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arXiv:2306.00722v1 [gr-qc] 1 Jun 2023
A Cosmological Unicorn Solution to Finsler Gravity
Sjors Heefer∗
Department of Mathematics and Computer Science,
Eindhoven University of Technology, Eindhoven 5600MB, The Netherlands
Christian Pfeifer†
ZARM, University of Bremen, 28359 Bremen, Germany
Antonio Reggio‡and Andrea Fuster§
Department of Mathematics and Computer Science,
Eindhoven University of Technology, Eindhoven 5600MB, The Netherlands
(Dated: June 2, 2023)
We present a new family of exact vacuum solutions to Pfeifer and Wohlfarth’s field equation in
Finsler gravity, consisting of Finsler metrics that are Landsbergian but not Berwaldian, also known
as unicorns due to their rarity. Interestingly we find that these solutions have a physically viable
light cone structure, even though in some cases the signature is not Lorentzian but positive definite.
We furthermore find a promising analogy between our solutions and classical FLRW cosmology.
One of our solutions in particular has cosmological symmetry, i.e. it is spatially homogeneous and
isotropic, and it is additionally conformally flat, with conformal factor depending only on the timelike
coordinate. We show that this conformal factor can be interpreted as the scale factor, we compute
it as a function of cosmological time, and we show that it corresponds to a linearly expanding (or
contracting) Finsler universe.
I. INTRODUCTION
The interest in Finsler-geometric modified theories of
gravity has picked up in recent years, and rightly so. It
has become clear that small, natural modifications in the
classic axiomatic approach by Ehlers, Pirani and Schild
(EPS) [1] naturally lead to Finsler spacetime geometry;
[2–4]; it has also become clear that modified dispersion
relations (MDRs), usually discussed in the context of
quantum gravity phenomenology [5] generically induce
a Finsler geometry on spacetime [6–9] and it has been
conjectured that Finsler spacetime geometry describes
the gravitational field of a kinetic gas more accurately
compared to its usual treatment in the Einstein-Vlasov
system, [10,11]. These results, to name just a few,
show clearly that in certain situations, for instance in
Planckian regimes or for certain type of matter, it is
to be expected that Finsler geometry should be the
proper way to model spacetime. Beside the applications
in gravitational physics, see also [12], there are several
other instances in physics, such as the description of the
propagation of waves in media, where Finsler geometry
seems to be the appropriate tool [13].
Various physically subclasses of Finsler spacetimes
can be distinguished, and two particularly important
ones are the class of Berwald spacetimes and the class
of Landsberg spacetimes that may be thought of as
∗s.j.heefer@tue.nl
†christian.pfeifer@zarm.uni-bremen.de
‡antonio.reggio1@gmail.com
§a.fuster@tue.nl
incrementally non-(pseudo-)Riemannian, respectively
(precise definitions will follow later). Every Berwald
spacetime is also Landsberg, but whether or not the
opposite is true has been a long standing open question
in Finsler geometry. In fact, Matsumoto has stated in
2003 that this question represents the next frontier of
Finsler geometry [14], and as a token of their elusivity,
Bao [14] has called these non-Berwaldian Landsberg
spaces ‘[. . . ] unicorns, by analogy with those mythical
single-horned horse-like creatures for which no confirmed
sighting is available.’ Since 2006 some examples of
unicorns have been obtained by Asanov [15], Shen [16]
and Elgendi [17] by relaxing the definition of a Finsler
space. Even such examples of so-called y-local unicorns
are still exceedingly rare.
Here we present a new family of exact solutions to
Pfeifer and Wohlfarth’s Finslerian extension of Einstein’s
field equations [18,19] which is precisely such a unicorn.
It falls into one of the classes introduced by Elgendi.
Our solutions extend the very short list of known exact
solutions in Finsler gravity. Indeed, to the best of
our knowledge the only ones currently known in the
literature are the (m-Kropina type) Finsler pp-waves
[20] and their generalization as Very General Relativity
(VGR) spacetimes [21], the Randers pp-waves [22], and
the pp-waves of general (α, β)-metric type [23].
Interestingly we find that these solutions have a phys-
ically viable light cone structure, even though in some
cases the signature is not Lorentzian but positive defi-
nite. In fact, the Finslerian light cone turns out to be
equivalent to that of the flat Minkowski metric. Further-
more we find a natural cosmological interpretation of one
2
of our solutions and a promising analogy with classical
FLRW cosmology. In particular, our solution has cos-
mological symmetry, i.e. it is spatially homogeneous and
isotropic, and it is additionally conformally flat, with con-
formal factor depending only on the timelike coordinate.
