Muon travel distance vs Atmosphere Thickness?

[curiousburke]

TL;DR Summary

Does calculating the distance traveled by a muon (L' in the muons frame), using two equations for length contraction (one at each end point) to eliminate delta t and solve for L' in terms of L and gamma lead to a contradiction?

I'm having a discussion with a friend/family member about a paper by Radwan Kassir in which he calculates the distance traveled by a muon (atmosphere thickness) in a non-standard way that shows the atmosphere thickness in the muons frame (L') is γ times the atmosphere thickness in the Earth frame (L), which of course conflicts with the standard approach that results in L' = L/γ.

The basic idea is that you can use two equations for Δx' at two different places (both of which equal L'), eliminate Δt, and solve for L' in terms of γ and L, which is Δx. First, calculate Δx' of the muon = γ × ( Δx(muon) - v Δt ) in which Δt is the time between creation and hitting the Earth in the Earth frame. In the muons frame Δx'(muon)=0, so Δx(muon) = v Δt. Second, calculate Δx' of the Earth = L' = γ × ( Δx(Earth) - v Δt). In this case, Δx(Earth)=0, so L' = γ × ( -v Δt). From the first eqn vΔt = -Δx(muon)=L, so L' = -γ × L.

Okay, I got a sign error, but the idea is the same. What is happing here to show this contradiction, and how is it resolved?

 

[curiousburke]

Just to be clear, I understand how to solve for the atmosphere thickness from the muons frame the standard way and get the L'=L/γ. What I am asking here is what is it about this other way of solving for L' that gives a different answer.

 

[PeterDonis]

a paper by Radwan Kassir

That paper is not a reliable source. (ResearchGate in general is not a reliable place to look for papers.) The predictions of standard relativity have been confirmed by experiment for this case. So any other model that gives a different result is experimentally falsified.

 

[PeterDonis]

What I am asking here is what is it about this other way of solving for L' that gives a different answer.

It's nonsense which has been experimentally falsified. That should be enough.

 

[Ibix]

Generally, if you've got $\gamma L$ instead of $L/\gamma$ then you've used the simultaneity convention of the wrong frame. For example, if you use two events that are simultaneous in the unprimed frame and distance $L$ apart and measure their separation in the primed frame you'll get $\gamma L$, but the events aren't simultaneous.

I can't be bothered to check his maths, but that's the standard "I think I understand relativity better than I actually do" mistake.

 

[ersmith]

The "equations for length contraction" are valid only between points measured at the same time in the moving frame, which is not the case here. The full Lorentz transformation is required.

 

[curiousburke]

Generally, if you've got $\gamma L$ instead of $L/\gamma$ then you've used the simultaneity convention of the wrong frame. For example, if you use two events that are simultaneous in the unprimed frame and distance $L$ apart and measure their separation in the primed frame you'll get $\gamma L$, but the events aren't simultaneous.

I can't be bothered to check his maths, but that's the standard "I think I understand relativity better than I actually do" mistake.

Yes, I had a hunch that it has something to do with simultaneity, but was not able to work out the details.

 

[Dale]

In this case, Δx(Earth)=0

You have listed a bunch of variables with no definitions of what they mean, but I think the error is here. $\Delta x (earth)$ is probably $L$. Also, please use LaTeX if you want other people to read your equations and find errors.

The source is not credible. Any conflict with relativity is an indication that the author doesn't know how to do relativity.

 

[curiousburke]

The "equations for length contraction" are valid only between points measured at the same time in the moving frame, which is not the case here. The full Lorentz transformation is required.

If the mistake is simply that you are not allowed to calculate contraction from observations at two different times, then how is that restriction expressed or included in the transform? We have these transform equations, but it seems that they can only be manipulated in specific ways. I would like to be able to point at the specific mistake being made and say a particular step violates some postulate.

What good are the "Lorentz transformations of the differences" equations if this is the case?

 

[Dale]

how is that restriction expressed or included in the transform?

It isn't. The Lorentz transform is not restricted in that manner. It is the simplified length contraction equation that is restricted.

 

[Ibix]

how is that restriction expressed or included in the transform?

