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Mathematical foundations for field theories on Finsler spacetimes

Journal of Mathematical Physics

March 2022
63(3):032503

DOI:10.1063/5.0065944

Authors:

Nicoleta Voicu

Transylvania University of Brașov

Christian Pfeifer

University of Bremen

This paper introduces a general mathematical framework for action-based field theories on Finsler spacetimes. As most often fields on Finsler spacetime (e.g., the Finsler fundamental function or the resulting metric tensor) have a homogeneous dependence on the tangent directions of spacetime, we construct the appropriate configuration bundles whose sections are such homogeneous fields; on these configuration bundles, the tools of coordinate free calculus of variations can be consistently applied to obtain field equations. Moreover, we prove that the general covariance of natural Finsler field Lagrangians leads to an averaged energy–momentum conservation law that, in the particular case of Lorentzian spacetimes, is equivalent to the usual pointwise energy–momentum covariant conservation law.

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arXiv:2106.14965v2 [math-ph] 15 Mar 2022

Mathematical foundations for ﬁeld theories on Finsler spacetimes

Mathematical foundations for ﬁeld theories on Finsler

spacetimes

Manuel Hohmann,1, 2 Christian Pfeifer,3and Nicoleta Voicu4, 2

1)Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu,

Estonia

2)Lepage Research Institute, 17. novembra 1, 08116 Prešov, Slovakia

3)ZARM, University of Bremen, 28359 Bremen, Germany

4)Faculty of Mathematics and Computer Science, Transilvania University, Iuliu Maniu Str. 50,

500091 Brasov, Romania

(*Electronic mail: nico.voicu@unitbv.ro)

(*Electronic mail: christian.pfeifer@zarm.uni-bremen.de)

(*Electronic mail: manuel.hohmann@ut.ee)

The paper introduces a general mathematical framework for action based ﬁeld theories on Finsler

spacetimes. As most often ﬁelds on Finsler spacetime (e.g., the Finsler fundamental function or the

resulting metric tensor) have a homogeneous dependence on the tangent directions of spacetime,

we construct the appropriate conﬁguration bundles whose sections are such homogeneous ﬁelds; on

these conﬁguration bundles, the tools of coordinate free calculus of variations can be consistently

applied to obtain ﬁeld equations. Moreover, we prove that general covariance of natural Finsler

ﬁeld Lagrangians leads to an averaged energy-momentum conservation law which, in the particular

case of Lorentzian spacetimes, is equivalent to the usual, pointwise energy-momentum covariant

conservation law.

Keywords: Finsler spacetime, projectivized tangent bundle, ﬁbered manifold, Euler-Lagrange operator,

energy-momentum distribution tensor

MSC2020: 83D05,58A20,53B40

CONTENTS

I. Introduction 2

II. Pseudo-Finsler spaces and Finsler spacetime manifolds 4

A. The notion of Finsler spacetime 4

B. Geometric objects on Finsler spacetimes 7

C. Homogeneous geometric objects on TM 10

III. The positively projectivized tangent bundle PT M +13

A. Deﬁnition and structure over general manifolds 13

1. Deﬁnition and structure 13

2. From PT M+to T M and back 15

B. Over Finsler spacetimes (M,L)16

1. Finsler Geometry on PT M +and volume forms 16

2. Integration on PTM+and integration on observer space 19

IV. Fibered manifolds and ﬁelds over a Finsler spacetime 20

A. Fibered manifolds over PT M+20

Mathematical foundations for ﬁeld theories on Finsler spacetimes 2

B. Fibered automorphisms 22

C. Homogeneous geometric objects on TM as sections 24

V. Finsler ﬁeld Lagrangians, action, extremals 27

A. Actions for ﬁelds as sections of PT M+27

B. Finsler gravity sourced by a kinetic gas 29

1. Finsler gravity Lagrangian 29

2. Kinetic gas Lagrangian 30

VI. Energy-momentum distribution tensor 32

A. Generally covariant Lagrangians 32

B. The energy momentum distribution tensor and the energy momentum density 33

VII. Summary and Outlook 39

Acknowledgments 40

Data Availability Statement 40

A. Jet bundles and the coordinate-free calculus of variations 40

1. Fibered manifolds and their jet prolongation 40

2. Horizontal and contact forms 41

3. Lagrangians and ﬁrst variation formula 43

I. INTRODUCTION

(Pseudo-)Finsler geometry is the most general geometry admitting a parametrization-invariant arc length

of curves. It generalizes Riemannian geometry by using as its fundamental, geometry-deﬁning object, a

general line element - which does not necessarily arise as the square root of any quadratic expression in

the velocity components, but is just a homogeneous expression of degree one in these. Historically, already

Riemmann himself introduced this concept in his habilitation lecture1,2, however only Finsler investigated it

more deeply3. Nowadays Finsler geometry is an established ﬁeld in mathematics4,5.

