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universe
Article
On the Non Metrizability of Berwald
Finsler Spacetimes
Andrea Fuster 1,†, Sjors Heefer 1,†, Christian Pfeifer 2, *,† and Nicoleta Voicu 3,†
1Department of Mathematics and Computer Science, Eindhoven University of Technology, Groene Loper 5,
5612AZ Eindhoven, The Netherlands; a.fuster@tue.nl (A.F.); s.j.heefer@tue.nl (S.H.)
2Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1,
50411 Tartu, Estonia
3Faculty of Mathematics and Computer Science, Transilvania University, Iuliu Maniu Str. 50, 500091 Brasov,
Romania; nico.voicu@unitbv.ro
*Correspondence: christian.pfeifer@ut.ee
† These authors contributed equally to this work.
Received: 28 February 2020; Accepted: 27 April 2020; Published: 1 May 2020
Abstract:
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.
Keywords: Finsler geometry; Berwald spaces; Berwald spacetimes; Szabo’s theorem; metrizability
1. Introduction
Pseudo-Finsler geometry is a promising mathematical framework for an improved geometric
description of the gravitational interaction, beyond the pseudo-Riemannian geometry employed in
general relativity [
1
–
11
]. Most recently, Finsler spacetime geometry was suggested as the optimal
mathematical language to describe the gravitational field of kinetic gases [
12
]. While positive definite
Finsler geometry, as an extension of Riemannian geometry, is a long-standing, well-established field in
mathematics [13–15], pseudo-Finsler geometry is still in the process of being developed [16–22].
The (pseudo-)Finslerian manifolds that can be regarded as closest to the (pseudo-)Riemannian
manifolds are the so-called Berwald spaces, resp. spacetimes [
23
–
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?
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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.
2. 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
,
˙
x)∈TM
is labeled by
the coordinates
(xa
,
˙
xa)
given by the decomposition of the vector
˙
x=˙
xa∂a∈TxM
, where
xa
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,˙
x)TM
and
T∗
(x,˙
x)TM
, of the tangent bundle are
{∂a=∂
∂xa
,
˙
∂a=∂
∂˙
xa}
and
{dxa
,
d˙
xa}
. In the following, unless
elsewhere specified, by smooth, we will always understand C∞-smooth.
2.1. Finsler Spaces
A Finsler space is a pair
(M
,
F)
, where
M
is a smooth
n
-dimensional manifold and the Finsler
function F:TM →Ris continuous on T M and smooth on TM \ {0}, and (see [14]):
•Fis positively homogeneous of degree one with respect to ˙
x:F(x,α˙
x) = αF(x,˙
x)for all α∈R+,
•the matrix:
gF
ab =1
2˙
∂a˙
∂bF2(1)
defines the Finsler metric tensor
gF
and is positive definite at any point of
TM \ {
0
}
, in one (and
then, in any) local chart around that point.
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The length of a curve
γ:R⊃[t1
,
t2]→M
on a Finsler space is defined by the parametrization
invariant length integral:
S[γ] = Zt2
t1
dτF(γ,˙
γ). (2)
To discuss Szabo’s theorem later, we need the building blocks of the geometry of a Finsler space,
the geodesic spray coefficients:
Ga=1
4gFaq (˙
xm∂m˙
∂qF2−∂qF2). (3)
They are defined on each coordinate chart of
TM \ {
0
}
and characterize the geodesic equation,
which, for arc length parametrized curves, is:
¨
xa+2Ga(x,˙
x) = 0 . (4)
Further objects we need for the arguments in this article are the horizontal derivative operators:
δa=∂a−˙
∂aGb˙
∂b, (5)
the local coordinate expressions of the Chern–Rund connection coefficients:
Γabc =1
2gFaq (δbgF
cq +δcgF
bq −δqgF
bc), (6)
the hh-Chern–Rund curvature:
Rcadb =δdΓcab −δbΓcad +Γcds Γsab −Γcbs Γsad , (7)
and its corresponding horizontal Ricci tensor:
Rab =Rmamb =δmΓmab −δbΓmam +Γmms Γsab −ΓmbsΓsam . (8)
It is important to notice that in general, Rab is not symmetric:
Rab −Rba =δaΓmbm −δbΓmam =Rcd ab ˙
xdCc. (9)
Here,
Ca=Cbab
are the components of the trace of the Cartan tensor
Cabc =1
2˙
∂agF
bc
. The last
equality in (9) can most easily be proven by introducing the function:
f=ln q|det gL
ab|. (10)
By direct calculation, one finds,
δaf=1
2|det gL
cd|δa|det gL
cd|=1
2gLmnδagL
mn =Γmam , (11)
˙
∂af=1
2|det gL
cd|˙
∂a|det gL
cd|=1
2gLmn ˙
∂agL
mn =Ca(12)
and therefore:
δaΓmbm −δbΓmam =δaδbf−δbδaf= [δa,δb]f=Rcdab ˙
xd˙
∂cf. (13)
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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}.
