Content uploaded by Nicoleta Voicu
Author content
All content in this area was uploaded by Nicoleta Voicu on Oct 25, 2024
Content may be subject to copyright.
Available via license: CC BY 4.0
Content may be subject to copyright.
arXiv:2410.18197v1 [math-ph] 23 Oct 2024
Cosmological Landsberg Finsler spacetimes
Annam´aria Friedl-Sz´asz,∗Elena Popovici-Popescu,†and Nicoleta Voicu‡
Faculty of Mathematics and Computer Science, Transilvania University,
Iuliu Maniu Str. 50, 500091 Brasov, Romania
Christian Pfeifer§
ZARM, University of Bremen, 28359 Bremen,
Germany and Faculty of Mathematics and Computer Science,
Transilvania University, Iuliu Maniu Str. 50, 500091 Brasov, Romania
Sjors Heefer¶
Department of Mathematics and Computer Science,
Eindhoven University of Technology, Eindhoven 5600MB, The Netherlands
We locally classify all possible cosmological homogeneous and isotropic Landsberg-type
Finsler structures, in 4-dimensions. Among them, we identify viable non-stationary Finsler
spacetimes, i.e. those geometries leading to a physical causal structure and a dynamical
universe. Noting that any non-stationary Landsberg metric must be actually non-Berwaldian
(i.e., it should be a so-called unicorn), we construct the unique Finsler, non-Berwaldian
Landsberg generalization of Friedmann-Lemaitre-Robertson-Walker geometry.
I. INTRODUCTION
The gravitational fields of physical systems that are described in terms of kinetic gases, can be
understood from a new perspective, namely in terms of Finsler gravity [1,2].
In the standard (Einstein-Vlasov [3,4]) description of the gravitational field of a kinetic gas in
general relativity (GR), a lot of the information of the gas is lost.
More precisely, the kinematics and dynamics of multi-particle systems described as a kinetic
gas are encoded into the 1-particle distribution function (1PDF) ϕ, which lives, depending on the
formulation, on the tangent or on the cotangent bundle (i.e., the space of all positions and velocities,
respectively, the space of all positions and momenta of particles) of spacetime. The 1PDF contains
all the information about the kinetic gas. In general relativity (GR), the gravitational field of
∗szasz.annamaria@unitbv.ro
†popovici.elena@unitbv.ro
‡nico.voicu@unitbv.ro
§christian.pfeifer@zarm.uni-bremen.de
¶s.j.heefer@tue.nl
2
kinetic gases is derived from the Einstein equations, coupled to an energy-momentum tensor that
is obtained from the 1PDF by averaging over the normalized 4-velocities in the following way:
Tab
KG =∫Sx
dΣx
˙xa˙xb
g(˙x, ˙x)ϕ(x, ˙x),(1)
where Sxis the set of normalized 4-velocities ˙xand dΣxis a suitable volume form [3,5–7]. Using
this kind of coupling between the kinetic gas and the geometry of spacetime, only specific aspects
of the kinetic gas are taken into account in the derivation of its gravitational field, namely the
second moments of the 1PDF with respect to the 4-velocities of the gas particles.
Two immediate questions arise immediately:
•How do all other moments of the 1PDF
Ta1...an(x) = ∫Sx
dSx˙xa1.... ˙xanϕ(x, ˙x),(2)
which certainly contain non-trivial information about the kinetic gas, contribute to its grav-
itational field? Why are they neglected in the coupling to the Einstein equations?
•How much information is lost in the averaging procedure?
Finsler gravity offers the opportunity to answer these questions. Finsler geometry is a straight-
forward generalization of (pseudo-)Riemannian geometry [8–10], based on a general geometric
length measure for curves. It describes the geometry of spacetime in terms of geometric objects
(a canonical nonlinear connection and its curvature) that naturally live on the tangent bundle of
spacetime. There exist various applications of Finsler geometry in physics [11]. In particular, it
leads to a natural extension of general relativity, generally known as Finsler gravity [12–15]. It
turns out that the action-based canonical formulation of Finsler gravity introduced in [16,17] can
very naturally be coupled to the full 1PDF of the kinetic gas [1,2,11]. This leads to a canonical
variational Finsler gravity equation, of the the form:
G(x, ˙x) ∶= R(x, ˙x) +P(x, ˙x) = κϕ(x, ˙x),(3)
where R=R(x, ˙x)and P=P(x, ˙x)are scalar functions built from the Finsler curvature tensor,
respectively, from a Finslerian quantity called the Landsberg tensor, and κis a constant (to be
determined).
The goal of Finsler cosmology—the study of Finsler gravity in homogeneous and isotropic
symmetry—is to describe the evolution of the universe in accordance with observation, without
the need for 95% of dark (matter and energy) constituents of the universe. The conjecture is that
3
the gravitational field of the neglected parts of the 1PDF of the cosmological kinetic gas account for
(at least parts of) the effects that are typically associated with dark matter and dark energy, when
their contribution to the geometry of spacetime is properly taken into account [1]. Independently
of this conjecture relating dark matter and dark energy to properties of kinetic gases and canonical
action-based Finsler gravity, there exist other approaches to relate dark matter and dark energy
to Finsler geometry [18–25].
In general, the above Finsler gravity equation is difficult to solve and so far, only several
solutions have been found [26–30], see [31] for an overview. A particular difficulty in solving the
Finsler gravity equation is introduced by the P-term, involving the Landsberg tensor. Intuitively,
the Landsberg tensor measures the rate of change of the ‘non-Riemannianity’ of the geometry as
one moves through spacetime. It is natural to first look for solutions to the Finsler gravity equation
whose Landsberg tensor vanishes (called Landsberg spacetimes) because in that case, the equation
simplifies significantly and the departure from pseudo-Riemannian geometry is in some sense mild.
Landsberg manifolds admit a particular subclass, called Berwald manifolds, [32], for which the
canonical (in general nonlinear) Finsler connection is a linear (affine) connection on spacetime.
These are the simplest Finsler manifolds, closest to pseudo-Riemannian geometry, and the most
investigated ones in the literature [26–29,33–41]. The various classes of Finsler spacetimes can be
summarized concisely by their departure from pseudo-Riemannian geometry as follows:
pseudo-Riemannian ⊂Berwald ⊂Landsberg ⊂general Finsler .(4)
Importantly however, in the presence of cosmological symmetry it was proven that Berwald struc-
tures are too simple to describe the evolution of the universe beyond general relativity, as any
solution to the field equation of this type must be either a pseudo-Riemannian Friedmann-Lemaitre-
Robertson-Walker (FLRW) geometry, or a stationary one [35]. Thus, the next more interesting
class to be investigated is the class of Landsberg spacetimes which are non-Berwaldian. Since these
are notoriously difficult to find (actually, a full classification thereof is an important open problem
in mathematics), [42–45]), non-Berwaldian Landsberg space(time)s have been called unicorns [45].
In this article, we locally classify all 4-dimensional homogeneous and isotropic Finsler structures
of Landsberg type (in particular, all those that are Landsberg, but non-Berwald).
Furthermore, we prove that among these, there is a unique class that admits a well-defined
causal structure (featuring both a well-defined, convex future-pointing timelike cone and a past
one, at each point) and hence represents a direct Finsler, non-Berwaldian Landsberg generalization
of FLRW geometry. The obtained Landsberg-type generalization of the FLRW metric can then
4
be used as an ansatz in the Finsler gravity equation, to derive the Finsler gravity analogue of the
Friedmann equations (however, this is a task for the future). Moreover, our findings solve the
mathematical question of the full classification of non-Berwaldian Landsberg Finsler geometries in
4 dimensions and under the assumption of homogeneous and isotropic symmetry.
