Explain this - the Muon experiment

[Michio Cuckoo]

Muons are formed from cosmic bombardment and they fall towards the Earth.

So here's the problem: Muon's have really short lifetimes. So short that they won't be able to reach the Earth in time before they wink out. However, we detect more muons than we should be.

The solution:

On Earth, we witness Time Dilation of the muon, which explains its longer lifetime.

From the Muon's frame, its lifetime is the same, but the distance between itself and the Earth is length contracted. Simple, right?


Now here's the conundrum: Why should the distance between the muon and the Earth be length contracted and not the other way round? Doesn't length contraction work both ways?

I'll leave you to think about it.


"Unthinking respect for authority is the greatest enemy of truth." - Albert Einstein

 

[HallsofIvy]

Muons are formed from cosmic bombardment and they fall towards the Earth.

So here's the problem: Muon's have really short lifetimes. So short that they won't be able to reach the Earth in time before they wink out. However, we detect more muons than we should be.

The solution:

On Earth, we witness Time Dilation of the muon, which explains its longer lifetime.

From the Muon's frame, its lifetime is the same, but the distance between itself and the Earth is length contracted. Simple, right?


Now here's the conundrum: Why should the distance between the muon and the Earth be length contracted and not the other way round? Doesn't length contraction work both ways?

I'll leave you to think about it.


"Unthinking respect for authority is the greatest enemy of truth." - Albert Einstein

What do you mean by "the other way around"? What other distance is there to be contracted?

 

[Michio Cuckoo]

What do you mean by "the other way around"? What other distance is there to be contracted?

Well, from our POV, that is. We on Earth.

Why is the distance between us and the muon not length contracted >>> from our POV.

Why is length contraction only considered from the muon's POV?

 

[Histspec]

Now here's the conundrum: Why should the distance between the muon and the Earth be length contracted and not the other way round? Doesn't length contraction work both ways?

You have to distinguish between "rest length" and "contracted length".
a) The "Rest length" is given when the endpoints of a distance are mutually at rest the whole time, that is, they are at rest in the same inertial frame. This length is the greatest possible between these endpoints.
b) The "contracted length" is the length of the same distance in every other inertial reference frame in relative motion, that is, the endpoints of this distance are both in motion with the same speed.

Now, all of the muons are produced in the atmosphere at a specific distance from Earth's surface.
So in the inertial frame in which the atmosphere is at rest, the distance between the point where the muons are generated and Earth's surface is by definition the rest length, which is the largest in all inertial frames - thus definition a) applies here.

But in the muon frame, the atmosphere is in motion. Thus definition b) applies here - the distance between the point where the muons are generated and Earth's surface, is contracted compared to the rest length.

In other words: Relativity doesn't say that every possible length is contracted, it only says that the "rest length" is contracted - we have to step from a) to b).
On the other hand, this specific length measured in the muon frame is not further contracted when you transform back to the atmosphere frame, because in this case we have to step from b) to a), by which the length is actually getting larger.

 

[ghwellsjr]

Well, from our POV, that is. We on Earth.

Why is the distance between us and the muon not length contracted >>> from our POV.

Why is length contraction only considered from the muon's POV?

Length contraction is also considered from the Earth's POV, it's just that the only thing moving at high speed are the muons and they are length contracted so instead of being perfect spheres, they are oblate, compressed along the direction of motion.

 

[Michio Cuckoo]

Length contraction is also considered from the Earth's POV, it's just that the only thing moving at high speed are the muons and they are length contracted so instead of being perfect spheres, they are oblate, compressed along the direction of motion.

From our eyes, the muon's are oblate, if they even have a shape.

From the muon's "eyes", Earth becomes even more oblate. So far so good.

But doesn't the distance between the speeding muon and our Earth also appear length contracted from our POV?

 

[Michio Cuckoo]

You have to distinguish between "rest length" and "contracted length".
a) The "Rest length" is given when the endpoints of a distance are mutually at rest the whole time, that is, they are at rest in the same inertial frame. This length is the greatest possible between these endpoints.
b) The "contracted length" is the length of the same distance in every other inertial reference frame in relative motion, that is, the endpoints of this distance are both in motion with the same speed.

Now, all of the muons are produced in the atmosphere at a specific distance from Earth's surface.
So in the inertial frame in which the atmosphere is at rest, the distance between the point where the muons are generated and Earth's surface is by definition the rest length, which is the largest in all inertial frames - thus definition a) applies here.

