On the metrizability of m -Kropina spaces with closed null one-form

Abstract

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.

Full text

J. Math. Phys. 64, 022502 (2023); https://doi.org/10.1063/5.0130523 64, 022502
© 2023 Author(s).
On the metrizability of m-Kropina spaces
with closed null one-form
Cite as: J. Math. Phys. 64, 022502 (2023); https://doi.org/10.1063/5.0130523
Submitted: 12 October 2022 • Accepted: 12 January 2023 • Published Online: 07 February 2023
Sjors Heefer, Christian Pfeifer, Jorn van Voorthuizen, et al.

Journal of
Mathematical Physics ARTICLE scitation.org/journal/jmp
On the metrizability of m-Kropina spaces
with closed null one-form
Cite as: J. Math. Phys. 64, 022502 (2023); doi: 10.1063/5.0130523
Submitted: 12 October 2022 •Accepted: 12 January 2023 •
Published Online: 7 February 2023
Sjors Heefer,1,a) Christian Pfeifer,2, b) Jorn van Voorthuizen,1,c) and Andrea Fuster1, d)
AFFILIATIONS
1Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands
2ZARM, University of Bremen, 28359 Bremen, Germany
a)Author to whom correspondence should be addressed: s.j.heefer@tue.nl
b)Electronic mail: christian.pfeifer@zarm.uni-bremen.de
c) Electronic mail: jornvanvoorthuizen@gmail.com
d)Electronic mail: a.fuster@tue.nl
ABSTRACT
We investigate the local metrizability of Finsler spaces with m-Kropina metric F=α1+mβ−m, where βis a closed null one-form. We show that
such a space is of Berwald type if and only if the (pseudo-)Riemannian metric αand one-form βhave a very specific form in certain coordinates.
In particular, when the signature of αis Lorentzian, αbelongs to a certain subclass of the Kundt class and βgenerates 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.
©2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0130523
I. INTRODUCTION
The study of differences and similarities between positive definite Finsler geometry and indefinite Finsler geometry is still in its begin-
nings 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) non-
linear 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
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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 Rela-
tivity (GVSR)35 in order to account for spacetime curvature, leading to physical predictions from curved m-Kropina spacetime geodesics36 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
αand a one-form βthat 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 αand βhave a very specific form in local coordinates. In particular, when the signature of αis Lorentzian, αbelongs to a certain subclass
of the Kundt class and βgenerates 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 Mwe assume that some chart ϕ:U⊂M→Rnis provided, and we identify any p∈Uwith its image (x1,...,xn)=ϕ(p)∈Rn. For
p∈U, each Y∈TpMin the tangent space to Mat pcan be written as Y=yi∂ip, where the tangent vectors ∂i≡∂
∂xifurnish the chart-induced
basis of TpM. This provides natural local coordinates on the tangent bundle TM via the chart
˜
ϕ:˜
U→Rn×Rn,˜
U=⋃
p∈U{p}×TpM⊂TM,˜
ϕ(p,Y)=(ϕ(p),y1,...,yn)=:(x,y).
These local coordinates on TM in turn provide a natural basis of its tangent spaces T(x,y)TM,
∂i≡∂
∂xi,¯
∂i≡∂
∂yi. (1)
A. Finsler spaces
For our purposes, a Finsler space is triple (M,A,F), where Mis a smooth manifold, Ais a conic subbundle of TM{0}(i.e., a non-empty
open subset A⊂TM{0}such that for any (x,y)∈Ait follows that (x,λy)∈Afor any λ>0) with non-empty fibers and F, the so-called
Finsler metric, is a continuous map F:TM{0}→R, smooth on A, that satisfies the following axioms:
●Fis positively homogeneous of degree one with respect to y,
F(x,λy)=λF(x,y),∀λ>0. (2)
●The fundamental tensor, with components gij =¯
∂i¯
∂j1
2F2, is nondegenerate on A.
