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# Thomas precession

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### Thomas precession

In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion. It can be understood geometrically as a consequence of the fact that the space of velocities in relativity is hyperbolic, and so parallel transport of a vector (the gyroscope's angular velocity) around a circle (its linear velocity) leaves it pointing in a different direction, or understood algebraically as being a result of the non-commutativity of Lorentz transformations. Thomas precession gives a correction to the spin–orbit interaction in quantum mechanics, which takes into account the relativistic time dilation between the electron and the nucleus of an atom.

The composition of two non-collinear Lorentz boosts, results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation. This rotation is called Thomas rotation, Thomas–Wigner rotation or Wigner rotation. The rotation was discovered by Thomas in 1926,[1] and derived by Wigner in 1939.[2] If a sequence of non-collinear boosts returns an object to its initial velocity, then the sequence of Wigner rotations can combine to produce a net rotation called the Thomas precession.[3]

There are still ongoing discussions about the correct form of equations for the Thomas precession in different reference systems with contradicting results.[4] Goldstein:[5]

The spatial rotation resulting from the successive application of two non-collinear Lorentz transformations have been declared every bit as paradoxical as the more frequently discussed apparent violations of common sense, such as the twin paradox.

Einstein's principle of velocity reciprocity (EPVR) reads[6]

We postulate that the relation between the coordinates of the two systems is linear. Then the inverse transformation is also linear and the complete non-preference of the one or the other system demands that the transformation shall be identical with the original one, except for a change of v to −v

With less careful interpretation, the EPVR is seemingly violated in some models.[7] There is, of course, no true paradox present.

## Contents

• History 1
• Introduction 2
• Definition 3
• Basic phenomena, their cause and explanation 4
• Discrete rotation using three reference frames 4.1
• Set-up 4.1.1
• Reversed configuration 4.1.2
• Explanation 4.1.3
• Continuous rotation 4.2
• Boosts from velocities 5.1
• Finding the Thomas rotation 6
• Applications 7
• In electron orbitals 7.1
• In a Foucault pendulum 7.2
• Remarks 9
• Notes 10
• References 11
• Textbooks 12

## History

Thomas precession in relativity was already known to Ludwik Silberstein,[8] in 1914. But the only knowledge Thomas had of relativistic precession came from de Sitter's paper on the relativistic precession of the moon, first published in a book by Eddington.[9]

In 1925 Thomas relativistically recomputed the precessional frequency of the doublet separation in the fine structure of the atom. He thus found the missing factor 1/2, which came to be known as the Thomas half.

This discovery of the relativistic precession of the electron spin led to the understanding of the significance of the relativistic effect. The effect was consequently named "Thomas precession".

## Introduction

Thomas precession is a kinematic effect in the flat spacetime of special relativity. In the curved spacetime of general relativity, Thomas precession combines with a geometric effect to produce de Sitter precession. Although Thomas precession (net rotation after a trajectory that returns to its initial velocity) is a purely kinematic effect, it only occurs in curvilinear motion and therefore cannot be observed independently of some external force causing the curvilinear motion such as that caused by an electromagnetic field, a gravitational field or a mechanical force, so Thomas precession is usually accompanied by dynamical effects.[10]

If the system experiences no external torque, e.g., in external scalar fields, its spin dynamics is determined only by the Thomas precession. A single discrete Thomas rotation (as opposed to the series of infinitesimal rotations that add up to Thomas precession) is present in situations whenever you have three or more inertial frames in non-collinear motion, as can be seen using Lorentz transformations.

## Definition

Consider a physical system moving through Minkowski spacetime. Assume that there is at any moment an inertial system such that in it, the system is at rest. This assumption is sometimes called the third postulate of relativity.[11] This means that at any instant, the coordinates and state of the system can be Lorentz transformed to the lab system through some Lorentz transformation.

Let the system be subject to external forces that produce no torque with respect to its center of mass in its (instantaneous) rest frame. The condition of "no torque" is necessary to isolate the phenomenon of Thomas precession. As a simplifying assumption one assumes that the external forces bring the system back to its initial velocity after some finite time. Fix a Lorentz frame O such that the initial and final velocities are zero.

