# LOG#048. Thomas precession.

LORENTZ TRANSFORMATIONS IN NON-STANDARD FORM

Let me begin this post with an uncommon representation of Lorentz transformations in terms of “uncommon matrices”. A Lorentz transformation can be written symbolically, as we have seen before, as the set of linear transformations leaving invariant

$ds^2=d\mathbf{x}^2-c^2dt^2$

Therefore, the Lorentz transformations are naively $X'=\mathbb{L}X$. Let $\mathbf{A}, \mathbf{B}$ be 3-rowed column matrices and let $M, R, \mathbb{I}$ represent $3\times 3$ matrices and $T$ will be used (unless it is stated the contrary) to denote the matrix transposition ( interchange of rows and columns in the matrix).

The invariance of $ds'^2=ds^2$ implies the following results from the previous definitions:

$\gamma^2-\mathbf{B}^2=1$

$M^TM =\mathbf{A}\mathbf{A}^T+\mathbb{I}$

$M^T\mathbf{B}=\gamma \mathbf{A}\leftrightarrow \mathbf{B}^T M=\gamma \mathbf{A}^T$

Then, we can write the matrix for a Lorent transformation (boost) in the following non-standard manner:

$\boxed{\mathbb{L}=\begin{pmatrix}\gamma & -\mathbf{A}^T\\ -\mathbf{B} & M\end{pmatrix}}$

and the inverse transformation will be

$\boxed{\mathbb{L}^{-1}=\begin{pmatrix}\gamma & \mathbf{B}^T\\ \mathbf{A} & M^T\end{pmatrix}}$

Thus, we have $\mathbb{L}\mathbb{L}^{-1}=\mathbb{I}_{4x4}\equiv \mathbb{E}$, where we also have

$\gamma^2-\mathbf{A}^2=1$

$M\mathbf{A}=\gamma \mathbf{B}$

$MM^T=\mathbf{B}\mathbf{B}^T+\mathbb{I}_{3x3}$

Let us define, in addition to this stuff, the reference frames $S, \overline{S}'$, corresponding to the the coordinates $\mathbf{X}$ and $\overline{\mathbb{X}}'$. Then, the boost matrix will be recasted, if the velocity read $\mathbf{v}=\mathbf{A}/\gamma$, as

$L_{v}=\begin{pmatrix}\gamma & -\gamma \mathbf{v}^T\\ -\gamma \mathbf{v} & \mathbb{I}+\frac{\gamma^2}{1+\gamma}\mathbf{v}\mathbf{v}^T\end{pmatrix}=\begin{pmatrix}\gamma & -\mathbf{A}^T\\ -\mathbf{A} & \mathbb{I}+\frac{\mathbf{A}\mathbf{A}^T}{1+\gamma}\end{pmatrix}$

Remark: a Lorentz transformation will differ from boosts only by rotations in the general case. That is, with these conventions, the most general Lorentz transformations include both boosts and rotations.

For all $\gamma>0$, the above transformation is well-defined, but if $\gamma<0$, then it implies we will face with transformations containing the reversal of time ( the time reversal operation T, please, is a different thing than matrix transposition, do not confuse their same symbols here, please. I will denote it by $\mathbb{T}$ in order to distinguish, althoug there is no danger to that confusion in general). The time reversal can be written indeed as:

$\mathbb{T}=\begin{pmatrix}-1 & \mathbf{0}^T\\ \mathbf{0} & \mathbb{I}\end{pmatrix}$

In that case, ($\gamma<0$), after the boost $L_{v}$, we have to make the changes $\gamma \rightarrow \vert \gamma\vert$ and $\mathbf{A}\rightarrow -\mathbf{A}$. If these shifts are done, the reference frames $\overline{S}$ and $\overline{S}'$ can be easily related

$\overline{X}'=LX=LL^{-1}_{v}\overline{X}$

in such a way that

$LL^{-1}_{v}=\begin{pmatrix}1 & \mathbf{0}\\ \mathbf{0} & R\end{pmatrix}=L_R$

where the rotation matrix is given formally by the next equation:

$R=M-\dfrac{\mathbf{B}\mathbf{A}^T}{1+\gamma}$

R must be an orthogonal matrix, i.e., $R^TR=\mathbb{I}_{3x3}$. Then $(\det (R))^2=1$, or $det R=\pm 1.$. For $\det R=-1$ we have the parity matrix

$\mathbb{P}=\begin{pmatrix}1 & \mathbf{0}^T\\ \mathbf{0} & -\mathbb{I}_{3x3}\end{pmatrix}$

and it will transform right-handed frames to left-handed frames $\overline{S}$ or $\overline{S}'$. The rotation vector $\alpha$ can be defined as well:

