# Does Thomas Precession violate angular momentum conservation?

## Main Question or Discussion Point

Suppose we have an accelerated rigid body that possesses angular momentum. Then the body will undergo Thomas precession, and if the angular momentum vector of the body is not aligned with the axis of the Thomas precession, the angular momentum vector of the body will precess when viewed in an inertial frame. The angular momntum thus changes in time in the absence of any externally-applied torque. This would seem to me to be in conflict with the principle of conservation of angular momentum.

I did a limited literature search and found one paper related to this question, by Muller, "Thomas Precession: Where's the Torque?", Am. J. Phys. 60(4). Muller says that Thomas precession actually does cause a torque, and so angular momentum conservation is not violated. Seems to me though that since the Thomas "torque" is not externally applied, it does not remedy the situation.

Am I missing something obvious here? Does anybody know of additional discussion of this in the literature?

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Jonathan Scott
Gold Member
I've not worked this through in detail, but there seems to be a fairly obvious explanation.

Consider how you would accelerate a fast-moving rigid body in a direction that is not parallel to its motion. A force applied simultaneously to the front and back of the object in its own rest frame would not act exactly simultaneously in the frame where it is fast-moving (and vice versa).

I assume that this tiny difference would impart the necessary rotation to conserve angular momentum. This would also mean that there would be a reaction back on the source of the accelerating force because the tiny lag would mean that the force would be very slightly out of alignment with the center of mass of the object, creating a torque on it.

I've not worked this through in detail, but there seems to be a fairly obvious explanation.

Consider how you would accelerate a fast-moving rigid body in a direction that is not parallel to its motion. A force applied simultaneously to the front and back of the object in its own rest frame would not act exactly simultaneously in the frame where it is fast-moving (and vice versa).

I assume that this tiny difference would impart the necessary rotation to conserve angular momentum. This would also mean that there would be a reaction back on the source of the accelerating force because the tiny lag would mean that the force would be very slightly out of alignment with the center of mass of the object, creating a torque on it.
Thanks for the reply. It seems you take my point that Thomas precession at a minimum presents an appearance of mechanical angular momentum conservation violation.

Perhaps I shouldn't have said "rigid body" when I am actually thinking of the electron spin in spin-orbit interaction. I was hoping to avoid a discussion about the applicability of classical physics to the atomic domain. This is probably unavoidable and I am prepared to argue the case but I would like to put it off if possible.

Maybe what you conjecture is true for a macroscopic body. Like, if we had two macroscopic non-conducting rigid bodies with static charge density on their surface. Keeping the charges static would hopefully simplify things. I would certainly like to hear about it if you make some calculations. This type of problem even in the absence of strong acceleration and consequent Thomas precession is interesting in itself and there are possibly still some unsettled issues about it. Do you know the term "hidden momentum"? I think it might even be in the new Jackson (which I don't have - I have the 2nd ed), but there were a number of papers about it in the 90s in Am. J. Phys. by Hnizdo, Namias, and others. There is a recent one on the archive by Boyer.

I can't say I'm seeing how it would fix my problem though.

For the electron in a circular Rutherfordian orbit around a proton (this is legitimate at least in the limit of large quantum numbers - per Jackson and everyone) we really do have that Thomas precession directly affects the observed precession frequency in the lab frame. For g=2 it exactly halves it, of course, leading to the "Thomas" factor of a half in the spin-orbit interaction energy.

I have worked through this explicitly (will provide on request) and I get that the total angular momentum cannot be a constant of the motion, due explicitly to Thomas precession, for any orbital radius, if the electron spin axis is not aligned with the orbital angular momentum vector. If there were no Thomas precession, the total would be constant for any relative orientation.

I think the implications of this, if any, should be looked into.

Stingray
You have to be more careful about the meaning of momentum conservation in relativity. I don't know your background, but I think the misunderstanding disappears if momenta are introduced in a more geometric way than is standard in introductory treatments. Hopefully this will make sense.