But in the muon frame, the atmosphere is in motion. Thus definition b) applies here - the distance between the point where the muons are generated and Earth's surface, is contracted compared to the rest length.

In other words: Relativity doesn't say that every possible length is contracted, it only says that the "rest length" is contracted - we have to step from a) to b).
On the other hand, this specific length measured in the muon frame is not further contracted when you transform back to the atmosphere frame, because in this case we have to step from b) to a), by which the length is actually getting larger.

Nice explanation bro. I'm afraid I might have to take some time to think about it.

 

[Nugatory]

But doesn't the distance between the speeding muon and our Earth also appear length contracted from our POV?

No. Imagine that the muons are being created 100 miles up in the sky; the point of creation is one hundred miles away from us and not moving relative to us, so we can just measure the distance to it using ordinary uncontracted rulers. Sure, the muon is rushing away from that point at relativistic velocities, but we aren't, so that doesn't affect our measurements of where that point is.

 

[Dale]

Well, from our POV, that is. We on Earth.

Why is the distance between us and the muon not length contracted >>> from our POV.

Why is length contraction only considered from the muon's POV?

Because the atmosphere and the Earth are not moving from our POV so the thickness of the atmosphere is not length contracted from our POV. Plug v=0 into the length contraction formula.

 

[Michio Cuckoo]

No. Imagine that the muons are being created 100 miles up in the sky; the point of creation is one hundred miles away from us and not moving relative to us, so we can just measure the distance to it using ordinary uncontracted rulers. Sure, the muon is rushing away from that point at relativistic velocities, but we aren't, so that doesn't affect our measurements of where that point is.

so let's look at like this: imagine a giant ruler sticking out of the Earth. To us earthlings, nothing happens to the ruler. However, to a speeding muon the ruler appears contracted. so that seems to explain it.

But, "its a bird! Its a plane! No, its Supermuon!"

Being blessed with extreme super powers, Supermuon is able to carry an entire ruler with him. From his point of view, nothing happens to the ruler. To us, the ruler appears to be length contracted.

So it appears that the length contraction can be applied both ways, unlike the current explanation.

Starting to sound like a crank, somebody please knock me on the head.

 

[russ_watters]

so let's look at like this: imagine a giant ruler sticking out of the Earth. To us earthlings, nothing happens to the ruler. However, to a speeding muon the ruler appears contracted. so that seems to explain it.

But, "its a bird! Its a plane! No, its Supermuon!"

Being blessed with extreme super powers, Supermuon is able to carry an entire ruler with him. From his point of view, nothing happens to the ruler. To us, the ruler appears to be length contracted.

So it appears that the length contraction can be applied both ways, unlike the current explanation.

What you said above in no way contradicts the accepted explanation.

 

[Dale]

Being blessed with extreme super powers, Supermuon is able to carry an entire ruler with him. From his point of view, nothing happens to the ruler. To us, the ruler appears to be length contracted.

So it appears that the length contraction can be applied both ways

Correct.

 

[Nugatory]

So it appears that the length contraction can be applied both ways, unlike the current explanation.

Ah... that IS the current explanation. If the ruler is moving relative to you, it is contracted.

So supermuon's super ruler is contracted when viewed from the Earth but not when viewed by supermuon. And the 100-mile tall ruler on the surface of the Earth is contracted when viewed by supermuon but not when viewed from the earth.

 

[Michio Cuckoo]

What you said above in no way contradicts the accepted explanation.

i thought the accepted explanation is that time dilation from our pov enables the muon to have a longer lifespan, while length contraction from the muon's pov allows it to reach earth.

But if length contraction is applied both ways, then there would be no need to use time dilation to explain how the muon reaches earth.

Or is it a combination of length contraction and time dilation, both from our pov, that ensures the muon can get to earth?

 

[Histspec]

Being blessed with extreme super powers, Supermuon is able to carry an entire ruler with him. From his point of view, nothing happens to the ruler. To us, the ruler appears to be length contracted.

So it appears that the length contraction can be applied both ways, unlike the current explanation.

Ok, so in the muon frame we have a large ruler. Its left endpoint is at the location where the muon is generated, while its right endpoint touches Earth's surface simultaneously.