$1+2\cos \alpha=Tr (R)\rightarrow \cos\alpha=\dfrac{Tr R-1}{2}$

so $\alpha^\mu=\dfrac{1}{2}\epsilon^{\mu\nu\lambda}R^\nu_{\lambda}\dfrac{\alpha}{\sin\alpha}, \forall 0\leq \alpha<\pi$. The rotation acting on 3-rowed matrices:

$R\mathbf{A}=\mathbf{B}$

implies that $\overline{X}'=R\overline{X}$, and it changes $-\mathbf{A}/\gamma$ of the frame S into $\overline{S}$. Passing from one frame into another, $\overline{S}'$ to $S'$, it implies we can define a boost with $L_{-\mathbf{B}/\gamma}$. In fact,

$L_{-\mathbf{B}/\gamma}L=\begin{pmatrix}1 & \mathbf{0}^T\\ \mathbf{0} & R\end{pmatrix}=L_R$

Q.E.D.

Remark(I): Without the time reversal, we would get $L_{R\mathbf{v}}L_R=L=L_RL_{\mathbf{v}}$

with $\mathbf{v}=\mathbf{A}/\gamma$ and $R=M-\dfrac{\mathbf{BA}^T}{1+\gamma}$.

Remark (II): $L_RL_v\rightarrow L^T=L^T_vL_R^T=L_vL_{R^T}$. If $L^T=L=L_{R\mathbf{v}}L_R$, then the uniqueness of $R\mathbf{v}$ provides that $R=R^T=R^{-1}$, i.e., that R is an orthogonal matrix. If R is an orthogonal matrix and a proper Lorentz transformation ( $det R=+1$), then we would get $\sin\alpha=0$, and thus $\alpha=0$ or $\alpha=\pi$, and so, $R=I$ or $R=2\mathbf{n}\mathbf{n}^T-1$, with the unimodular vector $\mathbf{n}$, i.e., $\vert \mathbf{n}\vert=1$. That would be the case $\forall \mathbf{v}\neq 0$ and $\mathbf{n}=\mathbf{v}/\vert\vert \mathbf{v}\vert\vert$. Otherwise, if $\mathbf{v}=0$, then $\mathbf{n}$ would be an arbitrary vector.

ADDITION OF VELOCITIES REVISITED

The second step previous to our treatment of Thomas precession is to review ( setting $c=1$) the addition of velocities in the special relativistic realm. Suppose a point particle moves with velocity $\overline{w}$ in the reference frame $\overline{S}$. Respect to the S-frame (in rest) we will write:

$\mathbf{x}=\overline{\mathbf{x}}+\dfrac{\gamma^2}{\gamma+1}(\overline{\mathbf{x}}\mathbf{v})\mathbf{v}+\gamma \mathbf{v}\overline{t}$

and

$t=\gamma \overline{t}+\gamma (\mathbf{v}\overline{\mathbf{x}})$

and with $\overline{x}=\overline{\mathbf{w}}\overline{t}$ we can calculate the ratio $\mathbf{u}=\mathbf{x}/t$:

$\mathbf{u}=\dfrac{\dfrac{\overline{\mathbf{w}}}{\gamma}+\dfrac{\gamma}{1+\gamma}(\mathbf{v}\overline{\mathbf{w}})\mathbf{v}+\mathbf{v}}{1+\mathbf{v}\overline{\mathbf{w}}}$

and thus

$\mathbf{u}\equiv \dfrac{\mathbf{v}+\mathbf{w}_\parallel+(\mathbf{w}_\perp/\gamma)}{1+\mathbf{v}\overline{\mathbf{w}}}$

where we have defined:

$(\mathbf{w}_\perp/\gamma)\equiv\dfrac{\overline{\mathbf{w}}}{\gamma}$

and

$\mathbf{w}_\parallel\equiv \dfrac{\gamma}{1+\gamma}(\mathbf{v}\overline{\mathbf{w}})\mathbf{v}$

Comment: the composition law for 3-velocities is special relativity is both non-linear AND non-associative.

There are two special cases of motion we use to consider in (special) relativity and inertial frames:

1st. The case of parallel motion between frames (or “parallel motion”). In this case $\overline{\mathbf{w}}=\lambda \mathbf{v}$, i.e., $\mathbf{w}\times \mathbf{v}=0$. Therefore,

$\mathbf{u}=\dfrac{\mathbf{v}+\overline{\mathbf{w}}}{1+\mathbf{v}\overline{\mathbf{w}}}$

This is the usual non-linear rule to add velocities in Special Relativity.