In the positive definite setting (meaning that gij 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 consen-
sus 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 fis positively
homogeneous of degree r, i.e., f(λy)=λrf(y)for all λ>0, then yi∂f
∂yi(y)=r f (y). In particular, this implies the identity
gij(x,y)yiyj=F(x,y)2. (3)
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Finsler geometry reduces to (pseudo-)Riemannian geometry in the case that A=TM{0}and F2is quadratic in the fiber coordinates yior
equivalently when gij =gij(x)depends only on the base manifold. Then, gij 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 Fand not the tensor gij. 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 gij(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
Ni
j(x,y)=1
4¯
∂jgikyl∂l¯
∂kF2−∂kF2. (4)
Torsion-freeness is the property that ¯
∂iNk
j=¯
∂jNk
i, and metric-compatibility is the property that δiF2=0, in terms of the horizontal derivative
induced by the connection, δi≡∂i−Nj
i¯
∂j. (5)
Alternatively, metric-compatibility can be defined as the property that ∇gij ≡ykδkgij −Nk
igkj −Nk
jgki =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
Rijk(x,y)=−[δj,δk]i=δjNi
k(x,y)−δkNi
j(x,y), (6)
Ric(x,y)=Riij(x,y)yj, (7)
Rij(x,y)=1
2¯
∂i¯
∂jRic. (8)
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
Ni
j(x,y)=Γi
jk(x)yk(9)
for a set of smooth functions Γi
jk :M→R. (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 Ni
j, it follows that
the functions Γi
jk 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. ¯
Ri
l jk =2∂[jΓik]l+2Γi
m[jΓm
k]l,¯
Rlk =¯
Ri
l ik, (10)
where we have employed the notation T[ij]=1
2(Tij −Tji)and T(ij)=1
2(Tij +Tji)for (anti-)symmetrization. We will refer to these as the affine
curvature tensor and the affine Ricci tensor, respectively. We note that ¯
Ri
l jk 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:
Rjkl =¯
Rj
i kl(x)yi, Ric =¯
Rij(x)yiyj,Rij =1
2¯
Rij(x)+¯
Rji(x). (11)
It is appropriate to stress here that, although Rij and ¯
Rij 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 ¯
Rij 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
1. The Finsler Ricci Tensor Rij is constructed from the canonical nonlinear connection associated with Faccording to Eqs. (6)–(8). The
Finsler Ricci Tensor
●always exists;
●is symmetric, by definition; and
●contains the same information as the Finsler Ricci scalar—more precisely, Ric =Rijyiyjand Rij =1
2¯
∂i¯
∂jRic.
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2. The affine Ricci Tensor ¯
Rij is constructed from the affine connection associated with Faccording to Eq. (10). The affine Ricci Tensor
●exists 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); and
●is 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
Theorem 1 (Szabo’s metrization theorem). Any positive definite Berwald space is metrizable by a Riemannian metric.
The proof of this theorem relies on averaging procedures47 for which it is essential that the Finsler metric Fis defined everywhere on
TM{0}. In the case of Finsler spacetimes, however, the domain where Fis 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=α1+mβ−m, where α=aijyiyjis constructed from a (pseudo-)Riemannian metric a=aij(x)dxidxj,β=biyiis constructed from a one-form
b=bi(x)dxi, and mis a real parameter. By a slight abuse of terminology, one also refers to αand βsimply as the (pseudo-)Riemannian metric
and the one-form, respectively. We also introduce the notation b2≡b2=aijbibjfor the squared norm of βwith respect to α. Throughout
the remainder of this article, all indices are raised and lowered with aij.
In the physics literature, spacetimes with metric of m-Kropina type have been dubbed Very General Relativity (VGR) spacetimes34 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 αis 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 metric29 F=α2β.
A. The Berwald condition
The Berwald condition for m-Kropina spaces F=α1+mβ−mformulated by Matsumoto in Ref. 48 states that such a space is of Berwald
type if and only if there exists a vector field fion Msuch that
∇jbi=m(fkbk)aij +bifj−mfibj. (12)
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 βin 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 b2≠0 in the proof is to
guarantee that α2and βare co-prime as polynomials in y, i.e., that α2is not the product of βwith another polynomial, α2=βγ. However, as
long as dim M>2, this is not possible anyway, irrespective of the value of b2. To see this, note that γ=ci(x)yiis necessarily a one-form due to
homogeneity. Then, it follows by differentiating twice that aij =1
2(bicj+bjci), showing that aij has rank ≤2. However, since aij is assumed to be
nondegenerate, it must have maximal rank, so this implies that dimM≤2. Hence for dim M>2 the assumption that b2=0 is not necessary
and Eq. (12) is also valid when the one-form is null.
In the special case that βis a closed and hence locally exact one-form, any fksatisfying this condition can always be written as fk=cbk
for some function con the base manifold and the condition reduces to the simpler one obtained in Ref. 34, namely,
∇jbi=cmb2aij +(1−m)bibj, (13)
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where we remark that our cis related to C(x)in Ref. 34 by C(x)=(1+m)c2 and that that our power mis related to the power nin Ref. 34
by n=−2m(1+m). To see this, assume that Matsumoto’s Berwald condition (12) holds. We have (db)(∂i,∂j)=∂ibj−∂jbi=∇ibj−∇jbi
=(1+m)(fibj−fjbi), so if biis locally exact then this expression vanishes and hence fibj=fjbimust hold for all i,j, which is only possible
if fiis proportional to bi(this can be checked easily at any given point in Mby choosing coordinates in which bihas only one nonvanishing
component at that point). In other words, fk=cbk. In this case, (12) reduces to (13). Note that the opposite holds (trivially) as well: The latter
condition implies that βis locally exact.