In physics, pseudo-Riemannian geometry is used to describe one of the four fundamental interactions,

gravity. In general relativity, gravity is encoded in the Lorentzian geometry of the four-dimensional space-

time manifold, which is determined by the matter content of spacetime via the Einstein equations6. The idea

to use geometry based on non-quadratic line elements to describe physical interactions goes far back, at least

to Randers7, who used, in addition to a metric, a 1-form to search for a uniﬁed geometric description of grav-

ity and electromagnetism. Since then, numerous applications of Finsler geometry in physics emerged8,9, for

example in the geometric description of ﬁelds in media10–15, to study non-local Lorentz invariant extensions

of fundamental physics16–26, and to ﬁnd extensions and modiﬁcations of general relativity for an improved

description of gravity27–32 that might explain dark matter or dark energy geometrically33–38.

From the mathematical point of view, a major difﬁculty in the formulation of pseudo-Finsler geometry

as generalization of peudo-Riemannian geometry is the existence, in each tangent space, of vectors along

which the geometry deﬁning function is either non-smooth or leading to a degenerate metric tensor. One of

the ﬁrst attempts to construct mathematically well deﬁned Lorentz-Finsler spacetimes goes back to Beem39.

It turned out that Beem’s deﬁnition was to restrictive to cover all cases one is interested in physics and

numerous extensions and reﬁnements have been discussed8,28,40–44 .

Mathematical foundations for ﬁeld theories on Finsler spacetimes 3

On the basis of the improved modern Finsler spacetime deﬁnitions, it has recently been suggested that the

gravitational ﬁeld of kinetic gases can be described by Finsler spacetime geometry31.

Motivated by the coupling of the kinetic gas to Finsler spacetime geometry and by the recent example

of a Finslerian vacuum action,43, we introduce a general framework for mathematically consistent action-

based ﬁeld theories over Finsler spacetimes. The main difﬁculty for such a construction is that, most often,

ﬁelds on a Finsler spacetime (e.g., the fundamental function L, or the resulting metric tensor) have a non-

trivial homogeneous dependence on the tangent directions of spacetime. For such homogeneous geometric

objects, the naive formulation of the calculus of variations - on conﬁguration manifolds sitting over the

tangent bundle - is not well deﬁned, since the variation itself possesses the same homogeneity as the original

ﬁeld; thus, imposing the variation to vanish on the boundary of the arbitrary compact integration domain

would typically force the variation to identically vanish, also inside this domain. Therefore, in order to

correctly apply the apparatus of the calculus of variations, one must ﬁrst carefully choose the ﬁber bundles

serving as conﬁguration manifolds for the theory, in such a way as to naturally and consistently accommodate

homogeneity.

The main goal of this article is to provide a general construction of conﬁguration bundles, deﬁne ac-

tion integrals for homogeneous ﬁelds, in the just explained sense, over Finsler spacetimes and to apply the

coordinate free calculus of variations to obtain ﬁeld equations for the ﬁelds in a mathematically rigorous

way. Also, we prove that invariance of the corresponding ﬁeld Lagrangians under lifted spacetime diffeo-

morphisms leads to an averaged energy-momentum conservation law, which generalizes the (pointwise)

energy-momentum conservation law in general relativity.

To achieve this, we proceed as follows:

Section II reviews the notion of Finsler spacetimes,45 and the geometry of Finsler spacetimes, on which

our later construction is based. We also give a brief discussion on the different deﬁnitions, which can be

found in the literature. Most importantly, we discuss the homogeneity properties of the appearing geometric

objects.

Section III presents the positively projectivized tangent bundle PT M+(also called in the literature on

positive deﬁnite Finsler spaces, the projective sphere bundle,4), which will serve as base manifold for action

integrals for ﬁeld theories on Finsler spacetimes. We ﬁrst discuss in detail the general concept of PT M +

without any geometric ﬁelds on the manifold in consideration, and then, how Finsler geometry can be under-

stood on PT M+. Albeit this is still a preparatory topic, it is treated in quite some detail, since a systematic

analysis of PT M+and of the various structures it gives rise to, both on general manifolds and on pseudo-

Finsler spaces, seems to be missing in the literature.

Having set the stage, we use PT M+as base manifold for general physical ﬁelds having a homogeneous

dependence on the direction; these ﬁelds are modeled as sections into conﬁguration bundles over PTM+in

Section IV. We introduce the corresponding conﬁguration bundles and ﬁbered automorphisms thereof, that

serve in deforming sections and thus give rise to variations.

Eventually, Section Vcombines all the previous concepts to write down the general form of well deﬁned

action integrals for homogeneous ﬁelds on Finsler spacetime. Once this is done, their ﬁeld equations are

then obtained by the standard techniques of calculus of variations - discussed here in a coordinate-free form.

In Section VI, we derive the response of Lagrangians to compactly supported diffeomorphisms on the

spacetime manifold, which leads to the novel notion of an energy-momentum distribution tensor. It satisﬁes

an averaged covariant conservation law and can be integrated to an energy-momentum tensor density on

spacetime. Only in very special cases, in particular in the case of a Lorentzian spacetime geometry, this

energy-momentum tensor density can be "un-densitized" to yield an energy-momentum tensor on the base

manifold.

The necessary notions of geometric calculus of variations (jet bundles over ﬁbered manifolds, ﬁbered

Mathematical foundations for ﬁeld theories on Finsler spacetimes 4

automorphisms, the ﬁrst variation formula in terms of differential forms and their Lie derivatives) are brieﬂy

presented in Appendix A.

