# Raychaudhuri Equations, Tidal Forces, and the Weak-Field Limit in Schwarzshild–Finsler–Randers Spacetime

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## Abstract

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## 1. Introduction

## 2. Geometrical Structure of the Model

#### 2.1. The Lorentz Tangent Bundle

#### 2.1.1. The Adapted Basis

#### 2.1.2. Metric Structure on TM

- F is continuous on $TM$ and smooth on $\tilde{TM}\equiv TM\setminus \left\{0\right\}$, i.e., the tangent bundle minus the null set $\left\{\right(x,y)\in TM|F(x,y)=0\}$.
- F is positively homogeneous to the first degree on its second argument:$$F({x}^{\mu},k{y}^{a})=kF({x}^{\mu},{y}^{a}),\phantom{\rule{2.em}{0ex}}k>0$$
- The form$${f}_{ab}(x,y)=\pm {\displaystyle \frac{1}{2}}{\displaystyle \frac{{\mathsf{\partial}}^{2}{F}^{2}}{\mathsf{\partial}{y}^{a}\mathsf{\partial}{y}^{b}}}$$$$det\left[{f}_{ab}\right]\ne 0$$

#### 2.1.3. Connection

#### 2.1.4. Curvature and Torsion

#### 2.1.5. Hilbert-like Action

#### 2.2. The SFR Model

#### 2.3. The Newtonial Limit

## 3. Generalized Deviation of Geodesics and Paths

#### 3.1. General Equations

#### 3.2. Application of the Weak-Field Limit

#### 3.3. The Schwarzschild–Finsler–Randers spacetime

## 4. Generalized Raychaudhuri Equations

#### 4.1. Horizontal Equations

#### 4.2. Vertical Equations

#### 4.3. Application to the SFR Model

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Triantafyllopoulos, A.; Kapsabelis, E.; Stavrinos, P.C.
Raychaudhuri Equations, Tidal Forces, and the Weak-Field Limit in Schwarzshild–Finsler–Randers Spacetime. *Universe* **2024**, *10*, 26.
https://doi.org/10.3390/universe10010026

**AMA Style**

Triantafyllopoulos A, Kapsabelis E, Stavrinos PC.
Raychaudhuri Equations, Tidal Forces, and the Weak-Field Limit in Schwarzshild–Finsler–Randers Spacetime. *Universe*. 2024; 10(1):26.
https://doi.org/10.3390/universe10010026

**Chicago/Turabian Style**

Triantafyllopoulos, Alkiviadis, Emmanuel Kapsabelis, and Panayiotis C. Stavrinos.
2024. "Raychaudhuri Equations, Tidal Forces, and the Weak-Field Limit in Schwarzshild–Finsler–Randers Spacetime" *Universe* 10, no. 1: 26.
https://doi.org/10.3390/universe10010026