We show that this conformal factor can be interpreted as
the scale factor, we compute the scale factor as a function
of cosmological time, and we show that it corresponds to
a linearly expanding (or contracting) Finslerian universe.
II. FINSLER GEOMETRY
Before we recall the basic notions of Finsler geom-
etry below we first introduce some notation. Given
a (spacetime) manifold M, which we assume to be
4-dimensional, and given some coordinates on M, we
will always consider the natural induced coordinates
on its tangent bundle T M . More precisely, given a
coordinate chart (U, x)on M,U⊂M, we obtain a
coordinate chart (T U, (x, y)) on T M ,T U ⊂T M of T M,
where a point Y=yµ∂µ∈T M is labeled by (x, y).
By a slight abuse of notation we will generally identify
any point in Mwith its expression in coordinates, and
similarly for points in T M , i.e. Y=y. We denote the
coordinate basis vectors of the tangent spaces of T M by
∂µ=∂/∂ xµand ¯
∂µ=∂/∂ yµ, where µ= 0, ..., 3.
A Finsler space is a smooth manifold Mendowed with
a Finsler metric, i.e. a smooth map F:T M \0→Rsuch
that
•Fis (positively) homogeneous of degree one with
respect to y:
F(x, λy) = λF (x, y),∀λ > 0 ; (1)
•the fundamental tensor
gµν =¯
∂µ¯
∂ν1
2F2(2)
is non-degenerate.
The fundamental tensor gµν depends generally on
both xand y. When Fis quadratic in y, or equiv-
alently when gµν depends only on x, then gµν is a
(pseudo-)Riemannian metric and the theory reduces to
(pseudo-)Riemannian geometry.
In order to describe spacetime geometry, one usually
demands that the signature of gµν be Lorentzian, at least
in some conic open subset of T M , which one might hope
to identify with the cone of timelike directions. More-
over, in applications one very often encounters Finsler
structures1that are only properly defined (smooth, non-
1In the literature one finds various other, more stringent, defini-
tions of Finsler spacetimes, going back to the original definition
by Beem [24]. They vary in their precise technical details, de-
pending on the scope of the application, see e.g. [25–27] and
references therein.
degenerate) on a subset of T M \0. Such Finsler metrics
are sometimes referred to as y-local, as opposed to y-
global [14]. In particular, the unicorn solution that we
will present here is of y-local type.
A. The nonlinear connection and geodesic spray
The Cartan non-linear connection is the unique homo-
geneous (in general non-linear) Ehresmann connection on
T M that is smooth on T M \ {0}, torsion-free and com-
patible with F. It may therefore be viewed as a gener-
alization of the Levi-Civita connection. For details we
refer e.g. to [28]. Its connection coefficients are given by
Nµ
ν=1
4¯
∂νgµρyσ∂σ¯
∂ρF2−∂ρF2.(3)
The nonlinear connection induces the horizontal deriva-
tives
δµ=∂µ−Nν
µ¯
∂ν,(4)
that, together with the ¯
∂µ, span each tangent space
T(x, ˙x)T M . The (geodesic) spray coefficients can then be
defined as
Gµ≡Nµ
νyν=1
2gµρ yσ∂σ¯
∂ρL−∂ρL,(5)
where the second equality follows from Euler’s theorem
for homogeneous functions. It immediately follows that
we also have Nµ
ν=1
2¯
∂νGµ. The importance of the spray
coefficients comes from the fact that the geodesics of F
are given by
¨xµ+Gµ(x, ˙x) = 0,(6)
which coincidentally is also the autoparallel equation of
the nonlinear connection Nµ
ν.
The curvature of the nonlinear connection is defined
via Rρµν ¯
∂ρ=−[δµ, δν], which implies that
Rρµν =δµNρ
ν−δνNρ
µ.(7)
From the nonlinear curvature one may define the Finsler
Ricci scalar and Ricci tensor as follows
Ric =Rρρµyµ, Rµν =1
2¯
∂µ¯
∂νRic.(8)
A Finsler space is said to be Ricci-flat if Ric = 0, or
equivalently, Rµν = 0. We remark that the Finsler Ricci
scalar is not to be confused with the scalar curvature
usually defined in (pseudo-)Riemannian geometry as R=
gµν Rµν , also sometimes called the Ricci scalar.
