We investigate the local metrizability of Finsler spaces with *m*-Kropina metric *F* = *α*^{1+m}*β*^{−m}, where *β* is a closed null one-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric *α* and one-form *β* have a very specific form in certain coordinates. In particular, when the signature of *α* is Lorentzian, *α* belongs to a certain subclass of the Kundt class and *β* generates the corresponding null congruence, and this generalizes in a natural way to arbitrary signature. We use this result to prove that the affine connection on such an *m*-Kropina space is locally metrizable by a (pseudo-)Riemannian metric if and only if the Ricci tensor constructed from the affine connection is symmetric. In particular, we construct all counterexamples of this type to Szabo’s metrization theorem, which has only been proven for positive definite Finsler metrics that are regular on all of the slit tangent bundle.

## I. INTRODUCTION

The study of differences and similarities between positive definite Finsler geometry and indefinite Finsler geometry is still in its beginnings and far from complete.^{1–3} The most prominent application of indefinite (to be precise Lorentzian) Finsler geometry is the one of Finsler spacetimes in classical and quantum gravitational physics,^{4–16} which recently put (pseudo-)Riemannian geometry and its applications into the focus of interest.^{17–25} Hence, a better understanding of the properties of indefinite Finsler geometry would be of great interest for physics as well as for mathematics.

Berwald spaces constitute an important class of Finsler spaces. They can be defined by the property that the canonical (Cartan) nonlinear connection reduces to a linear connection on the tangent bundle.^{26} It is natural to ask under what conditions this linear connection is (Riemann) *metrizable*, in the sense that there exists a (pseudo-)Riemannian metric that has the given linear connection as its Levi-Civita connection. In positive definite Finsler geometry, the answer to this question was given in 1988 by Szabo’s well-known metrization theorem,^{27} which guarantees that in this case the connection is *always* metrizable. In the more general context, where the fundamental tensor is allowed to have arbitrary, not necessarily positive definite, signature, the situation is more complex. It only became clear very recently that Szabo’s metrization theorem cannot be extended in general to arbitrary signatures.^{28} In other words, there exist Finsler metrics of Berwald type (most examples being not positive definite and not smooth on the entire slit tangent bundle) for which the affine connection is not metrizable by a (pseudo-)Riemannian metric.

It would be of great interest to know the precise conditions for metrizability in this more general context. As a first step in this direction, we investigate in this article the metrizability of a specific class of Finsler metrics, namely, *m*-Kropina metrics with a closed null one-form. The main result in this article, Theorem 6, states that the affine connection of such a space is metrizable if and only if the Ricci tensor constructed from the affine connection is symmetric and gives a second equivalent characterization in terms of the local expression of the defining (pseudo-)Riemannian metric and one-form, showing in particular that certainly not all such spaces are metrizable. This contrasts the situation for one-forms that are not null. In this case it is known that such an m-Kropina space is always metrizable by a metric conformal to α (see Ref. 49).

*m*-Kropina metrics, also called generalized Kropina metrics, were introduced by Hashiguchi *et al.* in Ref. 28 as a generalization of the standard Kropina metric.^{29} While the original Kropina metric has found a wide range of applications, *m*-Kropina metrics gained some popularity in the physics literature when it was discovered that they can be used to describe a modification of special relativity with local anisotropy,^{30,31} named very special relativity (VSR)^{32,33} and later generalized to *Very General Relativity* (VGR)^{34} or *General Very Special Relativity* (GVSR)^{35} in order to account for spacetime curvature, leading to physical predictions from curved *m*-Kropina spacetime geodesics^{36} and pp-waves.^{37}

The structure of this article is as follows: We start in Sec. II by recalling the basic notions of Finsler geometry that are relevant for our purpose and Szabo’s metrization theorem for positive definite Berwald spaces. In Sec. III, we recall the definition of *m*-Kropina metrics and the precise necessary and sufficient condition under which they are of Berwald type (Sec. III A). In fact, we provide a new proof of this Berwald condition in the Appendix. Subsequently, in Sec. III B, we specialize to *m*-Kropina metrics constructed from a (pseudo-)Riemannian metric *α* and a one-form *β* that is null with respect to this metric and closed. We first prove Lemma 3, stating that such a space is of Berwald type if and only if *α* and *β* have a very specific form in local coordinates. In particular, when the signature of *α* is Lorentzian, *α* belongs to a certain subclass of the Kundt class and *β* generates the corresponding null congruence. This construction generalizes in a natural way to arbitrary signature. The coordinates introduced in this lemma allow us to find a simple expression for the linear connection coefficients and the skew-symmetric part of the affine Ricci tensor. We then prove our main result, Theorem 6, providing two equivalent necessary and sufficient conditions for metrizability: symmetry of the affine Ricci tensor and a local condition for the coordinate expressions of the (pseudo-)Riemannian metric *α*. We end with a conclusion and discussion of the work in Sec. IV.

