# On the Non Metrizability of Berwald Finsler Spacetimes

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## Abstract

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## 1. Introduction

## 2. Finsler Geometry

#### 2.1. Finsler Spaces

- F is positively homogeneous of degree one with respect to $\dot{x}$: $F(x,\alpha \dot{x})=\alpha F(x,\dot{x})$ for all $\alpha \in {\mathbb{R}}^{+}$,
- the matrix:$$\begin{array}{c}\hfill {g}_{ab}^{F}={\textstyle \frac{1}{2}}{\dot{\partial}}_{a}{\dot{\partial}}_{b}{F}^{2}\phantom{\rule{0.166667em}{0ex}}\end{array}$$

#### 2.2. Finsler Spacetimes

- $\pi \left(\mathcal{Q}\right)=M$, where $\pi :TM\to M$ is the canonical projection;
- conic property: if $(x,\dot{x})\in \mathcal{Q}$, then for any $\lambda >0:$$(x,\lambda \dot{x})\in \mathcal{Q}$.

- L is positively homogeneous of degree two with respect to $\dot{x}$: $L(x,\lambda \dot{x})={\lambda}^{2}L(x,\dot{x})$ for all $\lambda \in {\mathbb{R}}^{+}$,
- on $\mathcal{A}$, the vertical Hessian of L, called the L-metric, is nondegenerate,$$\begin{array}{c}\hfill {g}_{ab}^{L}={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial}^{2}L}{\partial {\dot{x}}^{a}\partial {\dot{x}}^{b}}}\end{array}$$
- there exists a conic subset $\mathcal{T}\subset \mathcal{A}$ such that on $\mathcal{T}$, $L>0$, g has Lorentzian signature $(+,-,-,-)\phantom{\rule{3.33333pt}{0ex}}$ and, on the boundary $\partial \mathcal{T}$, L can be continuously extended as ${L|}_{\partial \mathcal{T}}=0$.1

- $\mathcal{A}$: the subbundle where L is smooth and ${g}^{L}$ is nondegenerate, with fiber ${\mathcal{A}}_{x}=\mathcal{A}\cap {T}_{x}M$, called the set of admissible vectors,
- $\mathcal{N}$: the subbundle where L is zero, with fiber ${\mathcal{N}}_{x}=\mathcal{N}\cap {T}_{x}M$,
- ${\mathcal{A}}_{0}=\mathcal{A}\backslash \mathcal{N}$: the subbundle where L can be used for normalization, with fiber ${\mathcal{A}}_{0x}={\mathcal{A}}_{0}\cap {T}_{x}M$,
- $\mathcal{T}$: a maximally connected conic subbundle where $L>0$ and the L-metric exists and has Lorentzian signature $(+,-,-,-)$, with fiber ${\mathcal{T}}_{x}=\mathcal{T}\cap {T}_{x}M$.

## 3. Berwald Spacetime Geometry and Metric-Affine Spacetime Geometry with Non-Metricity

**Theorem**

**1**

#### 3.1. A Necessary Condition for the Metrizability of Berwald Spacetimes

**Theorem**

**2**

#### 3.2. Non-Metrizable Berwald–Finsler Spacetimes

**Proposition**

**1.**

#### 3.3. Affine Structure of Berwald Spacetimes

**Proposition**

**2.**

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Theorem 2

## Appendix B. Generalized Bogoslovsky/Kropina–Finsler Lagrangians

- (1)
- $p>0$: $L=0\iff \zeta (\dot{x},\dot{x})=0$, and L is not defined for $\beta \left(\dot{x}\right)=0$;
- (2)
- $0>p>-1$: $L=0\iff \zeta (\dot{x},\dot{x})=0$ or $\beta \left(\dot{x}\right)=0$;
- (3)
- $p<-1$: $L=0\iff \beta \left(\dot{x}\right)=0$, and L is not defined for $\zeta (\dot{x},\dot{x})=0$.

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1. | It is possible to formulate this property equivalently with the opposite sign of L and the metric ${g}^{L}$ of signature $(-,+,+,+)$. We fixed the signature and sign of L here to simplify the discussion. |

2. | We will elaborate on this in forthcoming work. |

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**MDPI and ACS Style**

Fuster, A.; Heefer, S.; Pfeifer, C.; Voicu, N.
On the Non Metrizability of Berwald Finsler Spacetimes. *Universe* **2020**, *6*, 64.
https://doi.org/10.3390/universe6050064

**AMA Style**

Fuster A, Heefer S, Pfeifer C, Voicu N.
On the Non Metrizability of Berwald Finsler Spacetimes. *Universe*. 2020; 6(5):64.
https://doi.org/10.3390/universe6050064

**Chicago/Turabian Style**

Fuster, Andrea, Sjors Heefer, Christian Pfeifer, and Nicoleta Voicu.
2020. "On the Non Metrizability of Berwald Finsler Spacetimes" *Universe* 6, no. 5: 64.
https://doi.org/10.3390/universe6050064