B. The Chern-Rund connection
In addition to the canonical nonlinear connection, vari-
ous canonical linear connections can be introduced. How-
ever, the price one has to pay for linearity is that the
3
linear connections do not in general live on the vector
bundle T M but rather on its pull-back π∗T M by the
canonical projection π:T M →M. The pull-back bun-
dle π∗T M is considered as a vector bundel over T M \0
and sections of this vector bundle may be thought of sim-
ply as vector fields on Mwith a dependence on both x
and y6= 0. Since the manifold T M \0has dimension 2n,
we get in general two sets of linear connection coefficients,
namely
∇δµ∂ν= Γρ
νµ ∂ρ,∇¯
∂µ∂ν=¯
Γρ
νµ ∂ρ.(9)
The Chern-Rund connection is the unique linear connec-
tion on π∗T M that is torsion-free and almost metric com-
patible. For details we refer e.g. to [29]. These conditions
imply that ¯
Γρ
µν = 0 and
Γρ
µν =1
2gρσ (δµgσν +δνgµσ −δσgµν ).(10)
Notice the similarity to the formula for the Levi-Civita
Christoffel symbols of a (pseudo-)Riemannian metric.
From this it is immediately clear that the Chern-Rund
connection reduces to the Levi-Civita connection when
Fis (pseudo-)Riemannian.
C. Berwald and Landsberg spaces
Next we introduce two important classes of Finsler
spaces: Berwald spaces and Landsberg spaces. First, if
the spray is quadratic in y, i.e. ¯
∂µ¯
∂ν¯
∂σGρ= 0 then Fis
said to be of Berwald type. What this means geometri-
cally is that the Chern connection may be understood
as an affine connection on M, i.e. equivalently, a space
is Berwald if and only if the connection coefficients Γρ
µν
of the Chern connection depend only on x.
And second, introducing the Landsberg curvature
Sµνσ =−1
4yρ¯
∂µ¯
∂ν¯
∂σGρ,(11)
and the mean Landsberg curvature Sσ=gµν Sσµν , we
say that a space is (weakly) Landsberg if the (mean)
Landsberg curvature vanishes identically. The geomet-
rical significance of the Landsberg tensor is somewhat
more difficult to state in simple terms without introduc-
ing more machinery, so instead we refer e.g. to [14].
It is immediately obvious from the definitions that any
Berwald space is a Landsberg space. Also, a (pseudo-
)Riemannian space is always Berwald, hence in particular
any (pseudo-)Riemannian space is Landsberg.
D. Unicorns in Finsler geometry
As observed above, we have the following inclusions:
(pseudo-)Riemannian ⊂Berwald ⊂Landsberg.
It has been a long standing open question whether the
last inclusion is strict. Do there exist Landsberg space
that are not Berwald? In the y-global case the answer is
unknown. For y-local spaces some examples are known,
but these are exceedingly rare. As such, non-Berwaldian
Landsberg spaces are referred to as unicorns [14]. We
recommend [14,30] for reviews on the unicorn problem.
The first unicorns were found by Asanov [15] in 2006
and his results were generalized by Shen [16] a few years
later. These were the only known examples of unicorns
until Elgendi very recently provided some additional ex-
amples of unicorns [17]. One of the families of unicorns
introduced by Elgendi will be central in this work.
III. THE FINSLERIAN FIELD EQUATIONS
Although various proposals for Finslerian field equa-
tions in vacuum can be found in the literature [18,31–40],
it seems fair to say that Pfeifer and Wohlfarth’s field
equation [18] has the most robust foundation. It is
obtained as the Euler-Lagrange equation of the natural
Finsler generalization of the Einstein Hilbert action
[18,19], and furthermore it has been shown recently
that the equation is the variational completion of Rutz’s
equation [31], Ric = 0. The latter is arguably the
simplest and cleanest proposal, and well physically
motivated, but it cannot be obtained by extremizing
an action functional, complicating the coupling of the
theory to matter. For reference, Einstein’s vacuum
equation in the form Rµν −1
2gµν R= 0 is also precisely
the variational completion of the simpler equation
Rµν = 0 [41]. While in the GR case the completed
equation happens to be equivalent to the former, this is
not true any longer in the Finsler setting.
Pfeifer and Wohlfarth’s field equation in vacuum reads
Ric −L
3gµν Rµν
−L
3gµν ¯
∂µ˙
Sν−SµSν+∇δµSν= 0,(12)
where ˙
Sν≡yρ∇δρSν. For (pseudo-)Riemannian metrics,
(12) reduces to Einstein’s field equation in vacuum. From
the general expression (12) it becomes immediately ap-
parent that for weakly Landsberg spaces, characterized
by the defining property that Si= 0, the field equation
in vacuum attains the relatively simple form
Ric −L
3gµν Rµν = 0.(13)
Recalling the definition (8) of Rµν we have the following
immediate result:
Proposition 1. Any Ricci-flat, Ric = 0, weakly Lands-
berg space is a solution to the field equations (12)
In other words, any weakly Landsberg solution to the
Rutz equation is automatically a solution to (12).