## II. FINSLER GEOMETRY

Finsler geometry is a natural extension of Riemannian geometry.^{39–40} Given the philosophy that the length of a curve is obtained by integrating the norm of the tangent vector along the curve, Finsler geometry provides the most general way of assigning, smoothly, a length to curves on a smooth manifold. While in Riemannian geometry the length of a tangent vector is given by a quadratic (metric-induced) norm, Finsler geometry relaxes this quadratic requirement.

First of all, some remarks about notation are in order. Throughout this work, we will usually work in local coordinates, i.e., given a smooth manifold *M* we assume that some chart $\varphi :U\u2282M\u2192Rn$ is provided, and we identify any *p* ∈ *U* with its image $(x1,\u2026,xn)=\varphi (p)\u2208Rn$. For *p* ∈ *U*, each *Y* ∈ *T*_{p}*M* in the tangent space to *M* at *p* can be written as $Y=yi\u2202ip$, where the tangent vectors $\u2202i\u2261\u2202\u2202xi$ furnish the chart-induced basis of *T*_{p}*M*. This provides natural local coordinates on the tangent bundle *TM* via the chart

These local coordinates on *TM* in turn provide a natural basis of its tangent spaces *T*_{(x,y)}*TM*,

### A. Finsler spaces

For our purposes, a Finsler space is triple $(M,A,F)$, where *M* is a smooth manifold, $A$ is a conic subbundle of *TM*\{0} (i.e., a non-empty open subset $A\u2282TM\{0}$ such that for any $(x,y)\u2208A$ it follows that $(x,\lambda y)\u2208A$ for any *λ* > 0) with non-empty fibers and *F*, the so-called Finsler metric, is a continuous map $F:TM\{0}\u2192R$, smooth on $A$, that satisfies the following axioms:

*F*is positively homogeneous of degree one with respect to*y*,(2)$F(x,\lambda y)=\lambda F(x,y),\u2200\lambda >0.$The

*fundamental tensor*, with components $gij=\u2202\u0304i\u2202\u0304j12F2$, is nondegenerate on $A$.

In the positive definite setting (meaning that *g*_{ij} is assumed to be positive definite), one usually requires that $A=TM\{0}$. In the more general setting, however, this would exclude almost all interesting examples that have been studied in the literature. In fact there is no consensus on a standard definition of Finsler space when the signature is indefinite (see, e.g., Refs. 6, 17, and 41–44). A fundamental result essential for doing computations in Finsler geometry is Euler’s theorem for homogeneous functions, which states that if a function *f* is positively homogeneous of degree *r*, i.e., *f*(*λy*) = *λ*^{r}*f*(*y*) for all *λ* > 0, then $yi\u2202f\u2202yi(y)=rf(y)$. In particular, this implies the identity

Finsler geometry reduces to (pseudo-)Riemannian geometry in the case that $A=TM\{0}$ and *F*^{2} is quadratic in the fiber coordinates *y*^{i} or equivalently when *g*_{ij} = *g*_{ij}(*x*) depends only on the base manifold. Then, *g*_{ij} is a (pseudo-)Riemannian metric on *M*. To avoid confusion, we stress again that, in contrast to (pseudo-)Riemannian geometry, the term *Finsler metric* refers to the scalar *F* and not the tensor *g*_{ij}. This is standard in most Finsler geometry literature. On the other hand, we use the term *(pseudo-)Riemannian metric* in the standard way of (pseudo-)Riemannian geometry, referring to a *g*_{ij}(*x*) that is independent of *y*.

The coefficients of the *Cartan nonlinear connection*, the unique homogeneous (nonlinear) connection on *TM* that is smooth on $A$, torsion-free, and compatible with the Finsler metric can be expressed as

Torsion-freeness is the property that $\u2202\u0304iNjk=\u2202\u0304jNik$, and metric-compatibility is the property that *δ*_{i}*F*^{2} = 0, in terms of the *horizontal derivative* induced by the connection,

Alternatively, metric-compatibility can be defined as the property that $\u2207gij\u2261yk\delta kgij\u2212Nikgkj\u2212Njkgki=0$, in terms of the so-called dynamical covariant derivative ∇. For torsion-free homogeneous connections, the latter definition of metric-compatibility is equivalent to the former. The curvature tensor, Finsler Ricci scalar, and Finsler Ricci tensor of (*M*, *F*) are defined, respectively, as