All of the standard energy and momentum conservation laws in special relativity derive from symmetries of Minkowski spacetime. In particular, angular momentum conservation comes from invariance under rotations. This is most elegantly expressed by defining a component of some object's momentum "generated" by the Killing vector associated with some symmetry. Let

$$P = \int_\Sigma T^{a}{}_{b} \xi^b \mathrm{d} \Sigma_a$$

be that momentum. It will be angular if the Killing vector $\xi^a$ represents a rotation. This integral is to be carried out over some appropriate hypersurface going through the worldtube of a body with stress-energy tensor $T^{ab}$. Now stress-energy conservation $\nabla_a T^{ab} =0$ can be used to show that this quantity does not depend on the surface of integration. It is therefore a conserved quantity. The same thing could be directly derived from an action too. I just find this presentation more straightforward. In any case, the relativistic statements of linear and angular momentum conservation are of this form.

Now how does it relate to more standard treatments? It is common to define an explicit angular momentum tensor $S^{ab} = - S^{ba}$. This can be related to the usual vector angular momentum only in the body's center-of-mass frame. There is also a 4-momentum $p^a$. It turns out that the P I wrote down above is always a linear combination of these two objects. For some $A_a$ and $B_{ab} = - B_{ba}$ (these are actually very simple to relate to the Killing vector, but that's a detail), it happens that

$$P = p^a A_a + \frac{1}{2} S^{ab} B_{ab}.$$

If these tensors are defined along some (possibly accelerating) worldline $z(s)$, it may be shown that

$$0 = \dot{P} = \mathrm{d}P / \mathrm{d}s = \dot{p}^a A_a + \frac{1}{2} (\dot{S}^{ab} - 2 p^{[a} \dot{z}^{b]} ) B_{ab} .$$

This must be true for all possible coefficients, so we separately have the conservation laws $\dot{p}^a = 0$ and

$$\dot{S}^{ab} = 2 p^{[a} \dot{z}^{b]} .$$

Components of the angular momentum tensor are not generally conserved in an inertial reference frame. The extra term here represents Thomas precession. This is despite the fact that there is something that deserves to be called conservation of angular momentum: namely the fact that P's derived from rotational symmetries are conserved.

Edit: This is technically a little incomplete. Everything is more "standard" if an external force is introduced. This is easy to do, but I think what's written is enough to see the point I was trying to illustrate.

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Thanks for the reply, Stingray. I can't follow the first part but I think the second part I can get some meaning from.

I wasn't really expecting that the laws of relativity are being violated, and also I expect that this is only happening in a system where there is electromagnetic interaction going on and so radiative fields are carrying away any angular momentum imbalance.

I still think there is something interesting going on in the sense that it seems to me that we can have a system where in the laboratory frame the total mechanical angular momentum of the isolated system can be changing in time in the absence of any externally applied field and associated torque. The system here can be a hydrogen atom in an excited state where quantum effects are negligible, or a Rydberg atom, similarly. Would you say this is allowed according to your much deeper than my understanding of SR?

I guess, in the classical limit the mechanical angular momentum non-conservation would be negligible (though calculable and finite) compared to the classical radiative decay but transitioning into the quantum domain it would become significant.

You probably know that for large principle quantum number the radiation frequency and decay rate of hydrogen (or a Rydberg atom) is consistent with a circular-orbit Rutherford model, with (electric) dipole radiation due to the orbiting electron and decay due to radiation damping from the acceleration rate of the electron. I'm trying to extend classical physics inward a little further here by adding in another classical (i.e. non-quantum but not non-relativistic) effect. This would predict magnetic dipole radiation seems to me, which might be detectable in Rydberg atom experiments.

It's just a little hobby project for an engineer.

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my point ... Thomas precession at a minimum presents an appearance of mechanical angular momentum conservation violation.
I think you are correct and we have observations that show it. The key word being the “appearance” when evaluated in a classical manner.

Replacing your accelerating frame with a gravitational one, the orbital precession of Mercury is in principle a similar point that long demonstrated an angular momentum conservation violation that vexed science for decades. The solution (or at least a good mathematical analogy) that accurately describes both is of course GR. As detailed by stingray.

The biggest difference between the two is the Mercury issue comes with real world measurements.

