# Do atmospheric muons accelerate?

• nitsuj

#### nitsuj

<<Moderator note: This discussion was originally posted in another thread. It has been moved as it did not concern the OP of that thread.>>

The distance they travel in their own rest frame is zero. It is the Earth that travels, but as it is length contracted it needs to travel a shorter distance.

The muons never accelerate from A->B?

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The muons never accelerate from A->B?
For all practical purposes they are moving relative to the Earth's surface and atmosphere when they are created, and they continue to travel at that speed for their entire (short) lifetime.

For all practical purposes they are moving relative to the Earth's surface and atmosphere when they are created, and they continue to travel at that speed for their entire (short) lifetime.

What's "practical"? I was commenting on Orodruin's reply "The distance they travel in their own rest frame is zero." Which implicitly says the muon also doesn't proper accelerate. Since it claims the muon determines it had zero displacement. In either case you and Orodruin refer only to the observer dependent...which has what "practical" physical importance? none, it's relative. Either way your point or Orodruin's; it's moot. The proper acceleration part is of "practical" physical importance

The muon would be wrong to think that maybe the Earth moved to it during any part of it's observation, due to the first part

What's "practical"? I was commenting on Orodruin's reply "The distance they travel in their own rest frame is zero." Which implicitly says the muon also doesn't proper accelerate. Since it claims the muon determines it had zero displacement. In either case you and Orodruin refer only to the observer dependent...which has what "practical" physical importance? none, it's relative. Either way your point or Orodruin's; it's moot. The proper acceleration part is of "practical" physical importance

The muon would be wrong to think that maybe the Earth moved to it during any part of it's observation, due to the first part

No, the muon is perfectly valid in thinking the Earth is moving towards it. The muon essentially never accelerates, it is born with a velocity relative to the Earth and it decays with essentially the same velocity. The muon does have zero displacement in its rest frame by definition of the rest frame. In the muon rest frame, the Earth is moving.

The muon essentially never accelerates, it is born with a velocity relative to the Earth and it decays with essentially the same velocity. The muon does have zero displacement in its rest frame by definition of the rest frame. In the muon rest frame, the Earth is moving.

What does "essentially" mean? Doesn't that mean non-zero?

I presumed that the muon accelerates after it's "put" into existence. I'll trust what you're saying (the muon is put into existence while simultaneously having a velocity) if the muon is massless. in which case the "frame" (and in turn the "observations") of the muon are not physically valid anyways.

I interpret it as the muon would be correct in saying the length to Earth contracted. Which is physically valid & compatible to "our" frame of "the muon's clock ticked slower" & how Russ_watters described the two frames. What is not physically valid & compatible amongst all observations is "the Earth moved." Which makes it wrong...practically, and was your amendment.

Yes due to relative motion "In the muon rest frame, the Earth is moving." But that's not the whole story, just a physically symmetric however moot point of view between two frames of an asymmetric scenario/reality.

[moderator's note - edited to remove off-topic personal theory]

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I presumed that the muon accelerates after it's "put" into existence. I'll trust what you're saying (the muon is put into existence while simultaneously having a velocity) if the muon is massless.
The muon has initial velocity even given that it is massive. It is created with some velocity and being a fundamental particle it has a characteristic fixed mass.

The "essentially" and "practically" caveats that you are worried about are simply there because a muon is charged and there may be some small EM fields present around the earth. Ideally those are 0 and then the muon does not accelerate at all. If they are practically or essentially 0 then it practically and essentially doesn't accelerate. Also we are ignoring g which is essentially and practically OK.

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I presumed that the muon accelerates after it's "put" into existence. I'll trust what you're saying (the muon is put into existence while simultaneously having a velocity) if the muon is massless. in which case the "frame" (and in turn the "observations") of the muon are not physically valid anyways.

Why can the particle only be created with motion if it is massless?
Suppose I create a particle that is at rest relative to me in my lab (possible in principle, although it would be a fair amount of work in practice). How would you describe the motion of the particle if:
A) my mobile lab just happened to be aboard a truck driving past at 100 km/hr while you are standing alongside the road?
B) my mobile lab is still in the truck, but now it's parked by the side of the road?

The muon has initial velocity even given that it is massive. It is created with some velocity and being a fundamental particle it has a characteristic fixed mass.