4
IV. AN EXACT UNICORN SOLUTION TO
FINSLER GRAVITY
A. Elgendi’s class of unicorns
Elgendi recently introduced a class of unicorns [17]
with Finsler metric given by
F=aβ +pα2−β2e
aβ
aβ+√α2−β2,(14)
in terms of a real, nonvanishing constant aand
α=f(x0)p(y0)2+φ(ˆy), β =f(x0)y0,(15)
where fis a real-valued function and φ( ˆy) = φij ˆyiˆyj=
φij yiyjis a non-degenerate quadratic form on the space
spanned by ˆy= (y1, y2, y3), with constant, symmetric
coefficients φij . Here and in what follows, indices i, j, . . .
will run over 1,2,3, whereas greek induces µ,ν,... will
run over 0,1,2,3. From a gravitational physics perspec-
tive, the only degree of freedom of these Finsler functions
is the function f(x0). The geodesic spray of Fis given
by
G0=2f(x0)2(y0)2−α2
f(x0)2+a2−1
a2
α2−β2
f(x0)2f′(x0)
f(x0)
(16)
Gi=P yi,(17)
where
P= 2 y0+1
af(x0)pα2−β2f′(x0)
f(x0),(18)
and the Landsberg tensor vanishes identically. Note that
our Gkis twice the Gkin Elgendi’s paper [17], due to a
difference in convention. Explicitly then, Elgendi’s uni-
corns have the form
F=f(x0)y0+pφ(ˆy)e
y0
y0+√φ(ˆy).(19)
where we have absorbed the constant ainto a redefinition
of x0. For our purposes we will modify this expression
slightly, though.
B. The modified unicorn metric
The expression (19) defining the unicorn metric is only
well-defined whenever φ(ˆy)≥0. If φis positive definite,
this is necessarily the case, but in other signatures this
is not always true. In order to extend the domain of
definition of F, an obvious first approach would naturally
be to replace φby its absolute value, |φ|, i.e.
F=f(x0)y0+p|φ(ˆy)|e
y0
y0+√|φ(ˆy)|.(20)
From the physical point of view this is still not com-
pletely satisfactory, however. This can be seen by
considering the light cone corresponding to such a
Finsler metric F, given by the set of vectors for which
F= 0. Indeed, the light cone would be given by those
vectors satisfying y0=−p|φ|, which would imply that
the light cone is contained entirely within the half space
y0<0, which does not seem very realistic. To obtain
a viable light cone structure we consider a modified
unicorn metric, inspired by the construction of modified
Randers metrics in [23].
Thus, our starting point will be the following modified
unicorn metric:
F=f(x0)|y0|+sgn(φ)p|φ|e
|y0|
|y0|+sgn(φ)√|φ|.(21)
Below we’ll show that such a metric is still of the uni-
corn type, and indeed defines a physically reasonable cone
structure, under some conditions on φ. After that, we
will determine the signature of the fundamental tensor,
which will turn out to depend on the signature of φ, and
finally, in section IV C we determine the free function
f(x0)in (21) by application of the Finsler gravity equa-
tion. Afterwards we discuss the physical interpretation
and conclude.
1. Cone structure
First we observe that, regardless of the exact form or
signature of φ, our modified unicorn metrics have a light
cone structure that is equivalent to that of a pseudo-
Riemannian metric.
Proposition 2. The light cone of the modified unicorn
metric (21)is given by
y02+φ= 0.(22)
Proof. The result follows from the following sequence of
equivalences.
F= 0 ⇔sgn(φ)p|φ|+|y0|= 0 (23)
⇔sgn(φ)p|φ|=−|y0|(24)
⇔ |φ|=y02and φ < 0(25)
⇔φ=−y02(26)
⇔φ+y02= 0.(27)
Depending on the signature of φ, we can make a more
precise statement.
Proposition 3. Let Fbe the modified unicorn met-
ric (21)corresponding to some some non-degenerate
quadratic form φij. Then the following holds:
5
•For φij of signature (+,+,−)the null structure of
Fis identical to the Minkowski metric spacetime
lightcone structure of signature (+,+,+,−).
•If φij is negative definite, i.e. of signature
(−,−,−)then the light cone of Fis identical to the
Minkowski metric spacetime lightcone of signature
(+,−,−,−).
•If φij is positive definite, i.e. of signature (+,+,+)
then the light cone of Fis given by yµ= 0.
•For φij of signature (−,−, x)the null structure of
Fis identical to the one of a pseudo-Riemannian
metric manifold with signature (+,−,−,+).