In order to derive our main result—the identification of the non-Berwald Landsberg general-
ization of FLRW geometry (in equation (70))—we begin this article by recalling the necessary
mathematical concepts of Finsler geometry, the different classes (Berwald, Landsberg and weakly
Landsberg) of Finsler manifolds, and the form of homogeneous and isotropic Finsler spacetime
functions in Section II. Afterwards, in Section III, we derive and solve the Landsberg condition
and thus find all possible 4-dimensional homogeneous and isotropic non-Berwald Landsberg Fins-
lerian manifolds. Among them, we identify the only one which qualifies as a Finsler spacetime in
Section IV, after which we discuss our results and conclude in Section V.
II. PRELIMINARIES - HOMOGENEOUS AND ISOTROPIC FINSLER SPACETIME
GEOMETRY
This section introduces the mathematical preliminaries to Finsler spacetime geometry and its
application to cosmological homogeneous and isotropic symmetry. We start by introducing Finsler
spacetimes and Finsler geometry in Subsection II A. Then, we discuss different classes of Finsler
spacetimes and identify the relevant non-Berwald Landsberg type Finsler manifolds in Subsection
II B before we recall homogeneous and isotropic symmetry in Finsler geometry in Subsection II C.
Throughout this article, we consider Mto be a 4-dimensional connected, orientable smooth
manifold, (T M, π , M), its tangent bundle and ○
T M =T M /{0}the so-called slit tangent bundle,
i.e. the tangent bundle without its zero section. We will denote by (xa)a=0,3, the coordinates of a
point x∈U⊂Min a local chart (U, ϕ)and by (xa,˙xa), the naturally induced local coordinates of
points (x, ˙x) ∈ π−1(U)in the tangent bundle. Also, whenever there is no risk of confusion, we will
omit for simplicity the indices of the coordinates. Moreover we introduce the following shorthand
notations for derivatives:
∂a=∂
∂xa,˙
∂a=∂
∂˙xa.(5)
Indices a, b, c, ... will run over all spacetime dimensions 0,1,2,3 while indices α, β , ... will run
over the spatial dimensions 1,2,3.
5
A. Finsler spacetime geometry
Finsler spacetimes are a straightforward generalization of pseudo-Riemannian spacetimes, where
the geometry, and hence the description of gravity, is derived from a pseudo-norm instead of a
spacetime metric.
Following [46,47], let A⊂T M be a conic subbundle of T M , meaning that π(A) = Mand
(x, ˙x) ∈ A ⇔ (x, λ ˙x) ⊂ A.(6)
AFinsler spacetime (M , L)is a manifold Mequipped with a smooth, positively 2-homogeneous
Finsler Lagrangian L
L∶A → R, L(x, λ ˙x) = λ2L(x, ˙x),∀λ>0,(7)
such that:
•the Finsler metric tensor gab(x, ˙x), given by:
gab(x, ˙x) ∶= 1
2˙
∂a˙
∂bL(x, ˙x),(8)
is nondegenerate on A;
•there exists a smooth conic subbundle T⊂Awith connected fibers Tx=T∩TxM,∀x∈M,
such that on each Txwe have: L>0, gab has Lorentzian signature (+,−,−,−) and Lcan be
continuously prolonged as 0 to the boundary ∂Txof Tx.
This definition of Finsler spacetimes ensures the existence of the following physically necessary
structures:
•A large enough set of directions Aalong which the needed geometric objects are well defined.
•A set of non-trivial null directions N∶= {(x, ˙x) ∈ T M L(x, ˙x)=0}, interpreted as directions
along which light propagates.
•A set Tof convex cones of past or future pointing timelike directions.
•A set S∶= {(x, ˙x)∈TL(x, ˙x)=1}, of unit normalized directions in T.
These sets generalize the corresponding structures that are defined through the spacetime metric
on pseudo-Riemannian spacetimes.
6
Moreover, on a Finsler spacetime, we have the following parametrization-invariant1length mea-
sure for curves γ:
S[γ]=∫dτL(γ, ˙γ).(9)
For timelike curves, ˙γ∈Tγ, this length is physically interpreted as the proper-time measure, or
geometric clock. In this sense, Finsler spacetimes are spacetimes whose geometry is based on
general geometric clocks. The 1-homogeneous function F(x, ˙x)=L(γ, ˙γ), is called the Finsler
function and is the fundamental building block in the classical literature on positive definite Finsler
geometry [9,10].
When passing from positive definite to indefinite Finsler geometry, there exist various definitions
of Finsler spacetimes by different authors, which all differ slightly depending on the application
the authors are interested in [16,48–53]. Their main difference lies in the assumption on where the
Finsler Lagrangian is smooth and where the Finsler metric is nondegenerate. We will not enter
the discussion on Finsler spacetimes here, and refer the interested reader to the references and the
overview article [46]. For the purposes of the analysis of homogeneous and isotropic Finsler gravity,
we will work with the definition presented in this section.
The geometry of Finsler spacetimes is derived from the Finsler Lagrangian, in a very similar way
as pseudo-Riemannian spacetime geometry is derived from the spacetime metric, see for example
[9,10]. Relevant geometric notions in Finsler geometry, besides the Finsler metric (8), that we will
need throughout this article are:
•The Cartan tensor and its trace:
Cabc =1
4˙
∂a˙
∂b˙
∂cL=1
2˙
∂cgab, Ca=gbc Cabc .(10)
The Cartan tensor measures, at each point x∈M, how much the Finslerian metric tensor
ga,b(x, ˙x)differs from a pseudo-Riemannian one.
•The geodesic spray coefficients Ga, the canonical nonlinear connection coefficients Nab, and
the horizontal derivatives δa:
Ga=1
4gab(˙xc∂c˙
∂bL−∂bL), Nab=˙
∂bGa, δa=∂a−Nba˙
∂b.(11)
The geodesic spray defines the geodesic equation from the Euler Lagrange equations of the
Finsler length (9) in arclength parameterisation.
1By parameterization invariance we mean here invariance under orientation-preserving changes of the parameter.
7
•The curvature of the canonical nonlinear connection and the canonical curvature scalar (or
Finsler-Ricci scalar:
Rcab ˙
∂c=[δa, δb], R =Raab ˙xb.(12)
•The Chern-Rund covariant derivatives, defined on the adapted basis {δa,˙
∂a}of T(x, ˙x)T M as:
∇δaδb=Γcabδc=1
2gcd (δagbd +δbgad −δdgab)δc,∇δa˙
∂b=Γcab ˙
∂c,(13)
∇˙
∂aδb=0,∇˙
∂a
˙
∂b=0.(14)
•The dynamical covariant derivative - measuring the rate of variation of tensor fields along
geodesics of (M, L), is obtained from the Chern-Rund one by contraction with ˙xaand ex-
pressed by the symbol ∇ ∶= ˙xa∇δa.
•The Landsberg tensor and its trace
Pabc = ∇Cabc , Pa=gbc∇Cabc .(15)
They keep track of how much the ”non-Riemannianity” of our Finsler space varies as one
moves along its geodesics.
Having clarified these notions, we can classify Finsler geometries according to how much more
general they are compared to pseudo-Riemannian geometry.
B. Classes of Finsler geometries
In general, Finsler geometries are widely more general than pseudo-Riemannian ones. The
latter are defined in terms of the ten component functions of a metric tensor on a smooth manifold,
whereas the former are given by a 2-homogeneous function on the tangent bundle — which, from
the viewpoint of the base manifold, has infinitely many degrees of freedom. Hence, it is fair to say
that Finsler geometry gives us an enormous amount of additional freedom, compared to pseudo-
Riemannian geometry.
Among all possible Finsler geometries, there are subclasses which are closer to pseudo-
Riemannian geometry than others. We briefly recall them here in a ladder-structured way, from
particular to general. The classification below holds in arbitrary signature, even if we only focus
here on Finsler spacetimes:
8
•Pseudo-Riemannian manifolds: A Finsler spacetime (M, L)is pseudo-Riemannian if and
only if its Cartan tensor (10) vanishes, which means that the Finsler metric tensor (8) is
independent of ˙x, i.e. it is a pseudo-Riemannian metric.