The Pauli–Lubanski spin vector Sμ is defined to be (0, Si) in the system's rest frame, with Si the angular-momentum three-vector about the center of mass. In the motion from initial to final position, Sμ undergoes a rotation, as recorded in O, from its initial to its final value. This continuous change is the Thomas precession.[12]

## Basic phenomena, their cause and explanation

### Discrete rotation using three reference frames

The following is 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", described later.

#### Set-up

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′′ is rotated through angle ε about an axis parallel to u×v and then moves with velocity wd relative to Σ. Right: In frame Σ′′, Σ moves with velocity wd relative to Σ′′ and then moves with velocity wd relative to Σ.

When studying Thomas precession at the fundamental level, one typically uses a setup with three coordinate frames, Σ, Σ′ Σ′′. Frame Σ′ has velocity u relative to frame Σ, and frame Σ′′ has velocity v relative to frame Σ′.

The axes are, by construction, oriented as follows. Viewed from Σ′, the axes of Σ′ and Σ are parallel (the same holds true for the pair of frames when viewed from Σ.) Also viewed from Σ′, the spatial axes of Σ′ and Σ′′ are parallel (and the same holds true for the pair of frames when viewed from Σ′′.)[13] This is an application of EVPR: If u is the velocity of Σ′ relative to Σ, then u′ = −u is the velocity of Σ relative to Σ′. The velocity 3-vector u makes the same angles with respect to coordinate axes in both the primed and unprimed systems. This does not represent a snapshot taken in any of the two frames of the combined system at any particular time, as shold be clear from the detailed description below.

This is possible, since a boost in, say, the positive z-direction, preserves orthogonality of the coordinate axes. A general boost B(w) can be expressed as L = R−1(ez, w)Bz(w)R(ez, w), where R(ez, w) is a rotation taking the z-axis into the direction of w and Bz is a boost in the new z-direction.[14][15][16] Each rotation retains the property that the spatial coordinate axes are orthogonal. The boost will stretch the (intermediate) z-axis by a factor γ, while leaving the intermediate x-axis and y-axis in place.[17] The fact that coordinate axes are non-parallel in this construction after two consecutive non-collinar boosts is a precise expression of the phenomenon of Thomas precession.[nb 1]

The velocity of Σ′′ as seen in Σ is denoted wd = uv, where ⊕ refers to the relativistic addition of velocity (and not ordinary vector addition), to be formally introduced at a later stage.

#### Reversed configuration

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 is rotated through angle ε about an axis parallel to −(u×v) and then moves with velocity wi relative to Σ′′. Right: In frame Σ, Σ′′ moves with velocity wi relative to Σ and then is rotated through angle ε about an axis parallel to u×v.

Consider the reversed configuration, 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 and v → −v by EPVR. Then the velocity of Σ relative to Σ′′ is (−v) ⊕ (−u) ≡ −vu. By EPVR again, the velocity of Σ′′ relative to Σ is then wi = vu. (A)

One finds wdwi. While they are equal in magnitude, there is an angle between them. Which is the correct velocity of Σ′′ relative to Σ? Since this inequality may be somewhat unexpected and potentially breaking EPVR, this question is warranted.[nb 2]

#### Explanation

Comparison of velocity compositions wd and wi. Notice the same magnitudes but different directions.

The answer to the question lies in the Thomas precession, and that one must be careful in specifying which coordinate system is involved at each step. When viewed from Σ, the coordinate axes of Σ and Σ′′ are not parallel. While this can be hard to imagine since both pairs (Σ, Σ′) and (Σ′, Σ′′) have parallel coordinate axes, it is easy to explain qualitatively mathematically. The generators of boosts, K1, K2, K3, in different directions do not commute.[18] This has the effect that two consecutive boosts is not a pure boost in general, but a rotation preceding a boost

B(\mathbf v)B(\mathbf u) = B(\mathbf u \oplus \mathbf v)R(\mathbf u, \mathbf v)\,,

or equivalently a boost followed by a rotation

B(\mathbf v)B(\mathbf u) = R(\mathbf u, \mathbf v)B(\mathbf v \oplus \mathbf u)\,,

