theory 

(3) Analysis of length expansion theory

Logically, I will yield one more time. From the muon's perspective, let's assume that the Earth ran at 0.998c. There is one thing Hoffman has missed decisively. Muon is falling to earth at this moment. Muon is flying to Earth from various directions. As Hoffmann says, let's assume that the muon is now approaching from the Arctic. If so, the Earth will have to run towards the Arctic and toward the Muon at 0.998c. At this time, muon is approaching from the Antarctic side at the same time. If so, should the Earth run to the North Pole? Or should I run to Antarctica? If neither of these two cases is happening, will the Earth just explode?

Figure 4. Muon particles approaching simultaneously in the Antarctic and the Arctic and collapsing earth

The earth did not split because of muons approaching on both sides. We are still alive and living. This fact proves that the theory of length contraction is not correct. Thus, the muon paradox must be interpreted as a length expansion, not a length contraction.
 

This paradox is associated with several topics: 'Relations with twin paradoxes and muon paradoxes', 'Relations with companion rules', 'Muon in accelerator', 'GPS and muon paradox'. These topics will be discussed later.

There are those who claim that the earth shrinks when observed from the muon's standpoint. I will tell them once again. If you really think so, present a photo or video of the Earth's contraction or experimental evidence equivalent to that. Before presenting the evidence, you should not say "Earth has contracted."

Let's summarize one more time. If you claim that Muon's sea level reach is evidence of length contraction, you should answer a few questions below.

1. If the muon's sea level reach is evidence of length contraction, present experimental evidence (a picture of a flat earth like a pen cake).

2. I will present clear evidence of expansion of the length. The evidence is below.

David and Smith and James apparently observed muons on Earth.

Frisch, David H. and Smith, James H., “Measurement of the Relativistic Time Dilation Using μ-Mesons”, Am. J. Phys., 32. 342.
3. If a muon approaches the Antarctic and the Arctic at the same time, should the Earth collapse?

Lehgth Expansion(Amazon books)