•Berwald pseudo-Finsler spaces (M, L)(Berwald spaces): These are Finsler spaces whose
Cartan tensor (10) is non-vanishing, but whose nonlinear connection (11) is linear in ˙x:
Nab(x, ˙x)=Γabc(x)˙xc; (16)
equivalently, the canonical nonlinear connection arises from an affine connection with coef-
ficients Γabc(x)on the base manifold M[32]. Berwald-Finsler functions Lcan be found by
fixing a (symmetric) affine connection on spacetime and solving the equation
δaL=∂aL−Γcab(x)˙xb˙
∂cL=0.(17)
Berwald geometries are sometimes addressed as the non-trivial Finsler geometries closest to
pseudo-Riemannian geometry, and have been studied in depth from different angles, in both
mathematics and physics [26–29,33–41].
•Landsberg and weakly Landsberg geometries: A Finsler spacetime (M , L)is called Lands-
berg, respectively, weakly Landsberg, if its Lansberg tensor (15), respectively, its trace,
vanishes
Pabc =0, Pa=0.(18)
Landsberg geometries are those for which the Cartan tensor is preserved by parallel transport
along geodesics, i.e. the deviation from pseudo-Riemannian geometry ‘looks the same’ in each
tangent space. In weakly Landsberg geometries only the canonical volume form is preserved.
Weakly Landsberg and Landsberg geometries play an important role in Finsler gravity since,
for this class the gravitational field equations simplify tremendously [17,30], and it has been
shown that the Palatini and the metric formulation of action based Finsler gravity coincide
[14].
It was believed for a long time that all Landsberg spaces were automatically Berwald. Lands-
berg spaces that are not Berwald turned out to be extremely difficult to find, to the point it
was believed they might not exist at all; this is why the term unicorn geometries was coined
for them [45]. In 2006, however, some unicorns were found by Asanov [54] and his results
were generalized by Shen [55] a few years later. Just very recently, a systematic way of
constructing unicorns by conformal rescaling of Berwald metrics was introduced in [42–44].
9
The just presented classes of Finsler geometries can nicely be summarized by the following
inclusions:
pseudo-Riemannian ⊂Berwald ⊂Landsberg ⊂weakly Landsberg ⊂general Finsler .(19)
Since we are aiming to derive all possible homogeneous and isotropic Landsberg geometries in
4-dimensions, we now briefly recall the most important known results about homogeneous and
isotropic Finsler structures.
C. Homogeneous and isotropic symmetry
The Copernican and cosmological principle leads us to the conclusion that, on largest scales, the
universe is spatially homogeneous and isotropic [56]. In four dimensions (which is what we assume
for realistic spacetime geometries), this implies the existence of a global time function t∶M→R
and of six Killing vector fields, not only in the pseudo-Riemannian case but also in the general
Finsler setting [35].
More precisely, on each spatially homogeneous and isotropic 4-dimensional Finsler manifold
there exist local coordinates (t, r, θ, φ)such that the vector fields
X1=χsin θcos φ∂r+χ
rcos θcos φ∂θ−χ
r
sin φ
sin θ∂φ,(20a)
X2=χsin θsin φ∂r+χ
rcos θsin φ∂θ+χ
r
cos φ
sin θ∂φ,(20b)
X3=χcos θ∂r−χ
rsin θ∂θ,(20c)
X4=sin φ∂θ+cot θcos φ∂φ, X5= −cos φ∂θ+cot θsin φ∂φ, X6=∂φ,(20d)
generate diffeomorphisms on Mwhich leave the Finsler Lagrangian Linvariant. Technically, this
means that Lis constant along the flows of complete lifts of the above vector fields XC
I, I =1, ..6:
XC
I(L)=0.(21)
The complete lift of a vector field ξ=ξa∂aon Mto T M is given by ξC=ξa∂a=˙xc∂cξa˙
∂a. It can
now straightforwardly be shown that the Finsler Killing equations for X1to X6are satisfied if and
only if [57]
L(t, r, θ, φ, ˙
t, ˙r, ˙
θ, ˙
φ)=L(t, ˙
t, w), w2=˙r2
1−kr2+r2˙
θ2+sin2(θ)˙
φ2.(22)
The 2-homogeneity of the Finsler Lagrangian w.r.t. the ˙xvariables allows us to write down the
most general homogeneous and isotropic Finsler Lagrangian in the following, form based on a
10
function h(t, s)of just two variables,
L(t, ˙
t, w)=˙
t2L(t, 1, s)=∶ ˙
t2h(t, s)2, s =w
˙
t,(23)
which is the most convenient form to derive the Finsler geometric objects (this is done in Appendix
A)—in particular, the Landsberg tensor.
Thus, cosmological Finsler geometries have a very specific dependence on the coordinates. Ac-
tually, each spatial slice t=const. is a 3-dimensional Riemannian manifold of constant curvature k
with metric given by w2- just as in the FLRW case. The difference is that in the Finslerian case,
the intertwining between the time coordinate and the spatial part wof the Finsler Lagrangian is
much more general.
Returning to the classification of Finsler geometries in Section II B, the situation is as follows:
•All homogeneous and isotropic pseudo-Riemannian geometries are given by the FLRW met-
ric. In the language of this section, this means: h(t, s)=1−a(t)2s2.
•Homogeneous and isotropic Berwald geometries have been completely classified in [35]. All
non-Riemannian Berwald structures are given by h(t, s)=h(s)only. In particular, they are
stationary, hence never describe an dynamically evolving universe.
For non-Berwald Landsberg geometries (the so-called unicorns) the classification in homogeneous
and isotropic symmetry is not known. This is what we will provide in the next section of this
article.
III. SOLVING THE LANDSBERG CONDITION
In order to find all cosmologically symmetric (homogeneous and isotropic) Landsberg Finsler
Lagrangians L, we will solve the equation given by the vanishing of the Landsberg tensor (15) in
general
Pabc = ∇Cabc =0.(24)
To do this, it is convenient to first compute explicitly the relevant Finsler geometric objects (Car-
tan tensor, spray, nonlinear connection, etc.) for a generic homogeneous and isotropic Finsler
Lagrangian (23). This has been carried out in Appendix A. In what follows, we will frequently
refer to this Appendix whenever we use a result derived there.
11
The strategy to solve (24) is that we first identify some necessary conditions in section III A,
which then give rise to six branches as candidates for solutions. In section III B we solve each
branch and verify that only one of these branches leads to three families of non-Berwald solutions
of (24).