The fact that (12) and (13) do not agree for one-forms βthat are not closed has recently caused some confusion in the literature as it
was suggested in a recent article34 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 βwas overlooked in the proof given in
Ref. 34. Indeed in the Appendix, we reproduce the argument from34 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 βis 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 αΓk
ij for the Levi-Civita connection corresponding to αas
Γℓ
ij =αΓℓ
ij +maℓkaij fk−ajk fi−aki fj. (14)
When the one-form βis closed, and we write fk=cbkas before, this reduces to
Γℓ
ij =αΓℓ
ij +mcaijbℓ−δℓ
jbi−δℓ
ibj, (15)
which agrees with the result obtained in Ref. 34.
For clarity, we summarize the preceding discussion with the following proposition:
Proposition 2. Let F =α1+mβ−mbe 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 isatisfying
∇jbi=m(fkbk)aij +bifj−m f ibj. (16)
In this case, the affine connection coefficients of the Berwald space can be expressed in terms of the Christoffel symbols αΓk
ij for the
Levi-Civita connection corresponding to αas
Γℓ
ij =αΓℓ
ij +maℓkaij fk−ajk fi−aki fj. (17)
●If the one-form βis closed, F is of Berwald type if and only if there exists a smooth function c ∈C∞(M)satisfying
∇jbi=cmb2aij +(1−m)bibj. (18)
In this case, the affine connection coefficients of can be expressed as
Γℓ
ij =αΓℓ
ij +mcaijbℓ−δℓ
jbi−δℓ
ibj. (19)
●Conversely, Eq. (18) implies that βmust 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 db=0 and b2=aijbibj=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, det g
det a=(1+m)3α8mβ−2(1+4m)mb2α2+(1−m)β2, (20)
which vanishes identically when m=1 and b2=0.
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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 dudvfor the symmetrized tensor product of one-forms, e.g., dudv≡1
2(du⊗dv+dv
⊗du).
Lemma 3. F is Berwald if and only if around each p ∈M there exist local coordinates (u,v,x3,...,xn)such that
a=−2dudv+H(u,v,x)du2+Wa(u,x)dudxa+hab(u,x)dxadxb,b=du, (21)
with h some (pseudo-)Riemannian metric. In this case, the metric satisfies the Berwald condition (18) with
c=−∂vH
2(1−m). (22)
Proof. First, we may pick coordinates (v,x2,...,xn)around padapted to bin the sense that b=∂v, i.e., bi=δi
1. At this point, the metric
has the general form a=aijdxi⊗dxj. (Abusing notation a little bit, bsometimes 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 bmanifests as
the fact that a11 =avv =0 in these coordinates. Because bis closed and hence locally exact, we may write, locally, b=dufor some function
u(v,x2,...,xn). Equivalently, bi=∂iu. Note also that ∂iu=bi=aijbj=aijδj
1=ai1. Since a11 =0, it follows that ∂vu=∂1u=0. As b≠0 by
assumption, there must be some i≥2 such that ∂iu=ai1≠0 in a neighborhood of p. Order the coordinates x2,...,xnsuch that this is true for
i=2, i.e., assume without loss of generality that a21 ≠0. Next, define the map
x=(v,x2,...,xn)z→ ˜
x=(v,u(x2,...,xn),x3,...,xn).
Its Jacobian matrix and its inverse are given by
Jij=∂˜
xi
∂xj=













1 0 0 ... 0
0a21 a31 ... an1
0 0 1 ... 0
⋮ ⋮ ⋮ ...⋮
0 0 0 ... 1













,(J−1)ij=∂xi
∂˜
xj=













1 0 0 ... 0
0 1a21 −a31a21 ... an1a21
0 0 1 ... 0
⋮ ⋮ ⋮ ...⋮
0 0 0 ... 1













.
