# Talk:Thomas precession

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## error report

Seems to be an error on the canonical form, a (1/c squared) is hanging without a differential, I think it should be dropped Bbharim (talk) 09:42, 25 December 2007 (UTC) bbharim

Seems to be an error in the form of ds^2. I understand it should be:

(ds)^2=(c dt)^2 - (dr)^2 - (r d phi)^2 - (dz)^2

The present form is lacking the c scaling for dt. —Preceding unsigned comment added by Stephen Elliott (talkcontribs) 19:27, 3 November 2008 (UTC)

It's just using units in which c=1, as is commonly done in relativity.--76.167.77.165 (talk) 02:50, 16 October 2009 (UTC)

## poor choice of derivation

The derivation is basically copied directly, without credit, from the book by Rindler. It's also a poor choice of derivation. Rindler is introducing the reader to general as well as special relativity, so it makes sense for him to use GR in the derivation. For this article, it would be preferable to give an explanation in terms of special relativity, such as the one in Jackson's Classical Electromagnetism.--76.167.77.165 (talk) 00:25, 3 October 2009 (UTC)

The material has been removed as a copyvio. Charvest (talk) 01:14, 16 October 2009 (UTC)

## factor γ?

I appreciate efforts of all contributors to improve this WP article, but it seems to me that lengthy quotation of a paper that contradicts to long well-established results violates WP policies.

Concerning the spin precession rate, let me confirm that:

1. Two articles in External links Mathpages article on Thomas Precession and Alternate, detailed derivation of Thomas Precession are in agreement with Jackson's Classical Electromagnetism. Jackson is in agreement with Møller.

2. The recent works by Rhodes and Semon and Krivoruchenko are in agreement with C. Møller.

3. C. Møller is in agreement with the first paper on Thomas precession published by L. Föppl and P. J. Daniell, "Zur Kinematik des Born'schen starren Körpers", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 519 (1913).

4. G. B. Malykin suggests in Phys. Usp. 49, 83 (2006) that the spin precession rate is γ = 1/√(1 - v²/c²) times lower than that found by Föppl and Daniell, Møller and others.

I actually see neither conflicting results nor serious discussions among the experts. I thus remove the lengthy quotation of Malykin's conclusions and leave one neutral reference.

It was David Hilbert who recommended the work of Föppl and Daniell for publication. This work illustrates again the excellence of the Göttingen school.

--Gerasime (talk) 06:00, 2 May 2010 (UTC)

## Velocity-addition formula and Thomas precession suggestion (copied from Wikiproject physics and continued here)

Velocity addition in relativity is, in the usual presentation in a textbook something quite easy. It is captured in essence by the formulae ("standard configuration")

${\displaystyle v_{x}={\frac {v_{x}'+V}{1+{\frac {V}{c^{2}}}v_{x}'}},\quad v_{y}={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}v_{y}'}{1+{\frac {V}{c^{2}}}v_{x}'}},\quad v_{z}={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}v_{z}'}{1+{\frac {V}{c^{2}}}v_{x}'}}}$

and, when confined to a plane,

${\displaystyle v={\frac {\sqrt {v'^{2}+V^{2}+2Vv'\cos \theta '-({\frac {Vv'\sin \theta '}{c}})^{2}}}{1+{\frac {V}{c^{2}}}v'\cos \theta '}},\quad \tan \theta ={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}v'\sin \theta '}{v'\cos \theta '+V}}.}$

Derivations are two- or three-liners.

Velocity addition in relativity is, when taken to the full extent, as mathematically involved as you wish. Goldstein:

• "The decomposition process [describing successive pure Lorentz transformations as a pure Lorentz transformation preceded, or followed, by a space rotation] can be carried through on the product of two pure Lorentz transformations to obtain explicitly the rotation of the coordinate axes resulting from the two successive boosts [that is, the Thomas rotation]. In general, the algebra involved is quite forbidding, more than enough, usually, to discourage any actual demonstration of the rotation matrix"

Physically, it has implications that made people (non-cranks) see paradoxes (Macanu paradox). I believe the situation was not fully resolved until 1990.

Now we have these two articles where Velocity-addition formula actually treats a little advanced stuff truly belonging elsewhere, while (imo) failing to treat the simple stuff (not all formulae there, complicated proof) adequately, which should include the formulae (all of them), full easy proofs and applications, e.g. aberration of light. Thomas precession is entirely nontechnical today.