II. PSEUDO-FINSLER SPACES AND FINSLER SPACETIME MANIFOLDS

We begin this article by presenting a precise deﬁnition of Finsler spacetimes,45 on which we will base the

presentation and discussion of this article. We will comment on its relation to other deﬁnitions of Finsler

spacetimes given in the literature42,44,46,47, highlight the importance of the details in the deﬁnition which

ensure the existence of a well deﬁned causal structure and discuss some classes of examples. Moreover, we

brieﬂy review the geometric notions on Finsler spacetimes such as connections and curvature.

The notions presented in this section set the stage for the construction of action based ﬁeld theories on

Finsler spacetimes.

A. The notion of Finsler spacetime

Let Mbe a connected, orientable smooth manifold and T M , its tangent bundle with projection

TM :

T M →M. We will denote by xithe coordinates in a local chart on Mand by (xi,˙xi), the naturally induced

local coordinates of points (x,˙x)∈T M. Whenever there is no risk of confusion, we will omit the indices, i.e.,

write (x,˙x)instead of (xi,˙xi). Commas ,iwill mean partial differentiation with respect to the base coordinates

xiand dots ·ipartial differentiation with respect to the ﬁber coordinates ˙xi. Also, by ◦

TM =T M \{0}, we will

mean the tangent bundle of Mwithout its zero section.

Aconic subbundle of T M is a non-empty open submanifold Q⊂TM \{0}, with the following properties:

•

TM(Q) = M;

•conic property: if (x,˙x)∈Q, then, for any

>0 : (x,

˙x)∈Q.

Apseudo-Finsler space is,48, a triple (M,A,L),where Mis a smooth manifold, A⊂◦

T M is a conic

subbundle and L:A→Ris a smooth function obeying the following conditions:

1. positive 2-homogeneity: L(x,

˙x) =

2L(x,˙x),∀

>0,∀(x,˙x)∈A.

2. at any (x,˙x)∈Aand in one (and then, in any) local chart around (x,˙x),the Hessian:

gi j =1

∂

˙xi

∂

˙xj

is nondegenerate.

The conic subbundle A,where Lis deﬁned, smooth and with nondegenerate Hessian, is called the set of

admissible vectors. In the following, we will consider as A,the maximal set with these properties - hence,

we will write simply (M,L)instead of (M,A,L).

Another important conic subbundle in a pseudo-Finsler space is the set of non-null admissible vectors:

A0:=A\L−1{0}.(1)

This is the set where we can divide by Lin order to adjust the homogeneity degree of geometric objects in ˙x.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 5

Deﬁnition 1 (Finsler spacetimes) A Finsler spacetime is a 4-dimensional, connected pseudo-Finsler space

obeying the extra condition:

3. There exists a conic subbundle T⊂Awith connected ﬁbers Tx,x∈M, such that, on T:L>0,g

has Lorentzian signature (+,−,−,−)and L can be continuously extended as 0to the boundary

∂

The role of condition 3.is to ensure the existence of a proper causal structure for (M,L).

In the following, though we will not specify this explicitly, we will always consider that Lis continuously

prolonged as 0 on

∂

T; in particular, L(0) = 0.

The above deﬁnition is a minor modiﬁcation of the one introduced in45 to ensure that the cone of timelike

vectors is salient at each point. It recovers, in the particular case A:=T, the deﬁnition of improper Finsler

spacetimes in441. The existence and uniqueness of geodesics with given initial conditions (x,˙x)∈¯

T, which

was explicitly required in an older version of this deﬁnition in43 , follows from the axioms 1−3 above, see44,

and thus the deﬁnition presented here also covers the Finsler spacetimes discussed in46,47 .

In principle it would be possible to include directions in Tthat are not in A,but just in A, see e.g.,49,50

yet, for our purposes, it will be more convenient to assume that T⊂A, in order to avoid unnecessary

complications in variational procedures involving Tor subsets thereof.

Timelike vectors and the observer space.

For the application of Finsler spacetimes in physics, besides the sets of admissible (respectively, non-null

admissible) directions Aand A0, the following subsets of T M play an important role:

1. The conic subbundle T,called the set of future pointing timelike vectors.

2. The observer space, or set of unit future-pointing timelike directions

O:={(x,˙x)∈T|L(x,˙x) = 1},(2)

which satisﬁes the inclusion O⊂T⊂A0⊂A. Moreover, due to the homogeneity of L, we have at

any x∈M:

Tx= (0,∞)·Ox.(3)

3. The set N:=L−1{0}has the meaning of set of null or lightlike vectors. By continuously extending

Las zero to the boundary

∂

T, as speciﬁed above, we always have the inclusion

∂

T⊂N.(4)

It is important to notice that the null cone Nmight not be contained in A, but just in A.

The null boundary condition (4) ensures that, for every x∈M,the future timelike cone Txis actually an

entire connected component of L−1((0,∞)) ∩TxM. This leads to the following consequences.

Proposition 2 At any x ∈M,the observer space Oxis a connected component of the indicatrix Ix=L−1(1)∩

TxM.

1Physically, this ensures the existence of a global time orientation.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 6

FIG. 1. Future pointing timelike vectors Tx, observer space Oxand null directions Nxof a double null-cone Finsler

spacetime, in the tangent space at x∈M.