This singles out the (+,+,−)and (−,−,−)signatures
of φij as the ones that are physically reasonable. In the
first case it leads to the interpretation of the coordinate
x3as timelike coordinate, while in the second case x0
would be the timelike coordinate.
2. Signature of the fundamental tensor
Next we investigate the signature of our modified uni-
corn metrics.
Proposition 4. Consider a modified unicorn metric F
as in (21)and let S(φ)be the set of all ˆy= (y1, y2, y 3)
that are φ-spacelike and T(φ)the set of all ˆythat are
φ-timelike.
•If φis positive definite or negative definite then
gµν (x, y)is positive definite on its entire domain
of definition.
•If φis Lorentzian then gµν (x, y)is of Lorentzian
signature (+,+,+,−)for all y∈R× S(φ)and
gµν (x, y)is of signature (+,−,−,+) for all y∈
R× T (φ).
Before we give the proof, let us make some interesting
observations. Consider the two physically reasonable
scenario’s we identified below Prop. 3as a result of
their good cone structure, i.e. φhaving (+,+,−)or
(−,−,−)signature. In both cases the light cone is
equivalent to that of Minkowski space. Surprisingly,
inside the interior of this cone it is easy to see from
the previous proposition, that the signature of gis
not Lorentzian. Indeed, if φhas signature (+,+,−),
then ghas signature (+,−,−,+) inside the cone,
while if φis negative definite then gis positive defi-
nite inside the cone. In the later case this does not
contradict the existence of the lightcone structure due
to the reduced smoothness of the Finsler function F(21).
In many definitions of Finsler spacetimes [25,27], one
requires that there exists a cone inside which the funda-
mental tensor has Lorentzian signature, in order to guar-
antee the existence of a physical cone structure. Here,
however, we found that there exist Finsler geometries,
which do have a satisfactory cone structure, without this
property. This is an interesting new observation about
Finsler geometry in its own right: apparently even in pos-
itive definite signature, a cone structure may arise due to
irregularities (i.e. non-smoothness) of the Finsler met-
ric. It is therefore not immediately obvious whether the
positive definite signature poses a fundamental problem
in the application of the unicorn metrics in spacetime
physics or not. In the context of standard static Finsler
spacetimes, a lack of smoothness of the Finsler function
inside the cone was also considered acceptable [26,42].
Proof of Prop. 4.We may choose coordinates such that
φ=φ(ˆy) = ε1(y1)2+ε2(y2)2+ε3(y3)2. Since the space-
time dimension is fixed to 4, wherever Fis sufficiently
differentiable the calculation of the determinant of the
fundamental tensor is in principle a straightforward ex-
ercise. It is given by
det g=sgn(φ(ˆy))ε1ε2ε3f(x0)8exp 8|y0|
|y0|+sgn(φ)√|φ|.
(28)
The determinant already gives us a pretty good idea of
what the possible signature of gµν can be. In particular,
since gµν is a four-dimensional matrix, it has Lorentzian
signature, either of type (+,−,−,−)or (−,+,+,+), if
and only if its determinant is negative.
a) Assume φij is positive definite, then all ǫiand
sgn(φ(ˆy)) are positive, and hence det gis.
b) Assume φij is negative definite, then all ǫiand
sgn(φ(ˆy)) are negative, and hence det gis positive.
c) Assume φij is of Lorentzian signature (+,+,−), then
detg is negative whenever sgn(φ(ˆy)) >0, i.e. on S(φ),
and detg is positive whenever sgn(φ(ˆy)) <0, i.e. on
T(φ).
d) Assume φij is of Lorentzian signature (+,+,−), then
detg is negative whenever sgn(φ(ˆy)) <0, i.e. on S(φ),
and detg is positive whenever sgn(φ(ˆy)) >0, i.e. on
T(φ).
This already shows that gµν is Lorentzian if and only
if φis Lorentzian and y∈R× S(φ). But the sign of the
determinant does not suffice to determine whether this
signature is mostly plus or mostly minus. Similarly it
does not tell us much about the signature of gµν when φ
is positive or negative definite. In order to found, we’ll
distinguish the following cases.
Case 1: φLorentzian and y∈R× S(φ)
We first consider the case that φis Lorentzian.
Without loss of generality (wlog) we set φ(ˆy) =
ǫ(y1)2+ǫ(y2)2−ǫ(y3)2, where ǫ=±1. The choice of
the sign ǫselects if we are in case c) or d) from above.
6
Now note that given a vector y∈TxMwhich is
φ-spacelike, it follows from the symmetries of the
Finsler metric and in particular from the 3-dimensional
Lorentz symmetry of φthat for we may always change
coordinates, without changing the form of φ(and F),
such that y2=y3= 0.