(succeeding operators act on the left in any composition of operators), where B(u) is a boost with velocity u, and R(u, v) is a rotation in the plane spanned by u and v, through an angle directed from u to v. The inverse of the boost B(u) is given by reversing the velocity to obtain B(−u) ≡ B−1(u), so that B(u)B(−u) = 1. Similarly the inverse of the rotation R(u, v) is given by switching the velocities to reverse the sense of the angle to obtain R(v, u) ≡ R−1(u, v), so that R(u, v)R(v, u) = 1. Notice also R(−u, −v) ≡ R(u, v).

In the original set-up,the sequence of boosts and the rotation, corresponding to the motion of Σ′′ relative to Σ, is

B(\mathbf{v})B(\mathbf{u}) = B(\mathbf{w}_d)R(\mathbf{u},\mathbf{v})\,,

and by reciprocity the motion of Σ relative to Σ′′ is

B(-\mathbf{u})B(-\mathbf{v}) = R(\mathbf{v},\mathbf{u}) B(-\mathbf{w}_d) \,.

In the inverse set up, the sequence of boosts and the rotation, corresponding to the motion of Σ relative to Σ′′, is

B(-\mathbf{u})B(-\mathbf{v}) = B(-\mathbf{w}_i)R(-\mathbf{v},-\mathbf{u})\,,

and by reciprocity the motion of Σ′′ relative to Σ is

B(\mathbf{v})B(\mathbf{u}) = R(\mathbf{u},\mathbf{v})B(\mathbf{w}_i)\,.

Equating the results leads to the relation between the two boosts given by a rotation, in the form of a similarity transformation:

B(\mathbf{w}_i)= R^{-1}(\mathbf{u},\mathbf{v}) B(\mathbf{w}_d)R(\mathbf{u},\mathbf{v}) \,.

Now, realizing that wd is calculated in Σ and wi is calculated in Σ′′, one finds that wd and wi are supposed to be different. Both are correct! In system Σ the velocity wd applies, and in system Σ′′ it is wi. The fallacious point in the original discussion is (A). While everything is in order with reciprocity, and the result for wi is true, the coordinates in which it is true are those of Σ′′.

### Continuous rotation

Returning to the continuous case, 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.[19]

Relativistic velocity addition is given by[20]

\mathbf u \oplus \mathbf v = \frac{1}{1 + \frac{\mathbf u \cdot \mathbf v}{c^2}}\left[\mathbf u + \mathbf v + \frac{1}{c^2}\frac{\gamma_u}{1+\gamma_u}\mathbf u \times(\mathbf u \times \mathbf v)\right] \, ,

(VA 1)

or, in a form sometimes more suitable for interpretation and generalization to higher dimension,

\mathbf u \oplus \mathbf v = \frac{1}{1 + \frac{\mathbf u \cdot \mathbf v}{c^2}} \left[\left(1 + \frac{1}{c^2}\frac{\gamma_u}{1+\gamma_u} \mathbf u \cdot \mathbf v \right)\mathbf u + \frac{1}{\gamma_u} \mathbf v \right],

(VA 2)

where

\gamma_u = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}}

is the Lorentz factor, u can be thought of the velocity of a frame Σ′ relative to a frame Σ, and v is the velocity of an object, say a particle or another frame Σ′′ relative to Σ′.

In the present context, all velocities are best thought of as relative velocities of frames unless otherwise specified. The result w = uv is then the relative velocity of frame Σ′ relative to a frame Σ.

Velocity addition has the following properties.

• Norms are equal under interchange of velocity vectors:
|\mathbf u \oplus \mathbf v| = |\mathbf v \oplus \mathbf u|
\mathbf u \oplus \mathbf v \ne \mathbf v \oplus \mathbf u
\mathbf u \oplus (\mathbf v \oplus \mathbf w) \ne (\mathbf u \oplus \mathbf v) \oplus \mathbf w
• The composition of negative velocities is the negative of the composition:
(-\mathbf{u}) \oplus (-\mathbf v) = -(\mathbf u \oplus \mathbf v)

More detail and other properties of no direct concern here can be found in the main article.