### B. Berwald spaces

A Finsler space is said to be of Berwald type if the Cartan nonlinear connection defines a linear connection on *TM* or, in other words, an affine connection on the base manifold, in the sense that the connection coefficients are of the form

for a set of smooth functions $\Gamma jki:M\u2192R$. (See Ref. 45 for an overview of the various equivalent characterizations of Berwald spaces and Ref. 46 for a more recent one in terms of a first order partial differential equation.) From the transformation behavior of $Nji$, it follows that the functions $\Gamma jki$ have the correct transformation behavior to be the connection coefficients of a (torsion-free) affine connection on *M*. We will refer to this affine connection as the associated affine connection or simply *the* affine connection on the Berwald space. In addition to the curvature tensors defined in Eqs. (6)–(8), one may define additional curvature tensors for Berwald spaces: the ones associated with the uniquely defined affine connection.

where we have employed the notation $T[ij]=12Tij\u2212Tji$ and $T(ij)=12Tij+Tji$ for (anti-)symmetrization. We will refer to these as the *affine curvature tensor* and the *affine Ricci tensor*, respectively. We note that $R\u0304ljki$ coincides (up to some reinterpretations) with the *hh*-curvature tensor of the Chern–Rund connection. A straightforward calculation reveals the following relation between the different curvature tensors:

It is appropriate to stress here that, although *R*_{ij} and $R\u0304ij$ coincide in the positive definite setting and more generally whenever the Finsler metric is defined on all of $A=TM\{0}$,^{28} this is *not* true in general, as $R\u0304ij$ need not be symmetric. As this distinction is essential for our results, we end this section with a schematic overview of some important properties of the two Ricci tensors.

**Ricci tensors**

The

*Finsler Ricci Tensor**R*_{ij}is constructed from the canonical nonlinear connection associated with*F*according to Eqs. (6)–(8). The Finsler Ricci Tensoralways exists;

is symmetric, by definition; and

contains the same information as the

*Finsler Ricci scalar—m*ore precisely, Ric =*R*_{ij}*y*^{i}*y*^{j}and $Rij=12\u2202\u0304i\u2202\u0304jRic$.

The

*affine Ricci Tensor*$R\u0304ij$ is constructed from the affine connection associated with*F*according to Eq. (10). The affine Ricci Tensorexists only for Berwald spaces because otherwise there is no uniquely defined affine connection;

coincides with the Ricci tensor constructed from any of the four well-known linear connections associated with

*F*(Chern–Rund, Berwald, Cartan, Hashiguchi); andis

*not*necessarily symmetric (except in the positive definite case)—its symmetrization coincides with the Finsler Ricci Tensor, see Eq. (11).

In this work, we are primarily concerned with the *affine Ricci tensor* and in particular its property of being in general *not* symmetric as it can be used to characterize whether a given Finsler space is metrizable or not.

### C. Szabo’s metrization theorem

Given a Finsler space of Berwald type, the Cartan nonlinear connection defines a linear connection on *TM* by definition. Hence, the natural question arises whether there exists a (pseudo-)Riemannian metric (desirably of the same signature) that has this connection as its Levi-Civita connection. Simply put, is every Berwald space metrizable? For positive definite Finsler spaces defined on all of *TM*\{0}, the answer is affirmative as proven by Szabo.^{27}

(Szabo’s metrization theorem). *Any positive definite Berwald space is metrizable by a Riemannian* *metric.*

The proof of this theorem relies on averaging procedures^{47} for which it is essential that the Finsler metric *F* is defined everywhere on *TM*\{0}. In the case of Finsler *spacetimes*, however, the domain where *F* is defined is typically only a conic subset of *TM*\{0} and hence the classical proof does not extend to this case. It was indeed shown in Ref. 28 that Szabo’s metrization theorem is in general not valid for Finsler spacetimes. The culprit behind all counterexamples known to the authors is the fact that the affine Ricci tensor is in general not symmetric. Clearly, the property that the affine Ricci tensor be symmetric is a necessary condition for metrizability. We will see (Theorem 6) that for *m*-Kropina spacetimes with closed one-form, this is in fact also a sufficient condition at least locally.

## III. *m*-Kropina metrics

An *m*-Kropina space (sometimes called generalized Kropina space) is a Finsler space of (*α*, *β*)-type with a Finsler metric of the form *F* = *α*^{1+m}*β*^{−m}, where $\alpha =aijyiyj$ is constructed from a (pseudo-)Riemannian metric *a* = *a*_{ij}(*x*)d*x*^{i}d*x*^{j}, *β* = *b*_{i}*y*^{i} is constructed from a one-form *b* = *b*_{i}(*x*)d*x*^{i}, and *m* is a real parameter. By a slight abuse of terminology, one also refers to *α* and *β* simply as the (pseudo-)Riemannian metric and the one-form, respectively. We also introduce the notation *b*^{2} ≡ ‖*b*‖^{2} = *a*_{ij}*b*^{i}*b*^{j} for the squared norm of *β* with respect to *α*. Throughout the remainder of this article, all indices are raised and lowered with *a*_{ij}.