I suggest we collect the advanced stuff somewhere, perhaps Thomas precession, perhaps a new article Thomas rotation, which would make sense because Thomas precession really is a physical phenomenon with mathematical root Thomas rotation. Lorentz transformation would make sense too, but I don't know whether people want to allow for that article to swell much more. Meanwhile, Velocity-addition formula should be reduced to the basics as given in textbooks. YohanN7 (talk) 14:50, 5 July 2015 (UTC)

Agreed. For supporting the "advanced stuff" in the Thomas precession article, along with Goldstein's classical mechanics and Jackson's classical electrodynamics (both in their second editions), Barut's book Electrodynamics and Classical Theory of Particles and Fields (uses classical methods) and Ryder's QFT (uses Lorentz group theory) could be good sources. Landau and Lifshitz volumes 2, 3, or 4 possibly may have something on the Thomas precession. Need to check these later. M∧Ŝc2ħεИτlk 10:18, 6 July 2015 (UTC)
Nothing in L&L. References will become a problem. The full results on Thomas rotation are fairly new and aren't present in any of the classics. The best references I could find are original research papers, the A. A. Ungar papers from 1988-89. Some of these are referenced in Velocity-addition formula. They are good, but there may still be problems because Ungar took these results and developed a completely new theory by abstraction, see Gyrovector space, an article I just discovered we have, see also my comment at Talk:Gyrovector space. So far nothing is wrong, it is perfectly natural. What seems strange is that it appears as a one-man show. Moreover, Ungar wrote a book on his new theory and the first few chapters can be found online (search his name and go to his university homepage):
• ...
• Chapter 3: The Einstein Gyrovector Space
• ...
• Chapter 5: The Ungar Gyrovector space
• ...
See the problem?
I'd like to see a reference to Ungar's 1988-89 results (we don't need the gyrovector stuff) from a third party. I personally think the results are legitimate, they look credible and they should easily be verifiable numerically, but full proofs aren't published (due to their complicated nature, I believe that actually, referring to Goldstein). YohanN7 (talk) 11:01, 6 July 2015 (UTC)
Of course there's something in L&L. The hyperbolic velocity space of Ungar figures in a problem (with solution) on p. 38. A bit brief for referencing though. This is further referenced in a modern paper as "perhaps the most intriguing approach", search for Relativistic velocity space, Wigner rotation, and Thomas precession (link was blocked). Ungar's book is mentioned in passing. YohanN7 (talk) 11:51, 6 July 2015 (UTC)
I'll copy this discussion to Talk:Thomas precession, and we can continue there. YohanN7 (talk) 12:13, 6 July 2015 (UTC)
"What seems strange is that it appears as a one-man show": There is nothing strange about this. It is a complicated way of rephrasing the much simpler, well-established tensor treatment, and adds nothing new. If Einstein had chosen to use the unwieldy mathematics of Ungar, his discovery general relativity would have been severely delayed. —Quondum 13:58, 6 July 2015 (UTC)
Let us skip what Ungar did after the 1988-89 papers. Those papers are the relevant ones for this article and they do matrix algebra only. He allegedly solved the problem of identifying what L in
${\displaystyle B_{1}(\mathbf {v} )B_{2}(\mathbf {u} )=L}$
precisely is. It is a combination of a pure boost and a rotation
${\displaystyle L=B(\mathbf {w} )R(\mathbf {u} ,\mathbf {v} ),}$
where
${\displaystyle \mathbf {w} =\mathbf {v} \oplus \mathbf {u} ={\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} }{c^{2}}}}}\left\{\mathbf {v} +{\frac {1}{\gamma _{\mathbf {v} }}}\mathbf {u} +{\frac {1}{c^{2}}}{\frac {\gamma _{\mathbf {v} }}{1+\gamma _{\mathbf {v} }}}(\mathbf {v} \cdot \mathbf {u} )\mathbf {v} \right\}.}$
The rotation axis of R(u, v) is parallel to u × v, and the rotation angle ε is given by
${\displaystyle \cos \epsilon ={\frac {(k+\cos \theta )^{2}-\sin ^{2}\theta }{(k+\cos \theta )^{2}+\sin ^{2}\theta }},\quad k^{2}={\frac {\gamma _{u}+1}{\gamma _{u}-1}}{\frac {\gamma _{v}+1}{\gamma _{v}-1}},}$
where θ is the angle between u and v. There is also neat expression for R(u, v) resembling Rodrigues' rotation formula that I don't feel like writing down now (might just as well, can copy into article later and save my breath then),
${\displaystyle R(\mathbf {u} ,\mathbf {v} )=I+\sin \epsilon {\frac {\Omega (\mathbf {u} ,\mathbf {v} )}{uv\sin \theta }}+(1-\cos \epsilon ){\frac {\Omega ^{2}(\mathbf {u} ,\mathbf {v} )}{|\mathbf {u} \times \mathbf {v} |^{2}}},}$
where
${\displaystyle \Omega ={\begin{bmatrix}0&-\omega _{3}&\omega _{2}\\\omega _{3}&0&-\omega _{1}\\-\omega _{2}&\omega _{1}&0\end{bmatrix}},}$
with
${\displaystyle {\boldsymbol {\omega }}=\mathbf {u} \times \mathbf {v} .}$

