# Lorentz Violation in Finsler Geometry

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## Abstract

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## 1. Introduction

## 2. Introduction to Finsler Geometry

#### 2.1. A First Glance at Finsler Geometry

- Regularity: $\mathrm{F}$ is ${C}^{\infty}$ on the entire slit tangent bundle $T{M}_{0}:=TM\setminus \left\{0\right\}$.
- Positive homogeneity: $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda >0$.
- Strong convexity: The $n\times n$ Hessian matrix$${g}_{ij}:={\left(\frac{1}{2}{F}^{2}\right)}_{{y}^{i}{y}^{j}}$$

#### 2.2. Mathematical Concepts of Finsler Geometry

## 3. Modified Dispersion Relations and Finsler Geometry

#### 3.1. The Bridge between Modified Dispersion Relations and Finsler Geometry

#### 3.2. Physical Influences from Finsler Geometry

#### 3.2.1. Time Dilation in Finsler Geometry

#### 3.2.2. Arrival Time Delay of Astroparticles in Finsler Geometry

#### 3.2.3. Transformation between Inertial Frames and Modified Composition Laws

## 4. Connections between Lorentz Violation Theories and Finsler Geometry

#### 4.1. Doubly Special Relativity and Finsler Geometry

#### 4.2. Standard-Model Extension and Finsler Geometry

#### 4.3. Very Special Relativity and Finsler Geometry

## 5. Summary and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DSR | Doubly special relativity |

FRW | Friedmann–Lemaître–Robertson–Walker |

GZK | Greisen–Zatsepin–Kuzmin |

LV | Lorentz violation |

MDR | Modified dispersion relation |

QG | Quantum gravity |

SME | Standard-Model Extension |

VSR | Very special relativity |

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Designation | Generators | Algebra |
---|---|---|

$\mathfrak{t}\left(2\right)$ | ${T}_{1},{T}_{2}$ | $\left[{T}_{1},{T}_{2}\right]=0$ |

$\mathfrak{e}\left(2\right)$ | ${T}_{1},{T}_{2},{J}_{z}$ | $\left[{T}_{1},{T}_{2}\right]=0,\left[{T}_{1},{J}_{z}\right]=-i{T}_{2},\left[{T}_{2},{J}_{z}\right]=i{T}_{1}$ |

$\mathfrak{hom}\left(2\right)$ | ${T}_{1},{T}_{2},{K}_{z}$ | $\left[{T}_{1},{T}_{2}\right]=0,\left[{T}_{1},{K}_{z}\right]=i{T}_{1},\left[{T}_{2},{K}_{z}\right]=i{T}_{2}$ |

$\mathfrak{sim}\left(2\right)$ | ${T}_{1},{T}_{2},{J}_{z},{K}_{z}$ | $\left[{T}_{1},{T}_{2}\right]=0,\left[{T}_{1},{K}_{z}\right]=i{T}_{1},\left[{T}_{2},{K}_{z}\right]=i{T}_{2},$ |

$\left[{T}_{1},{J}_{z}\right]=-i{T}_{2},\left[{T}_{2},{J}_{z}\right]=i{T}_{1},\left[{J}_{z},{K}_{z}\right]=0$ |

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**MDPI and ACS Style**

Zhu, J.; Ma, B.-Q.
Lorentz Violation in Finsler Geometry. *Symmetry* **2023**, *15*, 978.
https://doi.org/10.3390/sym15050978

**AMA Style**

Zhu J, Ma B-Q.
Lorentz Violation in Finsler Geometry. *Symmetry*. 2023; 15(5):978.
https://doi.org/10.3390/sym15050978

**Chicago/Turabian Style**

Zhu, Jie, and Bo-Qiang Ma.
2023. "Lorentz Violation in Finsler Geometry" *Symmetry* 15, no. 5: 978.
https://doi.org/10.3390/sym15050978