On the Non Metrizability of Berwald Finsler Spacetimes

: We investigate whether Szabo’s metrizability theorem can be extended to Finsler spaces of indeﬁnite signature. For smooth, positive deﬁnite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern–Rund connection deﬁnes an afﬁne connection on the underlying manifold), then it is afﬁnely equivalent to a Riemann space, meaning that its afﬁne connection is the Levi–Civita connection of some Riemannian metric. We show for the ﬁrst time that this result does not extend to general Finsler spacetimes. More precisely, we ﬁnd a large class of Berwald spacetimes for which the Ricci tensor of the afﬁne connection is not symmetric. The fundamental difference from positive deﬁnite Finsler spaces that makes such an asymmetry possible is the fact that generally, Finsler spacetimes satisfy certain smoothness properties only on a proper conic subset of the slit tangent bundle. Indeed, we prove that when the Finsler Lagrangian is smooth on the entire slit tangent bundle, the Ricci tensor must necessarily be symmetric. For large classes of Finsler spacetimes, however, the Berwald property does not imply that the afﬁne structure is equivalent to the afﬁne structure of a pseudo-Riemannian metric. Instead, the afﬁne structure is that of a metric-afﬁne geometry with vanishing torsion.

The (pseudo-)Finslerian manifolds that can be regarded as closest to the (pseudo-)Riemannian manifolds are the so-called Berwald spaces, resp. spacetimes [23][24][25][26]. For these geometries, the canonical nonlinear connection, which is the fundamental building block of the Finsler geometry under consideration, is actually linear in its dependence on the directional variable of the tangent bundle, and thus gives rise to an affine connection on the base manifold. Hence, the natural question to ask is: Under which conditions are Berwald geometries metrizable, i.e., under which conditions does there exist a (pseudo-)Riemannian metric such that the affine Berwald connection is the Levi-Civita connection of this metric? For Berwald spaces, i.e., in the positive definite case, this question is answered by Szabo's theorem [27]. It states that every Berwald space is Riemann metrizable. In other words, for every Berwald space, there exists a Riemannian metric such that the affine connection of the Berwald space is identical to the Levi-Civita connection of the metric. The desired Riemannian metric can be constructed explicitly from the Finsler metric of the Berwald space by averaging the Finsler metric over the indicatrix of the Finsler function [28]. For Finsler spacetimes, no extension of Szabo's theorem has been presented so far.
In this article, we show for the first time that it is not possible to extend Szabo's theorem to Finsler spacetimes, and thus, Berwald spacetimes are in general non-metrizable. Instead, the following statement holds. The affine connection of every Berwald spacetime is identical to a metric-affine geometry on the base manifold, with a torsion-free connection that is in general not metric compatible. More explicitly, we find examples of Berwald spacetime metrics such that the Ricci tensor of the affine connection is not symmetric. The origin of this lack of symmetry lies in the weaker smoothness assumptions on the Finsler Lagrangian compared to the Finsler function on Finsler spaces.
We present this result as follows. In Section 2, we introduce the necessary notions of the geometry of Finsler spaces and Finsler spacetimes before we present our main result, the non-metrizability of Berwald-Finsler spacetimes, in Section 3. We close this article with a short discussion of our results in Section 4.

Finsler Geometry
In Finsler geometry, the geometry of a manifold is derived from a general geometric length measure for curves, defined by a so-called Finsler function. The origin of this idea goes back to Riemann [29], but it was only systematically investigated by Finsler in his thesis [13]. For positive definite geometries, the generalization of Riemannian geometry to Finsler geometry has long been known in the literature. Applying the same kind of generalization from pseudo-Riemannian geometry to pseudo-Finsler geometry is not so straightforward.
We will recall the definition of Finsler spaces and Finsler spacetimes and point out the differences in the construction.
Throughout this article, we consider the tangent bundle TM of an n-dimensional manifold M, equipped with manifold induced local coordinates, as follows. A point (x,ẋ) ∈ TM is labeled by the coordinates (x a ,ẋ a ) given by the decomposition of the vectorẋ =ẋ a ∂ a ∈ T x M, where x a are the local coordinates of the point x ∈ M. If there is no risk of confusion, we will sometimes suppress the indices of the coordinates. The local coordinate bases of the tangent and cotangent spaces, T (x,ẋ) TM and T * (x,ẋ) TM, of the tangent bundle are {∂ a = ∂ ∂x a ,∂ a = ∂ ∂ẋ a } and {dx a , dẋ a }. In the following, unless elsewhere specified, by smooth, we will always understand C ∞ -smooth.

