Does Thomas Precession violate angular momentum conservation?

[DaveLush]

Are you denying that spin g and orbital g have different values.?

..

No, I am not saying that the g for orbital angular momentum is not unity.

I am saying that the g that appears in that equation has come from the electron spin and its resulting intrinsic magnetic moment.

g is not directly about precession frequencies but rather about how much magnetic moment a given amount of angular momentum produces. So, the description of the magnetic field that is causing the precession of the orbit must necessarily have a g in it if it is the electron's intrinsic angular momentum that is producing the field. That's all wrapped up in that equation.

As I said previously it's not totally trivial to get that result and perhaps when I can post it it will become more transparent.

 

[DaveLush]

In this post I will derive my equation 3 from post #23 of this thread:

\omega_L = \frac{ge^2s}{2c^2mR^3}\nonumber


\omega_L is the (average) precession frequency of the electron orbit in the laboratory frame, due to the electron intrinsic magnetic moment.

To begin, the magnetic field of a magnetic moment \stackrel{\rightarrow}{m}} is (from Jackson 5.56 (2nd ed for proper units))

\stackrel{\rightarrow}{B} = \frac{3\stackrel{\rightarrow}{n}\left(<br /> \stackrel{\rightarrow}{n} \cdot \stackrel{\rightarrow}{m} \right) -<br /> \stackrel{\rightarrow}{m}}{R^3}

where \stackrel{\rightarrow}{n}} is a unit vector from the magnetic dipole to the field point. The electron intrinsic magnetic moment will be represented as

\stackrel{\rightarrow}{\mu} = -\frac{ge}{2 m c} \stackrel{\rightarrow}{s}=<br /> -\frac{ges}{2 m_e c}<br /> \hat{s}

The torque on the proton orbit around the electron in the electron rest frame is then

\stackrel{\rightarrow}{\tau} = \stackrel{\rightarrow}{r}_p \times<br /> \left(\frac{e}{c}\stackrel{\rightarrow}{v}_p \times \frac{3\stackrel{\rightarrow}{n}<br /> \left( \stackrel{\rightarrow}{n} \cdot \stackrel{\rightarrow}{\mu} \right) -<br /> \stackrel{\rightarrow}{\mu}}{R^3} \right)

where \stackrel{\rightarrow}{v}_p is the proton velocity in the electron rest frame. The above can be reduced to

\stackrel{\rightarrow}{\tau} = \frac{2e}{cR^2}<br /> (\stackrel{\rightarrow}{n} \cdot \stackrel{\rightarrow}{\mu}) \stackrel{\rightarrow}{v}_p

Apart from Thomas precession, the proton velocity in the electron rest frame is the negative of the electron velocity in the lab frame. The effect of Thomas precession at a Bohr radius is negligible here due to the Thomas frequency being four orders of magnitude less than the orbital frequency (at the Bohr radius). The above torque is also time-varying over the orbital period. Rewriting in terms of the circular orbit electron velocity and averaging over the orbit obtains that

\langle \stackrel{\rightarrow}{\tau} \rangle = -\frac{e^2 }{c R^{5/2} \sqrt{m}} \stackrel{\rightarrow}{\mu} \times \hat{L}

Now, from post #23 we had that the torque on the orbit in the laboratory frame is equal to the torque on the orbit in the electron rest frame, apart from a factor of gamma. This allows us to write down for the equation of motion of the orbit in the lab frame (also using L=e*sqrt(mR)):

\langle \stackrel{\rightarrow}{\tau} \rangle = \langle\dot{ \stackrel{\rightarrow}{L}}\rangle = \stackrel{\rightarrow}{L} \times \frac{e}{c R^{3} m_e} \stackrel{\rightarrow}{\mu}

Rewriting the magnetic moment in terms of the spin magnitude, etc, as per the third equation of this post results in the equation of motion for precessional motion with angular velocity magnitude given by the first equation of this post.

I want to mention that I am bothered by the averaging that is required to obtain this result. This also constitutes an issue with respect to angular momentum conservation. The Muller paper mentioned upthread may provide a way to fix this as it claims Thomas precession is also time-varying over the orbit. I haven't yet tried to do it though.

 

[DaveLush]

I want to mention that m_e and m are the same above, meaning the electron mass (and not the magnitude of the magnetic moment. Tried to fix it but too late.