Scott’s point on “force applied simultaneously to the front and back of the object in its own rest frame would not act exactly simultaneously … “ is something I would view as an alternative to the relativistic one. Mainly because you would need to select a preferred frame of reference in nature to work it out. Might be interesting to work out the differences in the two approaches on your “electron spin in spin-orbit interaction” issue assuming Macroscopic scales for the non GR approach.

But working that alternate view out would be challenging as you would have to ignore some SR rules like Preferred Frame etc.

RandallB thanks for the reply but I am not quite seeing how precession of an orbit perihelion violates (or appears to violate) angular momentum conservation. I am still thinking about it though.

At the moment I just wanted to report that I found an interesting old paper that seems to directly address my question. It is "Factors of 2 in magnetic moments, spin-orbit coupling, and Thomas Precession", Am.J.Phys. 61(6), by H.C. Corben. He appears to be getting that relativistic angular momentum conservation requires that the g-factor for spin is two. That would seem to confirm that this is an interesting issue.

I'm going to study this paper for a while. I have a wild conjecture that it might be even more interesting than Corben shows, that it might be part of the reason for quantum behavior. I know this sounds crazy but I have already gotten one result that I believe is not previously known, from a semiclassical treatment that is a planetary atomic model but with spin, that the ground-state Bohr radius of hydrogen may be found exactly by equating the spin and orbit precession frequencies. I would be interested if anyone is already aware of this. It is not too difficult to show, but it is hard to interpret what it means, at least for me so far. BTW I am not keeping any secrets, and will be happy to share any and all of my hobby project work on request, if anyone is interested.

I think the article by Muller that you cite, "Thomas Precession: Where's the Torque?", Am. J. Phys. 60(4), more or less explains it all. What do you think is wrong with it?

I think the article by Muller that you cite, "Thomas Precession: Where's the Torque?", Am. J. Phys. 60(4), more or less explains it all. What do you think is wrong with it?

In classical mechanics we would have that the time derivative of the angular momentum is zero in the absense of an applied torque. For an isolated system, the total angular momentum is a constant of the motion unless there's an external torque. With Thomas precession, though, one can have a bound, but isolated system, with no obvious externally-applied torque, and the total mechancial angular momentum is precessing due to the Thomas precession. So, if there is an equivalent applied torque due to Thomas precession (as Muller says), then where is it coming from? Seems to me it has to come from electrodynamics, from the radiation reaction. Radiation reaction on a moving magnetic dipole isn't in Jackson but I've seen it published. It would be interesting to see if this is numerically equally to Muller's torque. Then the radiation fields will be guaranteed to carrying away the angular momentum imbalance due to the reaction torque, as well. I think it would be a completion of the story, if it adds up.

If not, then the question would still be how is angular momentum conserved in this system?

If it does add up also then this predicts that Thomas precession causes radiation and makes one wonder if this is detectable and can be reconciled with experiment or current knowledge about spectral intensities and atomic decay times and so forth.

Total Angular Momentum is Conserved

The total angular momentum of the body is conserved. The spin angular momentum of the orbiting body is converted to the orbital angular momentum and vice versa. This is discussed in section IV of the Muller article.

The total angular momentum of the body is conserved. The spin angular momentum of the orbiting body is converted to the orbital angular momentum and vice versa. This is discussed in section IV of the Muller article.
Thanks, electron0511, you have stimulated me to learn something new. It finally struck me last night that the Thomas precession I've been working with is only the average. I had been reading the Muller paper and seeing the words, but associating them with the orbital angular momentum which I am already used to averaging over an orbit. There is an average and a time-varying part there, too. This has always bothered me that this violates angular momentum constancy. But it appears that what Muller and Corben and Hestenes (I will find the Hestenes reference if you are interested (in his 2003 Am J Phys I think)) are saying is that the orbital and Thomas precession orbital-frequency oscillating terms are going to cancel. That will really help my paper on this I'm working on become more compelling, I think.