The "essentially" and "practically" caveats that you are worried about are simply there because a muon is charged and there may be some small EM fields present around the earth. Ideally those are 0 and then the muon does not accelerate at all. If they are practically or essentially 0 then it practically and essentially doesn't accelerate. Also we are ignoring g which is essentially and practically OK.

It seems curious to me that a massive particle can have a velocity upon it's creation from the rest frame of it's creation. Which you tell me is "in Motion" relative to the source of it's creation, yet had experienced no acceleration. Yes it's idealized there are no "surprise" variables accelerating the muon.

Why can the particle only be created with motion if it is massless?
Suppose I create a particle that is at rest relative to me in my lab (possible in principle, although it would be a fair amount of work in practice). How would you describe the motion of the particle if:
A) my mobile lab just happened to be aboard a truck driving past at 100 km/hr while you are standing alongside the road?
B) my mobile lab is still in the truck, but now it's parked by the side of the road?

Not so simple, this is a creation of a particle We have a very good reason to use that event as "at rest", and the destination is idealized to be "at rest" to the source of the muon creation. It's not the same as a muon is created while on a truck going 100 kph past me so I measure the velocity to be x +100kph

the rest frame of it's creation.
what frame are you referring to here. Its creation is an event and an event doesn't have a rest frame.

Muons are created having velocity in the rest frame of the earth. Muons typically come from the decay of a pion into a muon and a neutrino. Because the neutrino carries a small amount of momentum the muon rest frame is not the same as the pions rest frame.

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what frame are you referring to here. Its creation is an event and an event doesn't have a rest frame.

Muons are created having velocity in the rest frame of the earth. Muons typically come from the decay of a pion into a muon and a neutrino. Because the neutrino carries a small amount of momentum the muon rest frame is not the same as the pions rest frame.

I was suggesting the point where the muon comes into existence be used as 0,0 and that from it we see the muon has accelerated.

So the momentum of the energy was Already in motion compared to earth, it broke up into a neutrino and muon, so those two are created in motion from Earths frame. Wow, I see how it can really be that the Earth is moving towards the muon from start to finish. It actually never experiences proper acceleration.

Thanks for explaining that Dalespam.

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Ah so the energy was Already in motion compared to earth, it broke up into a neutrino and muon, so those two are created in motion from Earths frame.
Yes, the pion's momentum must be conserved in its decay products, so pion already moving wrt Earth -> muon and neutrino created already moving wrt earth, each other, and pion*. Otherwise, momentum would not be conserved.

*(or rather the inertial frame where the now deceased pion was at rest)

Yes, the pion's momentum must be conserved in its decay products, so pion already moving wrt Earth -> muon and neutrino created already moving wrt earth, each other, and pion*. Otherwise, momentum would not be conserved.

*(or rather the inertial frame where the now deceased pion was at rest)

I want to highlight this point made by DaleSpam above: The muon and neutrino are not going to be at rest in the initial pion rest frame. If they were, then energy would not be conserved. In order for energy to be conserved, it is necessary that the muon and neutrino have an initial velocity with respect to the initial pion rest frame.

I want to highlight this point made by DaleSpam above: The muon and neutrino are not going to be at rest in the initial pion rest frame. If they were, then energy would not be conserved. In order for energy to be conserved, it is necessary that the muon and neutrino have an initial velocity with respect to the initial pion rest frame.

Yea that is where I went wrong. I was presuming point "a" and "b" were at rest to each other, and the muon moves from "a" to "b". oddly here the starting point is the muon's creation and has a non-zero momentum compared to point "b". never seen that before and don't like it lol

I need to think it through until I see it just like Nugatory said. The pion is the lab in motion.

here the starting point is the muon's creation and has a non-zero momentum
It is a little odd at first glance, but then if you think about Nugatory's point above it makes sense. When it is created it has some value for momentum, even if that value is 0. And if you look at it from any other reference frame then the momentum will be different. So if it just randomly happens to be 0 in the first frame, then it will definitely be non-zero in the second.

Since there are an infinite number of frames where it is non-zero and only one frame where it is zero, just randomly you would expect things to usually be created with non-zero momentum.

I was presuming point "a" and "b" were at rest to each other, and the muon moves from "a" to "b". oddly here the starting point is the muon's creation and has a non-zero momentum compared to point "b".