2.2. 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:
•π(Q) = M, where π:T M →Mis the canonical projection;
•conic property: if (x,˙
x)∈ Q, then for any λ>0 : (x,λ˙
x)∈ Q.
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 to
˙
x
:
L(x
,
λ˙
x) = λ2L(x
,
˙
x)
for all
λ∈R+
,
•on A, the vertical Hessian of L, called the L-metric, is nondegenerate,
gL
ab =1
2
∂2L
∂˙
xa∂˙
xb(14)
•
there exists a conic subset
T ⊂ A
such that on
T
,
L>
0,
g
has Lorentzian signature
(+
,
−
,
−
,
−)
and, on the boundary ∂T,Lcan be continuously extended as L|∂T=0.1
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=p|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:
•A
: the subbundle where
L
is smooth and
gL
is nondegenerate, with fiber
Ax=A ∩ TxM
, called the
set of admissible vectors,
•N: the subbundle where Lis zero, with fiber Nx=N ∩ TxM,
•A0=A\N
: the subbundle where
L
can be used for normalization, with fiber
A0x=A0∩TxM
,
•T
: a maximally connected conic subbundle where
L>
0 and the
L
-metric exists and has
Lorentzian signature (+,−,−,−), with fiber Tx=T ∩ TxM.
A major difference between Finsler spacetimes and Finsler spaces, as defined above, is the
existence of these different nontrivial subbundles of
TM \ {
0
}
. For Finsler spaces in their classical
definition, these bundles become trivial, i.e., A=A0=TM \ {0},N={0}, and T=∅.
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
A0
[
19
]. On
A
,
the geodesic spray is given by:
Ga=1
4gLaq (˙
xm∂m˙
∂qL−∂qL). (15)
1
It is possible to formulate this property equivalently with the opposite sign of
L
and the metric
gL
of signature
(−
,
+
,
+
,
+)
.
We fixed the signature and sign of Lhere to simplify the discussion.
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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.
3. 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 in ˙
x:
Ga(x,˙
x) = 1
2Gabc(x)˙
xb˙
xc. (16)
In this case, the second
˙
x
-derivatives of the geodesic spray coefficients are affine connection
coefficients ˙
∂b˙
∂cGa=Gabc(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:
Ga(x,˙
x) = 1
2Γabc(x,˙
x)˙
xb˙
xc. (17)
This means that on a Berwald space, or spacetime, the Chern–Rund connection coefficients are
independent of
˙
x
and the affine connection on
M
, defined by the connection coefficients
Gabc(x)
, is
precisely given by the Chern–Rund connection, i.e., Γabc(x,˙
x) = Γabc(x) = Gabc (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
gF
by
an averaging procedure over its ˙
xdependence [28].
Next we demonstrate that Szabo’s theorem can in general not be extended to Berwald spacetimes.
3.1. A Necessary Condition for the Metrizability of Berwald Spacetimes
Let
(M
,
L)
be a Berwald spacetime with Chern–Rund affine connection coefficients
Γabc =Γabc (x)
,
as defined in
(6)
. Then, the horizontal Chern–Rund Ricci tensor (see
(8)
) is independent of
˙
x
and takes
in every coordinate chart the form:
Rab =∂mΓmab −∂bΓmam +ΓmmsΓsab −Γmbs Γsam . (18)
It can be regarded as the Ricci tensor of the affine connection with coefficients
Γabc(x)
on
M
.