For any choice of epsilon, by direct calculation we find,
using that ǫ2= 1 and |ǫ|= 1, that gµν is of the form
gµν =e2|y0|
|y0|+ε|y1|f(x0)2
M0 0
0 1 0
0 0 −1
,(29)
where Mis an (ε-dependent) positive definite 2×2
matrix2. Hence we conclude that gµν is of the mostly
plus type (+,+,+,−).
Case 2: φLorentzian and y∈R× T (φ)
In this case we may wlog choose coordinates such that
φ(ˆy) = −ǫ(y1)2+ǫ(y2)2+ǫ(y3)2, where ǫ=±1, and such
that y2=y3= 0. Again by direct calculation we find
that gµν is of the form
gµν =e2|y0|
|y0|+ε|y1|f(x0)2
M0 0
0−1 0
0 0 −1
,(30)
where Mis a positive definite 2×2matrix. Hence we
conclude that gµν is in this case of signature (+,−,−,+).
Case 3: φpositive or negative definite
In this case we may wlog choose coordinates such that
φ(ˆy) = ǫ(y1)2+ǫ(y2)2+ǫ(y3)2, where ǫ=±1, and such
that, for any given y∈TxM, we have y2=y3= 0. Again
by direct calculation we find that gµν is of the form
gµν =e2|y0|
|y0|+ε|y1|f(x0)2
M0 0
0 1 0
0 0 1
,(31)
where Mis a positive definite 2×2matrix. Hence we
conclude that gµν is positive definite.
C. Solving the Finsler gravity equation
Next, we seek to determine the form of the free function
f(x0)of (21) from the Finsler gravity equations (13).
2Mmust be either positive definite or negative definite, since we
have already shown that gij has Lorentzian signature. The fact
that Mabvavb= ((y0)2+(y1)2)/(|y0|+ε|y1|)2>0for va= (0,1)
thus shows that Mis positive definite.
The geodesic spray of Fis given explicitly by
G0=(y0)2− |φ|f′(x0)
f(x0)(32)
Gi=P yi, i = 1,2,3(33)
where
P= 2 |y0|+sgn(φ)p|φ|sgn(y0)f′(x0)
f(x0)(34)
Proposition 5. Fis Ricci-flat if and only if fhas the
form f(x0) = c1exp c2x0.
Proof. By definition, and using homogeneity and the fact
that Nµ
ν=1
2¯
∂νGµ, we have
Ric =Rµµν yν= (δµNµ
ν−δνNµ
µ)yν(35)
=yν((∂µ−Nρ
µ¯
∂ρ)Nµ
ν−(∂ν−Nρ
ν¯
∂ρ)Nµ
µ)(36)
=yν∂µNµ
ν−yν∂νNµ
µ(37)
−yνNρ
µ¯
∂ρNµ
ν+yνNρ
ν¯
∂ρNµ
µ(38)
=1
2yν∂µ¯
∂νGµ−yν∂ν¯
∂µGµ(39)
−1
4yν¯
∂µGρ¯
∂ρ¯
∂νGµ−yν¯
∂νGρ¯
∂ρ¯
∂µGµ(40)
=∂µGµ−1
2yν∂ν¯
∂µGµ(41)
−1
4¯
∂µGρ¯
∂ρGµ−2Gρ¯
∂ρ¯
∂µGµ.(42)
Using the identities
¯
∂0P= 2f′/f, ¯
∂0G0= 2y0f′/f,
¯
∂2
0G0= 2f′/f, ¯
∂0¯
∂iG0=¯
∂0¯
∂iP=¯
∂2
0P= 0,
yi¯
∂iG0=−2|φ|f′/f, yi¯
∂iP= 2sgn(φy0)p|φ|f′/f,
(43)
one finds after some slightly tedious manipulations that
the last two terms in the expression for the Ricci tensor
can both be expressed as
¯
∂µGρ¯
∂ρGµ= 2Gρ¯
∂ρ¯
∂µGµ=nP 2,(44)
where n= dim M= 4 in our case. Hence these terms
cancel each other out precisely. Denoting G0=¯
G0f′/f
and P=¯
P f ′/f, so that ¯
G0and ¯
Pdo not depend on xµ,
and using the fact that ¯
∂µGµ=nP , one finds further-
more that
∂µGµ= (∂2
0log |f|)¯
G0,(45)
yν∂ν¯
∂µGµ=ny0(∂2
0log |f|)¯
P . (46)
Consequently we have
Ric =∂µGµ−1
2yν∂ν¯
∂µGµ(47)
= +(∂2
0log |f|)(48)
×(1 −n)(y0)2−n|y0|sgn(φ)p|φ| − |φ|,(49)
7
which in dimension n= 4 reduces to
Ric =−(∂2
0log |f|)(50)
×3(y0)2+ 4|y0|sgn(φ)p|φ|+|φ|.(51)
If Ric = 0 then we must in particular have
0 = ¯
∂2
0Ric =−6(∂2
0log |f|).(52)
It thus follows that Ric = 0 if and only if ∂2
0log |f|= 0,
the general solution to which is given by the stated form
of f.