Moreover, since det J=a21 ≠0, this matrix is invertible, xz→ ˜
xis a local diffeomorphism at p. It remains to find the form of the metric in the
new coordinates. We have
˜
aij =∂xk
∂˜
xi
∂xℓ
∂˜
xjakℓ, i.e. ˜
a=J−1TaJ−1. (23)
Therefore,
˜
a11 =(J−1T)1iaij(J−1)j1=a11 =0, (24)
˜
a12 =(J−1T)1iaij(J−1)j2=1, (25)
˜
a1b=(J−1T)1iaij(J−1)jb=a12(−ab1a21)+a1b=0, b=3, ...,n. (26)
This shows that a=˜
aij d˜
xid˜
xj=2dudv+Hdu2+Wbdudxb+hbcdxbdxcfor certain functions H,Wa,hab, and hence after a redefinition v→
−vwe may write the metric in the form
a=−2dudv+Hdu2+Wbdudxb+hbcdxbdxc. (27)
It follows from the easily checked fact that det h=−deta≠0 that hab is itself a (pseudo-)Riemannian metric of dimension n−2.
Our arguments thus far are independent of whether the m-Kropina space is of Berwald type or not. All we have used is that the (pseudo-)
Riemannian metric aadmits 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 Waand hab 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 (b2=0)as well, this condition
reduces to
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∇ibj=c(1−m)bibj. (28)
The m-Kropina space is Berwald if and only if there exists a function con Msuch that this condition is satisfied. On the other hand, computing
∇ibjexplicitly in the new coordinates, using the fact that bi=δu
iand gui =−δi
vand giv=0, yields
∇ibj=−1
2
∂aij
∂v . (29)
Combining Eqs. (28) and (29), using again that bi=δu
i, yields c(1−m)δu
iδu
j=−∂vaij2, or equivalently,
c=−∂vH
2(1−m)and ∂vWa=∂vhab =0. (30)
From this, it follows that Fis Berwald if and only if ∂vWa=∂vhab =0 and that cis 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 cinto Eq. (19) and using that bi=δu
iand consequently
bℓ=aℓkbk=aℓkδu
k=aℓu=−δℓ
v, we obtain the following.
Corollary 4. In the coordinates (u,v,x3,...,xn), the affine connection coefficients can be expressed in terms of the Levi-Civita Christoffel
symbols αΓk
ij of the (pseudo-)Riemannian metric αas
Γk
ij =αΓk
ij +ΔΓk
ij ≡αΓk
ij +m
2(1−m)∂vHaijδk
v+δk
jδu
i+δk
iδu
j. (31)
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.
Lemma 5. In the coordinates (u,v,x3,...,xn), the skew-symmetric part of the affine Ricci tensor is given by
¯
R[ij]=−mn
4(1−m)(δu
i∂j∂vH−δu
j∂i∂vH). (32)
Proof. From the definition (10) of the affine Ricci tensor of a Berwald space, it follows that its skew-symmetric part can be written as
¯
R[ij]≡1
2¯
Rij −¯
Rji=∂[kΓk
j]i−∂[kΓk
i]j. (33)
We now use the expression for the connection coefficients found in Corollary 4. Note that
ΔΓk
kj =m
2(1−m)∂vHavj+δu
j+nδu
j=mn
2(1−m)∂vHδu
j. (34)
Substituting this in the skew-symmetric part of the affine Ricci tensor, we obtain
¯
R[ij]=∂[kΓk
j]i−∂[kΓk
i]j=∂[kΔΓk
j]i−∂[kΔΓk
i]j=1
2−∂jΔΓk
ki +∂iΔΓk
kj(35)
=mn
4(1−m)δu
j∂i∂vH−δu
i∂j∂vH, (36)
where we have used the fact that the Ricci tensor corresponding to αis symmetric. ◻
Let us now prove our main result.
Theorem 6. Let (M,F=α1+mβ−m)be an m-Kropina space of Berwald type with closed null 1-form βand with dimM>2. The following
are equivalent:
(i) The affine connection is locally metrizable by a (pseudo-)Riemannian metric.
(ii) The affine Ricci tensor is symmetric, ¯
Rij =¯
Rji.
(iii) There exist local coordinates (u,v,x3,...,xn)such that b =du and
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a=−2dudv+˜
H(u,x)+ϕ(u)vdu2+Wa(u,x)dudxa+hab(u,x)dxadxb, (37)
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,x3,...,xn), by the following (pseudo-)
Riemannian metric: ˜
a=em
1−m∫uϕ(˜
u)d˜
ua. (38)
Before we present the proof, we want to point out two things. First, we note that if ϕ=0 then ˜
a=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 βis parallel with respect to α. It is a
well-known result that any (α,β)-metric for which βis parallel with respect to αis of Berwald type and that its affine connection coincides with
the Levi-Civita connection of α. Second, since ˜
ais 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 Fsatisfying any (and hence all) of the
equivalent conditions of Theorem 6 coincide with the null geodesics of the defining (pseudo-)Riemannian metric α.
