# Cosmological Finsler Spacetimes

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## Abstract

**:**

## 1. Introduction

## 2. Berwald Finsler Spacetime Geometry

- ${\pi}_{TM}\left(\mathcal{Q}\right)=M$;
- 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 L-metric, is non-degenerate,$$\begin{array}{c}\hfill {g}_{ab}^{L}={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial}^{2}L}{\partial {\dot{x}}^{a}\partial {\dot{x}}^{b}}};\end{array}$$
- there exists a conic subset $\mathcal{T}\subset \mathcal{A}$ such that on $\mathcal{T}$, $L>0$, ${g}^{L}$ 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 defined, smooth and ${g}^{L}$ is nondegenerate, with fiber ${\mathcal{A}}_{x}=\mathcal{A}\cap {T}_{x}M$, called the set of admissible vectors;
- $\mathcal{T}$: the set of future pointing timelike directions, a maximally connected conic subbundle where $L>0$, the L-metric exists and has Lorentzian signature $(+,-,-,-)$, with fiber ${\mathcal{T}}_{x}=\mathcal{T}\cap {T}_{x}M$;
- $\mathcal{N}=\left\{(x,\dot{x})\right|L(x,\dot{x})=0\}$: the subbundle where $L=0$, with fiber ${\mathcal{N}}_{x}=\mathcal{N}\cap {T}_{x}M$.

## 3. The Cosmological Principle on Finsler Spacetimes

#### 3.1. The Symmetry and the Isotropy Group

- spatially homogeneous, that is, for each fixed value T of the time function t and for any two points ${q}_{1}$ and ${q}_{2}$ in ${\Sigma}_{T}$, there exists a diffeomorphism of M which maps ${q}_{1}$ to ${q}_{2}$ and preserves the Finsler Lagrangian L. In other words, there exists a Lie Group G of isometries of $(M,L)$ acting transitively on each slice ${\Sigma}_{T}$;
- spatially isotropic, that is, around each $p\in M$ there exists a congruence of observer curves $\gamma (t,s)$ with tangent vector field $\frac{\partial \gamma}{\partial t}\in \mathcal{T}$, such that for each two lines $\left[{Z}_{1}\right]$ and $\left[{Z}_{2}\right]$ in ${T}_{p}{\Sigma}_{T}$ there exists a diffeomorphism of M preserving p, $\frac{\partial \gamma}{\partial t}$ and L while mapping $\left[{Z}_{1}\right]$ to $\left[{Z}_{2}\right]$ (the equivalence relation $\left[\right]$ is defined as $Z\sim \overline{Z}$ if $Z=\lambda Z,\lambda \in {R}^{\ast}$). In other words, the stabilizer ${G}_{p}=\{\psi \in G|\psi \left(p\right)=p\}$ at $p\in M$ acts transitively on the projective tangent spaces $P{T}_{p}{\Sigma}_{T}$ of ${\Sigma}_{T}$.

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Proof.**

- Under the above assumption that G is connected, the isotropy group ${G}_{p}$ is also connected. To see this, let us assume that ${G}_{p}$ is the disjoint union of at least two connected components, say, ${G}_{p}=A\bigsqcup B$, where A is the component of the identity. Since $P{T}_{p}{\Sigma}_{T}$ is connected and ${G}_{p}$ acts transitively on it, it follows that A also acts transitively on $P{T}_{p}{\Sigma}_{T}$ and see Reference [29] (p. 395),$$\begin{array}{c}\hfill {G}_{p}=A{H}_{v},\end{array}$$
- Further, we apply Cartan’s classification theorem ([29], p. 389 ), to ${G}_{p}$. The Theorem states that any connected Lie group is the direct product between one of its maximal compact subgroups, say $\mathcal{H}$, and a Euclidean space. Since ${G}_{p}$ acts transitively on the compact manifold $P{T}_{p}{\Sigma}_{T}$ (which, as we have seen above, has finite fundamental group), its maximally compact subgroup $\mathcal{H}$ also acts transitively on $P{T}_{p}{\Sigma}_{T}$. But, the smallest compact group that can act transitively on $P{T}_{p}{\Sigma}_{T}$ is 3-dimensional (more precisely, $SO\left(3\right)$), that is, $dim\mathcal{H}\ge 3$. Taking into account that the dimension of ${G}_{p}$ itself is 3, it means that ${G}_{p}=\mathcal{H}$, that is ${G}_{p}$ itself is compact.