A necessary, but not sufficient, condition for the connection defined by
Γabc
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:
Rab(x)−Rba (x) = Rcdab (x)˙
xdCc(x,˙
x), (19)
where we expressed the explicit dependence on variables
x
and
˙
x
to highlight that this equation
encodes much information about the geometry of Berwald spacetimes. In particular, it gives rise to the
following theorem:
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Theorem 2
(Symmetric Ricci tensor on smooth Berwald spacetimes)
.
If
(M
,
L)
is a Berwald spacetime
with A=TM \ {0}, then Rba (x) = Rab (x).
The proof of Theorem 2will be presented in Appendix A.
In other words, if
L
is smooth and
gL
is non-degenerate on
TM \ {
0
}
, the Ricci tensor of a
Berwald–Finsler geometry is symmetric. For Finsler spaces, as we defined them in this article, these
conditions are satisfied by definition; see Section 2.1. Yet, in the class of Finsler spacetimes, there are
many interesting classes of examples for which
A
is not entirely
TM \ {
0
}
, but usually only a subset.
This is the origin of the existence of Berwald spacetimes with a non-symmetric Ricci tensor.
If one weakens the definition of Finsler spaces and allows for Finsler functions that are not smooth
everywhere on
TM \ {
0
}
, as for example conic Finsler geometries, introduced in [
18
], with a positive
definite Finsler metric, then also these admit examples of Berwald type for which the Ricci tensor
is not symmetric and hence provide Berwald spaces that are not metrizable. The same arguments
we presented for Finsler spacetimes hold in the positive definite case, since the main point is the
non-smoothness of the Finsler function/Finsler Lagrangian on TM \ {0}.
Next, we explicitly present a class of Finsler spacetimes that are, in general, not metrizable, since
their Ricci tensor is not symmetric.
3.2. 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(˙
x
,
˙
x) = αab(x)˙
xa˙
xb
be constructed from a pseudo-Riemannian metric
α
,
β(˙
x) = βa(x)˙
xa
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(˙
x)2
α(˙
x,˙
x). Consider the (α,β)-Finsler Lagrangian:
L(x,˙
x) = α(˙
x,˙
x)s−p(c+m s)p+1. (20)
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
–
7
,
23
,
32
–
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 βwith respect to the Levi–Civita connection of αsatisfies:
∇aβb=H[c(1−p) + mα−1(β,β)]βaβb+c pα−1(β,β)αab , (21)
for an arbitrary function H=H(x)on M. The resulting geodesic spray is:
Ga(x,˙
x) = 1
2Γabc(x)˙
xb˙
xc=1
2γabc(x)−H(pc(δa
bβc+δa
cβb)−βa(mβbβc+pcαbc ))˙
xb˙
xc, (22)
and thus, the resulting affine connection coefficients are:
Γabc(x) = γabc (x)−H(cp(δa
bβc+δa
cβb)−βa(mβbβc+cpαbc )). (23)
Here,
γabc
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:
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Rbd =R[γ]bd +cpαbdH2α−1(β,β)(c+3c p +mα−1(β,β)) + βa∂aH
+βbβd2cpH2(c+mα−1(β,β)) + mβa∂aH
−βb∂dH(mα−1(β,β)−3cp)−cpβd∂bH. (24)
Thus, for the skew-symmetric part,
1
2(Rab −Rba )=1
2(4cp −mα−1(β,β))(βa∂bH−βb∂aH). (25)
We summarize these findings as follows.
Proposition 1.
Let
(M
,
L)
be a Finsler spacetime defined by a Finsler Lagrangian of the type
L(x
,
˙
x) =
α(˙
x
,
˙
x)s−p(c+m s)p+1
as in Equation
(20)
. Then, the skew-symmetric part
A(R)
of the Chern–Rund Ricci
tensor is given by:
A(R) = fβ∧dH , (26)
where
f=1
2(
4
cp −mα−1(β
,
β))
is a scalar function. If
f6=
0and
β∧dH 6=
0, then
(M
,
L)
is non-metrizable.