Proof. (i) trivially implies (ii). For (ii) ⇒(iii), we use the preferred coordinates introduced in the lemma above. By Lemma 5, the only
nonvanishing skew-symmetric components of the affine Ricci tensor are
¯
R[uj]=−mn
4(1−m)∂j∂vH,j=2,...,n. (39)
Note that the fact that there is an index uon the LHS and an index von 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 uvcomponent therefore yields
∂2
vH=0 and the remaining components yield ∂v∂aH=0, a=3,...,n. In other words, Hmust be linear in vand the corresponding linear
coefficient can depend only on the coordinate u. That is,
g=−2dudv+˜
H(u,x)+ϕ(u)vdu2+Wa(u,x)dudxa+hab(u,x)dxadxb. (40)
This proves (ii)⇒(iii). For the last implication (iii)⇒(i), recall from Corollary 4 that the affine connection coefficients can be
expressed as
Γk
ij =αΓk
ij +ΔΓk
ij ≡αΓk
ij +m
2(1−m)ϕ(u)aijδk
v+δk
jδu
i+δk
iδu
j. (41)
On the other hand, an elementary calculation shows that the Levi-Civita Christoffel symbols of a (pseudo-)Riemannian metric ˜
a=eψ(u)a
can be expressed in terms of the original Christoffel symbols as
˜
aΓℓ
ij =αΓk
ij +1
2ψ′(u)aijδℓ
v+δℓ
jδu
i+δℓ
iδu
j. (42)
Hence, since ψ′(u)=−2mc =m
1−mϕ(u)for the (pseudo-)Riemannian metric ˜
aindicated in the theorem, it follows that the connection
coefficients of ˜
acoincide with the affine connection coefficients of our 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
F=−2dudv+˜
H(u,x)+ϕ(u,x)vdu2+Wa(u,x)dudxa+δab dxadxb(1+m)/2(du)−m. (43)
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 ∂aϕ=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(u,x)=0, ϕ(u,x)=u, and Wa(u,x)=0. If we relabel the
coordinates xaas xand y, this leads to the Finsler metric
F=−2dudv+uvdu2+dx2+dy2(1+m)/2(du)−m, (44)
which has an affine connection given by following nonvanishing affine connection coefficients:
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Γu
uu =1+m
2(1−m)u,Γv
uu =−1−m−u2
2(1−m)v, (45)
Γv
uv=−u
2,Γv
xx =Γv
yy =Γx
ux =Γy
uy =−−m
2(1−m)u. (46)
As indicated by Eq. (38) in Theorem 6, this connection is metrizable by the Lorentzian metric
˜
g=emu2
2(1−m)−2dudv+uvdu2+dx2+dy2. (47)
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.
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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=α1+mβ−m, which also serves as extension for the proof
presented in Ref. 34, where it was overlooked that the one-form βneed not be closed. The derivations in this section have been performed
with the help of the xAct extension of Mathematica.54
The Finsler metric Lfor m-Kropina spaces is given by F=(aij(x)yiyj)1+m
2(bk(x)yk)−m. Using the decomposition
∇ibj=Aij +Sij, (A1)
where Aij =A[ij](x)is the antisymmetric and Sij =S(ij)(x)is the symmetric part of the covariant derivative, we find a geodesic spray Gj=Nj
iyi
of the form
Gk=αΓk
ij(x)yiyj+yiAkiα2m
β(m+1)(A2)
−bkα2m(β(m+1)yiyjSij −2mα2biyjAij)
2β(m+1)(β2(m−1)−b2α2m)(A3)
+ykm(β(m+1)yiyjSij −2mα2biyjAij)
(m+1)(β2(m−1)−b2α2m). (A4)
Indices are raised and lowered with the components of the (pseudo-)Riemannian metric defining α. In order to be of Berwald type, the
components Gkneed to be quadratic functions of y. This is the case, since for a Berwald space, Nj
i(x,y)=Γj
ik(x)ykand so Gj(x,y)=Nj
i(x,y)yi
=Γj
ik(x)ykyi.