A. Deriving the necessary and sufficient Landsberg conditions
Taking into account that the time function tis globally well defined (and so is the spatial metric
w), we can construct the following well-defined tensor fields from the “spatial metric” wintroduced
in (23):
wαβ =1
2˙
∂α˙
∂βw2,˙
∂αw=wα=1
wwαβ ˙xβ, wa=(−s, wα), T =˙
tw . (25)
These expressions make it convenient to display the Cartan tensor of a homogeneous and isotropic
Finsler Lagrangian as a sum of two tensor fields (see A 1 for the proof):
2Cabc =h(h′−sh′′)−sh′2Tabc +1
˙
thh′′′ +3h′h′′wawbwc,(26)
where primes denote derivatives with respect to sand Tabc =˙
∂a˙
∂b˙
∂cT. Next, the Landsberg tensor
is easily obtained in terms of the dynamical covariant derivative using (15) and the relations
∇wa=p(t, s)˙
twa(the precise expression of p(t, s), together with the proof of these equalities, are
also presented in the Appendix, in (A22)), we find:
4Pabc = ∇h(h′−sh′′)−sh′2Tabc+∇(hh′′′ +3h′h′′)
˙
t+3hh′′′ +3h′h′′p(t, s)wawbwc.(27)
Since T00c=˙
∂0˙
∂0˙
∂cT=0, the equations P000 =0 and P00α=0 are equivalent to each other and give
a first necessary condition on hin order for Lto be Landsberg:
∇(hh′′′+3h′h′′ )
˙
t= −3hh′′′ +3h′h′′p(t, s).(28)
Consequently, the second term in (27) vanishes and the remaining equations to be solved are:
∇(Tabc h(h′−sh′′)−sh′2)=0.(29)
The dynamical covariant derivative of Tabc has the remarkable property that it is of the form:
∇Tabc =˙
tq(t, s)Tabc (see (A26) for the expression of q(t, s)), meaning that the remaining part of
the Landsberg condition is equivalent to:
Tabc ˙
tq(t, s)h(h′−sh′′)−sh′2+∇h(h′−sh′′)−sh′2]=0,(30)
12
leading directly to the second necessary condition on hin order for Lto be Landsberg:
∇h(h′−sh′′)−sh′2= −˙
th(h′−sh′′)−sh′2q(t, s).(31)
Together, the two above conditions are also sufficient for the vanishing of Pabc. Briefly, the following
equivalence holds: Pabc =0 if and only if (28) and (31) hold.
Now there are two cases to be distinguished, since the term
D=hs(−sh′2+hh′+sh′′))∂th′(32)
will appear as a denominator.
•Case I: D=0, leads to two branches of candidates for being Landsberg geometries
0=∂th′⇔h=α(t)+β(s),(Branch 1)
0= −sh′2+hh′+sh′′⇔h=sc1(t)c2(t).(Branch 2)
•Case II: D≠0, will lead to four branches of candidates for being Landsberg geometries. In
this case equation (31) can be algebraically solved for h′′′(t, s)=Q(t, s)as a function of the
lower order derivatives of h, and (28) can be algebraically solved for h′′′′(t, s)=P(t, s)also
as a function of the lower order derivatives. Thus, a consistency (integrability) condition
for the system (28) and (31) is given in this case by Q′=P. Evaluating this consistency
condition with Wolfram Mathematica yields
0=h′′ sh′2+h−h′+sh′′sh′′∂th+h−sh′∂th′
hs−sh′2+hh′+sh′′∂th′(33)
×h′′h′sh′2+hh′−sh′′∂th+2sh′′h−h′2+hh′′∂th′−hh′−sh′2+hh′+sh′′∂th′′.
(34)
This leads to the remaining branches, deemed Branch 3 - Branch 6. Branch 3 and 4 can be
solved immediately:
0=h′′ ⇔h=c1(t)+c2(t)s , (Branch 3)
0=sh′2+h−h′+sh′′⇔h=c2(t)s2+c1(t),(Branch 4)
while Branch 5
0=sh′′∂th+h−sh′∂th′,(Branch 5)
13
and Branch 6
0=h′h′′sh′2+hh′−sh′′∂th+2sh′′h−h′2+hh′′∂th′
−hh′−sh′2+hh′+sh′′∂th′′ ,(Branch 6)
need further investigation.
The Finsler Lagrangian generated by (Branch 3) is degenerate, so not viable, while the one gen-
erated by (Branch 4) leads to a pseudo-Riemannian geometry (which is trivially Landsberg and
Berwald). We thus focus on the remaining four branches (Branch 1), (Branch 2), (Branch 5) and
(Branch 6). These require a more detailed discussion, which follows in the next section.
B. Solving the necessary and sufficient Landsberg conditions
The remaining task is to solve (28) and (31) for the four branches (Branch 1), (Branch 2),
(Branch 5) and (Branch 6) of the integrabillity condition (33).
1. Three branches without unicorns
As we will show below, the first three of the remaining branches (Branch 1), (Branch 2) and
(Branch 5) all lead to either a degenerate fundamental tensor, or a non-Riemannian Berwald, thus
stationary, geometry. Therefore, these branches do not lead to any candidates for a physically
interesting model for the evolution of the universe.
•Branch 1, h=α(t)+β(s): For this specific form of h, we find
P000 =s3˙α
4(α+β)3β′β′′ −(α+β)β′′′=0.(35)
This term only vanishes if either ˙α=0, which implies that Litself is independent of t(and
thus of Berwald type, see [35]), or if (3β′β′′ −(α+β)β′′′)=0. Differentiating the latter
with respect to timplies ˙αβ′′′ =0. Since, as already noted, ˙α=0 does not lead to any
convenient (non-Berwald) solution, the only nontrivial remaining possibility is that β′′′ =0.
When substituted back into the given equation an immediate consequence is β′β′′ =0. In
turn, this implies that β=c1(t)+c2(t)s. In other words we find h(t, s)=α(t)+c1(t)+c2(t)s,
being of the same form as (Branch 3) which yields a degenerate Finsler Lagrangian.
14
•Branch 2, h=sc1(t)c2(t): Evaluating the Landsberg tensor for the second branch (Branch 2),
the Landsberg tensor vanishes if c1(t)=c1is constant. We thus obtain the Finsler Lagrangian
L=˙
t2sc1c2(t),(36)
which is of Berwald type, since it is of the form identified in [35] (and the t-dependence can
be eliminated by a convenient redefinition of the tcoordinate). Moreover, it never defines a
Finsler spacetime structure, since L=0 does not lead to well-defined convex causal cones.
•Branch 5,sh′′∂th+(h−sh′)∂th′=0: Let us first assume that h−sh′≠0, which allows us
to rewrite the above equation as:
∂s∂th
h−sh′=0,(37)
which leads to c(t)(h−sh′)=∂th. Defining q=hs, we obtain
∂sq=∂sh
s=sh′−h
s2= − ∂th
s2c(t)= − 1
sc(t)∂tq , (38)
that is: s∂sq+c(t)−1∂tq=0. The latter can be solved by the methods of characteristics
giving
q(t, s)=q(u(t, s)), u(t, s)=se−∫t
0c(t′)dt′,(39)
and thus
h(t, s)=q(u(t, s))s . (40)
Unfortunately, the corresponding Finsler Lagrangian,
L=˙
t2h(u(t, s))2e2∫t
0c(t′)dt′, u(t, s)=se−∫t
0c(t′)dt′,(41)
is not only Landsberg, but of Berwald type, since the apparent t-dependence through the
factor e(t)∶= e−∫t
0c(t′)dt′can be absorbed by a redefinition of the t-coordinate as ˙η∶= ˙
te(t),
and then the results from [35] apply. The case when h−sh′=0 leads to: h=c1(t)s, which
is degenerate.
2. Branch 6: The unicorn branch
Having analyzed the above branches, we realize that 4-dimensional homogeneous and isotropic
unicorns (if any) must be given by (Branch 6). In order to solve the Landsberg condition for this
15
branch, we employ the following strategy. First we solve (Branch 6) for ∂th′′, which gives:
∂th′′ =h′′h′sh′2+hh′−sh′′∂th+2hs−h′2+hh′′∂th′
hh′−sh′2+hh′+sh′′.(42)
Second, we also solve (31) for ∂th′′, and get as alternative expression
∂th′′ =h′′sh′2+hh′−sh′′∂th
h−sh′2+hh′+sh′′+
h′2h2−s2h′2+hs−h′+sh′′
sh−sh′2+hh′+sh′′+h′′′
h′′
∂th′(43)
Equating these two necessary expressions for ∂th′′ gives a simplified necessary condition
2
s+h′
h−2h′′
h′+h′′′
h′′ ∂th′=0.(44)
The case when ∂th′=0 on some open subset of the domain of definition of hwas already discussed
in (Branch 1) and it leads (assuming nondegeneracy) to a Berwald metric. Physically speaking, this
would correspond to the existence of a region of spacetime where the Finsler metric is stationary.