This shows that the Ricci tensor corresponding to the Finsler Lagrangian
(20)
may not be
symmetric when
H6=
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 αand the one-form βbe given by:
α=2 dudv+vφ(x,y)du2+dx2+dy2,β=du, (27)
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
c6=
0 and
p6=
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:
1
2(Rux −Rxu )=p
p−1∂xφ, (28)
1
2Ruy −Ryu =p
p−1∂yφ. (29)
We see that they are independent of the parameters
c
and
m
and do not vanish for non-constant
φ
and
p6=
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
2We will elaborate on this in forthcoming work.
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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.
3.3. 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
–
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
γabc[g]
be the Christoffel symbols of its Levi–Civita
connection. Then:
Γabc(x) = γabc [g] + Dabc , (30)
where
D
is a
(
1, 2
)
-tensor field on
M
. By construction, the
Γabc(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
Dabc
defines the
non-metricity
Qabc =∇agbc =−Dsac gsb −Dsab gsc
of the connection. In view of this argument, we can
formulate the following proposition:
Proposition 2.
Let
(M
,
L)
be a Berwald–Finsler spacetime, and let
Γabc
be the induced affine connection
coefficients on
M
. Moreover, choose any pseudo-Riemannian metric
g
. The affine structure of the Finsler
spacetime
(M
,
L)
is equivalent to the affine structure of the metric-affine geometry of
(M
,
g
,
Γ)
, where the
connection defined by the connection coefficients
Γabc
is torsion free, but in general not metric compatible.
A Berwald–Finsler spacetime is metrizable if and only if there exists a pseudo-Riemannian metric
g
such that the
non-metricity Qabc vanishes.
4. 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.
Funding:
C.P. was supported by the Estonian Ministry for Education and Science through the Personal Research
Funding Grant PSG489, as well as the European Regional Development Fund through the Center of Excellence
TK133“The Dark Side of the Universe”. The work of A.F. is part of the research program of the Foundation for
Fundamental Research on Matter (FOM), which is financially supported by the Netherlands Organisation for
Scientific Research (NWO).
Acknowledgments:
The authors would like to acknowledge networking support by the COST Action
QGMM(CA18108), supported by COST (European Cooperation in Science and Technology).
Universe 2020,6, 64 9 of 12
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:
If (M,L)is a Berwald spacetime with A=TM \ {0}, then Rba (x) = Rab(x).
We now provide the proof of the theorem. The starting point of the proof is Equation (9):
Rab −Rba =δaΓmbm −δbΓmam =Rcd ab ˙
xdCc, (A1)
which on Berwald spacetimes implies that the expression:
φ(x,˙
x) = Rcdab (x)˙
xdCc(x,˙
x) = Rcdab (x)˙
xd˙
∂cf(x,˙
x)(A2)
is actually independent of
˙
x
. 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 q|det gL
ab|
, as defined in
(10)
, is zero-homogeneous in its dependence
on ˙
x, which means it naturally lives on the positive projective tangent bundle PT M+; see [19].
Recall that
PT M+
is defined as the set of equivalence classes
[(x
,
˙
x)]
, where
(x
,
˙
x)∼(x0
,
˙
x0)
if and
only if
(x0
,
˙
x0) = (x
,
λ˙
x)
for some positive real
λ
. This makes
PT M+
a 2
n−
1 dimensional manifold
with coordinate charts built as follows. Consider a coordinate chart
(U
,
ϕ)
on
M
and define the open
subsets
V+
i={(x
,
˙
x)∈TU|˙
xi>
0
}
and
V−
i={(x
,
˙
x)∈TU|˙
xi<
0
}
on
TM
. Then, for each
[(x
,
˙
x)]
with (x,˙
x)∈V±
i, we define the coordinates:
(xa,uα) = x0, ...xn,˙
x0
˙
xi, ..., ˙
xi−1
˙
xi,˙
xi+1
˙
xi, ..., ˙
xn−1
˙
xi, (A3)
where
a=
0, ...,
n−
1 and
α=
0, ...,
n−
2. More conveniently, one can use so-called homogeneous
coordinates
(xa
,
˙
xa)
, which are nothing but the coordinates on
TM
of an arbitrary representative
of the equivalence class
[(x
,
˙
x)]
. Homogeneous coordinates are only defined up to a scaling
factor.