To reach this goal, the first term in (A4) must either cancel with one of the other terms appearing or the contraction ybAkbmust lead to a
term proportional to β. Hence, the free index on ybAkbmust be on yk,bk, or Zk=Zk(x), where Zkare the components of another vector field
Zon M, in the following way:
yjAkj=βZk+ykT+bkUiyi, (A5)
for T=T(x)being a function and Ui=Ui(x)being the components of a one-form on M. These are the only possible terms, since by con-
struction, ybAkbis a linear function in y, and so the RHS must be as well. Factoring the linear dependence in yon both sides of the equation
leads to
Akj=bjZk+Tδk
j+bkUj, (A6)
which then implies by the antisymmetry Aij =A[ij]that
Aij =bi(Uj−Zj)−bj(Ui−Zi). (A7)
Defining fj=(Uj−Zj), 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 βis determined by biand an additional one-form with components fj. Using this in (A1) we get
∇ibj=(m+1)(bifj−bjfi)+Sij, (A8)
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
Gk=αΓk
ij(x)yiyj+mα2fk(A9)
+mα2bk2(mα2bi−(m−1)βyi)fi−Sijyiyj
2((m−1)β2−mb2α2)(A10)
+myk2mα2(b2yi−βbi)fi+βSijyiyj
((m−1)β2−mb2α2). (A11)
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
B(y,y)C(y,y)−S(y,y)
D(y,y), (A12)
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where each term X(y,y)=Xijyiyj,X=B,C,D,S,P, denotes a quadratic polynomial in yand B(y,y)=α2. In order for this function to be
quadratic in y, it must satisfy
B(y,y)C(y,y)−S(y,y)
D(y,y)=P(y,y)⇔C(y,y)−S(y,y)=P(y,y)D(y,y)
B(y,y), (A13)
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
dimM>2, it follows by the argument given right below Eq. (12) that B(y,y)=α2is 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
2(mα2bi−(m−1)βyi)fi−Sijyiyj=h(2((m−1)β2−mb2α2)). (A14)
Taking two derivatives with respect to y, we find
Sij =maij(hb2+2bkfk)−(m−1)(bifj+bjfi+hbibj). (A15)
Redefining fias fi=1
2(˜
fi−bih)and combining all expressions for the covariant derivative of βfinally gives the desired expression as follows:
∇ibj=m(˜
fkbk)aij +bj˜
fi−mbi˜
fj. (A16)
One can easily check that this condition on bjleads to a geodesic spray given by
Gk=αΓk
ij(x)yiyj+m
2α2˜
fk−mykyi˜
fi, (A17)
which indeed is quadratic, and so the m-Kropina space subject to condition (16) is indeed Berwald.
For m=1 and b2≠0, the first term in line (A11) is quadratic in yfor any tensor components Sij and we must investigate the second term
of that line, which becomes −myk2α2(b2yi−βbi)fi+βSijyiyj
(b2α2), (A18)
and it can only be quadratic in yif and only if βSijyiyj
(b2α2)=Qiyi(A19)
for some one-form on Mwith components Qi=Qi(x). The only way to achieve this is if Sij =qaij for some function q=q(x)on M, which
then must satisfy
biq=Qib2⇒Qi∼bi,q∼b2. (A20)
Thus, for m=1, Sij ∼aijb2. (A21)
For m=1 and b2=0, the determinant of the metric gvanishes globally, and hence this situation does not define a Finsler space or spacetime.
REFERENCES
1N. Voicu, “Conformal maps between pseudo-Finsler spaces,” Int. J. Geom. Methods Mod. Phys. 15, 1850003 (2018).
2A. Fuster, S. Heefer, C. Pfeifer, and N. Voicu, “On the non metrizability of Berwald Finsler spacetimes,” Universe 6, 64 (2020); arXiv:2003.02300 [math.DG].
3M. A. Javaloyes, E. Pendás-Recondo, and M. Sánchez, “An account on links between Finsler and Lorentz geometries for Riemannian geometers,” arXiv:2203.13391
[math.DG] (2022).
4R. K. Tavakol and N. Van den Bergh, “Viability criteria for the theories of gravity and Finsler spaces,” Gen. Relativ. Gravitation 18, 849–859 (1986).
5N. Voicu, “New considerations on Hilbert action and Einstein equations in anisotropic spaces,” AIP Conf. Proc. 1283, 249–257 (2010); arXiv:0911.5034 [gr-qc].
6C. Pfeifer and M. N. R. Wohlfarth, “Finsler geometric extension of Einstein gravity,” Phys. Rev. D 85, 064009 (2012); arXiv:1112.5641 [gr-qc].
7C. Lämmerzahl and V. Perlick, “Finsler geometry as a model for relativistic gravity,” Int. J. Geom. Methods Mod. Phys. 15, 1850166 (2018); arXiv:1802.10043 [gr-qc].