Finsler Spaces
A Finsler space is a pair (M, F), where M is a smooth n-dimensional manifold and the Finsler function F : TM → R is continuous on TM and smooth on TM \ {0}, and (see [14]): • F is positively homogeneous of degree one with respect toẋ: F(x, αẋ) = αF(x,ẋ) for all α ∈ R + , • the matrix: defines the Finsler metric tensor g F and is positive definite at any point of TM \ {0}, in one (and then, in any) local chart around that point.
The length of a curve γ : R ⊃ [t 1 , t 2 ] → M on a Finsler space is defined by the parametrization invariant length integral: To discuss Szabo's theorem later, we need the building blocks of the geometry of a Finsler space, the geodesic spray coefficients: They are defined on each coordinate chart of TM \ {0} and characterize the geodesic equation, which, for arc length parametrized curves, is: Further objects we need for the arguments in this article are the horizontal derivative operators: the local coordinate expressions of the Chern-Rund connection coefficients: the hh-Chern-Rund curvature: and its corresponding horizontal Ricci tensor: It is important to notice that in general, R ab is not symmetric: Here, C a = C b ab are the components of the trace of the Cartan tensor C abc = 1 2∂ a g F bc . The last equality in (9) can most easily be proven by introducing the function: By direct calculation, one finds, and therefore: An alternative proof of (13) can be obtained in terms of the trace of Equation (47) of [30] or the use of Equation (3.4.5) of [14].
The geometric objects introduced in this section make sense on local charts of TM \ {0}. Details on the construction and properties of these objects in Finsler geometry can be found in [14,15].
Turning to Finsler spacetimes, we will consider the same geometric objects and see that in general, these only make sense on subbundles of TM \ {0}.

Finsler Spacetimes
To discuss Finsler spacetimes properly, we recall the notion of a conic subbundle Q of TM [18], which is a non-empty open submanifold Q ⊂ TM\{0}, with the following properties: By a Finsler spacetime, we will understand in the following a pair (M, L), where M is a smooth n-dimensional manifold and the Finsler Lagrangian L : A → R is a smooth function on a conic subbundle A ⊂ TM, such that: • L is positively homogeneous of degree two with respect toẋ: on A, the vertical Hessian of L, called the L-metric, is nondegenerate, • there exists a conic subset T ⊂ A such that on T , L > 0, g has Lorentzian signature (+, −, −, −) and, on the boundary ∂T , L can be continuously extended as This is a refined version of the definition of Finsler spacetimes in [19] and basically covers, if one chooses A = T , the improper Finsler spacetimes defined in [31].
The one-homogeneous function F, which defines the length measure (2), is derived from the Finsler Lagrange function as F = |L| and interpreted as the proper time integral of observers. For clarity, we list some important sets appearing on Finsler spacetimes and comment on their meaning: The geometry of Finsler spacetimes is only well defined on A. Some operations, like integration with a canonical zero-homogeneous length measure, can even only be performed on A 0 [19]. On A, the geodesic spray is given by: 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.
All further geometric objects, which we introduced in the context of Finsler spaces, are defined by the same expressions (5), (6), (7), (8), and (9), but, just as the geodesic spray, only make sense on A.
The property that in general, the geometry of a Finsler spacetime is not defined on all of TM \ {0} is crucial in our following finding, that, in general, Szabo's theorem cannot be extended to Finsler spacetimes.

Berwald Spacetime Geometry and Metric-Affine Spacetime Geometry with Non-Metricity
A Finsler space or Finsler spacetime is said to be of Berwald type, or simply a Berwald space or spacetime, if the geodesic spray is quadratic inẋ: In this case, the secondẋ-derivatives of the geodesic spray coefficients are affine connection coefficients∂ b∂c G a = G a bc (x) on the base manifold. A standard result in Finsler geometry is that, in general, the geodesic spray can be expressed in terms of the Chern-Rund connection coefficients (6) as: This means that on a Berwald space, or spacetime, the Chern-Rund connection coefficients are independent ofẋ and the affine connection on M, defined by the connection coefficients G a bc (x), is precisely given by the Chern-Rund connection, i.e., Γ a bc (x,ẋ) = Γ a bc (x) = G a bc (x). For Berwald spaces, it is known that (see [27]): Theorem 1 (Szabo's theorem). Let (M, F) be a Finsler space of Berwald type. Then, there exists a Riemannian metric g on M such that the affine connection of the Berwald space is the Levi-Civita connection of g.
Thus, the affine structure of a Berwald space (M, F) is identical to the affine structure of a Riemannian manifold (M, g). The metric g can be constructed explicitly from the Finsler metric g F by an averaging procedure over itsẋ dependence [28].
Next we demonstrate that Szabo's theorem can in general not be extended to Berwald spacetimes.

