# Symmetry and Special Relativity

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## Abstract

**:**

## 1. Introduction

## 2. Inertial Frames

## 3. The Lorentz Transformations

- the x and ${x}^{\prime}$ axes are antiparallel, whereas the y and ${y}^{\prime}$ axes, as well as the z and ${z}^{\prime}$ axes, are parallel;
- the velocity of ${K}^{\prime}$ in K is a constant $\mathbf{v}\ne 0$ in the positive x direction ($\mathbf{v}=0$ is classical, not relativistic); and
- the origins O and ${O}^{\prime}$ correspond at time $t={t}^{\prime}=0$.

**(I)**- ${a}^{2}+abv=1$
**(II)**- $b(a+d)=0$
**(III)**- $av(a+d)=0$
**(IV)**- $abv+{d}^{2}=1$

**(I)**,we get $1+(b/a)v=1/{a}^{2}$. Let $\tilde{b}=b/a$. Then, ${a}^{2}=1/(1+\tilde{b}v)$, or

**(III)**and Equation (5) imply that $d=-a$. Thus,

**(1)**- ${\mu}^{2}{a}^{2}-{a}^{2}{v}^{2}={\mu}^{2}$
**(2)**- $2{\mu}^{2}{a}^{2}\tilde{b}+2{a}^{2}v=0$
**(3)**- ${\mu}^{2}{a}^{2}{\tilde{b}}^{2}-{a}^{2}=-1$.

**(1)**imply that

**(2)**yields

**(3)**. Therefore, there exists an invariant metric of the form Equation (10). Thus, from Equations (8) and (12), the matrix L of the transformation from ${K}^{\prime}$ to K is

## 4. Velocity Addition and a Universally Preserved Speed

**(V1)**- $-(\mathbf{v}\oplus \mathbf{u})=-\mathbf{v}\oplus (-\mathbf{u})$
**(V2)**- If $\mathbf{u}$ and $\mathbf{v}$ are parallel, then $\mathbf{u}\oplus (\mathbf{v}\oplus \mathbf{w})=(\mathbf{u}\oplus \mathbf{v})\oplus \mathbf{w}$.

## 5. The Velocity Ball as a Bounded Symmetric Domain

## 6. The Symmetric Velocity Ball as a Bounded Symmetric Domain

## 7. Symmetric Velocity Addition on a Complex Plane

## 8. Explicit Analytic Solutions When $\mathbf{E}\perp \mathbf{B}$

## 9. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Five uniformly spread discs ${\Delta}_{j}$, obtained by intersecting the three-dimensional velocity ball ${D}_{v}$ of radius 1 with $y-z$ planes at $x=0,\pm 1/3,\pm 2/3$. (

**b**) The images of the five discs under the action of $\phi \left(\mathbf{v}\right)$, with $\mathbf{v}=(1/3,0,0)$.

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**MDPI and ACS Style**

Friedman, Y.; Scarr, T.
Symmetry and Special Relativity. *Symmetry* **2019**, *11*, 1235.
https://doi.org/10.3390/sym11101235

**AMA Style**

Friedman Y, Scarr T.
Symmetry and Special Relativity. *Symmetry*. 2019; 11(10):1235.
https://doi.org/10.3390/sym11101235

**Chicago/Turabian Style**

Friedman, Yaakov, and Tzvi Scarr.
2019. "Symmetry and Special Relativity" *Symmetry* 11, no. 10: 1235.
https://doi.org/10.3390/sym11101235