8C. Pfeifer, “Finsler spacetime geometry in physics,” Int. J. Geom. Methods Mod. Phys. 16, 1941004 (2019); arXiv:1903.10185 [gr-qc].
9M. Hohmann, C. Pfeifer, and N. Voicu, “Finsler gravity action from variational completion,” Phys. Rev. D 100, 064035 (2019); arXiv:1812.11161 [gr-qc].
10M. Hohmann, C. Pfeifer, and N. Voicu, “Relativistic kinetic gases as direct sources of gravity,” Phys. Rev. D 101, 024062 (2020); arXiv:1910.14044 [gr-qc].
11I. P. Lobo and C. Pfeifer, “Reaching the Planck scale with muon lifetime measurements,” Phys. Rev. D 103, 106025 (2021); arXiv:2011.10069 [hep-ph].
J. Math. Phys. 64, 022502 (2023); doi: 10.1063/5.0130523 64, 022502-11
© Author(s) 2023

Journal of
Mathematical Physics ARTICLE scitation.org/journal/jmp
12A. Addazi et al., “Quantum gravity phenomenology at the dawn of the multi-messenger era—A review,” Prog. Part. Nucl. Phys. 125, 103948 (2022); arXiv:2111.05659
[hep-ph].
13E. Kapsabelis, P. G. Kevrekidis, P. C. Stavrinos, and A. Triantafyllopoulos, “Schwarzschild-Finsler-Randers spacetime: Dynamical analysis, geodesics and deflection
angle,” Eur. Phys. J. C 82, 1098 (2022).
14P. Carvalho, C. Landri, R. Mistry, and A. Pinzul, “Multimetric Finsler geometry,” arXiv:2208.03800 [math-ph] (2022).
15A. Garcia-Parrado and E. Minguzzi, “An anisotropic gravity theory,” Gen. Relativ. Gravit. 54, 150 (2022).
16J. Zhu and B.-Q. Ma, “Lorentz-violation-induced arrival time delay of astroparticles in Finsler spacetime,” Phys. Rev. D 105, 124069 (2022); arXiv:2206.07616 [gr-qc].
17C. Pfeifer and M. N. R. Wohlfarth, “Causal structure and electrodynamics on Finsler spacetimes,” Phys. Rev. D 84, 044039 (2011); arXiv:1104.1079 [gr-qc].
18E. Minguzzi, “The connections of pseudo-Finsler spaces,” Int. J. Geom. Meth. Modods. Phys. 11, 1460025 (2014); E. Minguzzi, “Erratum: The connections of pseudo-
Finsler spaces,” Int. J. Geom. Meth. Modods. 12(7), 1592001 (2015).
19A. G.-P. Gómez-Lobo and E. Minguzzi, “Pseudo-Finsler spaces modeled on a pseudo-Minkowski space,” Rep. Math. Phys. 82, 29–42 (2018); arXiv:1612.00829
[math.DG].
20E. Minguzzi, “Special coordinate systems in pseudo-Finsler geometry and the equivalence principle,” J. Geom. Phys. 114, 336–347 (2017).
21M. Javaloyes and M. Sánchez, “On the definition and examples of Cones and Finsler spacetimes,” RACSAM 114, 30 (2020).
22M. Hohmann, C. Pfeifer, and N. Voicu, “Cosmological Finsler spacetimes,” Universe 6, 65 (2020); arXiv:2003.02299 [gr-qc].
23N. Minculete, C. Pfeifer, and N. Voicu, “Inequalities from Lorentz-Finsler norms,” Math. Inequalities Appl. 373–398 (2021).
24A. B. Aazami, M. A. Javaloyes, and M. C. Werner, “Finsler pp-waves and the Penrose limit,” arXiv:2205.01162 [math.DG] (2022).
25M. Á. Javaloyes, M. Sánchez, and F. F. Villaseñor, “On the significance of the stress–energy tensor in Finsler spacetimes,” Universe 8, 93 (2022); arXiv:2202.10801
[gr-qc].
26L. Berwald, “Untersuchung der Krümmung allgemeiner metrischer Räume auf Grund des in ihnen herrschenden Parallelismus,” Math. Z. 25, 40–73 (1926).
27Z. Szabó, “Positive definite Berwald spaces,” Tensor 35, 25–39 (1981).
28M. Hashiguchi, S. H¯
oj¯
o, and M. Matsumoto, “On Landsberg spaces of two dimensions with (α,β)-metric,” J. Korean Math. Soc. 10, 17–26 (1973).