This shows that the following family of Finsler metrics
are exact vacuum solutions to Pfeifer and Wohlfarth’s
field equations in Finsler gravity:
F=c1ec2x0|y0|+sgn(φ)p|φ|exp |y0|
|y0|+sgn(φ)p|φ|!,
(53)
where φ=φ(ˆy) = φij yiyj, with φij being a three-
dimensional non-degenerate, symmetric bilinear form
with constant coefficients and signature (+,+,−)or
(−,−,−).
In fact it turns out that any solution of the type (21)
must have this form.
Proposition 6. A Finsler metric of the form (21)is a
solution to the Finslerian field equations in vacuum, Eq.
(13), if and only if it can be written locally as (53).
Before we give a sketch of the proof, it is important to
point out the exact meaning of the word ‘locally’ in the
proposition, as it will have essential consequences for the
physical viability of such solutions. Of course, the word
applies first and foremost to the x-coordinates in the
usual sense, but it also applies to the tangent space coor-
dinates yi: there is a priori no reason why one couldn’t
pick, at some point xµ, say, a certain φij in a certain
subset of TxMand a different φij in a different subset of
TxM. Of course this could in general result in not having
smoothness across the interface of the two regions, but
this does not necessarily have to be a problem, unless one
sets very strict smoothness requirements.
For consistency the different parts of T M must satisfy
some conditions. It is most natural to require that any
such part be a conic subbundle, i.e. an open conic subset
with non-empty fiber at each point in M.
Proof. In four spacetime dimensions the proof is straight-
forward and most easily performed in convenient coordi-
nates in which φis diagonal with all nonvanishing entries
equal to +1 or −1. From (21) on can directly compute
gµν and then its inverse gµν . From (51) together with (8)
one can immediately compute Rµν . We omit the inter-
mediate expressions because they are somewhat lengthy,
but plugging all of this into the field equation (13) leads
to
−∂2
0log |f|
3p|φ|−4sgn(φ)|y0|3−5y0p|φ|+|φ|3/2= 0.
(54)
Clearly this equation can only be satisfied for all yµin
an open set where the fundamental tensor has Lorentzian
signature if ∂2
0log |f|= 0, in which case (the last sentence
in the proof of ) Prop. 5shows that Fis in fact Ricci-flat
and therefore must have the form (53).
With this we found an exact solution to the Finsler
gravity equations (53), starting from a generalised ver-
sion of Elgendi’s unicorns (21).
D. Physical interpretation: A linearly expanding
universe
Having analyzed the mathematical properties of the
unicorn Finsler spacetimes (21), and found the exact uni-
corn vacuum solution (53) of the Finsler gravity vacuum
equations (13), we now turn to the physical interpreta-
tion of this solution. We find that these Finsler gravity
vacuum solutions yield a vacuum cosmology, with linear
time dependence of the scale factor.
To reach this conclusion we highlight the following
properties of the the unicorn Finsler spacetimes (21):
•Conformal flatness, with a conformal factor that is
only spacetime dependent
F(x, y) = f(x)F0(y).(55)
•Cosmological symmetry, for the case when φij has
signature (−,−,−), since then, by introducing spa-
tial spherical coordinates (r, θ, φ), we can write
F(x, y) = F(x0, y0, w),(56)
w2= (y1)2+ (y2)2+ (y3)2(57)
= (yr)2+r2(yθ)2+ sin2θ(yφ)2,(58)
which is precisely of the form of a spatial flat homo-
geneous and isotropic Finsler geometry [43]. This
construction does not work for φij with signature
(+,+,−), since then y3, and not y1would be the
timelike direction.
