These authors contributed equally to this work.

Applying the cosmological principle to Finsler spacetimes, we identify the Lie Algebra of symmetry generators of spatially homogeneous and isotropic Finsler geometries, thus generalising Friedmann-Lemaître-Robertson-Walker geometry. In particular, we find the most general spatially homogeneous and isotropic Berwald spacetimes, which are Finsler spacetimes that can be regarded as closest to pseudo-Riemannian geometry. They are defined by a Finsler Lagrangian built from a zero-homogeneous function on the tangent bundle, which encodes the velocity dependence of the Finsler Lagrangian in a very specific way. The obtained cosmological Berwald geometries are candidates for the description of the geometry of the universe, when they are obtained as solutions from a Finsler gravity equation.

To describe the evolution of the whole universe in cosmology, one applies the cosmological principle (CP), which states that there exists no preferred spatial position and no preferred spatial direction on large scales. Applying this principle to general relativity leads to the spatially homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) metric as the unique ansatz for the geometry of spacetime. It contains two free functions which depend only on time, the lapse function and the scale factor. The lapse function can be normalized to unity, by a suitable choice of the time coordinate. The scale factor remains the only free function to be determined as solution of the Einstein equations, sourced by a perfect fluid energy-momentum tensor. On the basis of this mathematical model for the universe, one has to conclude that only ∼5% of the Universe consists of standard model baryonic matter, while the rest is composed of what is nowadays called dark energy [

A promising approach for a geometric explanation of the dark matter and dark energy phenomenology is to use Finsler spacetime geometry for the description of the gravitational interaction, instead of pseudo-Riemannian geometry [

In this article we apply the cosmological principle to Finsler spacetime geometry. Starting from a symmetry group which acts transitively on spatial equal time surfaces and which contains a local isotropy group acting transitively on spatial directions at each point, we find that a cosmological homogeneous and isotropic Finsler geometry is defined by a Finsler Lagrangian with a very specific dependence on the tangent bundle coordinates. Yet, since the Finsler Lagrangian is a 2-homogeneous function in its dependence on the directional variable of the tangent bundle, the demand of cosmological symmetry leaves large classes of allowed Finsler Lagrangians; the symmetry demand does not provide a strong limitation in this regard. Specific choices of Finsler geometries have been investigated in their capability to explain aspects of the cosmological dark matter and dark energy phenomenology [

A class of Finsler spacetime geometries, which can be regarded as closest to pseudo-Riemannian geometry, are the so-called Berwald spacetimes [

We derive the most general cosmologically (spatially homogeneous and isotropic) Berwald spacetime geometry. It serves as simplest Finslerian candidate for the description of the geometry of the universe. The obtained Finsler Lagrangian contains one free function, which encodes the velocity dependence of the Finsler Lagrangian in a very specific way. The Berwald geometry we obtain is the minimal Finsler geometric extension of pseudo-Riemannian FLRW geometry.

We present our results in the following way. In

Throughout this article, we consider the tangent bundle

A conic subbundle of

By a Finsler spacetime we will understand in the following a pair

on

there exists a conic subset

This is a refined version of the definition of Finsler spacetimes in Reference [

The 1-homogeneous function

We like to point again the relation

An important building block of the geometry of Finsler spacetimes is the geodesic spray, locally given by the coefficients

It defines the Finsler geodesic equation in arclength parametrization

A Finsler spacetime is called of Berwald type [

This is equivalent to demanding that the canonical nonlinear connection coefficients are actually linear in their

Next, we will determine the most general cosmologically symmetric Finsler spacetimes from the CP, before we derive the most general homogeneous and isotropic Berwald spacetime in

We will prove in the following that, applying the CP to Finsler spacetime geometry singles out 6 symmetry generating vector fields, which characterize spatially homogeneous and isotropic Finsler spacetimes. More precisely, these symmetry generating vector fields are the same as the ones defining cosmological symmetry in general relativity, that is, the same as in the case of pseudo-Riemannian geometry. The key argument is that spatial homogeneity and isotropy ensure the existence of a maximally symmetric Riemannian metric on the 3-dimensional time slice hypersurfaces of a Finsler spacetime. The Finsler Lagrangians we will determine will involve this maximally symmetric Riemannian metric, albeit, in a non-trivial way.