Excluding such scenarios, we may assume that ∂th′≠0 and equate the bracket in (44) to zero,
2
s+h′
h−2h′′
h′+h′′′
h′′ − = 0.(45)
A first integral is,
c1(t)=s2hh′′
h′2⇔∂sh
h′=1−c1(t)
s2,(46)
which can in turn be integrated to
ln(h)=c3(t)+∫s
s2+c2(t)s+c1(t)ds (47)
⇔h=ec3(t)exp ∫s
s2+c2(t)s+c1(t)ds.(48)
In order to evaluate the integral, we distinguish three cases:
1. For c1(t)>c2(t)24, the integral can be solved to be
h=ec3(t)s2+c2(t)s+c1(t)exp
−c2(t)
4c1(t)−c2(t)2arctan
2s+c2(t)
4c1(t)−c2(t)2
.(49)
Using this expression in equation (28) we find that we obtain a non-Berwald Landsberg
Finsler geometry if and only if c1(t)=d1c2(t)2, where d1>14 is a constant. Thus, with
such a choice of d1,
L1=˙
t2e2c3(t)(d1c2(t)2+sc2(t)+s2)exp −2
√4d1−1arctan 2s+c2(t)
√4d1−1c2(t) (50)
16
defines a unicorn Finsler geometry. Defining a new conformal time coordinate ηvia ˙η=
d2c2(t)˙
t(and using ˜s=w˙η) transforms the unicorn to a conformal separated variables form
L1=˙η2Ω1(η)2f1(˜s)=˙η2e2c3(t(η)) d1
d2
2
+1
d2
˜s+˜s2exp −2
√4d1−1arctan 2˜sd2+1
√4d1−1.(51)
2. For c1(t)<c2(t)24, the integral is solved analogously to the previous case, just changing
arctan to arctanh:
h=ec3(t)s2+c2(t)s+c1(t)exp
−c2(t)
c2(t)2−4c1(t)arctanh
2s+c2(t)
c2(t)2−4c1(t)
(52)
=ec3(t)(s1(t)−s)s1(t)
s1(t)−s2(t)(s−s2(t)) s2(t)
s2(t)−s1(t).(53)
To get the second equality above, we used the relation exp(d1arctanh(X))=((1+X)(1−
X))d1/2and introduced
s1(t)s2(t)=c1(t), s1(t)+s2(t)= −c2(t).(54)
Evaluating again (28) for this solution of the necessary conditions, we find that we obtain
a Landsberg Finsler geometry if and only if s1(t)=d1s2(t), with d1∈R. If d= −1 , Lis
pseudo-Riemannian - i.e., it is the Finsler function of the FLRW metric. Yet, for d1≠ −1,
the obtained Finsler Lagrangian:
L2=˙
t2e2c3(t)(d1s2(t)−s)2d1
d1−1(s−s2(t)) 2
1−d1(55)
is non-Berwaldian, in other words, it defines a unicorn Finsler geometry. Again, a redefinition
of the time coordinate by ˙η=˙
td2s2(t)to conformal time η, where d2=const., casts it into a
separated variables form:
L2=˙η2Ω2(η)2f2(˜s)=˙η2e2c3(t(η))(d1−˜sd2)2d1
d1−1(˜sd2−1)2
1−d11
d2
2
.(56)
3. For c1(t)=c2(t)24 the integration in (47) yields
h=ec3(t)(2s+c2(t))exp c2(t)
2s+c2(t).(57)
Evaluating again (28) for this solution of the necessary conditions, we find immediately that
we obtain a non-Berwald Landsberg-Finsler geometry. Thus,
L3=˙
t2e2c3(t)(2s+c2(t))2exp 2c2(t)
2s+c2(t)(58)
17
defines a unicorn Finsler geometry. As for the other two unicorn classes, we can redefine
the time coordinate to conformal time ηby: ˙η=˙
tc2(t)d2(where d2=const.), to express the
Lagrangian as
L3=˙η2Ω3(η)2f3(˜s)=˙η2e2c3(t(η))(2˜sd2+1)2exp 2
2˜sd2+11
d2
2
.(59)
Having analyzed the Landsberg tensor for general 4-dimensional homogeneous and isotropic Finsler
Lagrangians, we have found three classes of unicorn geometries given by the Finsler Lagrangians
L1(50), L2(55) and L3(58).
Since we solved the equation Pabc =0 in all generality, there cannot exist any more classes. In
other words, we have found all 4-dimensional homogeneous and isotropic unicorns.
In earlier works, a different strategy was used to find unicorns. Starting from a Berwald Finsler
Lagrangian L0, unicorn Finsler geometries were constructed by a conformal rescaling
LU=eσ(x)L0,(60)
and solving the so-called σT condition [42–44]. This way, several unicorns were found. Due to
the ansatz LU=eσ(x)L0this method can, yet, not ensure that all possible unicorns of a certain
type are found, since it assumes the specific structure (60). In the 4-dimensional homogeneous and
isotropic scenario we study here, this ansatz would amount to assume a separated variable Finsler
Lagrangian from the beginning
LU=eσ(t)L0(s).(61)
Solving the σT condition for this form of the Finsler Lagrangian leads to the same unicorn geome-
tries L1,L2and L3that we found. We note that L1had already been found earlier in [55] and in
[58]. In addition, our approach proves that there cannot be others.
Hence our results completely classify and present explicitly all 4-dimensional homogeneous and
isotropic unicorn geometries.
Next we determine which of these classes of unicorns really define a Finsler spacetime.
IV. NON-BERWALD LANDSBERG FINSLER SPACETIMES
Having identified all homogeneous and isotropic non-Berwald Landsberg Finsler Lagrangians,
we are interested in the question which ones lead to a viable Finsler spacetime, i.e. which ones can
18
be used as spacetime geometry in cosmology? The answer to this question leads us to the unique
Finsler non-Berwald Landsberg (or Finsler unicorn) generalization of FLRW geometry.
In order to verify which Finsler Lagrangian satisfies the Finsler spacetime definition discussed
in the beginning of Section II A, we determine their null structure L=0 and verify the Lorentzian
signature of the Finsler metric by the sign of its determinant
det g=det wαβ
h4h′2
s2hh′′ .(62)
Let us begin with L1and L3, for which we immediately see that they do not yield any realistic
light cones, hence no Finsler spacetime structure. Indeed:
•The Finsler Lagrangian (49) can never vanish, since
s2+c2(t)s+c1(t)≠0,(63)
for the required constraints on the parameters c1(t)and c2(t). Thus, L1never defines a
Finsler spacetime.
•The Finsler Lagrangian of the type (59) can be expressed as
L3(η, ˜s)=˙η2e2c3(η)(2˜sd2+1)2exp 2
2˜sd2+11
d2
2
.(64)
A direct calculation of the determinant of the Finsler metric shows that it is positive. Thus,
L3does not lead to a Finsler spacetime structure2.
From the perspective of a physical spacetime structure, the only possibility remains thus L2
(56), which, introducing the constants e1=d1d2and e2=1d2with e1≠e2, can be expressed as
L2(η, ˜s)=˙η2e2c3(η)(e1−˜s)2e1
e1−e2(˜s−e2)−2e2
e1−e2.(65)
The lightlike directions of this geometry are given by ˜s=e1and ˜s=e2. In order to have both a
future and a past pointing lightcone, we need that e1e2<0. Moreover, we need that the exponents
2e1
e1−e2and −2e2
e1−e2are both positive. This is ensured by either choosing e1<0, e2>0 and e1−e2<0,
or choosing e1>0, e2<0 and e1−e2>0. Both choices lead to the same results.