This statement follows immediately from the connectedness and maximality of Tx.

As a consequence of the maximal connectedness of Ox, a result by Beem,39 ensures that Oxis a strictly

convex hypersurface of TxMand moreover, the set

Cx:=Tx∩L−1([1,∞)) = [1,∞)·Ox

is also convex. Based on this, we can state:

Proposition 3 In a Finsler spacetime as deﬁned above, all future timelike cones Tx,x∈M,are convex.

Proof. Fix x∈Mand consider some arbitrary u,v∈Tx,

∈[0,1].In order to show that w:= (1−

)u+

lies in Tx,we rescale it by

≥max(L(u)−1/2,L(v)−1/2); this way, the endpoints

vlie in Cxand, by

the convexity of Cx,we ﬁnd that

w∈Cx⊂Tx.The statement then follows from the conicity of Tx.

The above result generalizes a similar statement in43, which was proven under the more restrictive condi-

tion that Lis deﬁned and continuous on the entire T M. The convexity of the cones Txensures a well deﬁned

causal structure on a Finsler spacetime and that timelike geodesics locally extremize the length between

timelike separated points.

The Finslerian pseudo-norm, which deﬁnes the canonical geometric length measure for curves on a

Finsler spacetime, is deﬁned by

F:=p|L|,(5)

which implies

F2,

=sign(L).(6)

Examples of Finsler spacetimes

The above deﬁnition allows for Finsler spacetimes of:

1. Lorentzian type.If a:M→T0

2M,x7→ ax=ai j(x)dxi⊗dx jis a Lorentzian metric on M, then,

we can set A=◦

T M, as L:T M →R,L(x,˙x) = ax(˙x,˙x)is smooth on T M.Accordingly, F(x,˙x) =

q

ai j(x)˙xi˙xj

.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 7

2. Randers type,7, given by F(x,˙x) = p|ai j(x)˙xi˙xj|+bi(x)˙xi, where ais as above and b=bidxiis a

differential 1-form on M.We proved in43 that, if ai j bibj∈(0,1)then Fprovides a Finsler spacetime

structure on M.In the context of physics, these geometries are employed to study the motion of an

electrically charged particle in an electromagnetic ﬁeld, the propagation of light in static spacetimes51,

Lorentz violating ﬁeld theories from the standard model extension18,52,53 and Finsler gravitational

waves54. Recently also spinors have been constructed on Randers geometries in terms of Clifford

bundles55.

3. Bogoslovsky/Kropina type F= (|aij (x)˙xi˙xj|)1−q

2(bk(x)˙xk)q23,56, where q∈R; the conditions upon the

1-form b, such that Fdeﬁnes a spacetime structure depend on the value of q; a detailed discussion

is made in43. In physics, approaches to quantum ﬁeld theories and modiﬁcations of general relativity

that are only invariant under a subgroup of the Lorentz group, based on this geometry, have been

investigated under the name very special and very general relativity27,57–60.

4. Polynomial m-th root type F=|Ga1···am(x)˙xa1... ˙xam|1

m, which appear in physics for example in the

description of propagation in birefringent media, in the context of premetric electrodynamics and in

the minimal standard model extension14,15,18,61.

5. Anisotropic conformal transformations of pseudo-Riemannian geometry F=e

(x,˙x)p|ai j(x)˙xi˙xj|

have been studied in the context of an extension of the Ehlers-Pirani-Schild axiomatic to Finsler

geometry16,17,62, as well as examples for Finsler spacetimes according to Beem’s deﬁnition41.

6. General ﬁrst order perturbations of F=p|aij (x)˙xi˙xj|+

h(x,˙x)pseudo-Riemannian geometry, which

are often used in the study of the physical phenomenology of Planck scale modiﬁed dispersion

relations24–26,63,64.

7. Finsler spacetime metrics L(x,˙x) =

x(˙x)−ˆ

F2(x,˙x)obtained,42, by "signature-reversing" from posi-

tive deﬁnite Finsler ones ˆ

F:T M →Rusing a nonzero 1-form

∈Ω1(M). Their (2-homogeneous)

Finsler function Lis well deﬁned and smooth on the entire slit tangent bundle - which is quite a rare

feature among pseudo-Finsler metrics. The set T={(x,˙x):

x(˙x)≥ˆ

F(x,˙x)}satisﬁes the requests

for the set of future-pointing timelike vectors.

B. Geometric objects on Finsler spacetimes

Typical Finslerian objects in a Finsler spacetime (M,L)(more generally, in a pseudo-Finsler space) are

obtained similarly to the corresponding objects in positive deﬁnite Finsler spaces (M,F), see, e.g.,4,5,65, just

taking care that we have to restrict them to Aor, if necessary, to A0.

Apart from the Finslerian pseudo-norm F, the fundamental building blocks of the geometry of Finsler

spacetimes are obtained from partial derivatives of the Finsler Lagrangian function L. Below, we brieﬂy

present the coordinate expressions of the typical Finslerian geometric objects to be used in the following.