All that concerns me, and this article, is if this result is fully acknowledged. If it is, it is not cats pee. It is the velocity addition formula including the Thomas rotation explicitly. The fact that he subsequently abstracted his results and wrote a book and named things after himself and Einstein is something I wasn't aware of until very recently (until after I started this thread).
To be more precise about what surprises me; why isn't the above formula everywhere in new papers about Thomas precession? Oh, yes, they keep coming and the thing is still researched. The above ought quite rightly be called Ungar's formula (Christened so by not himself) if correct, and that should be the end of it. But it figures only in Ungar's paper and Wikipedia (R is called tom or gyr. YohanN7 (talk) 15:06, 6 July 2015 (UTC)
Could be. If so, it should be cited as a result in its own right, without all the rest. Though what would surprise me is that the factoring of the composition of two boosts into a boost and a rotation in a given frame of reference was not solved before, since every Lorentz transformation can presumably be factored that way. If Ungar's contribution is to be acknowledged, though, it is neither by Ungar not by WP that this should be. WP can cite other sources on the acknowledgement, but cannot make claims of originality by Ungar without suitable sources acknowledging his contribution. —Quondum 15:27, 6 July 2015 (UTC)
It surprises me too, but see Goldstein's quote. It should put Ungars later work in a little better light because each time you multiply two general LT you have the same problem. Goldstein, by the way, gives the algorithm to find w and R(u, v) from L. First find βw by looking at what L does to the origin in the primed system. Then construct B(−w). One has
${\displaystyle R(\mathbf {u} ,\mathbf {v} )=B(-\mathbf {w} )L.}$
The tricky part is to express this formally in terms of u, v and on a form manifestly a rotation matrix.
I can easily see it useful to work with a one-to-one parametrization of the Lorentz group consisting of pairs of boost vectors (a natural 3-vector b t w) and rotations. Ungar begins in that end in 1989. YohanN7 (talk) 15:56, 6 July 2015 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Here: The Relativistic Velocity Composition Paradox and the Thomas Rotation. The author (Constantin I. Mocanu, must be the same guy), by the way, have a paradox named after himself in conjunction with this. Mocanu, at least, acknowledges both the formula, and that the paradox is resolved by Ungar's work. I think we can give him credit. Had it not been for Ungar naming things after himself (and being the sole practitioner), and the debate over at Talk:Gyrovector space, none of us would have thought a split second about this, right? There are also scattered questions about Ungar's reliability floating around on the net for the same reason, one-man-show-promo, but the replies are along the lines "looks fair enough". YohanN7 (talk) 16:18, 6 July 2015 (UTC)

From the intro in the linked Mocanu paper, one gets the impression that the velocity composition "paradox" was only "resolved" recently by Ungar, but this article (Thomas precession) indicates that the Thomas precession was discovered by Thomas and derived by Wigner. That there was any "apparent paradox" that was not already resolved by Thomas and Wigner is not at all evident, and I am disinclined to believe it, since it already had the status of being recognized, acknowledged, and Lorentz transforms are well-studied. This suggests that Ungar and Mocanu are inclined to make claims of originality without adequately researching their claims. Any such claim should, at the very least, be accompanied by a careful analysis of what Thomas, Wigner and any subsequent authors on the topic did deal with, and to establish clearly what remained unresolved, if anything. Can we even describe what this supposedly unresolved paradox was?
The Goldstein quote essentially only makes the point that "the algebra involved [in the decomposition] is quite forbidding". Which I think is your main point. This assertion is counterintuitive to me, but I should not comment since I have never attempted it. What you say of Goldstein's method sounds straightforward (though I would have called it the observer's worldline rather than "the origin"), and does not sound forbidding. However, Goldstein's material seems reasonable to include in this article.
The idea of parameterising an arbitrary Lorentz transform with two boost vectors is perhaps interesting, but keep in mind that for a given frame of reference, not all Lorentz transforms can be decomposed into two boosts (a frame of reference is necessary, since the concept of a "pure boost" requires it). In particular, aside from the identity rotation, there is no rotation that can be decomposed as two boosts. This implies that such a parameterisation would be partial. A complete parameterisation in terms of two three-dimensional vectors could be done via a boost velocity vector and a Euler vector.
Also, on a side note, decomposition of a Lorentz transform into a boost and a rotation is presumably not commutative, so one presumably gets (again, for any given frame of reference) two different decompositions in general. —Quondum 17:37, 6 July 2015 (UTC)
No, it is pairs of boost vectors and rotation matrices. On the side note; correct. YohanN7 (talk) 18:16, 6 July 2015 (UTC)

Regrettably... >_< Barut is too brief to be useful (I really like his book EXCEPT for his short and useless description of the Thomas precession).

Jackson's EM 2nd edition is better, on pages 542-545 he does find the LT as a composition of two boosts equivalent to a boost and rotation, in terms of the boost generators K and rotation generators (uses S), and also goes on to find the Thomas precession angular velocity vector and the vectorial angle of rotation. If this is of interest I can paste the formulae for those without the book. M∧Ŝc2ħεИτlk 19:06, 7 July 2015 (UTC)

Is this different in the third edition you think (I have that one...)? YohanN7 (talk) 00:56, 8 July 2015 (UTC)
Apparently the 3rd edition is almost identical to the 2nd (minus the magnetohydrodynamics chapter in the 2nd edition, which is not relevant here). M∧Ŝc2ħεИτlk 14:49, 10 July 2015 (UTC)

## Images

@Maschen:, would you have the opportunity to manufacture an image or two? I'll try myself, but my artistic abilities let me at most draw a straight line. YohanN7 (talk) 15:22, 7 July 2015‎ (UTC)