For a stationary object you can get its length by measuring the x positions of the two ends and taking the difference. For a moving object you can do the same thing, but you must measure the end positions simultaneously because if you don't you don't measure length. There's no relativity in that statement. It's true of a totally mundane thing like a beetle walking on a ruler - if you measure the position of one end now and the other end a minute later that might be meters away. The bug is not that big.

Relativity makes it easy to do it by accident, though, because different frames have different ideas of what "at the same time" means. And if you carelessly use the simultaneity of one frame when making position measures in a different frame where the object you are measuring is moving, you can easily get a meter long bug.

There's nothing wrong with the measurements or the transforms. But the difference in position measurements made at different times is not the length. The error is almost in that last sentence.

 

[curiousburke]

You have listed a bunch of variables with no definitions of what they mean, but I think the error is here. $\Delta x (earth)$ is probably $L$. Also, please use LaTeX if you want other people to read your equations and find errors.

The source is not credible. Any conflict with relativity is an indication that the author doesn't know how to do relativity.

It's been years since I have done LaTex, but I will try.

$\Delta x(earth)$ refers to the position of Earths surface where the muon hits from the Earth reference frame. Maybe I see your point, but I'm not sure. As I understand it, he is assuming that change the muon position as seen from the muon frame $\Delta x'(muon)$, in which the prime indicates the muon frame, is always zero, and change in the Earth position as seen from the Earth frame $\Delta x(earth)$ is always zero. Is that not how the length contraction equation should be used?

L' in this case can be given by $\Delta x'(earth)$, the change in the Earth position as seen from the muon frame; L is given by $\Delta x(muon)$, the change in the muon position as seen from the Earth frame.

 

[Dale]

Δx(earth) refers to the position of Earths surface where the muon hits from the Earth reference frame

Interesting. Why is that labeled with a $\Delta$ ? That is a single event, so there is no change or difference or anything to subtract. I think that the author is probably confusing themselves with inconsistent notation. With a single event, the transform is going to be useless for establishing a length in either frame.

 

[curiousburke]

Interesting. Why is that labeled with a $\Delta$ ? That is a single event, so there is no change or difference or anything to subtract. I think that the author is probably confusing themselves with inconsistent notation. With a single event, the transform is going to be useless for establishing a length in either frame.

As I understand, it is two events: muon creation and muon hits Earth

 

[Dale]

As I understand, it is two events: muon creation and muon hits Earth

That was my original assumption. That is not zero as I said above. It is $L$.

It sounds like this author isn’t even setting up the problem coherently. There should not be any back and forth confusion about the meaning of any of the variables. They should have explicitly defined every single variable they used.

 

[curiousburke]

That was my original assumption. That is not zero as I said above. It is $L$.

It sounds like this author isn’t even setting up the problem coherently. There should not be any back and forth confusion about the meaning of any of the variables. They should have explicitly defined every single variable they used.

I don't understand how the Earth position as seen from the Earth reference frame can change, nor how the muon position as seen from the muon reference frame can change. They are each stationary in their respective frames. Therefore, the difference in Earth position at 2 times as seen from the Earth frame will always be zero. If this is wrong, then that is certainly the mistake being made.

 

[Dale]

I don't understand how the Earth position as seen from the Earth reference frame can change

It doesn’t.

That is not what $\Delta x$ represents according to your most recent post. The distance between the muon being created and the muon hitting the earth has nothing at all to do with the fact that the earth doesn’t move in its own frame.

 

[curiousburke]

Sorry, I think the notation is a source of confusion. $\Delta x$ without a prime is as seen from Earth reference frame. $\Delta x'$ is as seen from muon reference frame. $\Delta x(Earth)$ is the change in position of Earth as seen in Earth frame, which would be zero. $\Delta x'(Earth)$ is the change in position of Earth as seen from muon frame and would be $=L'$ in this calculation.

 

[curiousburke]

better notation might be: $\Delta x _e$ and $\Delta x_\mu$ to denote the change in position of Earth and the muon respectively as seen from the Earth frame

 

[Dale]

I think the notation is a source of confusion

Yup. So I think that is the problem. The author cannot have sensible mathematics at all if their symbols aren’t even consistent.

 

[ersmith]

The paper basically goes off the rails in one sentence:

"The altitude where the muons are created is the initial distance between the origins of the
relatively moving frames $K$ and $K'$ at the instant muons are created."