It's a longer-term momentum nonconstancy that I have been referring to, however. At least I still think it is a separate thing. The total angular momentum (that is, the sum of the spin and orbital angular momenta) cannot be constant if the spin and orbit are precessing at different rates. Saying it is nonconservation of angular momentum is too strong in the electrodynamic sense because radiation can balance the equation, but from a classical mechanics perspective I think it is fair to say it does, because there is no obvious external torque.

Muller, Corben, and Hestenes all talk about how this spin-orbit coupling is purely from relativistic kinematics and doesn't require magnetic moments to manifest. Corben and Hestenes, if I understand them, go on to say that if there is a magnetic moment associated with the spin, then angular momentum conservation will require that the gyromagnetic factor be two. The thing that I am referring to is also an electrodynamic efect that will only manifest if the spinning particle has a charge (and magnetic moment therefore), but I still believe it is a distinct phenomenon that has not been previously noted in the literature. Muller seems to have come the closest but there was some discussion in American Journal of Physics in the 90s by Hnizdo, Vraidman and others (citing back to a 1969 Furry paper) about "hidden momentum" and so forth that is related. Also, the 3rd edition of Jackson references these and Boyer has written about it recently. None of them note what I am referring to, though, so far as I can tell.

I am having a difficult time reconciling the Muller explanation of Thomas precession with the conventional one, such as in Jackson. I don't see how TP can be time-varying in the Jackson treatment as it is in the Muller version. Muller cites Misner Wheeler Thorne as the standard treatment and it does appear in there. It's just that the conventional version is just the cross product of the acceleration and velocity vectors and this is a constant vector for a circular orbit.

I'm confident still I will reconcile it because I have a strong motivation in that I think the time variation will fix up a problem with my project. I continue to try. However in investigating it I have come across something else that seems very interesting and worth mentioning in a post.

I've been studying Thomas's paper of 1927 (which has been scanned and posted on line with a lot of other great historic papers here: http://home.tiscali.nl/physis/HistoricPaper/Historic Papers.html ). Jackson references his section 4 but I am looking at section 6. There he determines a relation between spin and orbital angular momentum that must be satisfied if the angular momentum is to be a constant of the motion (it is at the bottom of page 15, not numbered). Thomas takes this to be a condition on the "electron angular momentum" (does he mean spin, or total?) but it has the orbital angular momentum in it and if you plug in the spin as magnitude hbar/2 then out will come quantized orbits. I wonder if this is or was generally recognized, or if Thomas thought of it this way. I can't see that he clearly did.

I have been trying to leave the discussion of my hobby project to my blog but this is exactly the sort of thing that I've been playing around with, and now what a surprise to discover Thomas was doing it, and of course doing a much better job than me. His expression for angular momentum constancy is similar to the one I got but it has an additional factor that is the "Sommerfeld relativity effect" of the precession of the perihelion. I have known there was such an effect for some time (it is in L&L Classical theory of Fields, at least) but I figured I could ignore it because until recently I've been restricting my analysis to circular orbits. So maybe this is the answer to the topic of this thread. Angular momtum in the presence of Thomas precession can be a constant of the motion provided that Thomas's condition is satisfied, that takes into account the Sommerfeld relativistic effect. I have to wonder though if the implications of this equation are fully recogniozed or have been overlooked. Seems to me that quantized orbits pop right out of it, if one is accustomed to thinking of the spin magnitude as a constant. Maybe Thomas didn't think of it this way, though.

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I retract my statement above that quantization can jump out of Thomas's equation for total angular momentum conservation. It doesn't after all when I tried to do it. The reason I thought that it would is because it appears at first glance to require that the orbital angular momentum magnitude be directly dependent on the spin angular momentum magnitude (K in his notation) but actually there is already a K in his "sigma" term (which is not a spin magnitude or a spin matrix but rather just a scalar constant equal to the rate of precession of the orbit perihelion) so the two Ks cancel out leaving a total angular momentum that doesn't care what the orbital angular momentum is.

Well this is how it is that a lot ideas that seem like they should work don't actually work so well in practice. I believe the important thing is to keep thinking them up and testing them (and keeping one's mouth shut until they have been tested, oops).