A Necessary Condition for the Metrizability of Berwald Spacetimes
Let (M, L) be a Berwald spacetime with Chern-Rund affine connection coefficients Γ a bc = Γ a bc (x), as defined in (6). Then, the horizontal Chern-Rund Ricci tensor (see (8)) is independent ofẋ and takes in every coordinate chart the form: It can be regarded as the Ricci tensor of the affine connection with coefficients Γ a bc (x) on M. A necessary, but not sufficient, condition for the connection defined by Γ a bc to be the Levi-Civita connection of a pseudo-Riemannian metric is that the Ricci tensor (18) is symmetric.
From (9), we find that for Berwald geometries, the skew-symmetric part is given by: where we expressed the explicit dependence on variables x andẋ to highlight that this equation encodes much information about the geometry of Berwald spacetimes. In particular, it gives rise to the following theorem: Next, we explicitly present a class of Finsler spacetimes that are, in general, not metrizable, since their Ricci tensor is not symmetric.

Non-Metrizable Berwald-Finsler Spacetimes
The following Finsler spacetimes are of Berwald type, but not metrizable, since their Ricci tensor is not symmetric. As we have seen in the previous section, the reason is that the set of admissible vectors A is smaller than TM \ {0}, which is rather the usual case for Finsler spacetimes, than the exception.
Let α(ẋ,ẋ) = α ab (x)ẋ aẋb be constructed from a pseudo-Riemannian metric α, β(ẋ) = β a (x)ẋ a a one-form on M evaluated on a generic tangent vector, and c, m, p real numbers. Moreover, define the zero-homogeneous variable s = β(ẋ) 2 α(ẋ,ẋ) . Consider the (α, β)-Finsler Lagrangian: A brief discussion about the causal properties of this type of Finsler Lagrangians can be found in Appendix B. Finsler spacetimes (M, L) are generalizations of Bogoslovsky/p-Kropina/very general relativity geometries [1,[5][6][7]23,[32][33][34][35], which we recover for c = 1, m = 0. The classical Kropina case is included by setting p = 1. In [36,Corollary 3], we have shown that (M, L) is a Berwald spacetime if and only if the covariant derivative of the one-form β with respect to the Levi-Civita connection of α satisfies: for an arbitrary function H = H(x) on M. The resulting geodesic spray is: and thus, the resulting affine connection coefficients are: Here, γ a bc are the Christoffel symbols of the pseudo-Riemannian metric α; R[γ] ab below is the corresponding Ricci tensor. For the Chern-Rund Ricci tensor (18), we find: Thus, for the skew-symmetric part, We summarize these findings as follows.  (20). Then, the skew-symmetric part A(R) of the Chern-Rund Ricci tensor is given by: This shows that the Ricci tensor corresponding to the Finsler Lagrangian (20) may not be symmetric when H = 0, in particular when the one-form β is not covariantly constant (cf. (21)). This is indeed often the case 2 , for example in the class of (α, β)-Kundt spacetimes introduced in [36], with Lagrangian (20). For the sake of conciseness, we present below a very simple geometry in this class for which the Ricci tensor is not symmetric, providing an explicit counterexample that shows that Szabo's theorem does not hold in general for Berwald spacetimes.
Let the Lorentzian metric α and the one-form β be given by: where u, v are so-called light-cone coordinates and φ is a scalar function. It is easy to see that β is a null one-form with respect to α, i.e., α −1 (β, β) = 0. Assuming c = 0 and p = 1, the pair α, β satisfies condition (21) with H = φ 2c(p−1) ; hence, the resulting Finsler Lagrangian L = αs −p (c + ms) p+1 is of Berwald type.
Plugging the function H in Equation (25), we find the following non-trivial components of the skew-symmetric part of the Ricci tensor: We see that they are independent of the parameters c and m and do not vanish for non-constant φ and p = 0. This simple counterexample shows that the affine connection of a Berwald spacetime (M, L) is not necessarily equivalent to the Levi-Civita connection of a (pseudo-)Riemannian metric. Hence, Szabo's theorem does not extend to general Finsler spacetimes. We would like to point out that if one would exchange the Lorentzian metric in the examples with a Riemannian metric, the same derivations can be made. Hence, if one allows for Finsler spaces with a Finsler function that is not smooth on all of TM \ {0}, as for instance conic Finsler geometries [18], one also finds counterexamples to Szabo's theorem for positive definite Finsler geometries.