In the physics literature, spacetimes with metric of *m*-Kropina type have been dubbed *Very General Relativity* (VGR) spacetimes^{34} or *General Very Special Relativity* (GVSR) spacetimes,^{35} introduced as generalizations of *Very Special Relativity* (VSR),^{32,33} which appears in the limiting case where *α* is flat. In the latter case, the corresponding *m*-Kropina metric is often referred to as the Bogoslovsky line element. When *m* = 1, the *m*-Kropina metric reduces to the standard Kropina metric^{29} *F* = *α*^{2}/*β*.

### A. The Berwald condition

The Berwald condition for *m*-Kropina spaces *F* = *α*^{1+m}*β*^{−m} formulated by Matsumoto in Ref. 48 states that such a space is of Berwald type if and only if there exists a vector field *f*^{i} on *M* such that

Here and throughout the remainder or the article, ∇ denotes the Levi-Civita connection corresponding to the (pseudo-)Riemannian metric *α*. In Ref. 48, the result is proven only for *non-null* one-forms *β* in Theorem 6.3.2.3 on p. 904, but as long as the dimension of the manifold is greater than 2 the proof is still completely valid for null one-forms also. Indeed, the only purpose of the assumption *b*^{2} ≠ 0 in the proof is to guarantee that *α*^{2} and *β* are co-prime as polynomials in *y*, i.e., that *α*^{2} is not the product of *β* with another polynomial, *α*^{2} = *βγ*. However, as long as dim *M* > 2, this is not possible anyway, irrespective of the value of *b*^{2}. To see this, note that *γ* = *c*_{i}(*x*)*y*^{i} is necessarily a one-form due to homogeneity. Then, it follows by differentiating twice that $aij=12(bicj+bjci)$, showing that *a*_{ij} has rank $\u22642$. However, since *a*_{ij} is assumed to be nondegenerate, it must have maximal rank, so this implies that dim *M* ≤ 2. Hence for dim *M* > 2 the assumption that *b*^{2}=0 is not necessary and Eq. (12) is also valid when the one-form is null.

In the special case that *β* is a closed and hence locally exact one-form, any *f*_{k} satisfying this condition can always be written as *f*_{k} = *cb*_{k} for some function *c* on the base manifold and the condition reduces to the simpler one obtained in Ref. 34, namely,

where we remark that our *c* is related to *C*(*x*) in Ref. 34 by *C*(*x*) = (1 + *m*)*c*/2 and that that our power *m* is related to the power *n* in Ref. 34 by *n* = −2*m*/(1 + *m*). To see this, assume that Matsumoto’s Berwald condition (12) holds. We have (d*b*)(*∂*_{i}, *∂*_{j}) = *∂*_{i}*b*_{j} − *∂*_{j}*b*_{i} = ∇_{i}*b*_{j} − ∇_{j}*b*_{i} = (1 + *m*)(*f*_{i}*b*_{j} − *f*_{j}*b*_{i}), so if *b*_{i} is locally exact then this expression vanishes and hence *f*_{i}*b*_{j} = *f*_{j}*b*_{i} must hold for all *i*, *j*, which is only possible if *f*_{i} is proportional to *b*_{i} (this can be checked easily at any given point in *M* by choosing coordinates in which *b*_{i} has only one nonvanishing component at that point). In other words, *f*_{k} = *cb*_{k}. In this case, (12) reduces to (13). Note that the opposite holds (trivially) as well: The latter condition implies that *β* is locally exact.

The fact that (12) and (13) do not agree for one-forms *β* that are not closed has recently caused some confusion in the literature as it was suggested in a recent article^{34} that the latter was the correct Berwald condition in full generality, i.e., also for non-closed one-forms. This, however, clearly contradicts the results obtained above. We have taken the opportunity here to resolve this issue. It turns out that the reason for the discrepancy is that the contribution of the antisymmetric part of the covariant derivative of *β* was overlooked in the proof given in Ref. 34. Indeed in the Appendix, we reproduce the argument from^{34} taking the antisymmetric part into account and we show that the resulting Berwald condition coincides with Matsumoto’s one, (12), as expected. Thus, we want to stress here again that (12) is the correct Berwald condition in general, whereas (13) only applies to the case in which the one-form *β* is closed.