 

[DaveLush]

For anyone interested, I want to report I am still working trying to reconcile Thomas's results in his 1927 paper with mine which are directly opposed. I think I may be getting close. He says his Eq. 6.72 may be obtained using the force (his 5.1) based on the Abraham spherical-shell-of-charge electron model. It is his 6.72 which differs from my equation of motion for the orbit (as above), and results in the conflict. I get mine from assuming the electron has an ordinary and constant-magnitude magnetic moment, and in the electron frame this creates a torque on the proton orbit. I mentioned above another way to do it by computing in the lab frame the torque on the orbit due to the anisotropy of the proton electric field, and that a moving magnetic dipole relativistically acquires an electric dipole moment, and hence there is a translational force on the orbiting electron in an anisotropic electric field, that on average is a torque on the orbit.

If I understand what Thomas did, he did it still another way, based on the force on the electron magnetic moment in the electron rest frame, according to Abraham's model, in the anisotropic magnetic field of the orbiting proton. This would be the reaction force, expected to be equal and opposite under Newton's third law, to the magnetic force on the moving proton in the electron intrinsic magnetic field. This reaction force on the electron in the electron rest frame is a "Stern-Gerlach" type of force, that is, the same sort of force that is used to sort spin-up from spin-down silver atoms in the Stern-Gerlach experiment. However in the Abraham model the resulting force law is different from what would be the force on an ordinary bar magnet, as Thomas puts it.

This is an illustration of how tricky and subtle classical electrodynamics can be. People are in fact still writing papers about this very area. Boyer posted one on the pre-print archive within the last year or so. I'm not sure if it's been published yet but previous ones by him and others (Hnizdo, Vraidman, others) were published particularly in Am Jour phys in the 90s. Some of these are cited in the 3rd edition of Jackson at the end of chapter 5. This is related to the somewhat controversial idea of "hidden" momentum.

So, anyhow, my plan here is to try to reproduce Thomas's result and if it holds up then determine what the corresponding force law on the moving proton would have to be. That can presumably be assessed for plausibility and further implications. I have to wonder whether something that has a different force law than a standard magnetic moment can still have the same potential energy energy of orientation in a magnetic field, as is required to match observed atomic spectra.

People are still working on classical electron models, as well, by the way. In particular, there is one by Kiessling and Appel. Personally I don't think any of these spherical-shell "fat" electron models are likely to be true (for obvious reasons and as noted by Thomas, e.g. that the equatorial speed of rotation has to be ~200c to yield the correct value of spin), but that doesn't mean they are not interesting and worth considering what are their dynamical behavior. Seems to me the zitterbewegung model that has a point-like electron acquiring spin due to runaway motion is the right one. Its full description awaits the full solution of the electrodynamic two-body problem. A good overview of this with historical background is the De Luca 2006 Phys Review E paper.

 

[DaveLush]

I have written a paper summarizing my findings with respect to Thomas's 1927 paper. I posted it on arXiv last week. Searching my name will turn it up.

I would like to submit it somewhere but I don't know where it would have a chance to get reviewed. I will welcome any recommendations.

People tell me it must be wrong but it seems to me it's so simple that it can't be.

I wonder if anyone would be willling to really go through it, as opposed to simply telling me I need to find the error. That would be great and I'll try to take criticism stoically. I'm not very confident I'll be able to find a journal that will send it out for review. The math is very elementary so it should not be difficult to go through. Thanks.

 

[DaveLush]

I came to what was for me a surprising realization the other day, that in spite of total angular momentum not being conserved in the presence of Thomas precession, the system can be nonetheless non-radiative, provided that the electron gyromagnetic ratio is twice the classically-expected value. Which of course, it is.

I got a Malykin review paper on Thomas precession which is very informative, and he says that Bagrov has a Russian-language monograph that proves Thomas precession does not cause radiation. This seemed impossible based on what I've been seeing, and it seemed difficult to get the Bagrov book and understand it, but then I realized I could calculate the magnetic dipole radiation of the system easily enough, so I did, and sure enough it did turn out to vanish provided g=2.

Then on further reflection this didn't seem surprising at all. Consider if we had two purely-classical magnetic moments forming an isolated system and interacting electromagnetically. Say, two superconducting coils loaded up with currents, in space, a little one inside a big one. These two coils mutually precess but angular momentum is conserved and they don't radiate. Both the total angular momentum and total magnetic moment are stationary.

Now suppose I come along and I magically make one coil twice as good at producing magnetic dipole moment as the other. In other words, I give it a g-factor of two. Now it is impossible to have both the total angular momentum and total magnetic moment of the two coils be stationary, if the two coils themselves are nonstationary. (I can prove this later easily enough if it's not obvious.) In the absence of Thomas precssion, then, we would have that the total angular momentum is a constant of the motion but the total magnetic moment would move, causing radiation. However, present Thomas precession, with g=2, the total magnetic moment is a constant of the motion, while the total angular momentum precesses. The Thomas precession introduces the famous "Thomas factor" of a half, which undoes the effect of the g-factor being two as far as the motion of the total magnetic moment is concerned. This suggests to me that possibly the g-factor of two is itself some sort of consequence of the Thomas precession.