There are two fundamental errors here. The phrase "at the instant muons are created" is frame dependent, whereas the author assumes this is invariant. More subtly, the author is speaking of the origin of the frames as though their distance can vary, but the origin has to be a single event in spacetime (i.e. must include the time as part of the definition). In other words, the author is falling into the two most common pitfalls of relativity deniers, neglecting the relativity of simultaneity and failing to understand reference frames.

Also, by making the origins of the two coordinate systems different the author has needlessly complicated things. It would be far better to use one single event (either the creation of the muon, or the detection of the muon in the lab would be the obvious choices) as the common origin.

On a meta note, it continually astounds me that relativity deniers believe that with some simple algebra they can uncover basic errors in relativity that have eluded the physics community for more than a century. As though the tens of thousands of Ph.D. level physicists and mathematicians who have studied relativity were incapable of doing arithmetic. Mathematicians of the caliber of David Hilbert, Emmy Noether, or Kurt Goedel didn't find any inconsistencies in relativity, but some rando on the internet did? Nope. The rando made a mistake, they just refuse to admit it.

 

[Dale]

by making the origins of the two coordinate systems different the author has needlessly complicated things

That also means the Lorentz transform doesn’t apply in its usual form.

 

[curiousburke]

Yup. So I think that is the problem. The author cannot have sensible mathematics at all if their symbols aren’t even consistent.

I don't think the symbols are inconsistent, but I apologize for not having explained them thoroughly from time start.

 

[curiousburke]

That also means the Lorentz transform doesn’t apply in its usual form.

yes, I agree about the origins not being the same, but changing them to be the same does not change their result as far as I can tell.

 

[Dale]

I don't think the symbols are inconsistent, but I apologize for not having explained them thoroughly from time start.

It sure seems like they are to me. You as a reader were not clear about what the variables meant and that is usually indicative that the author was not very clear either.

And honestly, that sort of thing is pretty common for these types of authors. If it is not some inconsistency then it is almost guaranteed to be a simultaneity mixup as @Ibix said.

 

[curiousburke]

The paper basically goes off the rails in one sentence:

"The altitude where the muons are created is the initial distance between the origins of the
relatively moving frames $K$ and $K'$ at the instant muons are created."

There are two fundamental errors here. The phrase "at the instant muons are created" is frame dependent, whereas the author assumes this is invariant. More subtly, the author is speaking of the origin of the frames as though their distance can vary, but the origin has to be a single event in spacetime (i.e. must include the time as part of the definition). In other words, the author is falling into the two most common pitfalls of relativity deniers, neglecting the relativity of simultaneity and failing to understand reference frames.

I don't think that is what was meant. I think he is saying that the distance between the two origins as defined can be viewed from either reference frame, so there are two values.

Also, by making the origins of the two coordinate systems different the author has needlessly complicated things. It would be far better to use one single event (either the creation of the muon, or the detection of the muon in the lab would be the obvious choices) as the common origin.

Admittedly, I'm no expert, but I worked it through shifting the origins to be the same, and it made no difference.

On a meta note, it continually astounds me that relativity deniers believe that with some simple algebra they can uncover basic errors in relativity that have eluded the physics community for more than a century. As though the tens of thousands of Ph.D. level physicists and mathematicians who have studied relativity were incapable of doing arithmetic. Mathematicians of the caliber of David Hilbert, Emmy Noether, or Kurt Goedel didn't find any inconsistencies in relativity, but some rando on the internet did? Nope. The rando made a mistake, they just refuse to admit it.

Okay, I agree on the extreme unlikelihood. Unfortunately, these arguments do nothing to prove them wrong. I understand that it's not your job to prove every person with a wacky theory wrong, but this author happens to have at least one paper in a peer reviewed journal and a book on the topic (maybe self published). I would simply like to know exactly where they have pulled the wool over my eyes such that I can see where the inconsistency they claim is formed.

 

[curiousburke]

It sure seems like they are to me. You as a reader were not clear about what the variables meant and that is usually indicative that the author was not very clear either.

And honestly, that sort of thing is pretty common for these types of authors. If it is not some inconsistency then it is almost guaranteed to be a simultaneity mixup as @Ibix said.

