# A Transformation Factor for Superluminal Motion That Preserves Symmetrically the Spacetime Intervals

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## Abstract

**:**

## 1. Introduction

#### 1.1. On the Evasive Tachyons

#### 1.2. On Imaginary Mass

#### 1.3. Definition of Tachyon

#### 1.4. The Lorentz Transformations

- If $\left|v\right|>\left|c\right|$, we must deal with imaginary numbers.
- If $\left|v\right|=\left|c\right|$, mathematically, this implies $\gamma =\infty $. This singularity constitutes a barrier with various implications; for example, the need for infinite energy for a massive particle to reach a speed of c.

#### 1.5. On the Mathematical Domain of the Lorentz Factor

## 2. Methods

#### Reverse Coordinates on the X Axis

Within aligned coordinates, the reverse transformation from x to ${x}^{\prime}$ is determined by $x=({x}^{\prime}+vt)$. If the same substitution is applied for reversed coordinates, we have

## 3. Derivation of the Transformation Factor Assuming That $\mathit{v}>\mathit{c}$

#### 3.1. Alternative Extended Transformation Factor

#### 3.2. The Special Case of AETF When $\alpha ={0}^{\circ}$ and $v<c$

#### 3.3. Special Case of AETF When $\alpha ={0}^{\circ}$ and $v>c$

#### 3.4. The Transformation between the Angles $\alpha $ and ${\alpha}^{\prime}$

#### 3.5. Special Case of AETF When $v<c$ and $\alpha >{0}^{\circ}$

#### 3.6. Special Case When $v>c$ and $\alpha >{0}^{\circ}$

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AETF | alternative extended transformation factor |

ETF | extended transformation factor |

LT | Lorentz transformations |

MLC | Minkowski light cone |

MST | Minkowski spacetime |

STR | special theory of relativity |

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**Figure 1.**Reverse frames. Location of an event P in the Minkowski light cone. MST where $\left|v\right|<\left|c\right|$.

**Figure 2.**Reverse frames. Location of a point (event) outside of the Minkowski light cone, where $\left|v\right|>\left|c\right|$.

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**MDPI and ACS Style**

Roldán, D.; Roldán-Aráuz, F.
A Transformation Factor for Superluminal Motion That Preserves Symmetrically the Spacetime Intervals. *Symmetry* **2023**, *15*, 1177.
https://doi.org/10.3390/sym15061177

**AMA Style**

Roldán D, Roldán-Aráuz F.
A Transformation Factor for Superluminal Motion That Preserves Symmetrically the Spacetime Intervals. *Symmetry*. 2023; 15(6):1177.
https://doi.org/10.3390/sym15061177

**Chicago/Turabian Style**

Roldán, Diego, and Francisco Roldán-Aráuz.
2023. "A Transformation Factor for Superluminal Motion That Preserves Symmetrically the Spacetime Intervals" *Symmetry* 15, no. 6: 1177.
https://doi.org/10.3390/sym15061177