2nd. The case of orthogonal motion between frames, where $\mathbf{v}\perp\mathbf{w}$. It means $\mathbf{v}\mathbf{w}=0$. Then,

$\mathbf{u}=\mathbf{v}+\mathbf{w}/\gamma= \mathbf{v}+\overline{\mathbf{w}}\sqrt{1-\mathbf{v}^2}$

This orthogonal motion to the direction of relative speed has an interesting phenomenology, since this inertial motion will be slowed down due to time dilation because the spatial distances that are orthogonal to $\mathbf{v}$ are equal in both reference frames.

Furthermore, we get also:

$\mathbf{u}^2=1-\dfrac{(1-\overline{\mathbf{w}}^2)(1-\mathbf{v}^2)}{(1+\mathbf{v}\overline{\mathbf{w}})}\leq 1$

Indeed, the condition $\mathbf{u}^2=1$ implies that $\overline{\mathbf{w}}^2=1$ or $\mathbf{v}^2=1$, and the latter condition is actually forbidden because of our interpretation of $\mathbf{v}$ as a relative velocity between different frames. Thus, this last equation shows the Lorentz invariance in Special relativity don’t allow for superluminal motion, although, a priori, it could be also used for even superluminal speeds since no restriction apply for them beyond those imposed by the principle of relativity.

THOMAS PRECESSION

We are ready to study the Thomas precession and its meaning. Suppose an inertial frame $\overline{\overline{S}}$ obtained from another inertial frame $\overline{S}$ by boosting the velocity $\overline{w}$. Therefore, $\overline{\overline{S}}$ owns the relative velocity $\mathbf{v}$ given by the addition rule we have seen in the previous section. Moreover, we have:

$\overline{\overline{x}}=L_{\overline{w}}\overline{x}=L_{\overline{w}}L_{\mathbf{v}}x$

Then, we get

$L_{\mathbf{v}}=\begin{pmatrix}\gamma_v & -\gamma_v \overline{\mathbf{v}}^T\\ -\gamma_v \mathbf{v} & \mathbf{1}+\dfrac{\gamma_v^2}{1+\gamma_v}\mathbf{v}\mathbf{v}^T\end{pmatrix}$

$L_{\overline{\mathbf{w}}}=\begin{pmatrix}\gamma_{\overline{\mathbf{w}}} & -\gamma_{\overline{\mathbf{w}}} \overline{\mathbf{w}}^T\\ -\gamma_{\overline{\mathbf{w}}} \overline{\mathbf{w}}^T & \mathbf{1}+\dfrac{\gamma_{\overline{\mathbf{w}}}^2}{1+\gamma_{\overline{\mathbf{w}}}}\overline{\mathbf{w}}\overline{\mathbf{w}}^T\end{pmatrix}$

where

$\gamma_{v}=\dfrac{1}{\sqrt{1-\mathbf{v}^2}}$

$\gamma_{\overline{\mathbf{w}}}=\dfrac{1}{\sqrt{1-\overline{\mathbf{w}}^2}}$

and then

$\boxed{L\equiv=L_{\overline{\mathbf{w}}}L_v=\begin{pmatrix}\gamma & -\mathbf{A}^T\\ -\mathbf{B} & M\end{pmatrix}}$

with

$\gamma (\mathbf{v},\overline{\mathbf{w}})=\gamma_v\gamma_{\overline{w}}(1+\mathbf{v}\overline{\mathbf{w}})\equiv \gamma (\overline{\mathbf{w}},\mathbf{v})$

$\mathbf{A}=\gamma (\mathbf{v},\overline{\mathbf{w}})\overline{\mathbf{w}}o \mathbf{v}$

$\mathbf{B}=\gamma (\overline{\mathbf{w}},\mathbf{v})\mathbf{v}o\overline{\mathbf{w}}$

$M=M(\overline{\mathbf{w}},\mathbf{v})=\mathbf{1}+\dfrac{\gamma_v^2}{1+\gamma_v}\mathbf{v}\mathbf{v}^T+\dfrac{\gamma_{\overline{\mathbf{w}}}^2}{1+\gamma_{\overline{\mathbf{w}}}}\overline{\mathbf{w}}\overline{\mathbf{w}}^T+\gamma_v\gamma_{\overline{\mathbf{w}}}\left( 1+\dfrac{\gamma_v\gamma_{\overline{\mathbf{w}}}}{(1+\gamma_v)(1+\gamma_{\overline{\mathbf{w}}})}\mathbf{v}\overline{\mathbf{w}}\right)\overline{\mathbf{w}}\mathbf{v}$

Here, we have defined:

$\boxed{\overline{\mathbf{w}}o \mathbf{v}\equiv \dfrac{\left( \gamma_{\overline{\mathbf{w}}}\gamma_v\mathbf{v}+\gamma_{\overline{\mathbf{w}}}\overline{\mathbf{w}}+\gamma_{\overline{\mathbf{w}}}\dfrac{\gamma_v^2}{1+\gamma_v}(\overline{\mathbf{w}}\mathbf{v})\right)}{\gamma (\mathbf{v},\overline{\mathbf{w}})}}$

Remark (I): The matrix L given by

$\begin{pmatrix}\gamma & -\mathbf{A}^T\\ -\mathbf{B} & M\end{pmatrix}$

is NOT symmetric as we would expect from a boost. According to our decomposition for the matrix $M$ it can be rewritten in the following way

$\boxed{R=R(\overline{\mathbf{w}},\mathbf{v})=M(\overline{\mathbf{w}},\mathbf{v})-\dfrac{\mathbf{B}\mathbf{A}^T}{1+\gamma}}$

This last equation is called the Thomas precession associated with the tridimensional 3-vectors $\mathbf{v},\overline{\mathbf{w}}$. We observe that R is a proper-orthogonal matrix from the multiplicative property of the determinants and the fact that all boosts have determinant one. Equivalently, from the condition $R=\pm 1$ for all orthogonal matrix R together with the continuous dependence of R on the velocities and the initial condition $R(0,0)=\mathbf{1}$.

Remark (II): From the definitions of M, and the vectors $\mathbf{A},\mathbf{B}$, we deduce that $\mathbf{v}\times \overline{\mathbf{w}}$ is an eigenvector of R with eigenvalue +1 and this gives the axis of rotation. The rotation angle $\alpha$ as calculated from $Tr R=1+2\cos\alpha$ is complicated expression, and only after some clever manipulations or the use of the geometric algebra framework, it simplifies to

$1+\cos\alpha=\dfrac{(1+\gamma_u+\gamma_v+\gamma_{\overline{w}})}{(1+\gamma_u)(1+\gamma_v)(1+\gamma_{\overline{w}})}>0$

In order to understand what this equation means, we have to observe that the components $\mathbf{v}$ and $\overline{\mathbf{w}}$ refer to different reference frames, and then, the scalar product $\mathbf{v}\mathbf{\overline{w}}$ and the cross product $\mathbf{v}\times\overline{\mathbf{w}}$ must be given good analitic expressions before the geometric interpretation can be accomplished. Moreover, if we want to interpret the cross product as an axis in the reference frame $S$, and correspondingly we want to split $L=L_{R\mathbf{v}}L_R$,  by the definition $\overline{\mathbf{w}}o\mathbf{v}$ we deduce that

$\mathbf{v}\times\mathbf{u}=\dfrac{\mathbf{v}\times\overline{\mathbf{w}}}{\gamma_v(1+\mathbf{v}\overline{\mathbf{w}})}$

and thus, the Thomas rotation of the inertial frame S has its axis orhtogonal to the relative velocity vectors $\mathbf{v},\mathbf{u}$ of the reference frame $\overline{\overline{S}}$, $\overline{\overline{S}}$ against S.

By the other hand, if we interpret the above last equation as an axis in the reference frame $\overline{\overline{S}}$, asociated to the split $L=L_RL_\mathbf{u}$, we would deduce that $L_{R\mathbf{u}}L_R$ implies the following consequence. The reference frame $\overline{\overline{S}}$ is got from boosting certain frame S’ obtained itself from a rotation of S by R. Then, $\overline{\overline{S}}$ obtains (compared with S or S’), a velocity whose components are $R\mathbf{u}$ in the inertial frame S’. Reciprocally, the components of the velocity of S or S’ against the frame $\overline{\overline{S}}$ are provided, in $\overline{\overline S}$, by $\overline{\overline{\mathbf{u}}}=-R\mathbf{u}$. Therefore, from the Thomas precession formula for R we observe that $R\mathbf{u}$ differs from $\mathbf{u}$ only by linear combinations of the vectors $\mathbf{v}$ and $\overline{\mathbf{w}}$. With all this results we easily derive:

$\overline{\overline{u}}\times \overline{\overline{\mathbf{w}}}=(-R\mathbf{u})\times (-\overline{\mathbf{w}})\propto \mathbf{v}\times \overline{\mathbf{w}}$

i.e., the axis for the Thomas rotation matrix of $\overline{\overline{S}}$ is orthogonal to the relative velocities $\overline{\overline{\mathbf{u}}}, \overline{\overline{\mathbf{w}}}$ of the inertial frames S, $\overline{S}$ against $\overline{\overline{S}}$. Finally, to find the rotation matrix, it is enough to restrict the problem to the case where $\overline{\mathbf{w}}$ is small so that squares of it may be neglected. In this simple case, R would become into:

$\boxed{R\approx \mathbf{1}+\dfrac{\gamma_v}{1+\gamma_v}\left(\overline{\mathbf{w}}\mathbf{v}^T-\mathbf{v}\overline{\mathbf{w}}^T\right)}$

and where the rotation angle is given by

$\boxed{\alpha\approx -\dfrac{\gamma_v}{1+\gamma_v}\mathbf{v}\times\overline{\mathbf{w}}\approx -\dfrac{\gamma_v^2}{1+\gamma_v}\mathbf{v}\times\mathbf{u}}$

In order to understand the Physics behind the Thomas precession, we will consider one single experiment. Imagine an inertial frame S in accelerated motion with respect to other inertial frame I. The spatial axes of S remain parallel at any time in the sense that the instantaneous reference frame coinciding with S at times $t+\Delta t$ are related by a pure boost in the limit $\Delta t\rightarrow 0$. This may be managed if we orient S with the aid of a very fast spinning torque-free gyroscope. Then, from the inertial frame I, S seems to be rotated at each instant of time and there is a continuous rotation of S against I since the velocity of S varies and changes continuously. This gyroscopic rotation of S relative to I IS the Thomas precession.  We can determine the angular velocity of this motion in a straightforward manner. During the small interval of time $\Delta t$ measured from I, the instantaneous velocity $\mathbf{v}$ of S changes by certain quantity $\Delta \mathbf{v}$, measured from I. In that case,

$\Delta \alpha=-\gamma_v^2\mathbf{v}\times\dfrac{\Delta\mathbf{v}}{(1+\gamma_v)}$

for the rotation vector during a time interval $\Delta t$. Thus, the angular velocity for the Thomas precession will be given by:

$\boxed{\omega_T=-\dfrac{\gamma^2}{1+\gamma_v}\mathbf{v}\times\dfrac{d\mathbf{v}}{dt}}$

or reintroducing the speed of light we get

$\boxed{\omega_T=-\dfrac{\gamma^2}{1+\gamma_v}\mathbf{v}\times\dfrac{1}{c^2}\dfrac{d\mathbf{v}}{dt}=\dfrac{\gamma^2}{1+\gamma_v}\dfrac{1}{c^2}\mathbf{a}\times\mathbf{v}}$

Remark(I): The special relativistic effect given by the Thomas precession was used by Thomas himself to remove a discrepancy and mismatch between the non-relativistic theory of the spinning electron and the experimental value of the fine structure. His observation was, in fact, that the gyromagnetic ratio of the electron calculated from the anomalous Zeeman effect led to a wrong value of the fine structure constant $\alpha$. The Thomas precession introduces a correction to the equation of motion of an electron in an external electromagnetic filed and such a correction induces a correction of the spin-orbit coupling, explaining the correct value of the fine structure.

Remark (II): In the framework of the relativistic quantum theory of the electron, Dirac realized that the effect of Thomas precession was automatically included!

Remark (III): Inside the Thomas paper, we find these interesting words

“(…)It seems that Abraham (1903) was the first to consider in any detail an electron with an axis. Many have since then considered spinning electron, ring electrons, and the like. Compton (1921) in particular suggested a quantized spin for the electron. It remained for Uhlenberg and Goudsmit (1925) to show ho this idea can be used to explain the anomalous Zeeman effect. The asumptions they had to make seemed to lead to optical and relativity doublet separations twice larger than those we observe. The purpose of the following paper, which contains the results mentioned in my recent letter to Nature (1926), is to investigate the kinematics of an electron with an axis on the basis of the restricted theory of relativity. The main fact used is that the combination of two Lorentz transformations without rotation in general is not of the same form(…)”.

From the historical viewpoint it should also be remarked that the precession effect was known by the end of 1912 to the mathematician E.Borel (C.R.Acad.Sci.,156. 215 (1913)). It was described by him (Borel, 1914) as well as by L.Silberstein (1914) in textbooks already 1914. It seems that the effect was even known to A.Sommerfeld in 1909 and before him, perhaps even to H.Poincaré. The importance of Thomas’ work and papers on this subject was thus not only the rediscovery but the relevant application to a virulent problem in that time, as it was the structure of the atomic spectra and the fine structure constant of the electron!

Remark (IV): Not every Lorentz transformation can be written as the product of two boosts due to the Thomas precession!

THE LORENTZ GROUP AS A QUASIDIRECT PRODUCT: QUASIGROUPS, LOOPS AND GYROGROUPS

Even though we have not studied group theory in this blog, I feel the need to explain some group theory stuff related to the Thomas precession here.