We begin by assuming that the spacetime in consideration possesses a smooth global time function

The demand of the existence of a symmetry group acting on the hypersurfaces

A Finsler spacetime geometry satisfies the CP if it is (see for example Wald [

Fix

Since the slice

We note that the homogeneity demand makes

We will now prove two lemmas to identify the dimension of the groups

Locally, in a coordinate chart around each

Using the Killing equation

Choose an arbitrary local chart such that the first coordinate is the time function

Let us properly understand the statement “the isotropy group

Any Lie group action on a manifold gives rise to an effective Lie group action on the respective manifold, by factorizing away the elements that provide trivial actions. In our case, assume the group

The group

Further, passing to the projectivised tangent spaces (that can be identified with the projective plane

There is actually a smaller subgroup of

The projectivized space

Finally, we use a result in Reference [

From Lemma 1 and Lemma 2 we find

We can actually state a much stronger result.

First we show that

Under the above assumption that

Further, we apply Cartan’s classification theorem ([

To explicitly determine the generators of the groups

Finally, solving the Finsler Killing equation

In order to find the desired Berwald Finsler spacetimes, we rewrite a generic Finsler Lagrangian in a specific way, which allows us to reduce the condition that a Finsler spacetime shall be Berwald, to a first order partial differential equation.

Every Finsler spacetime Lagragian

In case the expansion of

To avoid this problem observe that, if one has found one pseudo-Riemannian metric

To see this, let us expand

Using that

Hence, also

In the following, we will insert the most general

To evaluate the Berwald condition, we consider Finsler Lagrangians

The second ingredient in this condition is the

Using the above expressions in the Berwald condition (

Solving the Berwald condition for the most general combination of

Introducing the functions

We now analyze the first equation and find several cases in which we obtain trivial solutions, in the sense that the Finsler Lagrangian is pseudo-Riemannian or zero. The only case that provides proper Finslerian solutions is

Trivial solutions arise in the following situations:

If

Now it is helpful to introduce the function

We can thus rewrite Equation (

After multiplication with

Constructing the Finsler Lagrangian

If

From this analysis, we find that nontrivial cosmologically symmetric Berwald Finsler Lagrangians can only be obtained if

Demanding that

Having solved (

This equation intertwines the

In other words, the function

Using

After multiplication by

Recall from the definition (

The only remnant of the two different possible choices for the metric in the decomposition of the Finsler Lagrangian appears in the overall factor

One further step of simplification can be done by introducing the new coordinate

To summarize our findings we have proven the following theorem:

As an explicit example, one may consider

The cosmological principle assumes the existence of a symmetry group

Among the variety of possible Finsler geometric extensions of pseudo-Riemannian geometry as geometry of spacetime, Berwald spacetimes represent a most conservative generalization. Our discovery of the most general non-trivial cosmological, that is, spatially homogeneous and isotropic Berwald spacetimes reveals the class of geometries which extend the famous FLRW class of metrics into this realm. Most importantly, we found that cosmological Berwald geometries are parametrized by a free 0-homogeneous function on the tangent bundle, which intertwines the position and direction dependence of the Finsler Lagrangian in a very specific way. The resulting Finsler Lagrangian is

As the scale factor is determined by the Einstein equations on general relativity, the free function must be determined by suitable Finsler generalisations of the Einstein equations. Most of the suggested generalizations in the literature simplify significantly for Berwald geometries.

In particular, the ansatz (

The authors have all contributed substantially to the derivation of the presented results as well as analysis, drafting, review, and finalization of the manuscript. All authors have read and agreed to the published version of the manuscript.

C.P. and M.H. were supported by the Estonian Ministry for Education and Science through the Personal Research Funding Grants PSG489 (C.P.) and PRG356 (M.H.), as well as the European Regional Development Fund through the Center of Excellence TK133 “The Dark Side of the Universe”.

The authors would like to acknowledge networking support by the COST Actions QGMM (CA18108) and CANTATA (CA15117), supported by COST (European Cooperation in Science and Technology).

The authors declare no conflict of interest.

It is possible to equivalently formulate this property with opposite sign of