Hilbert form, Finsler metric tensor and Cartan tensor

On a Finsler spacetime (M,L)the Hilbert form

:A0→T0

1M, the Finslerian metric tensor g :A→T0

Mathematical foundations for ﬁeld theories on Finsler spacetimes 8

and the Cartan tensor C :A→T0

3Mare expressed, in every manifold induced local coordinate chart, as

(x,˙x):=F·idxi,F·i=

gi j ˙xj

F,(7)

g(x,˙x):=gi j(x,˙x)dxi⊗dx j,gi j :=1

2L·i·j(8)

C(x,˙x):=Ci jk (x,˙x)dxi⊗dx j⊗dxk,Ci jk :=1

4L·i·j·k.(9)

We note that the Hilbert form

is only deﬁned on A0(as it involves derivatives of F=p|L|, which are not

deﬁned at points where L=0), while the Finsler metric and the Cartan tensor are deﬁned on A.

A curve c:[a,b]→Mis called admissible if all its tangent vectors are in A.The arc length of a regular

admissible curve c:t∈[a,b]7→ c(t)on Mis calculated as

l(c) =

F(c(t),˙c(t))dt,(10)

where ˙c(t) = dc

dt (t).If, moreover, ˙c(t)is nowhere lightlike, i.e., (c(t),˙c(t)) ∈A0for all t, then l(c)can also

be expressed in terms of the Hilbert form as:

l(c) =

C∗

ImC

,(11)

where the symbol C:[a,b]→T M,t7→ (c(t),˙c(t)) denotes here the canonical lift of cto T M.

Proposition 4 (Finsler geodesics) , see, e.g.,5: Critical points of the length functional (10)are called

Finsler geodesics. In arc length parametrization, they are determined by the Finsler geodesic equation

¨xi(s) + 2Gi(x(s),˙x(s)) = 0,(12)

where ˙x(s) = dx

ds (s); the geodesic coefﬁcients are well deﬁned at all points (x,˙x)∈Aand given by

2Gi(x,˙x) = 1

2gih(L·h,j˙xj−L,h).(13)

Anonlinear connection will be understood as a connection on the ﬁbered manifold Ain the sense of66

(pp. 30-32), i.e. , as a splitting of the tangent bundle TAof A,

TA=HA⊕VA.

The vertical subbundle V A=ker d(

TM |A)and the horizontal subbundle HAare vector subbundles of the

tangent bundle (TA,

TA,A). The local adapted basis will be denoted by (

i,˙

∂

i),where

i:=

∂

xi−Gj

∂

˙xj,

∂

˙xiand its dual basis, by (dxi,

˙xi=d˙xi+Gijdxj). Here Gijare general local connection coefﬁcients.

We denote by hand vthe horizontal and, accordingly, the vertical projector determined by the nonlinear

connection; that is, for any vector X∈TA,locally written as X=Xi

i+˙

Xi˙

∂

iwe will have: hX=Xi

iand

vX=˙

Xi˙

∂

Mathematical foundations for ﬁeld theories on Finsler spacetimes 9

Cartan (canonical) nonlinear connection. The Cartan nonlinear connection Non a Finsler spacetime

(M,L)is deﬁned by the local connection coefﬁcients

Gij=Gi·j,(14)

Arc-length parametrized geodesics of the Finsler spacetime (M,L)are autoparallel curves of the canonical

nonlinear connection.

Nonlinear curvature tensor and Finsler Ricci scalar

The curvature tensor of the canonical nonlinear connection on a Finsler spacetime (M,L)is a tensor on

T M, which has the coordinate expression:

Rijkdx j∧dxk⊗˙

∂

i=dx j∧dxk⊗[

k] = (

kGij−

jGik)dx j∧dxk⊗˙

∂

i.(15)

The Finsler-Ricci scalar R0makes sense on A0and is given by

R0=1

LRiik ˙xk.

The way introduced it above is equal to minus the one employed in4(and denoted by Ric).

Besides the canonical nonlinear connection, it is possible to additionally deﬁne several linear connections

on A, which preserve the distributions generated by the nonlinear connection. In this article we will pick,

for simplicity, one of these linear connections as a mathematical tool to ensure that all objects we are dealing

with are well deﬁned tensors. Our particular choice of the linear connection is unessential, since it is just an

auxiliary tool. The whole construction is independent of the typical Finslerian linear connections that one

may use.

Chern-Rund linear connection

The Chern-Rund linear covariant derivative on a Finsler spacetime (M,L), deﬁned on A⊂T M, is locally

given by the relations

j=Γijk

i,D

k˙

∂

j=Γijk ˙

∂

i,D˙

∂

j=D˙

∂

j=0,(16)

where Γijk :=1

2gih(

kgh j +

jghk −

hgjk). We denote by |iD-covariant differentiation with respect to

The Chern-Rund linear covariant derivative allows us to introduce the dynamical covariant derivative in

a very simple way, namely, as ∇:Γ(TA)→Γ(TA)with ∇=˙xiD

i. An important remark is that, since

˙xiΓkji =Gkj,the dynamical covariant derivative only depends on the canonical nonlinear connection N,see5

(it can actually be introduced independently of Dor of any other additional structure).

The dynamical covariant derivative can be used to deﬁne a measure of the change of the Cartan tensor

along horizontal curves, called the Landsberg tensor, see4.