I have more to say about this but am thinking it will be more appropriate if I do a blog entry about it, maybe this weekend. The dynamical picture is richer than the kinematical one.

I stand by to be corrected, but I am becoming increasingly convinced Thomas's 1927 paper is seriously flawed. Not the derivation of Thomas precession, nor its existence or description, but rather his analysis of the hydrogen atom. Specifically, and I'll be interested here to know how I have it wrong, in his neglect of the orbit precession in the intrinsic magnetic field of the electron. He has that the electron spin axis precesses due to the proton charge, but states the following with respect to the orbit (on page 15, and I provided a link above to where the paper may be viewed online):

"The secular change of the orbit will be the relativity plus screening precession in its own plane, the Larmor precession of its normal about the external magnetic field, and the effect of the unknown second order terms in equations..."

Secular change just means the time derivative. The second part is the precession of the perihelion, not the orbit normal. Thomas proceeds to suppose that the third part due to the higher-order terms is just what's needed to balance out the angular momentum variation of the spin due to its precession, and assumes that it does. Out of this comes that the total angular momentum is a constant of the motion, that it is consistent to associate a non-orbital angular momentum with the electron, and that the spin axis and orbital plane precess around the total with constant angular velocity. When there is an external magnetic field then the total precesses around it and then the spin and orbit precess around the precessing total. This is exactly the picture that is found in many older and introductory quantum physics texts, as for example in Eisberg and Resnick. I have often wondered what was the original source of that picture, and now I'm confident it's Thomas 1927.

The easiest way to see that something is missing in this picture is to consider the situation in the electron rest frame where the proton is orbiting around the electron. Thomas's analysis is consistent with and uses that the electron spin axis is precessing in the magnetic field of the moving proton in this frame. There is more than just the Thomas precession in his equations. (That is, in his eq. 4.121, which is also in Jackson, the third term of the leading factor on the right, has both the Thomas contribution of one-half plus the g/2 part that is the spin magnetic moment precession in the absence of Thomas precession, in the field of the proton (magnetic or electric depending on frame of reference). So, it seems only necessary to me, ostensible kinematical treatment or no, that the torque on the orbit plane due to the electron intrinsic magnetic moment needs to be included as well. In the electron rest frame this is easy to see as due to the magnetic force on the proton charge moving through the electron's intrinsic magnetic field. In the laboratory frame it is not so easy to see and it mystified me for a year. However I now am fairly confident it is explainable as resulting from the electric dipole moment that is acquired by a moving (electron intrinsic in this case) magnetic dipole. The torque on the electron orbit is the result of the force on the electric dipole moment due to the anisotropy of the electric field of the proton. (When I say I am fairly confident I nean that I have been through the calculation already but haven't yet finished checking it especially for sign errors. But I think it will hold up and it has to under Lorentz covariance because there is clearly a torque there in the electron rest frame.)

It's important to note that this additional force is of first order in the field and velocity, and so not among his various higher-order terms.

I suppose I will have to try putting it into his equation while retaining his catch-all term that covers the higher-order terms to see how it comes out. Maybe these can still balance the total but adding just the electron-frame v-cross-B force on the proton will cause total angular momentum to be not a constant of the motion, which is why I started this thread in the first place.

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Suppose we have an accelerated rigid body that possesses angular momentum. Then the body will undergo Thomas precession, and if the angular momentum vector of the body is not aligned with the axis of the Thomas precession, the angular momentum vector of the body will precess when viewed in an inertial frame. The angular momentum thus changes in time in the absence of any externally-applied torque. This would seem to me to be in conflict with the principle of conservation of angular momentum.

Am I missing something obvious here? Does anybody know of additional discussion of this in the literature?

Very interesting discussion....Dave.

There is such a thing as torque free precession.,...which can arise when rotation is around an axis that is not the axis of symmetry. In such a system the angular momentum vector is not conserved, but the magnitude of the vector (and the kinetic energy) is conserved. The result is a polhode motion (similar to that of the Gravity Probe - B gyroscopes which are giving the NASA data crunchers fits - they think it is due to electrostatic 'patch' effects , but I think differently ;)).