Affine Structure of Berwald Spacetimes
In physics, metric affine theories of gravity are considered as an extension of general relativity [37,38] and an effective description of quantum gravity [39]. In these theories, a spacetime manifold is equipped with a spacetime metric and an independent affine connection, which in general is not metric compatible, possesses curvature and torsion. Special instances of metric affine gravity are those where the connection has either only torsion [40], is only not-metric compatible [41][42][43], has only curvature (the usual pseudo-Riemannian case), or possesses any possible combination of these three properties. In this section we briefly summarize that the geometry of non-metrizable Berwald spacetimes is equivalent to a metric affine geometry with a torsion free, non-metric compatible connection.
Pick any pseudo-Riemannian metric g, and let γ a bc [g] be the Christoffel symbols of its Levi-Civita connection. Then: where D is a (1, 2)-tensor field on M. By construction, the Γ a bc (x) are symmetric in their lower indices, and hence, they define a torsion-free affine connection. Thus, by the decomposition of affine connections into Levi-Civita, contorsion, and non-metricity parts, the tensor D a bc defines the non-metricity Q abc = ∇ a g bc = −D s ac g sb − D s ab g sc of the connection. In view of this argument, we can formulate the following proposition:

Discussion
Evidently, Finsler spacetime geometry has a very different behavior compared to that of Finsler spaces. The origin of this difference lies, on the one hand, in the weaker smoothness assumptions on the defining Finsler Lagrangian L and, on the other hand, in the fact that its indicatrix is necessarily non-compact in the indefinite case [44]. As a consequence, classical theorems that hold on Finsler spaces may not hold on Finsler spacetimes anymore. It was already known that Deicke's theorem [45] does not hold on generic Finsler spacetimes [44]. With this article, we demonstrated for the first time that the same is true for Szabo's theorem.
These findings call for a systematic study and classification of the geometric properties of Finsler spacetimes in general and of Berwald spacetimes in particular.

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

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Proof of Theorem 2
To identify the origin of the lack of symmetry of the Chern-Rund-Ricci tensor, which implies non-metrizability, we used Theorem 2, which we state here again: We now provide the proof of the theorem. The starting point of the proof is Equation (9): which on Berwald spacetimes implies that the expression: is actually independent ofẋ. It is clear that the Ricci tensor of a Berwald spacetime is symmetric if and only if φ identically vanishes, which we will now connect to the zeros of the derivatives of the function f . The function f = ln | det g L ab |, as defined in (10), is zero-homogeneous in its dependence onẋ, which means it naturally lives on the positive projective tangent bundle PTM + ; see [19].
where a = 0, ..., n − 1 and α = 0, ..., n − 2. More conveniently, one can use so-called homogeneous coordinates (x a ,ẋ a ), which are nothing but the coordinates on TM of an arbitrary representative of the equivalence class [(x,ẋ)]. Homogeneous coordinates are only defined up to a scaling factor. PTM + itself is a fiber bundle over M, with compact fibers, diffeomorphic to Euclidean spheres [46]. Functions on TM \ {0} can be understood as functions on PTM + if and only if they are zero-homogeneous inẋ; in homogeneous coordinates, calculus on PTM + is formally identical to the one on TM.
Assume that A = TM \ {0}. This implies that | det g| is smooth and nonzero on TM \ {0}, and so is f . Moreover, since they are zero-homogeneous inẋ, they can be regarded as functions on PTM + .
Fix an arbitrary x ∈ M and an arbitrary local chart around x, then f x (·) = f (x, ·) is defined on the fiber PT x M + , which is compact. Since f x is smooth, it admits at least a local extremum, say atẋ = v, and hence,∂ a f x |ẋ =v = 0. For the function φ(x,ẋ), this implies: On the other hand, on Berwald spacetimes, φ is independent ofẋ, and we can conclude that φ is identically zero for allẋ, which completes the proof.
As a remark, the crucial ingredient of the proof is that | det g L ab | is smooth and nonzero on the entire TM \ {0}. If | det g L ab | were only smooth and nonzero on a smaller subset, then f x would not be defined on the entire fiber PT x M + , and the conclusion of the theorem would fail.