Landsberg tensor

The Landsberg tensor P=Pijkdx j⊗dxk⊗

iis a tensor on T M , deﬁned, in any local chart, by:

Pijk =gmi∇Cm jk =Gi·j·k−Γijk .(17)

Its trace is denoted by Pi=Pjij .

The following identities will be useful when we consider action integrals and calculus of variations on

Finsler spacetimes:

iL=L|i=0,gi j|k=0,˙xi

|j=0,(18)

∇L=0,∇gi j =0,∇˙xi=0,(19)

Pijk ˙xk=0,Pi˙xi=0.(20)

Mathematical foundations for ﬁeld theories on Finsler spacetimes 10

They can all be proven by using the homogeneity properties of the tensors involved and the deﬁnition of the

canonical nonlinear connection in terms of the Finsler Lagrangian.

(Semi)-Riemannian geometry as Finsler geometry

Choosing L=gi j (x)˙xi˙xj, the geometry of a Finsler spacetime (M,L)becomes essentially the geometry of

the pseudo-Riemannian spacetime manifold (M,g). In this case, Gij=

ijk(x)˙xkand Rijk =ri

j kl ˙xl(where we

have denoted by small letters the geometric objects speciﬁc to Riemannian geometry). The relation between

the Finsler-Ricci scalar R0and the usual Riemannian one r=gi j rk

i jk is: gi j (LR0)·i·j=−2r.

C. Homogeneous geometric objects on TM

Homogeneity is a key concept in pseudo-Finslerian geometry, as the positive homogeneity of Lin ˙xentails

the positive homogeneity of all typical Finslerian geometric objects. We will brieﬂy present here some results

on homogeneous geometric objects deﬁned on conic subbundles Q⊂◦

TM.The results are straightforward

extensions of the results in5and67, referring to objects deﬁned on the whole slit tangent bundle.

Deﬁnition 5 (Fiber homotheties) By ﬁber homotheties on ◦

T M,we understand the mappings

χα

:◦

T M →

◦

T M,

χα

(x,˙x) = (x,

˙x),where

>0.

Fiber homotheties form a 1-parameter group of diffeomorphisms of ◦

T M,isomorphic to (R∗

+,·)and are

generated by the Liouville vector ﬁeld

C=˙xi˙

∂

We denote the corresponding group action by

,i.e.:

:◦

T M ×R∗

+→◦

T M,

((x,˙x),

) =

χα

(x,˙x).(21)

Deﬁnition 6 (Homogeneous tensor ﬁeld) Let T be a tensor ﬁeld over the conic subbundle Q⊂◦

TM. T is

called positively homogeneous of degree k ∈Rif and only if, for all

>0,its pullback along the restriction

χα

:Q→Qsatisﬁes

∗

kT.(22)

Theorem 7 A tensor ﬁeld T over Qis positively homogeneous of degree k ∈Rif and only if

LCT=kT .(23)

Proof. See67 (Lemma 4.2.9) for the proof in the special case of scalar functions and67 (Lemma 4.2.14)

for vector ﬁelds. In order to prove it in the general case, we momentarily reinterpret the multiplicative 1-

parameter group {

χα

}as the additive group R, by setting t:=log(

)∈Rand

t(x,˙x) = (x,et˙x) =

χα

(x,˙x),

for all (x,˙x)∈T M . Assume, ﬁrst, that Tis k-homogeneous, which means:

∗

tT=ekt T. Then,

LCT=d

dt (

∗

tT)



t=0

dt (ekt T)



t=0

=kT.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 11

Conversely, assume (23) holds. Differentiating the identity

∗

Twith respect to

=0, one

ﬁnds:

∗

tLCT=d

dt (

∗

tT),∀t.

Using (23), this leads to the differential equation d

dt (

∗

tT) = k

∗

tTin the unknown f(t) =

∗

tT. Integrating

this equation with the initial condition f(0) =

∗

0T=T, we ﬁnd

∗

tT=ekt T, which, reverting to the old

notation, is precisely

∗

kT.

In particular, positive 0-homogeneity in ˙x,i.e., invariance under the ﬁber rescalings

χα

>0,can be

treated as invariance under the ﬂow of C.

Note. In5,k-homogeneity of vector ﬁelds is deﬁned differently (it is, in our terms (k−1)-homogeneity).

We prefer, yet, this deﬁnition, which allows a uniﬁed treatment of tensor ﬁelds of any rank.

In the following, we will simply refer to positive homogeneity in ˙xas homogeneity. Some canonical

examples of homogenous structures on the tangent bundle are:

1. The Liouville vector ﬁeld Cis homogeneous of degree 0, since LCC= [C,C] = 0.

2. The vertical local basis vectors ˙

∂

iare homogeneous of degree -1, as [C,˙

∂

i] = −˙

∂

3. The natural tangent structure of T M,

J=dxi⊗˙

∂

i(24)

is a globally deﬁned, (−1)-homogeneous tensor of type (1,1). Homogeneity follows from:

LCJ=LC(dxi)⊗˙

∂

i+dxi⊗[C,˙

∂

i] = 0−dxi⊗˙

∂

i=−J,

where we have used 2. and LC(dxi) = diCdxi+iCddxi=0.