Sure, I'll try sometime this week if not tommorow. M∧Ŝc2ħεИτlk 19:06, 7 July 2015 (UTC)
Sorry to be sticking my nose into something that I'm not deeply involved in, but I thought I might mention something about the presentation of the detail, specifically with regard to frames of reference. I feel that a lot of misinterpretation results from the implicit Galilean thinking. For example, in the diagram, angles and velocities cannot be depicted without reference to the frame of reference. The angle between the velocities u and v as well as the velocity v itself are measured in the moving frame of reference in this case, but this is not clear at all from the diagram. The left diagram would make sense if drawn from the perspective of the frame boosted by u, and thus might even do better with a vector −u to the "unboosted" frame. Drawing like it is with head-to-tail is mathematically wrong in a sense, because there is no frame in which these vectors add, whereas in the boosted frame it is accurate to depict the vectors −u and v as independent vectors in the one frame where they both make sense. Juxtaposing the same picture in two frames of reference should work quite nicely. —Quondum 16:03, 10 July 2015 (UTC)
The figure was taken straight from fig.1 in the reference in the caption. Since the left diagram shows the set up (velocities, angles, coordinates and all) in the Σ' frame I'll negate u to −u as you suggest, but the head-to-tail arrangement of velocities does not even show vector addition, just the relative velocities. The right figure is in the Σ frame. Also, juxtapose what? If you mean overlapping the left and right figures into one, that would be impossible and become cluttered.
First I'll wait a while for other suggestions, then update in one go. M∧Ŝc2ħεИτlk 18:34, 10 July 2015 (UTC)
I think I understand what @Quondum: means, but I don't agree that the frames in which the quantities are defined is unclear, For vectors, their tails define the system. For angles, it is the frame of the two vectors involved. In this case of θ, one of the two vectors is "parallel transported" (or invoking EPVR if you want) from Σ to Σ′ as indicated by a dashed line. The home of ε is either Σ or Σ′′ depending on what comes first, boost or rotation, for the combined action of two boosts. Also, no addition (with +) is implied by head-to tail, but "Einstein addition" (with ⊕) is implied. As M says, this can be sourced, and basically the same figure can be found in other papers.
One thing that one must realize, is that parallel transport of these particular abstract 3-vectors is allowed. It is just not allowed (between Σ and Σ′′) in coordinates. If this wasn't the case, the Einstein velocity reciprocity relation would actually be violated (as Mocanu thought based on the numerical coordinate based discrepancy due to Thomas rotated systems) and we would have a true paradox.YohanN7 (talk) 21:24, 10 July 2015 (UTC)
If you can get one in ten readers, not fully familiar with special relativity, to determine this and explain within ten minutes of examining the article and picture, I would be highly surprised. And just what do you mean by parallel transport of these vectors between frames? Aside from perpendicular and parallel vectors, angles actually change as they are transported from one frame to the other due to contraction in one direction. —Quondum 05:04, 11 July 2015 (UTC)
I'll answer right away, but first, I'd like you to be very clear about what is unclear instead of saying that nobody will understand. Is it still unclear which system vectors refer to? Angles? I can see the need to explain more the text how EVRP = Einsteins velocity reciprocity principle works. But the article isn't nearly done yet. I got stuck with the velocity-addition formula, and don't have all the time in the world. I estimate that this article will double or triple in size by the time I'm done.
By parallel transport I mean essentially EVRP, v in one frame corresponds to v in the frame moving with velocity v wrt the first (with parallel axes). Then flip sign and you have parallel transported. These vectors are also not subject to Lorentz transformation (in this context). (As you can see from the example just given, there is no contraction.) A boost with β = 0.5 in the direction from the singularity of the black hole in the Milky way to the singularity of the black hole in the Andromeda galaxy is one such in every frame. These vectors do not live in spacetime, they are parameters of the boost part Lie algebra of the Lorentz group (modulo the map tanh I have mentioned elsewhere). If the frames are oriented the same way, just transport the vector. If not, we require a (passive) rotation of one of the frames before it works in coordinates. (This is actually the Thomas rotation in the case of Σ, Σ′′)
And no, I don't think I can get a single reader to fully grasp Thomas precession. Most of those who have actually studied it at one time in school do not understand it in full. Do you? That is a question, not part of an intricate insult. I'm just curious. As for me, I think I am getting there, but I am just not there yet. YohanN7 (talk) 06:48, 11 July 2015 (UTC)

────────────────────────────────────────────────────────────────────────────────────────────────────Could we please get back on the topic of the pictures? I have gone ahead and made the change since at least the left fig is relativistically correct then (Σ moves with -u as measured in Σ′). Also, to remove any confusion about (ordinary) velocity addition, the velocity vectors are thickened and no longer head to tail. Also tweaked colouring slightly. Let me know of anything else. I know YohanN7 said to wait, but these changes should not be problematic. Later I'll make an image for the two orderings, of boost then rotation, and rotation then boost. M∧Ŝc2ħεИτlk 07:44, 11 July 2015 (UTC)