The kinematical differences between Galilean and Einsteinian relativity theories is observed at many levels. The essential differences become apparent already on the level of the homogenous groups without reversals (inverses). Let me first consider the Galileo group. It is generated by space rotations $G_R=L_R$ and galilean boosts in any number and order. Using the notation we have developed in this post, we could write $X'=G_\mathbf{v}X$ in this way:

$G_\mathbf{v}=\begin{pmatrix}1 & \mathbf{0}^T\\ -\mathbf{v} & \mathbf{1}\end{pmatrix}$

The following relationships are deduced:

$G_RG_\mathbf{v}=G_{R\mathbf{v}}G_{R}$

$G_{R_1}G_{R_2}=G_{R_1R_2}$

$G_{\mathbf{v}_1}G_{\mathbf{v}_2}=G_{\mathbf{v}_1+\mathbf{v}_2}=G_{\mathbf{v}_2}G_{\mathbf{v}_1}$

In the case of the Lorentz group, these equations are “generalized” into

$L_RL_\mathbf{v}=L_{R\mathbf{v}}L_{R}$

$L_{R_1}L_{R_2}=L_{R_1R_2}$

$L_{\mathbf{v}_1}L_{\mathbf{v}_2}=L_{R(\mathbf{v}_1,\mathbf{v}_2)}L_{\mathbf{v}_1 o \mathbf{v}_2}$

where $R(\mathbf{v}_1,\mathbf{v}_2)$ is the Thomas precession and the circle denotes the nonlinear relativisti velocity addition. Be aware that the domain of velocities in special relativity is $\vert v\vert<1$, in units with c set to unity.

Both groups (Galileo and Lorentz) contain as a subroupt the group of al spatial rotations $G_R\equiv L_R$. The set of galilean or lorentzian boosts $G_v$ and $L_v$ are invariant under conjugation by $G_R=L_R$, since

$G_RG_vG_R^{-1}=G_{Rv}$

$L_RL_vL_R^{-1}=L_{Rv}$

are boosts as well. In the case of the Galileo group, the set of (galilean) boost forms an (abelian) subgroup and then, it provides an invariant group. We can calculate the factor group with respect to it and we will obtain an isomorphic group to the subgroup of space rotations. Using the group law for the Galileo group:

$\underbrace{G_{R_1}G_{v_1}}\underbrace{G_{R_2}G_{v_2}}=G_{R_1R_2}G_{R_2^{-1}v_1+v_2}=G_{R_3}G_{v_3}$

with $R_3=R_2R_1$ and $v_3=R_2^{-1}v_1+v_2$. As a consequence, the homogenous Galileo group (without reversals) is called a semidirect product of the rotation group with the Abelian group $\mathbb{R}^3$ of all boosts given by $\mathbf{v}$.

The case of Lorentz group is more complicated/complex. The reason is the Thomas precession. Indeed, the set of boost does NOT form a subgroup of the Lorentz group! We can define a product in this group:

$\boxed{L_{v_1} oL_{v_2}=L_{v_1 o v_2}}$

but, in the contrary to the result we got with the Galileo group, this condition does NOT define a group structure. In fact, mathematicians call objects with this property groupoids. The domain of velocities of the this lorentzian grupoid becomes a groupoid under the multiplication $v_1 o v_2$. It has dramatic consequences. In particular, the associative does not hold for this multiplication and this groupoid structure! Anyway, a weaker form of it is true, involving the Thomas precession/rotation formula:

$\boxed{(v_1 o v_2) o v_3=(R^{-1}(v_2,v_3)v_1) o (v_2 o v_3)}$

In an analogue way, the multiplication is not commuative in general too, but it satisfies a weaker form of commutativity. While in general groupoids require to distinguish between right and left unit elements (if any), we have indeed $\mathbf{v}=\mathbf{0}$ as a “two-sided” unit element for the velocity groupoid. In the same manner, while in general groupoids right and left inverses may differ (if any), in the case of Lorentz group, the groupoid associated to Thomas precession has a unique two-sided inverse $-\mathbf{v}$ for any $\mathbf{v}$ relative to the groupoid multiplication law. It is NON-trivial ( due to non-associativeness), albeit true, that the equation given by

$v_1 o v_2=v_3$

may be solved uniquely for $v_2$ and, provided we plug $v_2, v_3$, it may be solve uniquely for any $v_1$. A groupoid satisfying this property (i.e., a groupoid that allows such a uniqueness in the solutions of its equation) is called quasi-group.