This doesn't make sense. "Points" in space (which is what you appear to mean) are frame-dependent to start with; the muon doesn't move from point a to point b in space in any invariant sense, it only does so with respect to some particular frame. Also, points don't have momentum, and, as DaleSpam pointed out, objects don't have momentum in an invariant sense, they only have momentum with respect to some particular frame.

What you have in this problem are simply three different frames:

(1) In the frame in which the Earth is at rest, the pion has some velocity, and the muon and neutrino that are created from the pion have different velocities.

(2) In the frame in which the muon is at rest, the pion has some velocity, the neutrino has some different velocity, and the Earth has another different velocity.

(3) In the frame in which the pion is at rest, the muon has some velocity, the neutrino has a different velocity, and the Earth has yet another different velocity.

This doesn't make sense. "Points" in space (which is what you appear to mean) are frame-dependent to start with; the muon doesn't move from point a to point b in space in any invariant sense, it only does so with respect to some particular frame. Also, points don't have momentum, and, as DaleSpam pointed out, objects don't have momentum in an invariant sense, they only have momentum with respect to some particular frame.

What you have in this problem are simply three different frames:

(1) In the frame in which the Earth is at rest, the pion has some velocity, and the muon and neutrino that are created from the pion have different velocities.

(2) In the frame in which the muon is at rest, the pion has some velocity, the neutrino has some different velocity, and the Earth has another different velocity.

(3) In the frame in which the pion is at rest, the muon has some velocity, the neutrino has a different velocity, and the Earth has yet another different velocity.

My "points" "A" and "B" could be thought of as the ends of a very long ruler. A start and finish line is all.
The starting point of the observation ("A") is the same as the muons creation. Simply put I was wrong by not considering the "running start" of the muon's inherited momentum wrt Earth. I thought, for some unfounded reason, that it was implicit the muon is created & then accelerated; wrong on so many levels opps. But it instead, inherits the properties of the pion's comparative motion (less the neutrino, and some change in velocity wrt the original pion rest frame).

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I wonder if the basic misunderstanding here is applying classical intuitions to a quantum process. Classically, if you look at a bomb exploding with high speed photography, pieces of the shell start out at rest and go through all speeds before reaching their terminal speed. However, for radioactive decay, a nucleus at rest emits (say) a beta particle and neutrino at some speeds, and recoils, with none of the particles or nucleus having any 'intermediate' speed.

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Yes, weak nuclear is how was assuming this "muon being "put" into existence", thanks for identifying the source of "intuition" lol. So much so I scoffed at Nugatory's very accurate and elementary example, suggesting to him it's not so simple...but it is as he said.

I didn't even consider the whole process is in motion,

My "points" "A" and "B" could be thought of as the ends of a very long ruler.

And how is that ruler moving? In which frame is it at rest? You have to pick one.

If the ruler is at rest in the Earth's rest frame, then the pion has a very large velocity when it hits point A, where it turns into a muon and a neutrino; and the muon has a slightly different but still very large velocity as it leaves point A.

If the ruler is at rest in the pion's rest frame, then it's not "very long"; it's pretty short. The pion can be viewed as being at rest at point A in this frame; but at some instant it turns into a muon and a neutrino, and the muon has a fairly small velocity from A towards B. It doesn't take long to reach B because the distance from A to B, in this frame, is very short.

And how is that ruler moving? In which frame is it at rest? You have to pick one.

If the ruler is at rest in the Earth's rest frame, then the pion has a very large velocity when it hits point A, where it turns into a muon and a neutrino; and the muon has a slightly different but still very large velocity as it leaves point A.

At rest in Earth's frame, that said I presumed the creation of them was the sun (or our atmosphere) which is at rest (idealized) with Earth, which is wrong. The source of the muon is in motion wrt Earth to begin with.

My "points" "A" and "B" could be thought of as the ends of a very long ruler. A start and finish line is all.
The starting point of the observation ("A") is the same as the muons creation. Simply put I was wrong by not considering the "running start" of the muon's inherited momentum wrt Earth. I thought, for some unfounded reason, that it was implicit the muon is created & then accelerated; wrong on so many levels opps. But it instead, inherits the properties of the pion's comparative motion (less the neutrino, and some change in velocity wrt the original pion rest frame).
Just so we don't overlook the obvious: Although the muon is created with momentum from an Earth reference frame, it is also in continuous acceleration from an Earth reference frame - because it is falling through Earth gravity.