Next, studying the determinant of the Finsler metric for this Finsler Lagrangian, identifying
h(η, ˜s)=ec3(η)(e1−˜s)e1
e1−e2(˜s−e2)−e2
e1−e2from (65), yields
det g=det wαβ
h4h′2
˜s2hh′′
=det wαβ e2c3(η)(e1e2)(e1−˜s)4(e1+e2)
e1−e2(˜s−e2)−4e1+e2
e1−e2.(66)
2In [30] possible modifications of Finsler Lagrangians of the type L3have been discussed as solutions of the Finsler
gravity equation.
19
This determinant is clearly negative, since the sign is determined by the factor e1e2, and smooth
until it reaches the light cones. Thus, the Finsler metric has Lorentzian signature in the interior
of both, the future and the past pointing light cone. Depending on the exact values of e1or e2, it
vanishes at one of the light cones s=e1and blows up at the other s=e2.
In order to prove that signature of the metric tensor gof L2is (+,−,−,−)inside the timelike
cones, we note that all cosmologically symmetric Finsler metric are (α, β)-metrics, i.e. Finsler
Lagrangians that are functions L=L(α, β), where α=ec3(η)˙η2−w2is the FLRW metric (with
signature convention (+,−,−,−)written in conformal time and β=˙
t). Then, using Lemma 3 in
[59], it turns out that a necessary and sufficient condition for the signature condition of L2is that
det(g)<0, which is satisfied as we just have proven above.
It is remarkable that even if the determinant is not well defined on one of the light cones, the
geodesic equation, expressed in terms of the geodesic spray, is well defined. Thus, all lightlike
geodesics can be described without any problems. To see this, we display below the geodesic spray
explicitly, by evaluating the expression in the Appendix A 2 for (65),
G0=˙η2∂tc3(η)(e1e2−˜s2)
e1e2
,(67)
Gα=˜
Gα+˙
t˙xα∂tc3(η)
2(2e1e2−(e1+e2)˜s)
2e1e2
.(68)
We clearly see that G0and Gαare well defined for ˜s→e1or ˜s→e2.
As the Finsler Lagrangian is expressed in (65), the time coordinate can be interpreted as con-
formal cosmological time. Another redefinition of ηand the appearing constants defined by
˙η=a(η)˙
te2, a(η)=ec3(η), e1=fe2(69)
enables us to rewrite the physical Finsler Lagrangian in terms of standard cosmological time t
L2=˙
tf −a(t)w2f
f−1a(t)w−˙
t−2
f−1.(70)
In conclusion, among all the 4-dimensional homogeneous and isotropic non-Berwald Landsberg
(unicorn) geometries, we identified one which is physically viable. It possesses one dynamical
variable, a scale factor a(t), and two independent constants: the spatial curvature parameter k,
and a Finsler parameter f. For f= −1, the Finsler Lagrangian L2becomes pseudo-Riemannian
FLRW geometry.
With this discussion, we identified and introduced the unique Finsler non-Berwald Landsberg
(or Finsler unicorn) generalisation of FLRW geometry.
20
V. DISCUSSION AND CONCLUSION
In order to prove that the neglected contributions of the 1PDF of a kinetic gas to the gravi-
tational field of the kinetic gas can be the sources of dark matter and dark energy, the goal is to
solve the Finsler gravity equation (3) in homogeneous and isotropic symmetry. This endeavour
turns out to be very involved, even in this highly symmetric situation, due to the complexity of
the equations and the generally large number of degrees of freedom of a function h(t, s)(instead
of only a scale factor a(t)in general relativity) in Finsler geometry.
Our identification of the unique Finsler non-Berwald Landsberg (or Finsler unicorn) generali-
sation of FLRW geometry (70) is a significant step towards finding cosmological solutions of the
Finsler gravity equation.
For non-Berwald Landsberg geometries, the second part of the field equation (3) (which contains
the trace of the Landsberg tensor Pa) is identically zero, leaving only the non-trivial curvature
part. Moreover, the number of degrees of freedom that need to be determined by the equation is
drastically reduced to just one time-dependent scale factor, a(t), and a constant parameter f. We
demonstrated this by determining all possible homogeneous and isotropic unicorns directly from
the vanishing of the Landsberg tensor. To do so we used the geometric tensors of 4-dimensional
homogeneous and isotropic Finsler geometry. Their complete expressions are derived explicitly for
the first time and can be found in Appendix A. These will be extremely useful in the future for the
derivation of the Finsler Friedmann equations, in general and for the Finsler extension of FLRW
geometry presented here.
A detailed analysis of the physical consequences of the new Finslerian FLRW spacetime ge-
ometry will be presented in a follow-up article. One remarkable property is a difference in the
opening angles of the future and past light cones, i.e., a time asymmetry in the spacetime ge-
ometry. This might be connected to the direction of the flow of time in the future and deserves
thorough investigation.
In conclusion, with this work we are one step closer to understand the details of the description
of the gravitational field of kinetic gases in term of Finsler geometry instead of pseudo-Riemannian
geometry. Our work lays the foundation for solving the Finsler gravity equation sourced by the
1PDF of a kinetic gas in cosmology, to describe the evolution of the universe. Moreover, we gave a
complete mathematical classification of all 4-dimensional homogeneous and isotropic non-Berwald
Landsberg Finsler geometries.
21
ACKNOWLEDGMENTS
The authors would like to thank Andrea Fuster and Volker Perlick for useful discussions and
insights. CP acknowledges support by the excellence cluster QuantumFrontiers of the German Re-
search Foundation (Deutsche Forschungsgemeinschaft, DFG) under Germany’s Excellence Strategy
– EXC-2123 QuantumFrontiers – 390837967 and was funded by the Deutsche Forschungsgemein-
schaft (DFG, German Research Foundation) - Project Number 420243324 - and by the Transilvania
Felowships for Visiting Professors grant 2024 of the Transilvania University of Brasov. This work is
based upon collaboration within the COST Action CA 21136, “Addressing observational tensions
in cosmology with systematics and fundamental physics” (CosmoVerse).
Appendix A: Homogeneous and isotropic Finsler geometry
In this appendix, we explicitly list the geometric tensors of Finsler geometry in homogeneous
and isotropic symmetry. They build the foundation on which all derivations in this article are
based.
1. Finsler metric and Cartan tensor
In order to find the homogeneous and isotropic unicorns in Sec. III, we need the Finsler metric
and the Cartan tensor for cosmologically symmetric Finsler Lagrangians L, which are all described
by:
L=˙
t2h(t, s)2, s =w
˙
t, w2=˙r2
1−kr2+r2˙
θ2+sin2(θ)˙
φ2.(A1)
Introducing the notations,
wαβ =1
2˙
∂α˙
∂βw2=diag 1
1−kr2, r 2, r2sin2(θ),˙
∂αw=wα=1
wwαβ ˙xβ, wa=(−s, wα),(A2)
we first obtain:
˙
∂0s= −s
˙
t,˙
∂αs=wα
˙
t.(A3)
Denoting by primes derivatives with respect to s, we find:
22
•The components of the Finsler metric and its inverse:
g00 =h2−2shh′+s2(h′2+hh′′),(A4)
g0α=wαhh′−s(h′2+hh′′),(A5)
gαβ =hh′wαβ
s+sh′2+shh′′ −hh′wαwβ
s,(A6)
and
g00 =A , A =h′2+hh′′
h3h′′ ,
g0α=B˙xα
w, B =s(h′2+hh′′)−hh′
h3h′′ ,(A7)
gαβ =Cwαβ +D
w2˙xα˙xβ, C =s
hh′, D =(h−sh′)hh′−s(h′2+hh′′)
h3h′h′′ ,
where wαβ =diag(1−kr2,1
r2,1
r2sin2(θ))is the inverse of wαβ.
