# Visualization of Thomas–Wigner Rotations

## Abstract

**:**

## 1. Introduction

## 2. Method

- 〈1〉
- The grid $\mathcal{G}$ is Born-rigid.
- 〈2〉
- At the beginning and after completion of the ${N}^{\mathrm{th}}$ boost, $\mathcal{G}$ is at rest in frame $\left\{1\right\}$ and R returns to its starting position.
- 〈3〉
- R’s proper acceleration ${\alpha}_{R}$ and the boost’s proper duration $\Delta {\tau}_{R}$ are the same in all N sections.
- 〈4〉
- All boost directions and therefore all trajectories lie within the $xy$-plane.

## 3. Uniform Acceleration of a Born-Rigid Object

## 4. Sequence of Five Uniform Accelerations

## 5. Visualization

## 6. Discussion

#### 6.1. Derivation of Thomas–Wigner Angle

#### 6.2. Frame Boundaries

## 7. Conclusions

## Supplementary Materials

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MCIF | momentarily comoving inertial frame |

## Appendix A. Number of Boosts

#### Appendix A.1. Three Boosts

#### Appendix A.2. Four Boosts

## Appendix B. Computer Algebra Calculations

`vtwr3bst.py`(three boost case, see Appendix A.1) and

`vtwr5bst.py`(five boost case, see Section 4) are available online, for details see section “Supplementary Materials”. These scripts process Equations (A1), (A2), (16) and (17) and derive the results given in Equations (A5), (A6), (23), (26), (27), (39) and (40). The following paragraphs provide a few explanatory comments.

`vtwr5bst.py`calculates these matrices and their products in terms of ${T}_{12}$ and ${T}_{23}$ and inserts the result into Equation (17). The time component of Equation (17) yields the equation

`vtwr5bst.py`also shows that the result for the y-component of the equation ${\overrightarrow{P}}_{F}^{\left\{6\right\}}=0$ can be expressed as

`vtwr5bst.py`) leads to the product of two polynomials (expressions (26) and (27)), each of which is of the fourth order with respect to ${\left({T}_{12}\right)}^{2}$.

`vtwr5bst.py`evaluates the matrix elements ${R}_{1,1}$ and ${R}_{2,1}$ in terms of ${T}_{12}$ and ${T}_{23}$. Again, the resulting expressions are too unwieldy to reproduce them here.

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**Figure 1.**Trajectories of reference point R for $\gamma =2/\sqrt{3}\approx 1.15$ as seen from (laboratory) frame $\left\{1\right\}$ and frame $\left\{6\right\}$. The two frames are stationary with respect to each other, but rotated by a Thomas–Wigner angle of ${\theta}_{TW}\approx {14.4}^{\circ}$.

**Figure 2.**Trajectories of the reference point R as seen from the six frames $\left\{1\right\},\left\{2\right\},\dots ,\left\{6\right\}$. The switchover points are marked by ${X}_{i}^{\left\{k\right\}}$ with $X=\mathrm{A},\dots ,\mathrm{F}$. Corresponding switchover points are connected by dotted lines. The Lorentz factor is $\gamma =2/\sqrt{3}\approx 1.15$.

**Figure 3.**The angle between the boost direction vectors ${\widehat{e}}_{1}$ and ${\widehat{e}}_{2}$ in frame $\left\{1\right\}$ (solid line), and the angle between ${\widehat{e}}_{2}$ and ${\widehat{e}}_{3}$ in frame $\left\{2\right\}$ (dashed line) as a function of $\gamma $. The dotted line marks $+{180}^{\circ}$, the limit of ${\zeta}_{1,2}$ and ${\zeta}_{2,3}$ for $\gamma \to \infty $.

**Figure 4.**Boost directions for four different values of $\gamma =1/\sqrt{1-{\beta}^{2}}$. The boost direction in frame $\left\{1\right\}$, ${\widehat{e}}_{1}$ is assumed to point along the x-axis. In the relativistic limit $\gamma \to \infty $ (panel 4) the angles between ${\widehat{e}}_{k}$ and ${\widehat{e}}_{k+1}$ approach $+{180}^{\circ}$. The motion of the reference point R tends to be more and more restricted along the x-axis and the object’s trajectory transitions from a two- to a one-dimensional motion.

**Figure 5.**A series of grid positions as seen in the laboratory frame. The boost speed is taken to be $\beta =0.7$, resulting in a Thomas–Wigner rotation angle of about ${33.7}^{\circ}$. Coordinate time is displayed in the top right corner of each panel. The five boost phases are distinguished by colour. Evidently, switchovers between boosts do not occur simultaneously in the laboratory frame. The reference point (marked in red) moves along its trajectory counterclockwise, whereas the grid Thomas–Wigner rotates clockwise. For details, see text.

**Figure 6.**Thomas–Wigner rotation angle as a function of $\gamma $. For clarity, the angle is unwrapped and mapped to the range $[{0}^{\circ},+{360}^{\circ}]$.

**Figure 7.**Spacetime diagram of a one-dimensional grid consisting of seven points. The grid accelerates towards the positive x-direction. The trajectories are marked in blue/green, the mid point is taken as the reference R and its worldline is colored in red. Dots indicate the lapse of 0.1 s in proper time. After 0.6 s have passed on R’s clock, the acceleration stops and the points move with constant speed (green lines). Dashed and dotted lines connect simultaneous spacetime events in R’s comoving frame.

© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Beyerle, G. Visualization of Thomas–Wigner Rotations. *Symmetry* **2017**, *9*, 292.
https://doi.org/10.3390/sym9120292

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Beyerle G. Visualization of Thomas–Wigner Rotations. *Symmetry*. 2017; 9(12):292.
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**Chicago/Turabian Style**

Beyerle, Georg. 2017. "Visualization of Thomas–Wigner Rotations" *Symmetry* 9, no. 12: 292.
https://doi.org/10.3390/sym9120292