Three-line derivation of the Thomas precession

TL;DR

This paper presents a straightforward derivation of the Thomas precession, clarifying its origin and mathematical formulation in relativistic physics.

Contribution

It offers a concise, three-line derivation of the Thomas precession, simplifying understanding of this relativistic effect.

Findings

Clear derivation of Thomas precession

Enhanced understanding of relativistic spin effects

Simplified mathematical approach

Abstract

Instantaneous derivation of the Thomas precession.

Equations

$$\text{d}\mathbf{\Omega }=\frac{(\mathbf{v}+\text{d}{\mathbf{v}}^{\prime })\times (\mathbf{v}+\text{d}\mathbf{v})}{v^2}=\frac{1}{v^2}(\mathbf{v}\times \text{d}\mathbf{v}-\mathbf{v}\times \text{d}{\mathbf{v}}^{\prime }).$$

$$\mathbf{v}\times \text{d}{\mathbf{v}}^{\prime }=\mathbf{v}\times \frac{(\mathbf{v}+\text{d}\mathbf{v})\sqrt{1-v^2}}{1-(\mathbf{v}+\text{d}\mathbf{v})\cdot \mathbf{v}}=\frac{\mathbf{v}\times \text{d}\mathbf{v}}{\sqrt{1-v^2}},$$

$$\dot{\mathbf{\Omega }}=-\frac{1}{v^2}\left(\frac{1}{\sqrt{1-v^2}}-1\right)\mathbf{v}\times \dot{\mathbf{v}}.$$

Full text

Three-line derivation of the Thomas precession

Paweł Lewulis Supported by NCN Preludium 11, 2016/21/N/ST1/02599. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656, Warsaw, Poland

  

Andrzej Dragan

Institute of Theoretical Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore

Recently halfpage , a half-page derivation of the Thomas precession effect has been presented (see also all the references therein for the history and prehistory of the problem). In this short note we provide a much simpler, three-line derivation of that effect which, to our knowledge, is the simplest in the literature.

Consider Bob $B$ moving relative to Alice’s frame $A$ with a relativistic velocity $\bm{v}$, as shown in Fig. 1. If Bob’s velocity changes by an infinitesimal value $\text{d}\bm{v}^{\prime}$ relative to his initial reference frame $B$, Alice will observe his new velocity to be some $\bm{v}+\text{d}\bm{v}$. The new Bob’s frame $B^{\prime}$ is now rotated relative to Alice’s frame by a Thomas–Wigner angle $\text{d}\bm{\Omega}$. In the non-relativistic theory we have $\text{d}\bm{v}^{\prime}=\text{d}\bm{v}$ and the rotation is absent. Therefore, the Thomas–Wigner angle can be interpreted as an angle between the relativistic velocity of $B^{\prime}$ with respect to $A$, $\bm{v}+\text{d}\bm{v}$, and it’s non-relativistic approximation, $\bm{v}+\text{d}\bm{v}^{\prime}$:

[TABLE]

Let us use a relativistic transformation of the perpendicular velocity component, $\bm{u}^{\prime}_{\perp}=\frac{\bm{u}_{\perp}\sqrt{1-v^{2}}}{1-\bm{u}\cdot\bm{v}}$. We will consider a velocity transformation from $A$ to $B$. Here we substitute $\bm{u}\rightarrow\bm{v}+\text{d}\bm{v}$ and $\bm{u}^{\prime}\rightarrow\text{d}\bm{v}^{\prime}$. Next, we take the cross product of the resulting formula with $\bm{v}$:

[TABLE]

where we used the fact that the parallel (to $\bm{v}$) components of velocities do not contribute to the vector product with $\bm{v}$, and neglected higher order corrections from the denominator. Substituting Eq. (2) into Eq. (1) and dividing by an infinitesimal time $\text{d}t$ we obtain the Thomas precession rate:

[TABLE]