I made an update to my arxiv paper to incorporate the calculation. The comment for version 3 provides the section number.

Incidentally, I submitted the paper to Physical Review E, but they would not send it out for review. Well here is what they said:

We are sorry to inform you that your manuscript is not considered suitable for publication in Physical Review E. A strict criterion for acceptance in this journal is that manuscripts must convey new physics. To demonstrate this fact, existing work on the subject must be briefly reviewed and the author(s) must indicate in what way existing theory is insufficient to solve certain specific problems, then it must be shown how the proposed new theory resolves the difficulty. Your paper does not satisfy these requirements, hence we regret that we cannot accept it for publication and recommend that you submit it to a more appropriate journal, such as the American Journal of Physics.

 

[Hans de Vries]

Just a note:

Force on a magnetic dipole from a circular current (axial dipole):

F=\nabla\Big(\vec{m}\cdot B\Big)

Force on a magnetic dipole from two opposite monopoles (vector dipole):

F=\Big(\vec{m}\cdot\nabla\Big)B

There is a remark from Thomas on page 13 in which he expects the second
equation for the force on a small magnet in a non-uniform field while it should
be the first actually.

(The difference is this "hidden momentum" in your equation 10 and further)


Regards, Hans

 

[Hans de Vries]

Written in explicit matrix notation:

Lorentz force on an electric Vector dipole \vec{\mu} in an E field:

<br /> \begin{array}{l l |l l l| l | l | l}<br /> &amp; &amp; \partial_x \textsf{E}_x &amp; \partial_y \textsf{E}_x &amp; \partial_z \textsf{E}_x &amp; &amp; \mu_x &amp; \\<br /> \vec{F}_{elec} &amp; =\ - &amp; \partial_x \textsf{E}_y &amp; \partial_y \textsf{E}_y &amp; \partial_z \textsf{E}_y &amp; \cdot &amp; \mu_y &amp; \quad =\ -\,\partial_j E_i\, \mu_j \\<br /> &amp; &amp; \partial_x \textsf{E}_z &amp; \partial_y \textsf{E}_z &amp; \partial_z \textsf{E}_z &amp; &amp; \mu_z \\<br /> \end{array}<br />Lorentz force on a magnetic Axial dipole \vec{\mu} in a B field:

<br /> \begin{array}{l l |l l l| l | l | l}<br /> &amp; &amp; \partial_x \textsf{B}_x &amp; \partial_x \textsf{B}_y &amp; \partial_x \textsf{B}_z &amp; &amp; \mu_x &amp; \\<br /> \vec{F}_{mag} &amp; =\ +\, &amp; \partial_y \textsf{B}_x &amp; \partial_y \textsf{B}_y &amp; \partial_y \textsf{B}_z &amp; \cdot &amp; \mu_y &amp; \quad =\ +\partial_i B_j\, \mu_j \\<br /> &amp; &amp; \partial_z \textsf{B}_x &amp; \partial_z \textsf{B}_y &amp; \partial_z \textsf{B}_z &amp; &amp; \mu_z \\<br /> \end{array}<br />

(Note the swapping of the i and j indices at the right hand sides)Regards, Hans

 

[DaveLush]

Just a note:

Force on a magnetic dipole from a circular current:
F=\nabla\Big(\vec{m}\cdot B\Big)

Force on a magnetic dipole from two opposite monopoles:
F=\Big(\vec{m}\cdot\nabla\Big)B

There is a remark from Thomas on page 13 in which he expects the second
equation for the force on a small magnet in a non-uniform field while it should
be the first actually.

Regards, Hans

Yes and I have puzzled over what was his basis for expecting that. Perhaps it was commonly written that way in textbooks and with an implicit assumption that the electric field was nonvarying.

Of course if you derive what the force should be, you get the first law, as Jackson does in all three editions. However when you add in the hidden momentum, an "effective" force is obtained that is the second case plus a term in m-dot. This is described in the Jackson 3rd edition, and it traces back to Shockley and James, and Furry.

Thomas's analysis of the first law is correct but he doesn't notice or mention that it breaks Newton's third law, and thus violates conservation of linear momentum, which is of course the problem that revealed the need for including the hidden momentum in the equation of motion. With the hidden momentum, linear momentum is conserved but not the (secular) angular momentum. Without the hidden momentum, secular angular momentum is conserved but not the linear momentum. This is only a natural consequence of starting with an equality and then adding a nonzero term to one side.