PT M+
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
PT M+
if and only if they are
zero-homogeneous in
˙
x
; in homogeneous coordinates, calculus on
PT M+
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 in
˙
x
, they can be regarded as functions on
PT M+
.
Fix an arbitrary
x∈M
and an arbitrary local chart around
x
, then
fx(·) = f(x
,
·)
is defined on the
fiber
PTxM+
, which is compact. Since
fx
is smooth, it admits at least a local extremum, say at
˙
x=v
,
and hence, ˙
∂afx|˙
x=v=0. For the function φ(x,˙
x), this implies:
φ(x,v) = Rcdab (x)vd˙
∂cf(x,v) = 0 . (A4)
On the other hand, on Berwald spacetimes,
φ
is independent of
˙
x
, and we can conclude that
φ
is
identically zero for all ˙
x, which completes the proof.
As a remark, the crucial ingredient of the proof is that
|det gL
ab|
is smooth and nonzero on the
entire
TM \ {
0
}
. If
|det gL
ab|
were only smooth and nonzero on a smaller subset, then
fx
would not be
defined on the entire fiber PTxM+, and the conclusion of the theorem would fail.
Universe 2020,6, 64 10 of 12
Appendix B. Generalized Bogoslovsky/Kropina–Finsler Lagrangians
The class of Finsler Lagrangians we considered as a counterexample to Szabo’s theorem are given
in Equation
(20)
, which we rewrite here in the best way to briefly investigate the causal structure of the
Finsler Lagrangian:
L(x,˙
x) = α(˙
x,˙
x)s−p(c+m s)p+1=α(˙
x,˙
x)p+1
β(˙
x)2pc+mβ(˙
x2)
α(˙
x,˙
x)p+1
(A5)
=ζ(˙
x,˙
x)p+1
β(˙
x)2p, (A6)
where we introduced the shorthand notation of an effective bilinear form
ζ(˙
x
,
˙
x) = cα(˙
x
,
˙
x) + mβ(˙
x)2
.
Hence, the Finsler Lagrangian in consideration is effectively of Kropina/Bogoslovsky/VGR, for which
the metric
ζ
is constructed from more fundamental building blocks. Depending on the properties of
the building blocks β,α,m, and c,ζcan have different signatures.
The causal structure of
L
can be characterized for three different ranges of values for the
parameter p:
1. p>0: L=0⇔ζ(˙
x,˙
x) = 0, and Lis not defined for β(˙
x) = 0;
2. 0 >p>−1: L=0⇔ζ(˙
x,˙
x) = 0 or β(˙
x) = 0;
3. p<−1: L=0⇔β(˙
x) = 0, and Lis not defined for ζ(˙
x,˙
x) = 0.
The third case never leads to a Finsler spacetime since the null set
β(˙
x) =
0 singles out a
hyperplane and never allows for the existence of a convex cone of timelike vectors.
For the other two cases, a necessary condition to obtain a Finsler spacetime is that the bilinear
form
ζ
is a pseudo-Riemannian metric of a Lorentzian signature [
19
, Appendix B], where the same
class of Finsler Lagrangians is studied for p=−q. This demand leads to conditions on β,α,m, and c,
from the determinant:
det ζab =c3det(αab )(c+mα−1(β,β)) , (A7)
which must be negative.
For a choice of
β
,
α
,
m
, and
c
such that this necessary requirement is satisfied, one can apply the
classification done in [
19
, Appendix B] to identify viable Finsler spacetimes. A main finding there is
that for
−
1
<p<
1 and
β
being
ζ
timelike, the cone of future pointing timelike vectors
T
is given by
the cone of future pointing timelike vectors of ζ.
A complete classification of the Finsler Lagrangians
(20)
goes beyond the scope of this article and
is left for future investigation.
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