Deﬁnition 8 (Homogeneous nonlinear connection) A no nlinear connection T Q=HQ⊕VQon the conic

subbundle Q⊂◦

T M,is called homogeneous, if ﬁber homotheties preserve the horizontal subbundle, i.e.,

(

χα

)∗X∈HQfor all

>0and all X ∈HQ.

As it has been shown in5(Prop. 2.10.1), or in68 (Cor. 7.5.10), that a nonlinear connection on T M is

homogeneous if and only if the almost product structure P=h−vis 0-homogeneous; the result holds

without modiﬁcations on Q⊂T M .

In coordinates, homogeneity of a connection is characterized by the fact that its coefﬁcients Gij=Gij(x,˙x)

are 1-homogeneous functions in ˙x. An example of a homogeneous nonlinear connection is the Cartan non-

linear connection, (14), of a Finsler spacetime.

Almost all Finsler geometric objects discussed above are anisotropic tensor ﬁelds, which thus deserve a

special mentioning here. These can be mapped into speciﬁc tensor ﬁelds on the tangent bundle, called dis-

tinguished tensor ﬁelds, or d-tensor ﬁelds; for the latter, homogeneity can be discussed in a natural manner.

Deﬁnition 9 ,69: An anisotropic tensor ﬁeld on the conic subbundle Q⊂◦

TM is a section of the pullback

bundle

∗

TM|Q(Tp

qM),i.e. , a smooth mapping:

T:Q→Tp

qM,(x,˙x)7→ T(x,˙x);

i.e., for any (x,˙x)∈Q,T(x,˙x)is a tensor on M,based at x =

TM (x,˙x).

Mathematical foundations for ﬁeld theories on Finsler spacetimes 12

Consequently, an anisotropic tensor ﬁeld will be locally expressed as: T(x,˙x)=Ti1...ip

j1... jq(x,˙x) (

∂

i1⊗...⊗

∂

ip⊗

dx j1⊗... ⊗dx jq)|x.

In the presence of a nonlinear connection Non T M the following deﬁnition makes sense.

Deﬁnition 10 ,5: A d-tensor ﬁeld on a conic subbundle Q⊂◦

T M (regarded as a manifold) is a tensor ﬁeld

T∈Tp

q(Q), obeying the condition:

1,...,

p,V1,...,Vq) = T(

1,...

p+1V1,...,

p+qVq),

for an arbitrarily ﬁxed choice of the projectors

1,..,

p+q∈ {h,v}.

For instance, if Vis an arbitrary vector ﬁeld on Q,its horizontal and vertical components hVand vV,

taken separately, are d-tensor ﬁelds (of type (1,0)), as each of them acts on a single speciﬁed component h

or v

of a 1-form

∈Ω1(Q), whereas their sum is typically, not a d-tensor ﬁeld.

With respect to the horizontal/vertical adapted local bases of TQand T∗Q,a d-tensor ﬁeld Twill be

expressed as a linear combination of tensor products of

i,˙

∂

i,dxiand

˙xi, i.e., T(x,˙x)=Ti1...ip

j1... jq(x,˙x) (

i1⊗

... ⊗˙

∂

ip⊗dx j1⊗... ⊗

˙xjq)|(x,˙x).

In the presence of a homogeneous nonlinear connection, the adapted basis elements

iare 0-homogeneous

(and, as we have seen above, ˙

∂

iare (−1)-homogeneous), hence the degree of homogeneity (if any) of a d-

tensor ﬁeld Tcan be established in local coordinates, by simply evaluating the ˙x-homogeneity degree of the

coefﬁcients Ti1...ip

j1... jq.

Note. Anisotropic tensor ﬁelds can be mapped into (multiple) d-tensor ﬁelds on Q⊂T M via horizontal

or vertical lifts determined by the nonlinear connection. Yet, when doing this, one must take into account

that using horizontal lifts

∂

i7→

i, one obtains a d-tensor of different degree of homogeneity, compared to

the one obtained via a vertical lift

∂

i7→ ˙

∂

i, due to the −1-homogeneity of ˙

∂

Examples of canonical homogeneous d-tensors are the Liouville vector ﬁeld Cand the tangent structure J.

Further examples of homogeneous d-tensors arise once we consider a pseudo-Finsler structure on M. For

instance, on a Finsler spacetime, all the tensor ﬁelds encountered in the previous section are homogeneous

d-tensor ﬁelds on A,of some degree m:

• the Finslerian metric tensor g=gi jdxi⊗dxj(k=0);

• the curvature R=Rijk dx j⊗dxk⊗˙

∂

iof the canonical linear connection (k=0);

• the Landsberg tensor P=Pijkdx j⊗dxk⊗

i(k=0).

Other d-tensor ﬁelds, such as the Hilbert form

=F·idxi,or the Reeb vector ﬁeld

ℓ=li

are only deﬁned on A0=A\L−1(0),since the functions li=˙xiF−1are only deﬁned on A0.Both

and ℓ

are 0-homogeneous in ˙xand will play a crucial role in the following, as we will see in Section III B 1.