Velocity addition does not provide a complete description of the relation between the frames. One must formulate the complete description in terms of Lorentz transformations corresponding to the velocities. It is clear that to each admissible velocity u (next section) there corresponds a pure Lorentz boost,

\mathbf u \leftrightarrow B(\mathbf u).

In terms of these boosts,

\mathbf u \oplus \mathbf v \leftrightarrow B(\mathbf u \oplus \mathbf v).

Velocity addition corresponds also to composition of boosts,

B(\mathbf v)B(\mathbf u),

in that order. The incompleteness of the description of relative velocities of frames is expressed by the fact that

B(\mathbf u \oplus \mathbf v) \ne B(\mathbf v)B(\mathbf u).

To make the description complete, it is necessary to introduce a rotation, either performed before the boosts, or after the boosts,

\begin{align}B(\mathbf v)B(\mathbf u) &= B(\mathbf u \oplus \mathbf v)R_b(\mathbf u, \mathbf v),\\ B(\mathbf v)B(\mathbf u) &= R_a(\mathbf u, \mathbf v)B(\mathbf u \oplus \mathbf v).\end{align}

This rotation (either of them) is the Thomas rotation.

### Boosts from velocities

The passage from a velocity to a boost is obtained as follows. An arbitrary boost is given by[21]

B(\boldsymbol \zeta) = e^{-\boldsymbol \zeta \cdot \mathbf K},

where ζ is a triple of real numbers serving as coordinates on the boost subspace of the Lie algebra so(3, 1) spanned by the matrices

(K_1, K_2, K_3) = \left( \left[ \begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0 \end{smallmatrix} \right], \left[ \begin{smallmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0 \end{smallmatrix} \right], \left[ \begin{smallmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0 \end{smallmatrix} \right] \right).

The vector ζ is called the boost parameter or boost vector. Its norm is the rapidity. Introduce

\boldsymbol \zeta = \frac{\boldsymbol{\beta}}{\beta} \tanh^{-1}\beta

where β is the velocity parameter, the magnitude of the vector β = u/c. While for ζ one has 0 ≤ ζ < ∞ the β is confined by 0 ≤ β < 1, and hence 0 ≤ u < c. Thus

B(\boldsymbol \zeta) = e^{-(\tanh^{-1}\beta) \frac{ \boldsymbol{\beta}}{\beta} \cdot \mathbf K} = e^{-{\tanh^{-1}\beta \over c\beta} \mathbf u \cdot \mathbf K} \equiv B(\mathbf u).

The set of velocities satisfying 0 ≤ u < c is an open ball in 3 and is called the space of admissible velocities in the literature. It is endowed with a hyperbolic geometry described in the linked article.[22]

## Finding the Thomas rotation

The decomposition process described (below) can be through on the product of two pure Lorentz transformations to obtain explicitly the rotation of the coordinate axes resulting from the two successive "boosts". In general, the algebra involved is quite forbidding, more than enough, usually, to discourage any actual demonstration of the rotation matrix
— Goldstein (1980, p. 286)

In principle, it is pretty easy. Since every Lorentz transformation is a product of a boost and a rotation, the consecutive application of two pure boosts is a pure boost, either followed by or preceded by a pure rotation. Thus suppose

\Lambda = B(\mathbf w)R.

The task is to glean from this equation the boost velocity w and the rotation R from the matrix entries of Λ.[23] The coordinates of events are related by

x'^\mu = {\Lambda^\mu}_\nu x^\nu.

Inverting this relation yields

{(\Lambda^{-1})^\nu}_\mu{\Lambda^\mu}_\rho x^\rho = {(\Lambda^{-1})^\nu}_\mu x'^\mu,

or

x^\nu = {\Lambda_\mu}^\nu x'^\mu.

