From an earlier version of the Wikipedia article on Thomas precession concerning TP and LP=Larmor precession, regarding the paper: G B Malykin, "Thomas precession: correct and incorrect solutions", Physics-Uspekhi 49 (8) 837-853 (2006)

In a 2006 survey of the literature Malykin notes that there are numerous conflicting expressions for Thomas precession. This is partly explained by the fact that different authors use "Thomas precession" to refer to different things, often without saying what they are referring to and subsequent authors then misinterpret the results and apply them to other things, but, even taking this into account, some of the expressions in the literature are just plain wrong.

Malykin explains the source of some of these errors: "We emphasize that Thomas considered the rotation of the axes of the coordinate system accompanying the electron in its motion rather than the electron spin rotation. Subsequently, this led to a misunderstanding and the emergence of incorrect work on the TP problem. It is possible to introduce three different reference frames accompanying the electron motion around a circular orbit and, in the most general case, along a curvilinear trajectory: (i) a reference frame whose coordinate axes remain parallel or retain their angular position relative to the axes of a laboratory IRF, (ii) a reference frame one of whose coordinate axes is always coincident with the electron velocity vector, and (iii) a reference frame in which the electron spin vector retains its orientation relative to the coordinate axes. It is evident that the electron spin vector precesses relative to the coordinate axes of the two first systems, but the angular velocity of its precession is different in these systems. ... In several papers concerned with the TP, calculations are performed in the first approximation in v^2/c^2, where v is the speed of an elementary particle in the laboratory IRF and c is the speed of light. In this case, all authors arrive at the same expression first derived by Thomas, this being so irrespective of whether they consider the relativistic rotation of the particle spin or the relativistic rotation of the axes of the coordinate system comoving with the particle. In the most general case, however, the expressions for the TP obtained by different authors are radically different. As noted above, the problem is complicated by the fact that different authors assign different meaning to this expression: some imply the relativistic rotation of the particle spin in the laboratory IRF, some in the comoving reference frame (in this case, as noted above, the rotation law for the axes of this system may be defined in three ways), while others refer to the relativistic rotation of the axes of the coordinate system accompanying the particle in motion. ... As noted above, the expression for the TP in Thomas's first paper was obtained in the first approximation in v^2/c^2 and is always correct when this condition is fulfilled. In his subsequent work, on performing calculations for an arbitrary electron velocity v, Thomas derived an expression that correctly describes the relativistic rotation of the axes of the comoving coordinate system relative to the rest-frame (laboratory) system. However, because the majority of authors use the term TP in reference to the precession of the spin of an elementary particle, this subsequently led to several errors and misunderstandings. ... In 1952, in his famous monograph [78], the Danish scientist C Møller (1904-1980), an acknowledged expert on the theory of relativity, derived an expression for the TP that coincides, up to a sign, with the corresponding Thomas expression and is correct in the comoving frame of reference. However, it was stated in Ref. [78] that this expression was written for the laboratory IRF, which is incorrect. Møller's immense scientific prestige played a negative role in this case: since then, the majority of authors of papers, monographs, and lecture courses started using the expression for the TP from Ref. [78] or, in the derivation of suchlike expressions, tried to make them coincident with that given in Ref. [78]. ... At the same time, because the experimentally observed particle spin precession is caused by the sum of two effects, the TP and the LP, it is possible to choose different expressions for each of these effects, only provided that these expressions add up to correspond to expression (14). we conclude that the preferred method is the above-discussed method of recording the TP with the aid of mechanical gyroscopes in their orbital motion because quantum mechanical effects in experiments on charged elementary particles partly complicate the interpretation of experimental data."

Question: Is Malykin's criticism of the literature correct ?

  • 7
    $\begingroup$ Sounds interesting but you should consider rephrasing that criticism into your own question. Otherwise you risk that people will skip (down-vote/close) this question because it's too long and too little effort has been put into it. $\endgroup$
    – Marek
    Apr 5, 2011 at 20:06
  • $\begingroup$ Agreed: some more effort would be laudable. $\endgroup$
    – Sklivvz
    Apr 5, 2011 at 20:14
  • $\begingroup$ @Marek, the question is "Is Malykin's criticism of the literature correct ? " $\endgroup$ Apr 5, 2011 at 21:02
  • $\begingroup$ For a shorter version read just the parts of the text in bold. I wonder if it would have made a difference if I had only posted the bold text. $\endgroup$
    – user2929
    Apr 6, 2011 at 13:25

3 Answers 3


Here's a new arxiv posting which addresses Malykin's paper: http://arxiv.org/abs/1302.5678

"An important question about the Thomas precession angle ϵ and its generating angle θ is whether or not ϵ and θ have equal signs. According to Malykin, some explorers claim that ϵ and θ have equal signs while some other explorers claim that ϵ and θ have opposite signs. Malykin claims that these angles have equal signs while, in contrast, we demonstrate here convincingly that these angles have opposite signs. Our demonstration is convincing since it accompanies a focal identity, that interested explorers can test numerically in order to corroborate our claim that ϵ and θ have opposite signs."


You can find some critical remarks showing that Malykin is wrong in K. Rębilas, “Comment on 'Elementary analysis of the special relativistic combination of velocities, Wigner rotation and Thomas precession'”, European Journal of Physics 34, L55-L61 (2013).

  • 1
    $\begingroup$ Please explain why this reference is relevant, rather than simply pointing to the source. $\endgroup$
    – Danu
    Mar 13, 2014 at 11:31

Malykin actually agrees with L. Thomas' original paper in 1927 available here: https://virgilio.mib.infn.it/~oleari/public/relativita/materiale_didattico/Thomas_precession.pdf

See his equation 4.122: the second term on the RHS is the Thomas precession. You need to know that Thomas used the Greek letter beta for what we now usually call gamma. And Thomas specialized to the atomic case, so rather than referring to the acceleration, he refers to the E field, etc.

As Malykin points out in his paper, if you use Moller's formula rather than Thomas' formula, you will get ludicrously wrong answers for particles moving at the speed of light.

Put the second full paragraph on page 869 of Malykin's paper through Google translate (unless your Russian is less rusty than my Russian!), and you get:

"In particular, in this case, it follows from [Moller's formula] that the spin [of a] photon, like any massless particle, will perform in laboratory [frame of reference] an infinitely large number [of] revolutions per revolution of a particle [moving in a circle]"

Yes and that proves that Moller's formula is wrong since the helicity vector for a massless particle always follows along in the direction of the velocity.

Anyway, the case is closed: Malykin and Thomas are right, Moller is wrong.

And, yes, I do have a very short and simple derivation myself that I did before I read Thomas and Malykin: I should probably publish it some day. But, anyway, experiment wins: massless particles behave as Thomas and Malykin claim, not Moller.

Dave Miller in Sacramento


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