I 100% agree that it is likely to be a simultaneity issue. I said the same to the person I was discussing it with; however, I want to figure out the details of the error.

 

[PeterDonis]

I think he is saying that the distance between the two origins as defined can be viewed from either reference frame

This is true, but...

so there are two values.

...this is not. The two origins are events in spacetime; the interval between two events in spacetime is invariant, i.e., it is the same in all frames.

However, that does not mean it will be a spatial "distance" in all frames. That will only be true in one frame, the frame in which the two events are simultaneous. They can't be simultaneous in both frames.

 

[Dale]

I don’t see how that is possible without clear definitions of each variable. You want us to find the specific wrong detail when there are no specific details to begin with. Don’t you see the inherent impossibility of that?

 

[curiousburke]

This is true, but...


...this is not. The two origins are events in spacetime; the interval between two events in spacetime is invariant, i.e., it is the same in all frames.

However, that does not mean it will be a spatial "distance" in all frames. That will only be true in one frame, the frame in which the two events are simultaneous. They can't be simultaneous in both frames.

In what frame would the muon creation and the muon hitting the Earth's surface be simultaneous?

 

[curiousburke]

I don’t see how that is possible without clear definitions of each variable. You want us to find the specific wrong detail when there are no specific details to begin with. Don’t you see the inherent impossibility of that?

Yes, I do. It's a compliment; I thought I could roughly describe the setup and the more SR knowledgeable minds here could simply point to the error. I'll try to write it up better and come back.

 

[PeterDonis]

In what frame would the muon creation and the muon hitting the Earth's surface be simultaneous?

None. The muon moves on a timelike worldline. So if the spacetime origins of the two frames are those two events, then it makes no sense at all to talk about the "distance" between them, because the interval between them is timelike, not spacelike.

 

[jbriggs444]

I thought I could roughly describe the setup and the more SR knowledgeable minds here could simply point to the error.

It is not possible, in general, to take an nonsensical argument, calculation or computer program and find a single compact "error" without which the original would be, at least, sensible.

If you do find such an "error", there is no guarantee that it is unique. There can be multiple sensible starting points from which drivel emerges.

Writing a compiler that produces decent error messages is hard. So is teaching.

 

[ersmith]

Let's take the origin to be the creation of the muons. Then the detection of the muons has coordinates $(t', x') = (\frac{L'}{v}, 0)$ in the muon frame, and $(t, x) = (\frac{L}{v}, L)$ in the lab frame (using units in which $c = 1$, and making the positive x axis point from creation to detection). Applying the Lorentz transformation from lab frame to muon frame we see:
$$(t', x') = \gamma(t - vx, x - vt) = \gamma(\frac{L}{v} - vL, L - v\frac{L}{v}) = (\gamma L(\frac{1}{v} - v), 0) = (\gamma L \frac{1-v^2}{v}, 0) = (\frac{L}{\gamma v}, 0)$$
So $L' = \frac{L}{\gamma}$.

You'll get the same result using any other origin, of course. The important thing is to always do calculations using one set of coordinates, never to mix the two (and never to assume *anything* about how the two are related: use only the Lorentz transformation to convert).

 

[curiousburke]

Let's take the origin to be the creation of the muons. Then the detection of the muons has coordinates $(t', x') = (\frac{L'}{v}, 0)$ in the muon frame, and $(t, x) = (\frac{L}{v}, L)$ in the lab frame (using units in which $c = 1$, and making the positive x axis point from creation to detection). Applying the Lorentz transformation from lab frame to muon frame we see:
$$(t', x') = \gamma(t - vx, x - vt) = \gamma(\frac{L}{v} - vL, L - v\frac{L}{v}) = (\gamma L(\frac{1}{v} - v), 0) = (\gamma L \frac{1-v^2}{v}, 0) = (\frac{L}{\gamma v}, 0)$$
So $L' = \frac{L}{\gamma}$.

You'll get the same result using any other origin, of course. The important thing is to always do calculations using one set of coordinates, never to mix the two (and never to assume *anything* about how the two are related: use only the Lorentz transformation to convert).

I have no doubt that we can solve for $L' = \frac{L}{\gamma}$. The question is: what is the mistake in his manipulation of the same equation such that he obtains $L' = \gamma L$ ?