Solving the Shorter Distance Puzzle with Spacetime

[Devil Moo]

I see an example from book stated that a muon that are created at high altitudes can travel to the ground because of time dilation about its lifetime. In the view of muon, it travels shorter distance compared with the view of the Earth.

Suppose there are two muons. In view of muon A, muon B is moving to it. And vice versa.
Is there still shorter distance?
I am really confused with the shorter distance.

One of the reasons is length contraction is a reciprocal effect. And length of an object is supposed to measure two points at one instant.
For the distance, it is not a reciprocal effect in the example above. And distance is measured by two points at different time.

To find out "Is there still shorter distance?", how do I solve it?
Using Spacetime?

 

[Orodruin]

In the view of muon, it travels shorter distance compared with the view of the Earth.

First order of business in understanding relativity is to get the setups straight. In the rest frame of the muon, the muon does not travel, the Earth does.

Second, length contraction, as you have realized, requires the notion of something being of a fixed size in a given frame, with respect to which it is at rest. To understand things in a better fashion, I recommend that you instead look at the full set of Lorentz transformations.

 

[Devil Moo]

For the Earth and muon example, it changes the Earth to muon.
Both of them have lifetime 2.2us at their own rest frame.
Suppose they move 0.98c.
In view of muon A, muon B moves to A in 0.98c.
In view of muon B, muon A moves to B in 0.98c.
Will they have shorter distance?
When does the shorten distance exist? In what situation?

 

[Nugatory]

Suppose there are two muons. In view of muon A, muon B is moving to it. And vice versa.
Is there still shorter distance?
I am really confused with the shorter distance.

In the case of the atmospheric muons, the distance that we're talking about is the distance from the top of the atmosphere where the muon is created to the surface of the earth. The top of the atmosphere and the surface of the Earth are at rest relative to an observer on the surface of the Earth so the distance between them is not contracted according to that observer. The top of the atmosphere and the surface of the Earth is moving relative to the inbound muon, so an observer traveling along with that muon will find that the distance between them is contracted.

If you just have two muons traveling towards one another, both will report the same distance between them. There is no length contraction because they're talking about the distance between two points that are are at rest relative to them: "Where I am, which isn't moving because I'm sitting still while the other guy is rushing towards me" and "Where the other guy was at the moment that I measured the distance, and that point isn't moving relative to me even though the other guy moved away from it even as I was making the measurement".

 

[Devil Moo]

In the case of the atmospheric muons, the distance that we're talking about is the distance from the top of the atmosphere where the muon is created to the surface of the earth. The top of the atmosphere and the surface of the Earth are at rest relative to an observer on the surface of the Earth so the distance between them is not contracted according to that observer. The top of the atmosphere and the surface of the Earth is moving relative to the inbound muon, so an observer traveling along with that muon will find that the distance between them is contracted.

If you just have two muons traveling towards one another, both will report the same distance between them. There is no length contraction because they're talking about the distance between two points that are are at rest relative to them: "Where I am, which isn't moving because I'm sitting still while the other guy is rushing towards me" and "Where the other guy was at the moment that I measured the distance, and that point isn't moving relative to me even though the other guy moved away from it even as I was making the measurement".

Is it difficult know that?
Especially for the space, how do I know the distance between two points is shorten or not?

 

[A.T.]

how do I know the distance between two points is shorten or not?

You apply the Lorentz transformation.

 

[Devil Moo]

Find delta x and delta x' ?

 

[Nugatory]

Is it difficult know that?
Especially for the space, how do I know the distance between two points is shorten or not?

Length contraction is always a comparison of measurements by different observers in motion relative to one another. If they come up with different lengths when they measure, then we say that the shorter measurement is contracted.

 

[Devil Moo]

Actually, I understand the length contraction but I am confused with the shorten distance.
Because the length contraction and time dilation are a reciprocal effect.
But that's not.

 

[Vanadium 50]

I understand the length contraction but I am confused with the shorten distance

I think there is a language problem. They mean the same thing.

 

[Mentz114]

First order of business in understanding relativity is to get the setups straight. In the rest frame of the muon, the muon does not travel, the Earth does.
Second, length contraction, as you have realized, requires the notion of something being of a fixed size in a given frame, with respect to which it is at rest. To understand things in a better fashion, I recommend that you instead look at the full set of Lorentz transformations.

@Devil Moo, pay attention to this.

Actually, I understand the length contraction but I am confused with the shorten distance.
Because the length contraction and time dilation are a reciprocal effect.
But that's not.

The shortening/lengthening is not relevant. The scenario can be set up as a spacetime diagram. I attach one showing the events in the lab frame and the muon frame. The clock times are the same in both, naturally.

 

[Devil Moo]

@Mentz114
How do you generate them?
What software?

 

[Mentz114]

The diagrams were made with a win32 application I wrote a few years ago. If you have a windows pc I can give you a url to download the installer.

There is similar tool here that runs in a browser http://ibises.org.uk/Minkowski.html

 

[Devil Moo]

@Mentz114 I have a windows pc. For the application, can I change to different frames?
Do you have relativistic momentum application?

 

[harrylin]

Actually, I understand the length contraction but I am confused with the shorten distance.
Because the length contraction and time dilation are a reciprocal effect.
But that's not.

Did you consider the effect of relativity of simultaneity? Length contraction and time dilation could not be a reciprocal effect without relativity of simultaneity. If someone here doesn't understand something related to time dilation or length contraction, perhaps half the time this is because that fundamental point is ignored.

 

[Orodruin]

If someone here doesn't understand something related to time dilation or length contraction, perhaps half the time this is because that fundamental point is ignored.

I would go as far as calling it closer to between 80 and 90 percent of the time ...

 

[Devil Moo]

@harrylin I am just considering inertial frame...

 

[harrylin]

@harrylin I am just considering inertial frame...

Exactly! If you want to understand how length contraction and time dilation can be reciprocal, you need to understand how, typically, distant clocks of an inertial frame are synchronized and what that implies for clock synchronization in frames that are in relative motion. See §1 of http://fourmilab.ch/etexts/einstein/specrel/www/ as well as the last sentences of §2.