See here for a description of such a torque free precession. http://en.wikipedia.org/wiki/Poinsot's_construction
You may want to read it completely .

(Although I am not real familiar with the deeper quantum aspects of relativistic angular momentum problems, you seem to be operating in a semi-classical way so I think this may be applicable)....and by analogy you may be able to see a similar principle operating here....without having to appeal to an 'external' torque. Of course, in your case it would be more complex since you are dealing with both spin and orbital components....but the principle remains without having to enlist an 'external' torque.

This begs the question as to why there would there be an 'off axis rotation'.
Possibly this may just be a by product of Thomas precession since it results from acceleration and thus may even entail a hint of non-Euclidean geometry ....(obviating the necessity of appealing to radiation reaction or external torque).

Just a thought.

Creator

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reilly
1. If there was a serious problem with Thomas precession, then the Dirac approach to the hydrogen atom would be wanting. It's not.

2. Thomas precession is a consequence of the fact that Lorentz transformations in different directions do not commute -- something to do with the non-commutivity of rotations. And, in spite of this lack of commutivity, angular momentum is conserved for isolated systems, but, in general, not for individual system components.

3. Review the theory of rigid body motion, like gyroscopes and spinning tops, as well as the behavior of angular momentum under spatial translations to get a bit of insight into the details appropriate for the problem discussed here.

Regards,
Reilly Atkinson

1. If there was a serious problem with Thomas precession, then the Dirac approach to the hydrogen atom would be wanting. It's not.
If you read my post carefully, you will see I took care to state that I am not doubting the existence or quantitative description of Thomas precession.

Also in any case I would dispute your categorical statement that a theory can't work if some part of it is in error or misconceived. Theories sometimes do work in spite of having errors in them (an example I can cite here is Sommerfeld's theory of the hydrogen atom and the anomalous Zeeman effect, see Bucher for identification of the errors he made and how it gives the right answer in spite of them) and if that's the case there is often a new and important insight to be gained. If such were true about a theory as important as Dirac's I think it would be quite remarkable.

2. Thomas precession is a consequence of the fact that Lorentz transformations in different directions do not commute -- something to do with the non-commutivity of rotations. And, in spite of this lack of commutivity, angular momentum is conserved for isolated systems, but, in general, not for individual system components.
Again I don't disagree with the existence or common derivations of Thomas precession, merely with the conventional wisdom that the system total angular momentum is a constant of the motion in spite of it. I expect angular momentum will still be conserved but only due to radiative fields.

Thanks a bunch Creator for pointing out the Poinsot's article to me. I had not heard of it previously. Yes I am trying to get to quantum theory starting from classical physics plus the existence of spin, so it may well be relevant to my thinking, at least.

For everyone, here is a direct link to the Thomas 1927 paper:

http://home.tiscali.nl/physis/HistoricPaper/Historic Papers.html

The paragraph I quoted above is just before his equation 6.5 on page 15.

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reilly
For the electron in a circular Rutherfordian orbit around a proton (this is legitimate at least in the limit of large quantum numbers - per Jackson and everyone) we really do have that Thomas precession directly affects the observed precession frequency in the lab frame. For g=2 it exactly halves it, of course, leading to the "Thomas" factor of a half in the spin-orbit interaction energy.

I have worked through this explicitly (will provide on request) and I get that the total angular momentum cannot be a constant of the motion, due explicitly to Thomas precession, for any orbital radius, if the electron spin axis is not aligned with the orbital angular momentum vector. If there were no Thomas precession, the total would be constant for any relative orientation.

I think the implications of this, if any, should be looked into.
So, lay it on us. Your conclusions, if correct, would have far reaching consequences for atomic physics. The devil is in the details. I reserve my right to think it very unlikely that you are correct, but I'm willing to be convinced that you might be.
Regards,
Reilly Atkinson

So, lay it on us. Your conclusions, if correct, would have far reaching consequences for atomic physics. The devil is in the details. I reserve my right to think it very unlikely that you are correct, but I'm willing to be convinced that you might be.
Regards,
Reilly Atkinson
Thanks for asking. Reilly. It's posted on arxiv.org. If you search on author "lush" you will find it uniquely. I would post a link but I have already been chastized for linking to unpublished papers (even not by me).