Finally, as also proven in Ref. 48, whenever condition (16) is satisfied, the affine connection coefficients of the Berwald space can be expressed in terms of the Christoffel symbols $\Gamma ijk\alpha $ for the Levi-Civita connection corresponding to *α* as

When the one-form *β* is closed, and we write *f*_{k} = *cb*_{k} as before, this reduces to

which agrees with the result obtained in Ref. 34.

For clarity, we summarize the preceding discussion with the following proposition:

*Let* *F* = *α*^{1+m}*β*^{−m} *be an* *m**-Kropina metric on a manifold* *M* *with dimension greater than two.*

*F**is of Berwald type if and only if there exists a smooth vector field**f*^{i}*satisfying*(16)$\u2207jbi=m(fkbk)aij+bifj\u2212mfibj.$*In this case, the affine connection coefficients of the Berwald space can be expressed in terms of the Christoffel symbols*$\Gamma ijk\alpha $*for the Levi-Civita connection corresponding to**α**as*(17)$\Gamma ij\u2113=\Gamma ij\u2113\alpha +ma\u2113kaijfk\u2212ajkfi\u2212akifj.$*If the one-form**β**is closed,**F**is of Berwald type if and only if there exists a smooth function**c*∈*C*^{∞}(*M*)*satisfying*(18)$\u2207jbi=cmb2aij+(1\u2212m)bibj.$*In this case, the affine connection coefficients of can be expressed as*(19)$\Gamma ij\u2113=\Gamma ij\u2113\alpha +mcaijb\u2113\u2212\delta j\u2113bi\u2212\delta i\u2113bj.$*Conversely**, Eq. (18)**implies that**β**must be**closed.*

### B. Metrizability of *m*-Kropina spaces with closed null one-form

From here onward, we will focus on *m*-Kropina metrics with closed null one-form and we will assume that *n* = dim *M* > 2. In other words, we will assume that d*b* = 0 and *b*^{2} = *a*_{ij}*b*^{i}*b*^{j} = 0. This will allow us to deduce the exact conditions for local metrizability. As a remark, we point out that the case *m* = 1 is excluded by our definition of Finsler space as can be seen from the expression of the determinant of the fundamental tensor,

which vanishes identically when *m* = 1 and *b*^{2} = 0.

The following lemma extends a result from Ref. 19. We will use the convention that indices *a*, *b*, *c*, … run from 3 to *n*, whereas indices *i*, *j*, *k*, … run from 1 to *n*, and we use the notation d*u*d*v* for the symmetrized tensor product of one-forms, e.g., $dudv\u226112(du\u2297dv+dv\u2297du)$.

*F*

*is Berwald if and only if around each*

*p*∈

*M*

*there exist local coordinates*(

*u*,

*v*,

*x*

^{3}, …,

*x*

^{n})

*such that*

*with*

*h*

*some (pseudo-)Riemannian metric. In this case, the metric satisfies the Berwald condition*

*(18)*

*with*

*v*,

*x*

^{2}, …,

*x*

^{n}) around

*p*adapted to

*b*in the sense that

*b*=

*∂*

_{v}, i.e., $bi=\delta 1i$. At this point, the metric has the general form

*a*=

*a*

_{ij}d

*x*

^{i}⊗ d

*x*

^{j}. (Abusing notation a little bit,

*b*sometimes denotes the one-form and sometimes the vector field uniquely corresponding to it via the isomorphism induced by

*a*. It should be clear from context which is meant.) The null character of

*b*manifests as the fact that

*a*

_{11}=

*a*

_{vv}= 0 in these coordinates. Because

*b*is closed and hence locally exact, we may write, locally,

*b*= d

*u*for some function

*u*(

*v*,

*x*

^{2}, …,

*x*

^{n}). Equivalently,

*b*

_{i}=

*∂*

_{i}

*u*. Note also that $\u2202iu=bi=aijbj=aij\delta 1j=ai1$. Since

*a*

_{11}= 0, it follows that

*∂*

_{v}

*u*=

*∂*

_{1}

*u*= 0. As

*b*≠ 0 by assumption, there must be some

*i*≥ 2 such that

*∂*

_{i}

*u*=

*a*

_{i1}≠ 0 in a neighborhood of

*p*. Order the coordinates

*x*

^{2}, …,

*x*

^{n}such that this is true for

*i*= 2, i.e., assume without loss of generality that

*a*

_{21}≠ 0. Next, define the map

*J*=

*a*

_{21}≠ 0, this matrix is invertible, $x\u27fcx\u0303$ is a local diffeomorphism at

*p*. It remains to find the form of the metric in the new coordinates. We have

*H*,

*W*

_{a},

*h*

_{ab}, and hence after a redefinition

*v*→ −

*v*we may write the metric in the form

*h*= − det

*a*≠ 0 that

*h*

_{ab}is itself a (pseudo-)Riemannian metric of dimension

*n*− 2.