So to answer the topic title, "yes".

Although the muon is created with momentum from an Earth reference frame, it is also in continuous acceleration from an Earth reference frame - because it is falling through Earth gravity.

This is true in principle, but the muon is traveling so fast that the time it takes for it to get from the upper atmosphere, where it's created, to near the surface of the Earth, where it's detected, is too short for the acceleration due to gravity to be significant in practice. So you can ignore the Earth's gravity when analyzing this scenario.

Just so we don't overlook the obvious: Although the muon is created with momentum from an Earth reference frame, it is also in continuous acceleration from an Earth reference frame - because it is falling through Earth gravity.

So to answer the topic title, "yes".

Dalespam stated the obvious "outside" influence of the scenario is obviously not considered; in post #6. And for the reason PeterDonis stated. My retort in post #8 was "Yes it's idealized there are no "surprise" variables accelerating the muon." "Surprise" being anything not mentioned in the discussion...obviously.

"The source of the muon is in motion wrt Earth to begin with." quoted from the last line of the post just before yours is an obvious answer to the Topic Title. Being the starter, who wrote that statement obviously means the thread is "complete"good fun

Oh and the answer is No*, just so we don't miss the obvious.

*within the context of this discussion which focused on the velocity of the muon wrt Earth and how the muon got that velocity...it's not from its creation but from its source. Neutrino is also created and part of the sum of what was a pion, their source. There is no moment of missing momentum...called the three M's...or simply mmm...'cause it's so good lol just kidding of course, they told me it's conservation of energy.

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This is true in principle

I would say false on principle (of general relativity :p). There is no proper acceleration of the muons due to gravity and an accelerometer following the muons would show zero apart from the effects of electromagnetic fields and collisions with the atmosphere (which again are not amounting to much).

ahaha fantastic point that I missed too. Proper acceleration is the acceleration referred to in the title of the thread and...kinda obviously lol.

There is no proper acceleration of the muons due to gravity

No, but in principle there is tidal gravity present, so in principle you can't say the muons are at rest in an inertial frame, even though they are in free fall. In practice the effects of tidal gravity are too small to matter. I should have made it clearer that that's really what I was referring to; the word "acceleration" was the wrong word to use.

Don't they accelerate in a direction opposite to their velocity, due to interaction with the atmosphere? Also known as "slow down"?

Sure and among many other variables idealized away in the scenario due to lack of any physical significance.

Cosmic ray muons lose about a third of their energy due to the atmosphere - not exactly insignificant.

Cosmic ray muons lose about a third of their energy due to the atmosphere - not exactly insignificant.

Wow, uh yea, the inquiry is more specific than you presume.

Cosmic ray muons lose about a third of their energy due to the atmosphere - not exactly insignificant.

Although in terms of speed, that's the difference between ##.9797c## and ##.9747c##, so assuming constant speed through the atmosphere is still a pretty good approximation.

Although in terms of speed, that's the difference between ##.9797c## and ##.9747c##, so assuming constant speed through the atmosphere is still a pretty good approximation.

Wow does that ever highlight the kinematics with respect to energy required to go just a little bit faster at near c velocities, or in this case the reverse.

Although in terms of speed, that's the difference between ##.9797c## and ##.9747c##, so assuming constant speed through the atmosphere is still a pretty good approximation.
Interesting...
... from the Earth's viewpoint, yes... but from the muon's initial frame I would think things look very different. They still experience a massive acceleration from the opposing/onrushing atmosphere. :)

from the muon's initial frame I would think things look very different. They still experience a massive acceleration from the opposing/onrushing atmosphere.

They experience proper acceleration (I haven't calculated its magnitude so I don't know if it is aptly described as "massive"), but that only results in a negligible change in the velocity of the Earth in this frame (which will have the same magnitude, but opposite direction, as the the velocity of the muon in the Earth's frame). So it depends on what kind of acceleration you're talking about, proper acceleration (may be significant) or coordinate acceleration (not significant). For purposes of using an inertial frame to model what's going on, the important quantity is coordinate acceleration, which is negligible. For other purposes, proper acceleration might be more important.