Another interesting thing he mentions is that Abraham arrived at a spinning electron model with g=2, based purely on classical electrodynamics (a spinning sphere of charge). That seems contrary to what is taught or what I have ever read. I thought g=2 is thought to be essentially non-classical.

I have a link to the Abraham 1903 paper, but I don't know any German, and I think, even if I did, it will still be hard to read his notation. However if anyone is interested in this, I would be interested to know if Thomas's remark can be substantiated, and can post the link after I hunt it up.

I suspect it is a mistake, though. We have to give Thomas a break because he was only an undergraduate at the time. That paper was submitted by Bohr though and reviewed by Pauli also.

 

[DaveLush]

Thanks for looking at my paper, Hans.

You are one of very few, I suspect.

 

[Hans de Vries]

Thanks for looking at my paper, Hans.

You are one of very few, I suspect.

It's a very interesting (and very fundamental) subject. I will go on reading it :^)
This is the kind of stuff which is handled in chapter 8 of my book in progress.

Regards, Hans

 

[Hans de Vries]

I have a link to the Abraham 1903 paper, but I don't know any German, and I think, even if I did, it will still be hard to read his notation. However if anyone is interested in this, I would be interested to know if Thomas's remark can be substantiated, and can post the link after I hunt it up.

This is always interesting. Google does optical character recognition of documents
scanned in as images. You might use google's translator to convert the text to English.

Regards, Hans

 

[Hans de Vries]

Just a tip on notation since you are using Jackson's formula's for the
Lorentz transform which are unnecessary complex:


E&#039; ~=~ \gamma E ~-~ \frac{\gamma^2}{\gamma+1}\,\vec{\beta}(\vec{\beta}\cdot E)

can be written simpler using the unit vector of \hat{\beta}

E&#039; ~=~ \gamma E ~-~ \frac{\beta^2\gamma^2}{\gamma+1}\,\hat{\beta}(\hat{\beta}\cdot E)

Using 1-\beta^2\gamma^2=\gamma^2 you can simplify this to

E&#039; ~=~ \gamma E ~+~ (1-\gamma)\,\hat{\beta}(\hat{\beta}\cdot E)

With E_\parallel as the component parallel to \hat{\beta} this becomes:

E&#039; ~=~ \gamma E ~+~ (1-\gamma)\, E_\parallel

and finally:

E&#039; ~=~ \gamma E_\bot ~+~ E_\parallelRegards, Hans

 

[DaveLush]

Here is the link to Abraham 1903, with thanks to Paul de Haas, whose historical physics papers site is linked upthread:

http://www.hep.princeton.edu/%7Emcdonald/examples/EM/abraham_ap_10_105_03.pdf

I had earlier spent a little bit of time trying to figure if the google translator could do a scanned document but didn't see how. Maybe I will try again later. Understanding what Abraham really did has gotten pushed down in priority compared to other things I'm currently doing, though. Also it seems so unlikely. I guess if I thought there was a serious chance Abraham really did derive g=2 classically I would be more inclined to spend the time.

Thanks for the reformulation of the field transformations, Hans, I wasn't aware it could be viewed so simply and that should be helpful.

Also on my first reading of your initial post I don't think I took your complete meaning and in future I should probably add the term "current-loop" to clarify the kind of magnetic dipole I'm referring to, but that has always been the current-loop kind.

I want to mention, I need to do an update to my paper as that last update was very hastily done upon my discovering that the total angular momentum could be moving around and yet the system be non-radiative, generally. Up until that point, I was trying to show that there was a unique orbital angular momentum that would lead to a non-radiative condition, and that would be the one with L= h-bar. Now that idea is dashed as clearly so long as g=2 the system is nonradiative (with respect to magnetic dipole radiation, I mean; I am not claiming electric dipole radiation does not occur) regardless of L. So anyhow, that new section VIIC and the one following are still a little mixed up about equating the mutual precession frequencies being possibly relevant to radiativity (if that is a word). I am hoping to do a more extensive update however pretty soon that will also address the energetics of the spin-orbit coupling. I did not really recognize it until after I had submitted it to Phys Rev E, but that paper raises the question about whether and how Thomas's contention that his relativity precession obtains the proper spin-orbit coupling holds up when the hidden momentum is incorporated. I had been thinking that that all would carry over but now I think there is no reason to think that. I also think that may be a way to figure out what is the meaning of that the spin and orbit mutual precession frequencies equate at L=h-bar, since "m dot B" for the energy of a magnetic dipole m in a magnetic field B goes over to "omega dot s" for the precession freqency omega of the angular momentum of the dipole, s.