Set x′ = (ct′, 0, 0, 0). Then xν will record the spacetime position of the origin of the primed system,

x^\nu = {\Lambda_0}^\nu x'^0,

or

x = \begin{pmatrix}ct \\ x_1 \\ x_2 \\ x_3\end{pmatrix} = \begin{pmatrix}{\Lambda_0}^0 ct' \\ {\Lambda_0}^1 ct' \\ {\Lambda_0}^2 ct' \\ {\Lambda_0}^3 ct'\end{pmatrix}

But

\Lambda^{-1} = (B(\mathbf w)R)^{-1} = R^{-1}B(\mathbf -w),

and[24]

B(-\mathbf w) = \begin{bmatrix} \gamma&\gamma\beta_x&\gamma\beta_y&\gamma\beta_z\\ \gamma\beta_x&1+(\gamma-1)\dfrac{\beta_x^2}{\beta^2}&(\gamma-1)\dfrac{\beta_x \beta_y}{\beta^2}&(\gamma-1)\dfrac{\beta_x \beta_z}{\beta^2}\\ \gamma\beta_y&(\gamma-1)\dfrac{\beta_y \beta_x}{\beta^2}&1+(\gamma-1)\dfrac{\beta_y^2}{\beta^2}&(\gamma-1)\dfrac{\beta_y \beta_z}{\beta^2}\\ \gamma\beta_z&(\gamma-1)\dfrac{\beta_z \beta_x}{\beta^2}&(\gamma-1)\dfrac{\beta_z \beta_y}{\beta^2}&1+(\gamma-1)\dfrac{\beta_z^2}{\beta^2}\\ \end{bmatrix}

(50)

Multiplying this matrix with a pure rotation will not affect the zeroth columns and rows, and

x = \begin{pmatrix}ct \\ x_1 \\ x_2 \\ x_3\end{pmatrix} = \begin{pmatrix}\gamma ct' \\ \gamma\beta_x ct' \\ \gamma\beta_y ct' \\ \gamma\beta_z ct'\end{pmatrix} = \begin{pmatrix}\gamma ct' \\ \gamma w_x t' \\ \gamma w_y t' \\ \gamma w_z t'\end{pmatrix} = \gamma \begin{pmatrix} ct' \\ w_x t' \\ w_y t' \\ w_z t'\end{pmatrix},

which could have been anticipated from the formula for a simple boost in the x-direction, and for the relative velocity vector

\frac{1}{ct}\mathbf x = \frac{\mathbf w}{c} = \boldsymbol \beta = \begin{pmatrix} \frac{x_1} {ct} \\ \frac{x_2} {ct} \\ \frac{x_3} {ct}\end{pmatrix} = \begin{pmatrix} \beta_x \\ \beta_y \\ \beta_z \end{pmatrix} = \begin{pmatrix}{\Lambda_0}^1/{\Lambda_0}^0 \\ {\Lambda_0}^2/{\Lambda_0}^0 \\ {\Lambda_0}^3/{\Lambda_0}^0 \end{pmatrix}.

Thus given with Λ, one obtains β and w by little more than inspection of Λ−1. (Of course, w can also be found using velocity addition per above.) From w, construct B(−w). The solution for R is then

R = B(-\mathbf w)\Lambda.

With the ansatz

\Lambda = RB(\mathbf w),

one finds by the same means

R = \Lambda B(-\mathbf w).

Finding a formal solution in terms of velocity parameters u and v involves first formally multiplying B(v)B(u), where both B(u) and B(v) are on the form of equation (50), formally inverting, then reading off βw form the result, formally building B(−w) according to equation (50) from the result, and, finally, formally multiplying B(−w)B(v)B(u). It should be clear that this is a daunting task, and it is difficult to interpret/identify the result as a rotation, though it is clear a priori that it is. It is these difficulties that the Goldstein quote at the top refers to. The problem has been thoroughly studied under simplifying assumptions over the years.

## Applications

### In electron orbitals

In quantum mechanics Thomas precession is a correction to the spin-orbit interaction, which takes into account the relativistic time dilation between the electron and the nucleus in hydrogenic atoms.

Basically, it states that spinning objects precess when they accelerate in special relativity because Lorentz boosts do not commute with each other.

To calculate the spin of a particle in a magnetic field, one must also take into account Larmor precession.