My post is very much on the topic of the pictures.
The new version has its own problems that are worse than those of the original. It indicates that the origins of Σ′ and Σ′′ do not coincide at the time of the second boost. This unnecessarily introduces a spatial translation that the mathematics (to be) in the text does not cover. The figure also does not describe uv. For that the perspective in the original figure is essential. (Nor does it describe v ⊕ −u.) Lastly, not having arrows emanating from origins leads into doubt what exactly is being boosted. The image is also not sourced anymore. YohanN7 (talk) 12:27, 11 July 2015 (UTC)
Ideally, there should be, in all, five figures. First, a tweaked version of the present left figure supporting that the pairs Σ, Σ′) and Σ′, Σ′′) are parallel. Then the previous pair. In addition, the pair representing the "inverse" configuration. It is also possible to add to both of the pairs two the "other choice" corresponding to the choice of "rotation first or last", for all together seven images, one singlet plus two triplets. YohanN7 (talk) 12:38, 11 July 2015 (UTC)
I can see I'm only complicating things. I'll note that the original picture on the left with the origin of the axes shown at the juncture of the vectors instead of at the tail of the first would give a direct hint of the frame of reference in which that entire lefthand picture (directions, magnitudes, angles) all make exact sense. But take this as you will.
My suggestion for understanding would be to translate everything about boosts (in 1+2 Minkowski space) into related statements about rotations in Euclidean 3-space to see what makes sense. They are formally equivalent under a suitable "twist" (introduction of the imaginary unit in the right places, thus changing hyperbolic to trigonometric functions and the like). If a statement makes sense in Euclidean geometry without time, there is a straightforward translation into the Minkowski context, and vice-versa. It helps to identify equivalences: in any "frame of reference" we have a preferred axis, let's say respectively t and z in the x, y, t Minkowski and x, y, z systems. Points correspond to points. Rotations in the xy plane correspond. Boosts correspond to rotations around an axis in the xy plane.
A velocity vector associated with a boost (a 4-velocity, with a dropped dimension, starts as a vector in the t direction, boosted by the Lorentz transform of the boost, then scaled so that the t component is 1, then that component dropped, leaves us with a 2-velocity) correspond to a kind of projection in the Euclidean case: start with a vector in the z-direction, apply the rotation, scale the vector to have z-component 1, then drop that component, leaving a vector in the xy plane that represents the "boost" type of rotation. In both case this splits a full Lorentz/rotation into a "pure boost" and "pure rotation". Velocity addition and Thomas precession can be accurately understood in this analogy via the nonabelian group nature of rotations. Obviously in Euclidean space, two vectors representing "pure boost" rotations do not add, and one gets Thomas precession when two "pure boost" rotations are around different axes. —Quondum 14:06, 11 July 2015 (UTC)
The purpose of this article is not to explain a single Lorentz boost to a newbie by passing to ict in order to hide the indefinite nature of the metric. (That practice is questionable anyhow, and totally out of fashion). It is also sort of naive to expect all difficulties with Thomas precession (mathematical and conceptual) vanish with a linear change of variables.
Your method of manufacturing the velocity vector corresponding to a boost by applying the Lorentz tranformation corresponding to the velocity vector to the (unit?) time direction is kindasorta going over the bridge for water. You have the velocity vector, it is u. Suppose not. Then you also can't define the Lorentz transformation that turns the t-axis into it, because the velocity vector itself is needed for that.
I have written several times ITT and elsewhere that these 3-vectors, u etc, are not subject to be boosted. They are invariants (modulo rotations) in this context. They do not represent the velocity of a particle. They parametrize a Lorentz transformation (modulo rotations) viz.
${\displaystyle B({\boldsymbol {\zeta }})=e^{-{\boldsymbol {\zeta }}\cdot \mathbf {K} },}$
where ς are coordinates on the boost subspace of the Lie algebra spanned by K = (K1, K2, K3) with norm = rapidity (strike out a factor i in linked article, different convention for exp), with
${\displaystyle {\boldsymbol {\zeta }}={\boldsymbol {\hat {\beta }}}\tanh ^{-1}\beta ,}$
where
${\displaystyle \mathbf {\boldsymbol {\beta }} ={{\boldsymbol {u}} \over c}.}$
Since
${\displaystyle {\boldsymbol {\hat {\beta }}}={{\boldsymbol {\beta }} \over \beta },}$
one has
${\displaystyle B({\boldsymbol {\zeta }})=e^{-\tanh ^{-1}\beta {\boldsymbol {\hat {\beta }}}\cdot \mathbf {K} }=e^{{-\tanh ^{-1}\beta \over \beta }{\boldsymbol {\beta }}\cdot \mathbf {K} }=e^{{-\tanh ^{-1}\beta \over c\beta }\mathbf {u} \cdot \mathbf {K} }\equiv B(\mathbf {u} ).}$
You do not perform
${\displaystyle \mathbf {u} '=B(\mathbf {u} )\mathbf {u} ,\quad {\text{or}}\quad \mathbf {u} =B(\mathbf {u} )\mathbf {(} 1,0,0,0)^{T}}$
or anything like it as you suggested, and
In both case this splits a full Lorentz/rotation into a "pure boost" and "pure rotation"
is a bold statement of unclear origin. I doubt that you can prove that (or supply a reference).
The entire problem of Thomas precession boils down to
${\displaystyle e^{{-\tanh ^{-1}\beta \over c\beta }(\mathbf {u} +\mathbf {v} )\cdot \mathbf {K} }\neq e^{{-\tanh ^{-1}\beta \over c\beta }\mathbf {u} \cdot \mathbf {K} }e^{{-\tanh ^{-1}\beta \over c\beta }\mathbf {v} \cdot \mathbf {K} }.}$
This problem does not go away by passing to ict for spacetime. YohanN7 (talk) 16:00, 11 July 2015 (UTC)
What I said was intended as a way of evaluating whatever is said in the article for validity by translating it to the more familiar Euclidean framework, not for inclusion in the article.
Gauging by the lack of understanding of what my intent was, and the energy you are putting into responses before getting the basics of understanding aligned, I deem this to be an appropriate point to step out of the discussion. —Quondum 16:55, 11 July 2015 (UTC)
I really did try to understand, hence my long and detailed reply, because I took you seriously and wanted to give you a serious reply (instead of following my first instincts and just tossing it off as rubbish).
Instead of acting like a [diva], please try once more to explain to me what you mean, preferably using mathematics instead of words. Not all are as quick as you in picking up new things. YohanN7 (talk) 17:12, 11 July 2015 (UTC)