•The Cartan tensor:
C000 = −1
2
s3
˙
t(hh′′′ +3h′h′′),(A8)
C00α=wα
s3
2w(hh′′′ +3h′h′′),(A9)
C0αβ =wαβ
2wh(h′−sh′′)−sh′2−wαwβ
2wh(h′−sh′′)−sh′2+s2(hh′′′ +3h′h′′),(A10)
Cαβγ = −(wβγ wα+wαγ wβ+wαβwγ−3wαwβwγ)h(h′−sh′′)−sh′2
2sw
+swαwβwγ(hh′′′ +3h′h′′)
2w.(A11)
Introducing the scalar variable T=˙
tw, this can be combined to a full Finsler spacetime
tensor as
Cabc =1
2h(h′−sh′′)−sh′2Tabc +(hh′′′ +3h′h′′)s
2wwawbwc,(A12)
where:
Tabc ∶= ˙
∂a˙
∂b˙
∂cT(A13)
=(δ0
cδβ
bδα
a+δ0
aδβ
bδα
c+δ0
bδβ
cδα
a)wαβ−wαwβ
w
+˙
t
w2δγ
cδβ
bδα
a(3wαwβwγ−wαβwγ−wαγ wβ−wγ β wα).(A14)
23
2. Geodesic spray components and nonlinear connection coefficients
In order to derive the Landsberg tensor in equation (27), we need to evaluate dynamical covariant
derivatives, which employ geodesic spray coefficients and nonlinear connection coefficients (11).
Here we display their explicit form for homogeneous and isotropic Finsler Lagrangians.
From the definition of the geodesic spray (11) we find:
G0=˙
t2h′′∂th−h′∂th′
2hh′′ (A15)
and
Gα=˜
Gα+1
2˙
t˙xαsh′′∂th+h−sh′∂th′
shh′′ ,(A16)
where the quantities ˜
Gαare the geodesic spray coefficients of the 3-dimensional Finsler function
w2, given by
˜
Gα=1
4wαβ ˙xγ˙
∂β∂γw2−∂βw2.(A17)
The nonlinear connection coefficients are then obtained by Nab=˙
∂bGa, as follows:
N0
0=˙
t
2h2h′′2h′′2(sh′+2h)∂th−h′(sh′+2)h′′ +shh′′′∂th′+sh′h′′∂th′′,(A18)
N0
α=˙
twαh′
2sh2h′′2−h′′2∂th+(h′h′′ +hh′′′)∂th′−hh′′∂th′′,(A19)
Nα0=˙xα
2h2h′′2sh′′2(h+sh′)∂th+h′′2h2−sh′(h+sh′)+shh′′′(h−sh′)∂th′
−sh2h′′(h−sh′)∂th′′,(A20)
Nαβ=Γαβγ ˙xγ+˙
tδα
β
2shh′′ sh′′∂th+h−sh′∂th′−wβ˙xα
2s2h′′2(h′′ +sh′′′)∂th′−s∂th′′
+wβ˙xαh′
2h2h′′2−h′′2∂th+(h′h′′ +hh′′′)∂th′−hh′′∂th′′.(A21)
where Γαβγ are the coefficients of the Levi-Civita connection of the spatial metric with components
wαβ .
By means of these expressions, one can directly determine the following terms, which we used
to derive the Landsberg tensor (27) and study the necessary equations (28) and (31), which imply
its vanishing:
24
•The dynamical covariant derivative of wa:
∇wa=˙xbδbwa−Nbawb(A22)
=˙xb∂bwa−Gb˙
∂bwa−Nbawb(A23)
=p(t, s)wa˙
t , (A24)
with
p(t, s)=1
2hh′′2−h′′2∂th+(h′h′′ +hh′′′)∂th′−hh′′∂th′′.(A25)
•The dynamical covariant derivative of Tabc :
∇Tabc =q(t, s)˙
tTabc ,(A26)
with
q(t, s)=1
2shh′′2−sh′′2∂th+h′′(2h+sh′)+shh′′′∂th′−shh′′∂th′′.(A27)
[1] M. Hohmann, C. Pfeifer, and N. Voicu, The kinetic gas universe, Eur. Phys. J. C 80, 809 (2020),
arXiv:2005.13561 [gr-qc].
[2] M. 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].
[3] H. Andreasson, The Einstein-Vlasov System/Kinetic Theory, Living Rev. Rel. 14, 4 (2011),
arXiv:1106.1367 [gr-qc].
[4] O. Sarbach and T. Zannias, Relativistic kinetic theory: An introduction,
AIP Conf. Proc. 1548, 134 (2013),arXiv:1303.2899.
[5] J. Ehlers, General-relativistc kinetic theory of gases, in Relativistic Fluid Dynamics, edited by C. Cat-
taneo (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011) pp. 301–388.
[6] O. Sarbach and T. Zannias, The geometry of the tangent bundle and the relativistic kinetic theory of
gases, Class. Quant. Grav. 31, 085013 (2014),arXiv:1309.2036 [gr-qc].
[7] O. Sarbach and T. Zannias, Relativistic Kinetic Theory: An Introduction, Proceedings, 9th Mexi-
can School on Gravitation and Mathematical Physics: Cosmology for the XXI Century: Inflation,
Dark Matter and Dark Energy (DGFM-SMF): Puerto Vallarta, Jalisco, Mexico, December 3-7, 2012,
AIP Conf. Proc. 1548, 134 (2013),arXiv:1303.2899 [gr-qc].
[8] P. Finsler, ¨
Uber Kurven und Fl¨achen in allgemeinen R¨aumen, Ph.D. thesis, Georg-August Universit¨at
zu G¨ottingen (1918).
25
[9] R. Miron and I. Bucataru, Finsler Lagrange geometry (Editura Academiei Romane, 2007).
[10] D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Finsler-Riemann Geometry (Springer, New York,
2000).
[11] C. Pfeifer, Finsler spacetime geometry in Physics, Int. J. Geom. Meth. Mod. Phys. 16, 1941004 (2019),
arXiv:1903.10185 [gr-qc].
[12] S. F. Rutz, A Finsler generalisation of Einstein’s vacuum field equations, General Relativity and Grav-
itation 25, 1139 (1993).
[13] A. Triantafyllopoulos, E. Kapsabelis, and P. Stavrinos, Gravitational Field on the Lorentz
Tangent Bundle: Generalized Paths and Field Equations, Eur. Phys. J. Plus 135, 557 (2020),
arXiv:2004.00356 [gr-qc].
[14] M. A. Javaloyes, M. Sanchez, and F. F. Villasenor, The Einstein–Hilbert–Palatini formalism in
pseudo-Finsler geometry, Advances in Theoretical and Mathematical Physics 26, 3563 – 3631 (2022),
arXiv:2108.03197 [math.DG].
[15] A. Garcia-Parrado and E. Minguzzi, An anisotropic gravity theory, Gen. Rel. Grav. 54, 150 (2022),
arXiv:2206.09653 [gr-qc].
[16] C. Pfeifer and M. N. R. Wohlfarth, Causal structure and electrodynamics on Finsler spacetimes,
Phys.Rev. D84, 044039 (2011), arXiv:1104.1079 [gr-qc].
[17] M. Hohmann, C. Pfeifer, and N. Voicu, Finsler gravity action from variational completion,
Phys. Rev. D 100, 064035 (2019),arXiv:1812.11161 [gr-qc].
[18] A. Kouretsis, M. Stathakopoulos, and P. Stavrinos, The General Very Special Relativity in Finsler
Cosmology, Phys.Rev. D79, 104011 (2009), arXiv:0810.3267 [gr-qc].
[19] N. E. Mavromatos, V. A. Mitsou, S. Sarkar, and A. Vergou, Implications of a Stochastic Microscopic
Finsler Cosmology, Eur. Phys. J. C72, 1956 (2012),arXiv:1012.4094 [hep-ph].
[20] G. Papagiannopoulos, S. Basilakos, A. Paliathanasis, S. Savvidou, and P. C. Stavri-
nos, Finsler–Randers cosmology: dynamical analysis and growth of matter perturbations,
Class. Quant. Grav. 34, 225008 (2017),arXiv:1709.03748 [gr-qc].