In conclusion, we can say that the Lorentz group IS, in sharp contrast to the Galileo group, in no way a semidirect product, being what mathematicians and physicists call a simple group, i.e., it is a noncommutative group having no nontrivial invariant subgroup! It is due to the fact that the multiplication rule of the Lorentz group without reversals makes it, in the sense of our previous definitions, the quasidirect product of the rotation group (as a subgroup of the automorphism group of the velocity groupoid)  with the so-called “weakly associative groupoid of velocities”. Here, weakly associative(-commutative) groupoid means the following: a groupoid with a left-sided unit and left-sided inverses with the next properties:

1. Weak associativeness: $R(\mathbf{0},\mathbf{v})=R(-\mathbf{v},\mathbf{v})=\mathbf{1}$

2. Loop property (from Thomas precession formula): $R(v_1,v_2)=R(v_1,v_1 o v_2)$

and where the automorphims group of the velocity groupoid is defined with the next equations

Definition (Automorphism group of the velocity groupoid): $(Sv_1)o(Sv_2)=S(v_1 o v_2)$

Note: an associative groupoid is called semigroup and and a semigroup with two-sided unit element is called a monoid.

This algebraic structure hidden in the Lorentz group has been rediscovered several times along the History of mathematical physics. A groupoid satisfying the loop property has been named in other ways. For instance, in 1988, A. A. Ungar derived the above composition laws and the automorphism group of the Thomas precession R. Independently, A. Nesterov and coworkers in the Soviet Union had studied the same problem and quasigroup since 1986. And we can track this structure even more. 20 years before the Ungar “rediscovery”, H. Karzel had postulated a version of the same abstract object, and it was integrated into a richer one with two compositions (laws). He called it “near-domain”, where the automorphims R (Thomas precessions) were to be realized by the (distributive) left multiplication with suitable elements of the near-domian ( the reference is Abh. Math.Sem.Uni. Hamburg, 1968).

However, Ungar himself developed a more systematic treatment and description for the Thomas precession “groupoid” that is behind all this weird non-associative stuff in the Lorentz-group in 3+1 dimensions. Accorging to his new approach and terminology, the structure is called “gyrocommutative gyrogroup” and it includes the Thomas precession as “Thomas gyration” in this framework. If you want to learn more about gyrogroups and gyrovector spaces, read this article

http://en.wikipedia.org/wiki/Gyrovector_space

Some other authors, like Wefelscheid and coworkers, called K-loops to these gyrogroups. Even more, there are two extra sources from this nontrivial mathematical structure.

Firstly, in Japan, M.Kikkawa had studied certain loops with a compatible differentiable structure called “homegeneous symmetric Lie groups” ( Hiroshima Math. J.5, 141 (1975)). Even though he did not discuss any concrete example, it is natural from his definitions that it was the same structure Karzel found. Being romantic, we can observe certain justice to call K-loops to gyrogroups (since Kikkawa and Karzel discovered them first!). The second source can be tracked in time since the same ideas were already known by L.Sabinin et alii circa 1972 ( Sov. Math. Dokl.13,970(1972)). Their relation to symmetric homogeneous spaces of noncompact type has been discussed some years ago by W. Krammer and H.K.Urbatke, e.g., in Res. Math.33, 310 (1998).

Finally, a purely algebraic loop theory approach (with motivations far way from geometry or physics) was introduced by D. A. Robinson in 1966. In 1995, A. Kreuzer showed thath it was indeed identical to K-loops, again adding some extra nomenclature ( Math.Proc.Camb. Phylos.Soc.123, 53 (1998)).

THOMAS PRECESSION: EASY DEDUCTION

We have seen that the composition of 2 Lorentz boosts, generally with 2 non collinear velocities, results in a Lorentz transformation that IS NOT a pure boost but a composition of a single Lorentz transformation or boost and a single spatial rotation. Indeed, this phenomenon is also called Wigner-Thomas rotation. The final consequence, any body moving on a curvilinear trajectory undergoes and experiences a rotational precession, firstly noted by Thomas in the relativistic theory of the spinning electron.

In this final section, I am going to review the really simple deduction of the Thomas precession formula given in the paper http://arxiv.org/abs/1211.1854

Imagine 3 different inertial observers Anna, Bob and Charles and their respective inertial frames A, B, and C attached to them. We choose A as a non-rotated frame with respect to B, and B as a non-rotated reference frame w.r.t. C. However, surprisingly, C is going to be rotated w.r.t. A and it is inevitable! We are going to understand it better. Let Bob embrace Charles and let them move together with constant velocity $\mathbf{v}$ w.r.t. Anna. In some point, Charles decides to run away from Bob with a tiny velocity $\mathbf{dv'}$ w.r.t. Bob. Then, Bob is moving with relative velocity $-\mathbf{dv'}$ w.r.t. C and Anna is moving with relative velocity $-\mathbf{v}$ w.r.t. B. We can show these events with the following diagram:

Now, we can write Charles’ velocity in the Anna’s frame by the sum $\mathbf{v+dv}$. Since the frame C is rotated with respect to the A frame, his velocity in the C frame will be $\hat{\mathbf{v}}$ will be calculated step to step as follows. Firstly, we remark that