In Earth's frame, the ruler is contracted as you correctly said - but as explained above, the distance (in the atmosphere) between the muon-creation-point and Earth's surface is much larger than the ruler's length. How is that possible? Well, due to relativity of simultaneity. When (in Earth's frame) the left end of the contracted ruler is at the point of the muon's origin, the right endpoint is still far away from Earth's surface. And when the right endpoint eventually reaches Earth, its left endpoint has already passed the muon-creation-point long ago.

But if length contraction is applied both ways, then there would be no need to use time dilation to explain how the muon reaches earth.

Length contraction is applied whenever an object at rest in one frame is viewed from a moving frame. You only have to look in which frame the considered object is at rest, and in which it is moving. That's all.

 

[ghwellsjr]

i thought the accepted explanation is that time dilation from our pov enables the muon to have a longer lifespan, while length contraction from the muon's pov allows it to reach earth.

But if length contraction is applied both ways, then there would be no need to use time dilation to explain how the muon reaches earth.

Or is it a combination of length contraction and time dilation, both from our pov, that ensures the muon can get to earth?

In any given Frame of Reference, things that are stationary are not time dilated or length contracted. Things that are moving are time dilated and length contracted, the faster they move, the greater the dilation and contraction.

As I pointed out earlier, in the Earth's FoR, the only thing moving at high speed are the muons so it is only time dilation that explains how they reach the Earth (although they are also length contracted). In the muon's FoR, everything in the solar system (except for the other muons) is moving at high speed, so the entire solar system including distances between the sun and planets and their moons are length contracted along the direction of motion so it is only [STRIKE]time dilation[/STRIKE] length contraction that explains how they reach the Earth (although all these other objects are also time dilated).

 

[Dale]

i thought the accepted explanation is that time dilation from our pov enables the muon to have a longer lifespan, while length contraction from the muon's pov allows it to reach earth.

But if length contraction is applied both ways, then there would be no need to use time dilation to explain how the muon reaches earth.

Or is it a combination of length contraction and time dilation, both from our pov, that ensures the muon can get to earth?

Length contraction and time dilation always occur for anything which is moving in any given inertial frame. So there is length contraction and time dilation in both frames in this example.

HOWEVER, in this scenario there is only one time that is of interest, the decay time of the muon. Similarly there is only one length that is of interest, the length of the atmosphere. Any lengths in the muon's frame are contracted in the Earth's frame, but they are not of interest to this problem. Any times in the Earth's frame are dilated in the muon's frame, but they are not of interest to this problem.

It is not that they don't occur, they are just not relevant for this specific scenario.

 

[Michio Cuckoo]

In any given Frame of Reference, things that are stationary are not time dilated or length contracted. Things that are moving are time dilated and length contracted, the faster they move, the greater the dilation and contraction.

As I pointed out earlier, in the Earth's FoR, the only thing moving at high speed are the muons so it is only time dilation that explains how they reach the Earth (although they are also length contracted). In the muon's FoR, everything in the solar system (except for the other muons) is moving at high speed, so the entire solar system including distances between the sun and planets and their moons are length contracted along the direction of motion so it is only [STRIKE]time dilation[/STRIKE] length contraction that explains how they reach the Earth (although all these other objects are also time dilated).

Got it. So since the [dist.between planets] is moving at high speed frome the muon's FoR, so it is length contracted.

But from the Earth's FoR, the muon isn't the only thing moving at high speed. The [distance between Earth and muon] is also moving at high speed and hence also appears length contracted, right?

 

[GAsahi]

The [distance between Earth and muon] is also moving at high speed and hence also appears length contracted, right?

The distance isn't moving. So, it doesn't "appear length contracted".

 

[ghwellsjr]

Got it. So since the [dist.between planets] is moving at high speed frome the muon's FoR, so it is length contracted.

But from the Earth's FoR, the muon isn't the only thing moving at high speed. The [distance between Earth and muon] is also moving at high speed and hence also appears length contracted, right?

No. What you are doing is jumping to a conclusion about how one FoR transforms into a second FoR without actually performing a Lorentz Transformation. This is how almost all of the so-called Paradoxes in SR come about.

 

[DrGreg]

The [distance between Earth and muon] is also moving at high speed and hence also appears length contracted, right?

The simple concept of length contraction applies only to the distance between two things that are at rest relative to each other. The Earth and the muon are in relative motion, so simple length contraction does not apply. (The Lorentz transformation does apply however.) You can apply length contraction to: the distance between the top and bottom of the atmosphere; or the distance between two muons following one behind the other.

(Ignoring gravitational tidal effects which are negligible for very high speed muons over this distance.)