*m*-Kropina space is of Berwald type or not. All we have used is that the (pseudo-)Riemannian metric

*a*admits a one-form that is null and closed. We will prove next that the

*m*-Kropina space is Berwald if and only if the functions

*W*

_{a}and

*h*

_{ab}do not depend on coordinate

*v*. To this end, we employ the Berwald condition (16). In fact, since the one-form is assumed to be closed we may use the simpler version, Eq. (18). Moreover, since the one-form is null (

*b*

^{2}= 0) as well, this condition reduces to

*m*-Kropina space is Berwald if and only if there exists a function

*c*on

*M*such that this condition is satisfied. On the other hand, computing ∇

_{i}

*b*

_{j}explicitly in the new coordinates, using the fact that $bi=\delta iu$ and $gui=\u2212\delta vi$ and

*g*

_{iv}= 0, yields

*F*is Berwald if and only if

*∂*

_{v}

*W*

_{a}=

*∂*

_{v}

*h*

_{ab}= 0 and that

*c*is in that case given by the desired expression, completing the proof.□

From here onward, we will assume our space is Berwald. Substituting the form of *c* into Eq. (19) and using that $bi=\delta iu$ and consequently $b\u2113=a\u2113kbk=a\u2113k\delta ku=a\u2113u=\u2212\delta v\u2113$, we obtain the following.

*In the coordinates*(

*u*,

*v*,

*x*

^{3}, …,

*x*

^{n})

*, the affine connection coefficients can be expressed in terms of the Levi-Civita Christoffel symbols*$\Gamma ijk\alpha $

*of the (pseudo-)Riemannian metric*

*α*

*as*

We can use the preceding results to analyze the (deviation from the) symmetry of the affine Ricci tensor, which has a very simple expression in these coordinates, as the following result shows.

*In the coordinates*(

*u*,

*v*,

*x*

^{3}, …,

*x*

^{n})

*, the skew-symmetric part of the affine Ricci tensor is given by*

*α*is symmetric.□

Let us now prove our main result.

*Let* (*M*, *F* = *α*^{1+m}*β*^{−m}) *be an* *m**-Kropina space of Berwald type with closed null 1-form* *β* *and with* dim *M* > 2*. The following are equivalent:*

*The affine connection is locally metrizable by a (pseudo-)Riemannian**metric.**The affine Ricci tensor is symmetric,*$R\u0304ij=R\u0304ji$.*There exist local coordinates*(*u*,*v*,*x*^{3}, …,*x*^{n})*such that**b*= d*u**and*(37)$a=\u22122dudv+H\u0303(u,x)+\varphi (u)vdu2+Wa(u,x)dudxa+hab(u,x)dxadxb,$*with**h**some (pseudo-)Riemannian metric of dimension**n*− 2*.*

*In this case, the affine connection is metrizable, in the chart corresponding to the coordinates*(

*u*,

*v*,

*x*

^{3}, …,

*x*

^{n})

*, by the following (pseudo-)*

*Riemannian metric:*

Before we present the proof, we want to point out two things. First, we note that if *ϕ* = 0 then $a\u0303=a$, i.e., the affine connection is metrizable by the defining (pseudo-)Riemannian metric *α*. This was to be expected, since in that case the one-form *β* is parallel with respect to *α*. It is a well-known result that any (*α*, *β*)-metric for which *β* is parallel with respect to *α* is of Berwald type and that its affine connection coincides with the Levi-Civita connection of *α*. Second, since $a\u0303$ is conformally equivalent to *a*, the two metrics have identical causal structure and, moreover, their null geodesics coincide (as unparameterized curves). This implies that the null geodesics of any *F* satisfying any (and hence all) of the equivalent conditions of Theorem 6 coincide with the null geodesics of the defining (pseudo-)Riemannian metric *α*.

*u*on the LHS and an index

*v*on the RHS is not a typo. The antisymmetric part of the

*uj*component of the Ricci tensor are determined by the

*vj*-derivative of

*H*. By assumption, the Ricci tensor is symmetric. The

*uv*component therefore yields $\u2202v2H=0$ and the remaining components yield

*∂*

_{v}

*∂*

_{a}

*H*= 0,

*a*= 3, …,

*n*. In other words,

*H*must be linear in

*v*and the corresponding linear coefficient can depend only on the coordinate

*u*. That is,

*ii*) ⇒ (

*iii*). For the last implication (

*iii*) ⇒ (

*i*), recall from Corollary 4 that the affine connection coefficients can be expressed as

*m*-Kropina metric. This completes the proof of the theorem.□

Theorem 6 provides necessary and sufficient conditions for an m-Kropina space with closed null one-form to be locally metrizable. In Sec. III C, we apply our results to an explicit example from the physics literature.