I tried copying some of it directly here but it doesn't compile correctly as written.

My paper has undergone peer review, and no particular errors have been pointed out, but publication has been denied. I have decided not to submit it further in its current form. I continue working on extending it.

I have posted some of the editor comments here on my blog. Foundations of Physics is the only journal that sent it out for review. At AJP and PLA editors reviewed it. I didn't post the Foundations of Physics reviewer comments here, although I have posted them elsewhere. I'll put them up on my blog here eventually. One of the two FOOP reviewers was quite favorable, I thought. The other was not, but did not say where I went wrong. That was version 1 and I later found a very bad error that led me badly astray in the end part but the initial part that I believe shows the angular momentum problem has continued to hold up. The end parts still need a lot of work and I probably should have held them back but oh well.

The nonconstancy of the total angular momentum is discovered in my section V, equation (36) of version 3.

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I guess the editing time window has closed for the above post. I was planning to delete and replace it with a commitment to get just what I am talking about in this thread up using tex, so nobody will have to read my pretentious paper. The paper covers a lot of other stuff and anyhow I acknowledge it is not generally suitable for discussion here and that was not my intent. I just wanted to make this bit available as soon as possible in response to Reilly's request.

It will take me a little while to get it to look right which I will do with previews and it may take a couple of days but possibly I will have some time to do it tonight.

OK here is how I come to the conclusion that Thomas precession leads to nonconstancy of total angular momentum despite a lack of external torque. My model is a quasiclassical hydrogen atom where the electron orbits the proton. The orbit here will be circular but it does not need to be so restricted necessarily. The proton and electron are point-like charges with usual mass and the electron only has spin angular momentum with a definite orientation, fixed magnitude nominally hbar/2, and an associated intrinsic magnetic moment of one Bohr magneton. The proton mass will be regarded as very large compared to the electron's, so that the proton is stationary in the inertial frame. In the electron rest frame the proton is orbiting the electron, and so the proton orbit will feel a torque in that frame due to the magnetic force on the proton as the proton charge traverses the electron's intrinsic magnetic field. This can be transformed using the standard transformation (i.e. as in Goldstein):

$$\left( \frac{d \stackrel{\rightarrow}{L}}{dt} \right)_{\text{lab}} = \left( \frac{d \stackrel{\rightarrow}{L}}{dt} \right)_{\text{elec}} + \stackrel{\rightarrow}{\omega}_{\text{T}} \times \stackrel{\rightarrow}{L} \nonumber$$

into a torque on the electron orbit in the laboratory frame. $$\stackrel{\rightarrow}{L}$$ above is the orbital angular momentum and $$\stackrel{\rightarrow}{\omega}_{\text{T}}$$ is the Thomas precession angular velocity (as per e.g. Jackson 2nd ed 11.119). The Thomas angular velocity is parallel to the orbital angular momentum, so the rate of change of the orbit orientation is the same in both frames if the relativstic gamma factor may be approximated as one. (Alternatively, as I mentioned above, (I am pretty sure) one can compute the torque on the orbit in the laboratory frame as being due to the translational force on the electron acquired electric diplole moment due to its motion, in the anisotropic electric field of the proton.) The result is the following equation of motion for the orbital angular momentum:

$$\dot{\stackrel{\rightarrow}{L}} = -{\stackrel{\rightarrow}{L}} \times \omega_L \hat{s} \nonumber$$

where $$\hat{s}$$ is the spin axis orientation vector and

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

where g is the electron g-factor, e is the fundamental charge, m is the electron mass, R is the orbital radius, c is the speed of light. The equation of motion for the spin may be gotten to even more easily or found in textbooks (e.g. Jackson 11.108) and may be written as

$$\dot{\stackrel{\rightarrow}{s}} = -{\stackrel{\rightarrow}{s}} \times \omega_s \hat{L} \nonumber$$

where $$\hat{L}$$ is a unit vector in the direction of the orbital angular momentum and for the circular orbit

$$\omega_s = \left( \frac{g}{2} - \frac{1}{2}\right) \left( \frac{e^3} {c^2 {m}^{3/2}R^{5/2}} \right)\nonumber$$