$\hat{\mathbf{v}}\neq -\mathbf{v}-d\mathbf{v}$

Secondly, the angle $d\mathbf{\Omega}$ of an infinitesimal rotation is given by:

$d\mathbf{\Omega}=-\dfrac{\hat{\mathbf{v}}}{\vert \hat{\mathbf{v}}\vert }\times \dfrac{\mathbf{v}+d\mathbf{v}}{\vert \mathbf{v}+d\mathbf{v}\vert}\approx -\dfrac{\hat{\mathbf{v}}}{v^2}\times (\mathbf{v}+ d\mathbf{v})\;\;\; (1)$

The precession rate in the A frame will be provided using the general nonlinear composition rule in SR. If the motion is parallel to the x-axis with velocity $V$, we do know that

$u'_x=\dfrac{u_x-V}{1-\dfrac{u_x V}{c^2}}$

$u'_y=\dfrac{u_y\sqrt{1-\dfrac{V^2}{c^2}}}{1-\dfrac{u_x V}{c^2}}$

$u'_z=\dfrac{u_z\sqrt{1-\dfrac{V^2}{c^2}}}{1-\dfrac{u_x V}{c^2}}$

and where $\mathbf{u}=(u_x,u_y,u_z)$ and $\mathbf{u}'=(u'_x,u'_y,u'_z)$ are the velocities of some object in the rest frame and the moving frame, respectively. For an arbitrary non-collinear, non-orthogonal, i.e., non parallel velocity $\mathbf{V}=(V_x,V_y,V_z)$ we obtain the transformations

$\boxed{\mathbf{u}'=\dfrac{\sqrt{1-\dfrac{V^2}{c^2}}\left(\mathbf{u}-\dfrac{\mathbf{u}\cdot\mathbf{V}}{V^2}\mathbf{V}\right)-\left( \mathbf{V}-\dfrac{\mathbf{u}\cdot\mathbf{V}}{V^2}\mathbf{V}\right)}{1-\dfrac{\mathbf{u}\cdot\mathbf{V}}{c^2}}\;\;\; (2)}$

and where the unprimed and primed frames are mutually non-rotated to each other. Using this last equation, (2), we can easily describe the transition from the frame A to the frame B. It involves the substitutions:

$\mathbf{V}\rightarrow \mathbf{v}$

$\mathbf{u}\rightarrow \mathbf{v}+d\mathbf{v}$

$\mathbf{u}'\rightarrow d\mathbf{v}'$

After leaving the first order terms in $d\mathbf{v}$, we can get the following expansion from eq.(2):

$d\mathbf{v}'\approx \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}\left(d\mathbf{v}-\dfrac{\mathbf{v}\cdot d\mathbf{v}}{v^2}\mathbf{v}\right)+\dfrac{1}{1-\dfrac{v^2}{c^2}}\dfrac{\mathbf{v}\cdot d\mathbf{v}}{v^2}\mathbf{v}\;\;\; (3)$

Using again eq.(2) to make the transition between the B frame to the C frame, i.e., making the substitutions:

$\mathbf{V}\rightarrow d\mathbf{v}'$

$\mathbf{u}\rightarrow -\mathbf{v}$

$\mathbf{u}'\rightarrow \hat{\mathbf{v}}$

and dropping out higher order differentials in $d\mathbf{v}'$, we obtain the next formula after we neglect those terms

$\boxed{\hat{\mathbf{v}}\approx -\mathbf{v}+\dfrac{\mathbf{v}\cdot d\mathbf{v}'}{c^2}\mathbf{v}-d\mathbf{v}'\;\;\; (4)}$

The final step consists is easy: we plug eq.(3) into eq.(4) and the resulting expression into eq.(1). Then, we divice by the differential $dt$ in the final formula to provide the celebrated Thomas precession formula:

$\boxed{\dot{\Omega}=\dfrac{d\Omega}{dt}=\omega_T=-\dfrac{1}{v^2}\left(\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}-1\right)\mathbf{v}\times \dot{\mathbf{v}}\;\;\; (5)}$

or equivalently

$\boxed{\dot{\Omega}=\dfrac{d\Omega}{dt}=\omega_T=-\dfrac{1}{v^2}\left(\gamma_{\mathbf{v}}-1\right)\mathbf{v}\times \mathbf{a}\;\;\; (6)}$

It can easily shown that these formulae is the same as the given previously above, writing $v^2$ in terms of $\gamma$ and performing some elementary algebraic manipulations.

Aren’t you fascinated by how these wonderful mathematical structures emerge from the physical world? I can say it: Fascinating is not enough for my surprised mind!