### C. An explicit example: Finsler VSI spacetimes

In this section, we apply our results to the Finsler VSI spacetimes presented in Ref. 34, with the four-dimensional Finsler metric

By Lemma 2, this spacetime is of Berwald type. It is in general not metrizable since the corresponding affine Ricci tensor is not symmetric. By Theorem 6, the exact condition for metrizability in this case is that *∂*_{a}*ϕ* = 0. The case *ϕ* = 0 provides a Finsler version of the gyratonic pp-wave metric,^{49,50} which according to Theorem 6 is metrizable by the Lorentzian gyratonic pp-wave metric.

A simple nontrivial locally metrizable example is provided by the case where $H\u0303(u,x)=0$, *ϕ*(*u*, *x*) = *u*, and *W*_{a}(*u*, *x*) = 0. If we relabel the coordinates *x*^{a} as *x* and *y*, this leads to the Finsler metric

which has an affine connection given by following nonvanishing affine connection coefficients:

As indicated by Eq. (38) in Theorem 6, this connection is metrizable by the Lorentzian metric

## IV. DISCUSSION

Recent developments around the non-metrizability of Berwald spaces of indefinite (in particular, Lorentzian) signature contrast the well-known metrizability theorem by Szabo for *positive definite* Berwald spaces. These findings inspired us to investigate the question of metrizability for *m*-Kropina Finsler metrics constructed from a (pseudo-)Riemannian metric and a closed null one-form in this article. While the analogous question for the case of not null (and not necessarily closed) one-forms is known to have a simple answer, namely that any such space is metrizable, the situation is different when null one-forms are considered. Our main result, Theorem 6, gives a necessary and sufficient condition for local metrizability: that the *affine Ricci tensor*—the Ricci tensor constructed from the affine connection, not to be confused with the more commonly discussed Finsler Ricci tensor—must be symmetric.

Moreover, in the coordinates introduced in Lemma 3, any Berwald *m*-Kropina metric attains a pretty simple form. It can then be seen at a glance whether a given geometry is locally metrizable or not. Moreover, in the metrizable case, our theorem gives the explicit form of a (nonunique) (pseudo-)Riemannian metric that “metrizes” the affine connection in terms of those coordinates.

The question of metrizability is not only a natural one from the mathematical point of view, but it is also of interest in the realm of physics, particularly in the field of Finsler gravity, which asserts that the spacetime geometry of our physical universe might be Finslerian. One of its postulates is that physical objects and light rays moving only under the influence of gravity follow Finslerian geodesics through spacetime. If the Finsler metric on spacetime were metrizable, this would imply that these trajectories reduce to the geodesics of a (pseudo-)Riemannian metric, precisely as is the case in Einstein gravity. Apart from obvious mathematical implications, it would be interesting to investigate the conceptual and physical consequences of this as well.

It would obviously be of great interest to have a generalization of Theorem 6 to arbitrary Finsler spaces of Berwald type. To this effect, we note that, curiously, all examples of non-metrizable Berwald spaces currently available in the literature, as well as all of the additional examples known privately to the authors, have an affine Ricci tensor that is not symmetric. Together with the results obtained in this article in the specific case of *m*-Kropina metrics, this leads us to hypothesize that perhaps a Berwald space is metrizable by a (pseudo-)Riemannian metric if and only if its affine Ricci tensor is symmetric. In fact, some general results about Riemann-metrizability of arbitrary symmetric affine connections are known.^{52–53} An affine connection is metrizable if and only if the holonomy group is a subgroup of the generalized (pseudo-)orthogonal group.^{53} Hence, a future project is to investigate the structure of the holonomy group of the affine connection corresponding to a Berwald space and how it relates to the geometry-defining Finsler metric.