29V. Kropina, “On projective two-dimensional Finsler spaces with special metric,” Trudy Sem. Vektor. Tenzor. Anal. 11, 277–292 (1961).
30G. Yu. Bogoslovsky, “A special-relativistic theory of the locally anisotropic space-time,” Il Nuovo Cimento B 40, 99 (1977).
31G. Yu. Bogoslovsky, “On a special relativistic theory of anisotropic space-time,” Dokl. Akad. Nauk SSSR 213, 1055–1058 (1973).
32A. G. Cohen and S. L. Glashow, “Very special relativity,” Phys. Rev. Lett. 97, 021601 (2006); arXiv:hep-ph/0601236 [hep-ph].
33G. Gibbons, J. Gomis, and C. Pope, “General very special relativity is Finsler geometry,” Phys. Rev. D 76, 081701 (2007); arXiv:0707.2174 [hep-th].
34A. Fuster, C. Pabst, and C. Pfeifer, “Berwald spacetimes and very special relativity,” Phys. Rev. D 98, 084062 (2018), arXiv:1804.09727 [gr-qc].
35A. P. Kouretsis, M. Stathakopoulos, and P. C. Stavrinos, “The general very special relativity in Finsler cosmology,” Phys. Rev. D 79, 104011 (2009); arXiv:0810.3267
[gr-qc].
36M. Elbistan, P. Zhang, N. Dimakis, G. Gibbons, and P. Horvathy, “Geodesic motion in Bogoslovsky-Finsler spacetimes,” Phys. Rev. D 102, 024014 (2020);
arXiv:2004.02751 [gr-qc].
37A. Fuster and C. Pabst, “Finsler pp-waves,” Phys. Rev. D 94, 104072 (2016); arXiv:1510.03058 [gr-qc].
38P. Finsler, “Über Kurven und Flächen in allgemeinen Räumen,” Ph.D. thesis, Georg-August Universität zu Göttingen, 1918.
39D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Finsler-Riemann Geometry (Springer, New York, 2000).
40J. Szilasi, Connections, Sprays and Finsler Structures (World Scientific, 2014).
41J. K. Beem, “Indefinite Finsler spaces and timelike spaces,” Can. J. Math. 22, 1035 (1970).
42G. S. Asanov, Finsler Geometry, Relativity and Gauge Theories (D. Reidel Publishing Company, 1985).
43M. A. Javaloyes and M. Sánchez, “Finsler metrics and relativistic spacetimes,” Int. J. Geom. Methods Mod. Phys. 11, 1460032 (2014).
44M. Angel Javaloyes and M. Sánchez, “On the definition and examples of Finsler metrics,” Ann. Sc. Norm. Super. Pisa, Class. Sci. 13, 813–858 (2014).
45J. Szilasi, R. L. Lovas, and D. C. Kertész, “Several ways to Berwald manifolds - and some steps beyond,” Extracta Math. 26, 89–130 (2011); arXiv:1106.2223 [math.DG].
46C. Pfeifer, S. Heefer, and A. Fuster, “Identifying Berwald Finsler geometries,” Differ. Geom. Appl. 79, 101817 (2021).
47M. Crampin, “On the construction of Riemannian metrics for Berwald spaces by averaging,” Houston J. Math. 40, 737–750 (2014).
48P. Antonelli et al.,Handbook of Finsler Geometry (Springer, 2003), Vol. 2, pp. 729–1437.
49J. Podolsky, R. Steinbauer, and R. Svarc, “Gyratonic pp-waves and their impulsive limit,” Phys. Rev. D 90, 044050 (2014); arXiv:1406.3227 [gr-qc].
50J. W. Maluf, J. F. da Rocha-Neto, S. C. Ulhoa, and F. L. Carneiro, “Kinetic energy and angular momentum of free particles in the gyratonic pp-waves space-times,”
Classical Quantum Gravity 35, 115001 (2018); arXiv:1801.04957 [gr-qc].
51E. Tanaka and D. Krupka, “On metrizability of invariant affine connections,” Int. J. Geom. Methods Mod. Phys. 09, 1250014 (2012); arXiv:1111.3009 [math-ph].
52L. Tamássy, “Metrizability of affine connections.” BJGA 1, 83–90 (1996).
53B. G. Schmidt, “Conditions on a connection to be a metric connection,” Commun. Math. Phys. 29, 55–59 (1973).
54J. M. Martín-García, “xAct: Efficient tensor computer algebra for mathematica,” http://xact.es/, 2002–2022.
J. Math. Phys. 64, 022502 (2023); doi: 10.1063/5.0130523 64, 022502-12
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