Combining these observations, we find that the unicorn
Finsler spacetimes (21) are of the form
F(x0, y0, w) = f(x0)F0(y0, w).(59)
This from reminds immediately at classical flat FLRW
spacetimes in conformal time η, which we identify here
8
with the x0coordinate. When written in the language of
Finsler geometry they are of the form (59) with
F0F LRW =p(−(y0)2+ (y1)2(y2)2+ (y3)2).(60)
A redefinition of the time coordinate via ∂˜x0
∂x0=f(x0),
which implies that ˜y0=f(x0)y0, then leads to the stan-
dard form of flat FLRW geometry
F=p(−(˜y0)2+f(˜x0)2((y1)2+ (y2)2+ (y3)2)) ,(61)
where the conformal factor is nothing but the usual cos-
mological scale factor and ˜x0is the usual cosmological
time t.
For the Finsler function (21) we employ the coordinate
change ∂˜x0
∂x0=f(x0), implying that ˜y0=f(x0)y0, so that
F=|˜y0|+sgn(φ)f(˜x0)p|φ|e
˜y0
˜y0+s gn(φ)f(˜x0)√|φ|.(62)
Hence, as in the classical FLRW geometry case, the con-
formal factor can be interpreted as scale factor of the
spatial universe, x0as conformal time ηand ˜x0as cos-
mological coordinate time t. For now we will adopt this
classical cosmology notation.
To be a solution of the Finsler gravity equations we
found that f(η) = c1eηc2, which implies from the coordi-
nate change between ηand tthat
dt =c1eηc2dη ⇔η(t) = 1
c2
ln c2
c1
(t−c3),(63)
where c3is a constant of integration. Thus, in cosmologi-
cal time, the scale factor of the vacuum Finsler cosmology
we find is
f(t) = c2(t−c3).(64)
Interestingly, it turns out that these solutions are not
only Ricci-flat and conformally flat (by their explicit
form), but flat, in the sence that all components of the
non-linear curvature tensor Rabc =δbNa
c−δcNa
avanish.
Nevertheless the spacetime has non-trivial geometric fea-
tures.
V. DISCUSSION
The solutions that we have presented above are, to
the best of our knowledge, the first non-Berwaldian
exact solutions to Pfeifer and Wohlfarth’s field equation.
Known exact solutions are scarce since in particular
the Landsberg tensor terms in the field equations
are diffucult to understand. Employing a unicorn
Ansatz, i.e. non-Berwaldian Landsberg spaces, makes
our solutions particularly special. We have shown
that there is a subclass of our solutions for which the
lightcone structure is physically viable. In fact, it is
equivalent to the lightcone of the flat special relativistic
Minkowski metric. Moreover, we have shown that one
of the solutions has cosmological symmetry, i.e. it is
spatially homogeneous and isotropic. Additionally, it is
conformally flat, with conformal factor depending only
on the timelike coordinate, and we have shown that this
conformal factor can be interpreted as the scale factor,
which then turns out to be a linear function of cosmo-
logical time, leading to the natural interpretation of
a linearly expanding (or contracting) Finslerian universe.
As an additional curiosity that we have found that
the requirement of a physically light cone structure does
not strictly speaking necessitate Lorentzian signature,
as is widely assumed. This is illustrated by one of our
solutions, which has positive definite signature, and yet
has a light cone that is equivalent to the lightcone of
flat Minkowski space. It is interesting and suprising that
such things are apparently possible in Finsler geometry,
and this paper shows the first explicite example of a
(positive definite) Finsler metric with this property,
which seems to be closely related to lack of smoothness
of the Finsler metric in certain nontrivial subsets of T M .
The results obtained in this paper motivate us to begin
a systematic search for cosmological Landsberg space-
times that solve the field equations, using recent results
characterizing cosmological symmetry in Finsler space-
times [43] and Elgendi’s machinery for constructing uni-
corns using conformal transformations [17,44]. Since
(properly Finslerian) cosmological solutions of Berwald
type are necessarily static [43] any interesting such
Landsberg spacetime must necessarily be a unicorn.
The exciting next step in the study of Finsler gravity,
is to study unicorn solutions of the field equation sourced
by the 1-particle distribution function of a kinetic gas, in
homogeneous and isotropic symmetry. This scenario de-
scribes a realistic universe, filled with a kinetic gas with a
non-trivial velocity distribution. As we already obtained
a non-trivial solution for vacuum Finsler unicorn cosmol-
ogy, more realistic, matter sourced solutions will help us
to further investigate the conjecture that an accelerated
expansion of the universe is caused by the contribution of
the velocity distribution of the cosmological gas, which
sources a Finslerian spacetime geometry.
ACKNOWLEDGMENTS
C.P. was funded by the cluster of excellence Quan-
tum Frontiers funded by the Deutsche Forschungsgemein-
schaft (DFG, German Research Foundation) under Ger-
many’s Excellence Strategy - EXC-2123 QuantumFron-
tiers - 390837967. The authors would like to acknowledge
networking support by the COST Action CA18108.
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