For now, I reverted to the previous version. I'll read through this discussion, the article, and other papers, and create the seven diagrams later this evening or tomorrow morning. M∧Ŝc2ħεИτlk 19:47, 11 July 2015 (UTC)

There is no hurry, and seven diagrams may be overkill. I'll myself let this rest for a couple of days. YohanN7 (talk) 20:07, 11 July 2015 (UTC)

### New images

Velocity composition and Thomas rotation in xy plane, velocities u and v separated by angle θ. Left: As measured in Σ′, the orientations of Σ and Σ′′ appear parallel to Σ′. Centre: In frame Σ, Σ′′ moves with velocity wd relative to Σ and is rotated through angle ε about an axis parallel to u×v. Right: In frame Σ′′, Σ is rotated through angle ε about an axis parallel to −(u×v) and moves with velocity wd relative to Σ′′.
Velocity composition and Thomas rotation in xy plane, velocities u and v separated by angle θ. Left: As measured in Σ′, the orientations of Σ and Σ′′ appear parallel to Σ′. Centre: In frame Σ′′, Σ moves with velocity wi relative to Σ′′ and is rotated through angle ε about an axis parallel to −(u×v). Right: In frame Σ, Σ′′ is rotated through angle ε about an axis parallel to u×v and moves with velocity wi relative to Σ.
Comparison of velocity compositions wd and wi. Notice the same magnitude but different directions, separated by an angle.

OK - here are the new images. As far as I can tell they are correct, although sadly the inverse configuration does not necessarily appear in the sources, see Talk:Thomas precession#About the inverse configuration... (not too much of a problem considering WP:CALC). M∧Ŝc2ħεИτlk 16:05, 12 July 2015 (UTC)

I'll check in detail, but I suspect, just by geometric construction while drawing these, that the angle between wd and wi is just ε, because of parallel transport YohanN7 mentioned above. Also, if the angle between the x-axis and wd is 2ε α say, and then the angle between the x-axis and wi is ε αε. M∧Ŝc2ħεИτlk 16:16, 12 July 2015 (UTC)
Above: I added underlined statements, and stroke-out the incorrect parts. M∧Ŝc2ħεИτlk
Maschen, please look at your diagrams with Lorentz contraction in mind, and you'll see that a space-like set of axes do not remain orthogonal under boosts that are not in the direction of one of the three axes. —Quondum 16:32, 12 July 2015 (UTC)
Quondum, of course they are not by the general LT matrix for velocity in any direction. These styles of diagrams are in the papers by Ungar and Mocanu and others (the inverse configuration is rarely clearly shown, unlike the "forwards configuration", but is not relevant here). These sketches just show the velocity compositions and the Thomas rotation angle, as calculated by the velocity addition and boost/rotation formulae. If they were shown to be "warped", then the pictures would become harder to understand because of the Thomas rotation. M∧Ŝc2ħεИτlk 16:49, 12 July 2015 (UTC)
A spacelike set of orthogonal axes do remain orthogonal. A general boost is expressible as
${\displaystyle L=R^{-1}BR}$
where R is a suitable rotation taking, say, the z-direction into the direction of the boost. Each operation preserves orthogonality of axes. YohanN7 (talk) 17:16, 12 July 2015 (UTC)

