I Length Contraction: Muon vs Earth Frames

[TheWonderer1]

http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/muon.html
https://newt.phys.unsw.edu.au/einsteinlight/jw/module4_time_dilation.htm#equivalence

I was looking at these links and got confused. It seems to me that they are saying the same thing but the UNSW link uses the clock in the other person's view. Thus, the time dilation seen in Jasper's Frame would require length contraction to compensate (only using one to keep things in perspective).

For the muon experiment, it experiences length contraction in it's rest frame which is expected.

What I'm confused about is the UNSW article seems to contradict the muon experiment. The Earth in its own frame doesn't have length contraction. I see this as similar to Jasper looking on as Zoe drives by in his rest frame.

It seems that the frames are getting confused. Although, I've seen it written like this before so why doesn't the Earth see the muon contraction. Let's say that the muon was a bowling ball and it was traveling at the same speed, would you see the length contract?

 

[Nugatory]

I currently understand that it works like this. If you are on Earth and a rocket ship leaves, your clock will show 3 secs in 9 Km and you will see a transmission saying the rocket went 1 sec but only 3 Km. In order for the velocity to stay the same you need to change the length.

Length contraction is not involved in that example, which is all about relativity of simultaneity. Both observers agree that the rocket clock reads one second when the distance between rocket and Earth is three kilometers and and three seconds when the distance between Earth and rocket is nine kilometers (and that the rocket is moving at only three km/sec, so the relativistic effects will be very small - the speed of light is 100,000 times greater).

However, because of the relativity of simultaneity they will not agree that the events "rocket is three kilometers away and rocket clock reads one second" happened at the same time as the event "earth clock reads one second". The rocket observer will find that the first of these events happened after the second (the Earth clock is slow relative to his own) while the Earth observer will find that the second event happened after the first (the rocket clock is slow relative to his own).

It is essential to nail down your understanding of the Lorentz transformations and relativity of simultaneity before you will be able to understand things in terms of length contraction and time dilation.

 

[Nugatory]

Although, I've seen it written like this before so why doesn't the Earth see the muon contraction. Let's say that the muon was a bowling ball and it was traveling at the same speed, would you see the length contract?

Using a frame in which the Earth is at rest, the bowling ball does contract but this has a negligible effect on the time it takes the bowling ball to reach the surface - it still has to pass through roughly 100 kilometers of air.

The best way to understand the muon measurements are:
1) As viewed from the Earth (using a frame in which the Earth and it atmosphere are at rest), the muon covers the uncontracted distance from top of atmosphere to Earth's surface. Divide the distance covered by the speed and you get the flight time, which is greater than the muon lifetime in this frame. However, the muon lives to make it to the surface of the Earth because its clock is running slow relative to the Earth clock.
2) As viewed from the muon (using a frame in which the muon is at rest and the Earth is rushing towards it) the muon's clock is ticking at its normal rate of one second per second. However, the thickness of the Earth's atmosphere is length contracted, so when we divide the distance covered by the speed, we find it takes less time for the surface of the Earth to cover that distance and reach the muon. Again, the muon survives.

 

[TheWonderer1]

Using a frame in which the Earth is at rest, the bowling ball does contract but this has a negligible effect on the time it takes the bowling ball to reach the surface - it still has to pass through roughly 100 kilometers of air.

The best way to understand the muon measurements are:
1) As viewed from the Earth (using a frame in which the Earth and it atmosphere are at rest), the muon covers the uncontracted distance from top of atmosphere to Earth's surface. Divide the distance covered by the speed and you get the flight time, which is greater than the muon lifetime in this frame. However, the muon lives to make it to the surface of the Earth because its clock is running slow relative to the Earth clock.
2) As viewed from the muon (using a frame in which the muon is at rest and the Earth is rushing towards it) the muon's clock is ticking at its normal rate of one second per second. However, the thickness of the Earth's atmosphere is length contracted, so when we divide the distance covered by the speed, we find it takes less time for the surface of the Earth to cover that distance and reach the muon. Again, the muon survives.

Ok this is very helpful. Also, I deleted the rocket bit bc I thought about it and what I wrote made no sense.

I do have a question: If a clock is going 1 seconds per 3 second, does this cause a length contraction? I want to be sure I get the basics. From the UNSW article, a clock moving more slowly would mean a shorter length.

By the way, I read the article carefully (it is changing frames). It is taking the information from the other person's clock and making length changes to keep the same velocity in all frames of reference. This might explain a few things and why I was confused.