#### 3.2. The Symmetry Generators

## 4. Homogeneous and Isotropic Berwald Spacetimes

#### 4.1. The Berwald Condition

#### The Cosmological Berwald Condition

#### 4.2. Solving the Cosmological Berwald Condition

#### 4.2.1. Trivial Solutions

- If N is different from zero, then we can divide the first equation in (24) by N,$$\begin{array}{c}\hfill \frac{M(t,s)}{N(t,s)}\Omega (t,s)+\frac{\partial}{\partial s}\Omega (t,s)=0\phantom{\rule{0.166667em}{0ex}}.\end{array}$$Now it is helpful to introduce the function$$\begin{array}{c}\hfill A(t,s)=\frac{1+\sigma {s}^{2}a{\left(t\right)}^{2}}{{a}^{\prime}\left(t\right)-a\left(t\right)\left[{s}^{2}(c\left(t\right)-\sigma a\left(t\right){a}^{\prime}\left(t\right))-d\left(t\right)\right]}\phantom{\rule{0.166667em}{0ex}}\end{array}$$$$\begin{array}{c}\hfill \frac{1}{A(t,s)}\frac{\partial}{\partial s}A(t,s)=\frac{M(t,s)}{N(t,s)}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$We can thus rewrite Equation (25) as$$\begin{array}{c}\hfill \frac{\Omega (t,s)}{A(t,s)}\frac{\partial}{\partial s}A(t,s)+\frac{\partial}{\partial s}\Omega (t,s)=0\phantom{\rule{0.166667em}{0ex}}.\end{array}$$After multiplication with $A(t,s)$ we find$$\begin{array}{c}\hfill \frac{\partial}{\partial s}\left(\Omega (t,s)A(t,s)\right)=0\phantom{\rule{1.em}{0ex}}\Rightarrow \phantom{\rule{1.em}{0ex}}\Omega (t,s)A(t,s)=f\left(t\right)\phantom{\rule{0.166667em}{0ex}},\end{array}$$$$\begin{array}{c}\hfill \Omega (t,s)=\frac{f\left(t\right)}{A(t,s)}=f\left(t\right)\frac{{a}^{\prime}\left(t\right)-a\left(t\right)\left[{s}^{2}(c\left(t\right)-\sigma a\left(t\right){a}^{\prime}\left(t\right))-d\left(t\right)\right]}{1+\sigma {s}^{2}a{\left(t\right)}^{2}}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$Constructing the Finsler Lagrangian $L={\dot{t}}^{2}(1-a{\left(t\right)}^{2}{s}^{2})\Omega (t,s)$ from this solution immediately yields$$\begin{array}{c}\hfill L=I\left(t\right){\dot{t}}^{2}+J\left(t\right){w}^{2}\phantom{\rule{0.166667em}{0ex}},\end{array}$$
- If $N=0$ and $M\ne 0$, then the first equation in (24) implies immediately that $\Omega (t,s)=0$ and thus the Finsler Lagrangian is $L=0$.

#### 4.2.2. Finslerian Solutions

**Theorem**

**2.**

- pseudo-Riemannian (quadratic in $\dot{x}$), in which case it is, up to t coordinate redefinition, given by the FLRW metric, or
- nontrivially Finslerian, in which case it is of the form (45).

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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

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Hohmann, M.; Pfeifer, C.; Voicu, N. Cosmological Finsler Spacetimes. *Universe* **2020**, *6*, 65.
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Hohmann M, Pfeifer C, Voicu N. Cosmological Finsler Spacetimes. *Universe*. 2020; 6(5):65.
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**Chicago/Turabian Style**

Hohmann, Manuel, Christian Pfeifer, and Nicoleta Voicu. 2020. "Cosmological Finsler Spacetimes" *Universe* 6, no. 5: 65.
https://doi.org/10.3390/universe6050065