### In a Foucault pendulum

The rotation of the swing plane of Foucault pendulum can be treated as a result of parallel transport of the pendulum in a 2-dimensional sphere of Euclidean space. The hyperbolic space of velocities in Minkowski spacetime represents a 3-dimensional (pseudo-) sphere with imaginary radius and imaginary timelike coordinate. Parallel transport of a spinning particle in relativistic velocity space leads to Thomas precession, which is similar to the rotation of the swing plane of a Foucault pendulum.[25] The angle of rotation in both cases is determined by the area integral of curvature in agreement with the Gauss–Bonnet theorem.

Thomas precession gives a correction to the precession of a Foucault pendulum. For a Foucault pendulum located in the city of Nijmegen in the Netherlands the correction is:

\omega \approx 9.5 \cdot 10^{-7}\, \mathrm{arcseconds} / \mathrm{day}.

## Remarks

1. ^ This preservation of orthogonality of coordinate axes should not be confused with preservation of angles between spacelike vectors taken at one and the same time in one system, which, of course, does not hold. The coordinate axes transform under the passive transformation presented, while the vectors transform under the corresponding active transformation.
2. ^ This is sometimes called the "Mocanu paradox". Mocanu himself didn't name it a paradox, but rather a "difficulty" within the framework of relativistic electrodynamics in a 1986 paper. He was also quick to acknowledge that the problem is explained by Thomas precession Mocanu (1992), but the name lingers on.

## Notes

1. ^ Thomas 1926
2. ^ Wigner 1939
3. ^ Rhodes & Semon 2005
4. ^ Rebilas 2013
5. ^ Goldstein 1980, p. 287
6. ^ Einstein 1922
7. ^ Mocanu 1992
8. ^ Silberstein 1914, p. 169
9. ^ Eddington 1924
10. ^ Malykin 2006
11. ^ Goldstein 1980
12. ^ Ben-Menahem 1986
13. ^ Ungar 1988
14. ^ Weinberg 2002, pp. 68–69
15. ^ Cushing 1967
16. ^ Sard 1970, p. 74
17. ^ Ben-Menahem 1985
18. ^ Ryder (1996, p. 37)
19. ^ Ben-Menahem 1986
20. ^ Ungar 1988, p. 60
21. ^ Jackson 1999, p. 547
22. ^ Landau & Lifshitz 2002, p. 38
23. ^ Goldstein 1980, p. 285
24. ^ Jackson 1999, Eq. (11.98)
25. ^ Krivoruchenko 2009

## References

• Ben-Menahem, A. (1985). "Wigner's rotation revisited". Am. J. Phys. 53: 62–66.
• Ben-Menahem, S. (1986). "The Thomas precession and velocityspace curvature". J. Math. Phys. 27: 1284 pp.
• Krivoruchenko, M. I. (2009). "Rotation of the swing plane of Foucault's pendulum and Thomas spin precession: Two faces of one coin". Phys. Usp. (IOPScience) 52 (8): 821–829.
• Malykin, G. B. (2006). "Thomas precession: correct and incorrect solutions". Phys. Usp. 49 (8): 83 pp.
• Mocanu, C.I. (1992). "On the relativistic velocity composition paradox and the Thomas rotation". Found. Phys. Lett. (Kluwer Academic Publishers-Plenum Publishers) 5 (5): 443–456.
• Rebilas, K. (2013). "Comment on Elementary analysis of the special relativistic combination of velocities, Wigner rotation and Thomas precession". Eur. J. Phys. (IOPScience) 34 (3): L55. (free access)
• Rhodes, J. A.; Semon, M. D. (2005). "Relativistic velocity space, Wigner rotation and Thomas precession". Am. J. Phys. 72: 943 pp.
• Silberstein, L. (1914). The Theory of Relativity. London:
• Ungar, A. A. (1988). "Thomas rotation and parameterization of the Lorentz group". Foundations of physics letters (

## Textbooks

• Sard, R. D. (1970). Relativistic Mechanics - Special Relativity and Classical Particle Dynamics. New York: W. A. Benjamin.
• R. U. Sexl, H. K. Urbantke (2001) [1992]. Relativity, Groups Particles. Special Relativity and Relativistic Symmetry in Field and Particle Physics. Springer. p. 38–43.