Incorrect. You are engaging in (invalid) OR. Get your facts straight. It is time to get others involved. —Quondum 17:35, 12 July 2015 (UTC)
See except from Cushing's article below. YohanN7 (talk) 20:18, 12 July 2015 (UTC)
Quondum - this is not OR, why have you taken this to Wikipedia talk:WikiProject Mathematics‎ ? M∧Ŝc2ħεИτlk 17:52, 12 July 2015 (UTC)
Am am not being given room to speak here. Comments above like "Instead of acting like a diva, ..." are not called for. This talk page interaction is getting out of hand, and I would like more eyes on edits, for example, discussed using terms like "parallel transport between frames" (WTF?). This is good enough reason, though I do have factual disputes. For example, the set of orthogonal space-like axes at rest as seen by an observer in their rest frame (actually world-sheets), when boosted in a direction other than one of the three axes, does not appear at right angles from a boosted frame. When I point this out, I just get contradicted. Hence more eyes would be good. —Quondum 18:24, 12 July 2015 (UTC)
The frames in the single figure involving wd and wi are parallel transported, there is no change in orientation. M∧Ŝc2ħεИτlk 18:42, 12 July 2015 (UTC)
From ref Vector Lorentz transformations by James T. Cushing. Good enough Q?
In addition to the fully referenced claims in the article, I am providing an except from Cushing's paper supporting my claims about parallel spatial coordinate axes. It is on the right. YohanN7 (talk) 20:18, 12 July 2015 (UTC)
Also, you did act exactly like a diva. I didn't take your ideas about boost velocities at face value, and therefore you threatened to leave, with the undertone I'm not intellectually equipped to understand the basics of your theories. I even fed you by asking you to stay. YohanN7 (talk) 19:01, 13 July 2015 (UTC)
They look great M. But the captions are slightly off. The difference between pics two and three in each triple is whether we Thomas rotate first and then boost, or vice versa. YohanN7 (talk) 16:36, 12 July 2015 (UTC)
Correct me if wrong but the centre pics in each triple are boosts then rotations, the right pics in each are rotations then boosts. Will tweak captions. M∧Ŝc2ħεИτlk 16:49, 12 July 2015 (UTC)
For "forwards configuration":
${\displaystyle B(\mathbf {v} )B(\mathbf {u} )=B(\mathbf {w} _{d})R(\mathbf {u} ,\mathbf {v} )}$
so the inverse must be
${\displaystyle B(-\mathbf {u} )B(-\mathbf {v} )=R(\mathbf {v} ,\mathbf {u} )B(-\mathbf {w} _{d})}$
because multiplying the two equations gives the identity.
Similarly for "inverse configuration":
${\displaystyle B(-\mathbf {u} )B(-\mathbf {v} )=B(-\mathbf {w} _{i})R(-\mathbf {v} ,-\mathbf {u} )}$
so the inverse must be
${\displaystyle B(\mathbf {v} )B(\mathbf {u} )=R(\mathbf {u} ,\mathbf {v} )B(\mathbf {w} _{i})}$
Is that right? M∧Ŝc2ħεИτlk 17:03, 12 July 2015 (UTC)
I corrected the above equations, see below Talk:Thomas precession#Order. M∧Ŝc2ħεИτlk 09:02, 13 July 2015 (UTC)

## Σ to Γ?

Why have we temporarily changed the names of Σ, Σ′, Σ′′, to Γ, Γ′, Γ′′, just because the relative velocities are reversed in direction? I find this more confusing to follow, and think we should just stick to Σ. We only need one name per frame. M∧Ŝc2ħεИτlk 22:15, 11 July 2015 (UTC)

Agreed! YohanN7 (talk) 22:25, 11 July 2015 (UTC)
Done. M∧Ŝc2ħεИτlk 07:41, 12 July 2015 (UTC)

This may seem obvious but I don't understand the velocity composition for the "inverse configuration". If the images are to be correct this needs to be cleared up.

"Consider the reversed configuration (temporarily change names of the frames to Σ,Σ′,Σ′′), namely, frame Σ moves with velocity u relative to frame Σ′, and frame Σ′, in turn, moves with velocity v relative to frame Σ′′. In short, u → − u, v → −v by reciprocity. Then the velocity of Σ relative to Σ′′ is vu. By reciprocity, the velocity of Σ′′ relative to Σ is wi = −vu."

According to the velocity addition formula, for the "forwards" composition, the velocity of Σ′′ relative to Σ is

${\displaystyle \mathbf {u} \oplus \mathbf {v} ={\frac {\mathbf {u} +\mathbf {v} }{1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}+{\frac {1}{c^{2}}}{\frac {\gamma _{u}}{1+\gamma _{u}}}{\frac {\mathbf {u} \times (\mathbf {u} \times \mathbf {v} )}{1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}.}$

since Σ′ moves with velocity u relative to Σ (first stage of composition), and Σ′′ moves with velocity v relative to Σ′ (second stage).

[ It follows

${\displaystyle (-\mathbf {u} )\oplus (-\mathbf {v} )=-(\mathbf {u} \oplus \mathbf {v} )\,\,]}$

For the inverse composition, the velocity of Σ relative to Σ′′ is surely

${\displaystyle (-\mathbf {v} )\oplus (-\mathbf {u} )=-(\mathbf {v} \oplus \mathbf {u} )}$

since Σ′ moves with velocity v relative to Σ′′ (first stage of composition), and Σ moves with velocity u relative to Σ′ (second stage). If this is correct, then velocity of Σ′′ relative to Σ is then

${\displaystyle \mathbf {v} \oplus \mathbf {u} }$

[ Aside... the axis of the Thomas rotation would surely correspond to

${\displaystyle (-\mathbf {v} )\times (-\mathbf {u} )=\mathbf {v} \times \mathbf {u} =-\mathbf {u} \times \mathbf {v} }$

meaning a rotation in by the same angle in the opposite sense because the axis is reversed.]