[21] X. Li, S. Wang, and Z. Chang, Anisotropic inflation in the Finsler spacetime,
Eur. Phys. J. C75, 260 (2015),arXiv:1502.02256 [gr-qc].
[22] M. Hohmann and C. Pfeifer, Geodesics and the magnitude-redshift relation on cosmologically symmetric
Finsler spacetimes, Phys. Rev. D95, 104021 (2017),arXiv:1612.08187 [gr-qc].
[23] Y. Akrami et al. (CANTATA), Modified Gravity and Cosmology. An Update by the CANTATA Network,
edited by E. N. Saridakis, R. Lazkoz, V. Salzano, P. Vargas Moniz, S. Capozziello, J. Beltr´an Jim´enez,
M. De Laurentis, and G. J. Olmo (Springer, 2021) arXiv:2105.12582 [gr-qc].
[24] E. Kapsabelis, E. N. Saridakis, and P. C. Stavrinos, Finsler–Randers–Sasaki gravity and cosmology,
Eur. Phys. J. C 84, 538 (2024),arXiv:2312.15552 [gr-qc].
[25] P. C. Stavrinos and A. Triantafyllopoulos, Cosmology Based on Finsler and Finsler-like Metric Structure
of Gravitational Field, arXiv:2405.19318 [gr-qc].
26
[26] A. Fuster and C. Pabst, Finsler pp-waves, Phys. Rev. D 94, 104072 (2016),arXiv:1510.03058.
[27] A. Fuster, C. Pabst, and C. Pfeifer, Berwald spacetimes and very special relativity,
Phys. Rev. D98, 084062 (2018),arXiv:1804.09727 [gr-qc].
[28] S. Heefer, C. Pfeifer, and A. Fuster, Randers pp-waves, Phys. Rev. D 104, 024007 (2021),
arXiv:2011.12969.
[29] S. Heefer and A. Fuster, Finsler gravitational waves of (α, β)-type and their observational signature,
Class. Quantum Gravity 40, 184002 (2023),arXiv:2302.08334.
[30] S. Heefer, C. Pfeifer, A. Reggio, and A. Fuster, A cosmological unicorn solution to finsler gravity,
Phys. Rev. D 108, 064051 (2023).
[31] S. Heefer, Finsler Geometry, Spacetime & Gravity – From Metrizability of Berwald Spaces to Exact
Vacuum Solutions in Finsler Gravity, Ph.D. thesis (2024), arXiv:2404.09858 [gr-qc].
[32] L. Berwald, Untersuchung der Kr¨ummung allgemeiner metrischer R¨aume auf Grund des in ihnen
herrschenden Parallelismus, Mathematische Zeitschrift 25, 40 (1926).
[33] Z. Szab´o, Positive definite Berwald spaces, Tensor, New Series 35, 25 (1981).
[34] M. Crampin, On the construction of riemannian metrics for berwald spaces by averaging, Houston Jour.
Math. 40, 737 (2014).
[35] M. Hohmann, C. Pfeifer, and N. Voicu, Cosmological Finsler Spacetimes, Universe 6, 65 (2020),
arXiv:2003.02299 [gr-qc].
[36] A. 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].
[37] C. Pfeifer, S. Heefer, and A. Fuster, Identifying Berwald Finsler geometries,
Differ. Geom. Appl. 79, 101817 (2021),arXiv:1909.05284 [math.DG].
[38] S. Cheraghchi, C. Pfeifer, and N. Voicu, Four-dimensional SO(3)-spherically symmetric Berwald Finsler
spaces, Int. J. Geom. Meth. Mod. Phys. 20, 2350190 (2023),arXiv:2212.08603 [math.DG].
[39] N. Voicu, C. Pfeifer, and S. Cheraghchi, Birkhoff theorem for Berwald-Finsler spacetimes,
Phys. Rev. D 108, 104060 (2023),arXiv:2306.07866 [math.DG].
[40] S. Heefer, C. Pfeifer, J. van Voorthuizen, and A. Fuster, On the metrizability of m-Kropina spaces
with closed null one-form, J. Math. Phys. 64, 022502 (2023),arXiv:2210.02718.
[41] S. Heefer, Berwald m-Kropina spaces of arbitrary signature: Metrizability and Ricci-flatness (2024),
arXiv:2407.00094 [math.DG].
[42] S. Elgendi, Solutions for the Landsberg unicorn problem in Finsler geometry,
Journal of Geometry and Physics 159, 103918 (2021).
[43] S. G. Elgendi and L. Kozma, (α, β)-metrics satisfying the T-condition or the σT -condition,
The Journal of Geometric Analysis 31, 7866 (2021).
[44] S. Elgendi, On the problem of non-Berwaldian Landsberg spaces,
Bulletin of the Australian Mathematical Society 102, 331 (2020).
[45] D. Bao, On two curvature-driven problems in Riemann-Finsler geometry, in Finsler Geometry, Sapporo
27
2005: In Memory of Makoto Matsumoto, Adv. Stud. Pure Math., Vol. 48 (Mathematical Society of
Japan, 2007) pp. 19–71.
[46] M. Hohmann, C. Pfeifer, and N. Voicu, Mathematical foundations for field theories on Finsler space-
times, J. Math. Phys. 63, 032503 (2022),arXiv:2106.14965 [math-ph].
[47] M. Hohmann, C. Pfeifer, and N. Voicu, Relativistic kinetic gases as direct sources of gravity, Phys.
Rev. D101, 024062,10.1103/PhysRevD.101.024062 (2020).
[48] J. K. Beem, Indefinite Finsler spaces and timelike spaces, Can. J. Math. 22, 1035 (1970).
[49] A. Bernal, M. A. Javaloyes, and M. S´anchez, Foundations of Finsler Spacetimes from the Observers’
Viewpoint, Universe 6, 55 (2020),arXiv:2003.00455 [gr-qc].
[50] C. Lammerzahl, V. Perlick, and W. Hasse, Observable effects in a class of spherically symmetric static
Finsler spacetimes, Phys. Rev. D86, 104042 (2012),arXiv:1208.0619 [gr-qc].
[51] W. Hasse and V. Perlick, Redshift in Finsler spacetimes, Phys. Rev. D100, 024033 (2019),
arXiv:1904.08521 [gr-qc].
[52] E. Caponio and A. Masiello, On the analyticity of static solutions of a field equation in Finsler gravity,
Universe 6, 59 (2020),arXiv:2004.10613 [math.DG].
[53] E. Caponio and G. Stancarone, Standard static Finsler spacetimes,
Int. J. Geom. Meth. Mod. Phys. 13, 1650040 (2016),arXiv:1506.07451 [math.DG].
[54] G. Asanov, Finsleroid-Finsler spaces of positive-definite and relativistic types,
Rep. Math. Phys. 58, 275 (2006).
[55] Z. Shen, On a class of Landsberg metrics in Finsler geometry,
Canadian Journal of Mathematics 61, 1357–1374 (2009).
[56] S. Weinberg, Gravitation and Cosmology (John Wiley and Sons, New York, 1972).
[57] C. Pfeifer and M. N. R. Wohlfarth, Finsler geometric extension of Einstein gravity, Phys.Rev. D85,
064009 (2012), arXiv:1112.5641 [gr-qc].
[58] G. Asanov, Finsleroid-finsler spaces of positive-definite and relativistic types,
Reports on Mathematical Physics 58, 275 (2006).
[59] N. Voicu, A. Friedl-Sz´asz, E. Popovici-Popescu, and C. Pfeifer, The Finsler Spacetime Condition for
(α,β)-Metrics and Their Isometries, Universe 9, 198 (2023),arXiv:2302.09937 [math.DG].