The leading factor on the right is the "Thomas factor" that accounts for the factor of a half in the anomalous Zeeman effect with g=2. Now we have what we need to evaluate the time derivative of the total angular momentum. In the absence of an external field, we expect this to be zero so let us write

$$\dot{\stackrel{\rightarrow}{J}} = \dot{\stackrel{\rightarrow}{L}} + \dot{\stackrel{\rightarrow}{s}} = -{\stackrel{\rightarrow}{L}} \times \omega_L \hat{s} - {\stackrel{\rightarrow}{s}} \times \omega_s \hat{L} = (L\omega_L - s\omega_s) \hat{s} \times \hat{L} = 0 \nonumber$$

For non-aligned spin and orbital angular momenta, this leads to allowed orbital angular momentum given by

$$L = \frac{\omega_s}{\omega_L} s \nonumber$$

Substituting for $$\omega_L\)$$ and $$$$\omega_s$$$$
and reducing yields

$$L = \left( \frac{g}{2} - \frac{1}{2}\right) \left(\frac{2 e {m}^{1/2} R^{1/2}}{g}\right) \nonumber$$

Also it is easy to get for circular orbits that

$$L = e \sqrt{mR} \nonumber$$

and so

$$L = \left( \frac{g}{2} - \frac{1}{2}\right) \left(\frac{2 L}{g}\right) \nonumber$$

which requires for nonzero L that

$$g = g - 1 \nonumber$$

in order for constancy of the vector total angular momentum to be achieved. This is a contradiction, for all finite values of g. Therefore, there exist no radii where angular momentum is constant, for circular orbits, where the spin and orbital angular momenta are not either parallel or antiparallel.

I want to mention that I have also done this accounting for a finite proton mass and the result is the same.

Also, BTW, if the spin and orbit angular velocity magnitudes are equated, the Bohr radius is obtained.

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This can be transformed using the standard transformation (i.e. as in Goldstein):

$$\left( \frac{d \stackrel{\rightarrow}{L}}{dt} \right)_{\text{lab}} = \left( \frac{d \stackrel{\rightarrow}{L}}{dt} \right)_{\text{elec}} + \stackrel{\rightarrow}{\omega}_{\text{T}} \times \stackrel{\rightarrow}{L} \nonumber$$
So where in your calculation is your value for Thomas precession frequency , omega(t) ???

Since Thomas Precession is a kinematic effect totally separate from the EM spin orbit interaction it must be added in seperately as shown in Goldstein eqn. Correct?

Where is your value for omega(t) in terms of radius etc . ?

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So where in your calculation is your value for Thomas precession frequency , omega(t) ???

Since Thomas Precession is a kinematic effect totally separate from the EM spin orbit interaction it must be added in seperately as shown in Goldstein eqn. Correct?

Where is your value for omega(t) in terms of radius etc . ?

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Thanks for reading it carefully, Creator.

The Thomas precession is the omega_T in the equation you copied, I think you realize.

If the relativistic gamma may be approximated as one then omega_T according to Jackson is (1/2c^2)*(vector acceleration cross vector velocity). For a circular orbit we will have that a=v^2/R and v = e/sqrt(mR) and perpendiculararity so the magnitude of the cross product is the product of the vector magnitudes. This will give omega_T = e^3/(2c^2m^(3/2)R^(5/2)). This appears in my equation above for omega_s. That equation is basically in agreement with the standard treatment, which is not usually done the same way but is equivalent and yields the same famous (g/2-1/2) factor that is the Zeeman anomaly.

Note, the Thomas precession doesn't affect the transformation of the orbital angular momentum from the electron to the lab frame, at least to first order, because the angular velocity vector is parallel to the orbital angular momentum. But the transformation equation for the spin has the omega_T crossed on the spin and these are not generally parallel so we end up with the minus a half in the (g/2-1/2) factor.