So... why is the reversed composition, the velocity of Σ relative to Σ′′, given by what is usually just stated in the papers (and this article) as it if were "obvious":

${\displaystyle \mathbf {v} \oplus \mathbf {u} \ldots \quad ?}$

and correspondingly the velocity of Σ′′ relative to Σ is

${\displaystyle \mathbf {w} _{i}=-\mathbf {v} \oplus \mathbf {u} \ldots \quad ??}$

Basically... What is the ordering of velocities in the inverse configuration. Thanks, M∧Ŝc2ħεИτlk 08:22, 12 July 2015 (UTC)

This probably boils down to notational abuse. vu = (−v)⊕(−u). Horrible, and I should never have adopted it without mention (think it is from Ungar). The ordering of boosts in the inverse configuration is of course first with v and then with u. YohanN7 (talk) 11:29, 12 July 2015 (UTC)
There is an error in the text as well. I'll fix it. YohanN7 (talk) 11:38, 12 July 2015 (UTC)

## Order

The order of boosts in the new addition looks wrong. I think uv corresponds to B(v)B(u). YohanN7 (talk) 20:59, 12 July 2015 (UTC)

How? I thought this equation (which you may have wrote):
${\displaystyle B(\mathbf {u} )B(\mathbf {v} )=B(\mathbf {u} \oplus \mathbf {v} )R(\mathbf {u} ,\mathbf {v} )}$
was correct, and remember it in a paper by Ungar, looking for it now... M∧Ŝc2ħεИτlk 21:03, 12 July 2015 (UTC)
Wait a second, what is the convention we are using to order successive boost and rotation operators? By right (A, AB, ABC, ...) or left (A, BA, CBA, ...) multiplication? M∧Ŝc2ħεИτlk 21:06, 12 July 2015 (UTC)
Rightmost operator acts first in a product of operators(by the most common convention and ours). How this is matched with velocity addition is an other matter of convention. I am tired so don't trust me on this. YohanN7 (talk) 21:12, 12 July 2015 (UTC)
The unfortunate problem is that
${\displaystyle B(\mathbf {u} )B(\mathbf {v} )=B(\mathbf {u} \oplus \mathbf {v} )R(\mathbf {u} ,\mathbf {v} )}$
appears in papers by e.g. Ungar (1988) and Mocanu (1992), while what is supposed to be the same thing
${\displaystyle B(\mathbf {v} )B(\mathbf {u} )=B(\mathbf {u} \oplus \mathbf {v} )R(\mathbf {u} ,\mathbf {v} )}$
appears in papers by e.g. Costella et al (2001).
I think we should just stick to the left-acting multiplication (equivalently right-most operator acts first), which is the second formula here. It fits better with matrix multiplication, or general operation composition. What do you think? (When ready, I can imagine you want to break for a while). M∧Ŝc2ħεИτlk 08:52, 13 July 2015 (UTC)

## Plans

• Section outlining how to find rotation in principle (based on Goldstein).  Done
• Section carrying though calculation where second boost is infinitesimal (based on either Jackson or Goldstein)
• Section briefly listing recent attempts (based on survey in paper by Rhodes & Semon + referenced papers).
• Section on Ungar's explicit formula.
• Section on Wigner rotation as applied to QFT (based on Wigner's 1939 paper and Weinberg's QFT I)

YohanN7 (talk) 21:04, 12 July 2015 (UTC)

Looks fine to me. M∧Ŝc2ħεИτlk 21:07, 12 July 2015 (UTC)

## Removal of content

Regrettably I had to remove this:

When studying Thomas precession at the fundamental level, one typically uses a setup with three coordinate frames, Σ, Σ′ Σ′′. Frame Σ′ has velocity v relative to frame Σ, and frame Σ′′ has velocity v + Δv relative to frame Σ′. Then calculate the composition of two boosts, as shown in Wigner rotation.

This provides a "discrete" version of the continuous Thomas precession tailored to study the Thomas rotation resulting from two consecutive finite boosts, whereas in the continuous process, the system can be thought of as being "continually boosted". The effect of the torqueless forces on Sμ between two instants of proper system time t, t + Δt, is to multiply it by a pure boost matrix (in an appropriate representation). The product of these matrices, a limiting case as Δt → 0, for the entire trajectory corresponds to a pure spatial rotation, and this is the common explanation of the precession.[1]

Yes, it has a reference. However...

The first paragraph is now basically superfluous, and only serves as a "intro/preview" of what's coming.

The second uses the proper time of the particle. The explanation I wrote is a rewrite of Jackson's Classical EM (2nd edition) p. 543-546 and uses the lab time for describing the accelerated motion, and transformations from the lab frame to the particle's frame at two instants of time as recorded in the lab frame. IMO this is easier to follow. So the quotation had to be cut out. Also, it is clear from the explanation that no forces or torques are required to obtain the precession.

However, including spin back in the explanation would be better, and I don't want to dismissively remove referenced material other people have written. Spin and the reference to Ben-Menahem (1986) should be worked back in, maybe give the explanation in two segments; one for the proper time of the particle, and another for the lab time measured by the observer. MŜc2ħεИτlk 22:35, 30 December 2015 (UTC)