# Confusion in proper length and how to use the formula of contraction of length

Suppose a muon particle (imagine it does not decay) is moving towards earth with velocity $$v=0.998c$$ from point M. Distance between earth and point M is 10km. Now, I want find the distance covered by the muon particle in its reference frame. We know, $$L_0=\gamma L$$. Here,$$\gamma=1/\sqrt{1-v^2/c^2}$$=15.82. Now which value is 10km? Is it $$L=10$$km or $$L_0=10$$km? I was taught that L0 is the distance in the reference frame where the distance is in. So what should be the value of $$L$$ and $$L_0$$? I am confused. And will the diameter of earth with respect to muon particle reduce too?

The answer depends on how the question should be interpreted: when you say that the "distance between earth and point M is $$10$$ km", are you implying that this is the distance for an "outside observer" or the distance measured by the muon?

Whatever the case, with $$L_0$$ we indicate the so-called proper length, while $$L$$ is the distance observed by an observer in motion relative to the object, the one for which lengths appear to be contracted.

Lastly, this effect is only valid on the direction in which the motion occurs, not on the perpendicular ones, so that the diameter of the earth doesn't change for the muon.

• By "distance between earth and point M is 10 km", I wanted to mean 10km is the distance by an observer staying on earth. My understanding is when I am evaluating the distance wrt muon, proper length is the distance that the muon particle observe (suppose it has an eye). And oppositely when the distance is measured wrt an observer on earth, proper length is what this observer on earth sees. Is there any problem in my understanding? If there's, what it is. And what's the solution of this dilemma? Oct 8, 2022 at 1:34

Proper length (or rest length) is the length measured in an inertial frame in which the endpoints of the experimental setup are constantly at rest. Contracted length is the length measured in an inertial frame in which the endpoints of the experimental are constantly in motion with same velocity.

So the only thing you have to find out is: which frame is the rest frame of the experimental setup. That's easy: The location of the upper endpoint of this experimental setup is designated by the location of the upper atmosphere particles with which cosmic radiation constantly collide in order to constantly produce muons at the same location. And the lower endpoint is the Earth's surface constantly reached by the muons. Since both points are evidently at rest in the rest frame of Earth, the rest length of the experimental setup is measured in the rest frame of the atmosphere and Earth.

In the muon rest frame, on the other hand, the experimental setup and its endpoints (the upper endpoint being the atmospheric particles and the lower endpoint being the Earth's surface defining the experimental setup) evidently are moving at same velocity, therefore the length of the experimental setup is contracted from the muons viewpoint.

The same of course is the case when you conduct muon experiments in an particle accelerator. The endpoints of the experimental setup are obviously defined by the endpoints that are constantly at rest with respect to the particle accelerator (that is, there is a place or endpoint where the muons are constantly created and there is another place or endpoint where they are constantly detected). So you only have to ask yourself: Which frame is the rest frame of the particle accelerator - well, since it is at rest on the Earth's surface, its rest frame is obviously the Earth frame. Then you know that this is the frame in which one measures the rest length between the endpoints of the experimental setup, being the largest length between the endpoints measured in any frame, whereas in any other frame that distance is smaller.

• Are you telling in muons frame, earth is coming towards it and so it is not the rest frame? If it is true, then I want to remind you that we were told to measure the distance which is passed by the muon in its frame. As the earth is moving towards the muon in muon's frame, the distance is changing. If we consider the distance after a very short period of time dt, the distance won't be the distance that we were told to measure. That's why, to me, it seems the distance from muon's and earth's frame to be the same. Can you please tell what's the problem in my understanding? Oct 8, 2022 at 13:05
• Previously, I used to think that only the length of the object contracts due to speed comparable to c. But do the distance really contract too? Will the diameter of earth seem to be contracted in muon's frame than what we measure by sitting on the earth? Oct 8, 2022 at 13:11