## ACKNOWLEDGMENTS

C.P. was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Project No. 420243324 and acknowledges support from cluster of excellence Quantum Frontiers funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC-2123 QuantumFrontiers—390837967. All of us would like to acknowledge networking support provided by the COST Action CA18108, supported by COST (European Cooperation in Science and Technology).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Sjors Heefer**: Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). **Christian Pfeifer**: Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). **Jorn van Voorthuizen**: Formal analysis (supporting); Writing – review & editing (supporting). **Andrea Fuster**: Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

### APPENDIX: PROOF OF THE BERWALD CONDITION FOR m-KROPINA METRICS

Here, we provide a proof of the Berwald condition (16) for *m*-Kropina spaces *F* = *α*^{1+m}*β*^{−m}, which also serves as extension for the proof presented in Ref. 34, where it was overlooked that the one-form *β* need not be closed. The derivations in this section have been performed with the help of the xAct extension of Mathematica.^{54}

The Finsler metric *L* for *m*-Kropina spaces is given by $F=(aij(x)yiyj)1+m2(bk(x)yk)\u2212m$. Using the decomposition

where *A*_{ij} = *A*_{[ij]}(*x*) is the antisymmetric and *S*_{ij} = *S*_{(ij)}(*x*) is the symmetric part of the covariant derivative, we find a geodesic spray $Gj=Nijyi$ of the form

Indices are raised and lowered with the components of the (pseudo-)Riemannian metric defining *α*. In order to be of Berwald type, the components *G*^{k} need to be quadratic functions of *y*. This is the case, since for a Berwald space, $Nij(x,y)=\Gamma ikj(x)yk$ and so $Gj(x,y)=Nij(x,y)yi=\Gamma ikj(x)ykyi$.

To reach this goal, the first term in (A4) must either cancel with one of the other terms appearing or the contraction $ybAkb$ must lead to a term proportional to *β*. Hence, the free index on $ybAkb$ must be on *y*^{k}, *b*^{k}, or *Z*^{k} = *Z*^{k}(*x*), where *Z*^{k} are the components of another vector field *Z* on *M*, in the following way:

for *T* = *T*(*x*) being a function and *U*_{i} = *U*_{i}(*x*) being the components of a one-form on *M*. These are the only possible terms, since by construction, $ybAkb$ is a linear function in *y*, and so the RHS must be as well. Factoring the linear dependence in *y* on both sides of the equation leads to

which then implies by the antisymmetry *A*_{ij} = *A*_{[ij]} that

Defining *f*_{j} = (*U*_{j} − *Z*_{j}), we see that a necessary condition for a m-Kropina space to be Berwald is that the antisymmetric part of the covariant derivative of the one-form *β* is determined by *b*_{i} and an additional one-form with components *f*_{j}. Using this in (A1) we get

where the factor (*m* + 1) was added in front of the antisymmetric part to display the following expressions more compactly. For the geodesic spray, one finds

The use of the derived expression for the antisymmetric part of the covariant derivative (A7) ensures that the second term in the geodesic spray above is quadratic in *y*. To achieve this for the third term for the case *m* ≠ 1, let us investigate the structure of the *y*-dependence of this term. It is of the type

where each term *X*(*y*, *y*) = *X*_{ij}*y*^{i}*y*^{j}, *X* = *B*, *C*, *D*, *S*, *P*, denotes a quadratic polynomial in *y* and *B*(*y*, *y*) = *α*^{2}. In order for this function to be quadratic in *y*, it must satisfy

for some second order polynomial *P*(*y*, *y*). Since the left-hand side is a second order polynomial in y, the right-hand side must be. Assuming dim *M* > 2, it follows by the argument given right below Eq. (12) that *B*(*y*, *y*) = *α*^{2} is an irreducible quadratic polynomial in *y*. As long as *m* ≠ 1, *D* ≠ *h*(*x*)*B*(*y*, *y*). Thus, *P*(*y*, *y*) must satisfy *P*(*y*, *y*) = *h*(*x*)*B*(*y*, *y*), for a solution of the equation to exist. Hence, the fraction in the first term of line (A11) must be proportional to an arbitrary function *h* = *h*(*x*) on *M*. This yields the equation

Taking two derivatives with respect to *y*, we find

Redefining *f*_{i} as $fi=12(f\u0303i\u2212bih)$ and combining all expressions for the covariant derivative of *β* finally gives the desired expression as follows:

One can easily check that this condition on *b*_{j} leads to a geodesic spray given by

which indeed is quadratic, and so the *m*-Kropina space subject to condition (16) is indeed Berwald.

For *m* = 1 and *b*^{2} ≠ 0, the first term in line (A11) is quadratic in *y* for any tensor components *S*_{ij} and we must investigate the second term of that line, which becomes

and it can only be quadratic in *y* if and only if

for some one-form on *M* with components *Q*_{i} = *Q*_{i}(*x*). The only way to achieve this is if *S*_{ij} = *qa*_{ij} for some function *q* = *q*(*x*) on *M*, which then must satisfy

Thus, for *m* = 1,

For *m* = 1 and *b*^{2} = 0, the determinant of the metric *g* vanishes globally, and hence this situation does not define a Finsler space or spacetime.