An important feature of both the Chern connection D(and more generally, of any of the typical Finslerian

connections in the literature) and of the dynamical covariant derivative ∇on TA,is that they preserve

the distributions generated by the canonical nonlinear connection Nand hence, they map d-tensors into d-

tensors,5. Moreover, the degree of homogeneity of d-tensors is preserved by D-covariant differentiation with

respect to 0-homogeneous vector ﬁelds (and is increased by 1 by dynamical covariant differentiation).

Mathematical foundations for ﬁeld theories on Finsler spacetimes 13

III. THE POSITIVELY PROJECTIVIZED TANGENT BUNDLE PT M+

The positively projectivized tangent bundle PT M+is essential for a mathematically well deﬁned calculus

of variations o n Finsler spacetimes. It also gives a nice way to understand positively homogeneous geometric

objects on T M, such as the Finsler function, or d-tensors, as sections of bundles sitting over PT M +, which

we will discuss in detail in Section IV.

We will ﬁrst introduce PT M+over general manifolds before we formulate the geometry of Finsler space-

times on PT M+. This reformulation is important to construct well deﬁned integrals of homogeneous func-

tions. We will show that integration on domains in PT M+is actually equivalent to integration over subsets

of the observer space O, with the advantage that PT M+is explicitly independent of the Finsler Lagrangian

L, whereas the observer space (and therefore, all its subsets, which one may use as integration domains) are

deﬁned in terms of L.

A. Deﬁnition and structure over general manifolds

We ﬁrst give the deﬁnition of the positively projectivized tangent bundle, before we analyze its structure

and point out how objects on PT M+are related to 0-homogeneous objects on T M. In the context of Finsler

spacetimes, the positively projectivized tangent bundle was brieﬂy discussed in43. In the literature on positive

deﬁnite Finsler geometry, PTM+is typically called the projective sphere bundle.

Actually, in positive deﬁnite Finsler geometry, this bundle is interchangeably used with the indicatrix

bundle, as the two bundles are globally diffeomorphic. But, in Lorentzian Finsler geometry, as we will see

below, this diffeomorphism does no longer exist, hence, in order to avoid any confusion, we preferred to

make a clear distinction by the used terminology.

1. Deﬁnition and structure

Deﬁnition 11 (The positive, or oriented, projective tangent bundle) Let M be a connected, orientable

smooth manifold of dimension n. The positive projective tangent bundle is deﬁned as the quotient space

PTM+:=◦

T M/∼(25)

where ∼is the equivalence relation on ◦

T M given by:

(x,˙x)∼x,˙x′⇔˙x′=

˙x for some

>0.(26)

In other words we identify the half-line {(x,

˙x)|

>0}as a single point. We denote by

+:◦

TM →PT M+,(x,˙x)7→ [(x,˙x)]

the canonical projection.

The usual projectivized tangent bundle PT M is obtained from PT M+by deleting the distinction between

positive and negative scaling factors, in other words:

PTM =PT M+/Z2.

Conversely, by attaching orientations to the lines representing points of PT M ,one gets PT M+.In other

words, PTM+is the canonical oriented double covering (also called orientation covering in (70,p. 394)) of

Mathematical foundations for ﬁeld theories on Finsler spacetimes 14

the (2n−1)-dimensional manifold PT M,in particular, it is always orientable. The above discussion can be

summarized as follows.

Proposition 12 If M is a connected smooth n-dimensional manifold, then PT M+is a smooth, orientable

manifold of dimension 2n−1.

The orientability of PT M +is essential when considering integrals on PTM+.

The smooth structure on PT M +is constructed as follows. Start with an atlas on T M, induced by an atlas

on Mand denote by (xi,˙xi)the corresponding coordinate functions; then, for each local chart domain U∈

T M and each i=0,...,n−1, deﬁne the open sets: Ui=(x,˙x)∈TU |˙xi>0,Ui′=(x,˙x)∈U|˙xi<0.

Then, for each U+∈ {

+(Ui),

+(Ui′)}and each [(x,˙x)] ∈U+, we deﬁne the diffeomorphisms

+:=

(xi,u

)as:

(xi,u

) = (x0,..., xn−1,˙x0

˙xi, ..., ˙xi−1

˙xi,˙xi+1

˙xi, ..., ˙xn−1

˙xi).(27)

The result is a differentiable atlas {(U+,

+)}on PTM+.

Using these charts, a quick direct computation shows that the projection

+:◦

TM →PT M+,(xi,˙xi)7→

(xi,u

)is a submersion. Since, obviously,

+is surjective, it follows that (◦

TM,

+,PT M+)is a ﬁbered

manifold; actually, it posseses an even richer structure, as has already been pointed out in43. Let us brieﬂy

recall this result:

Proposition 13 (The principal bundle (◦

T M,

+,PTM+,R∗

+))The slit tangent bundle ◦

TM is a principal

bundle over PT M+,with ﬁber (R∗

+,·).

Proof. Consider

, as deﬁned in (21), as the right action of the Lie group (R∗

+,·)on ◦

T M. This action

preserves the ﬁbers (

+)−1([x,˙x]) = (x,

˙x)|

∈R∗

+of

+,i.e., the half-lines with direction (x,˙x).

Moreover, each of the ﬁbers of

+is obviously homeomorphic to R∗

Actually, PTM+is nothing but the space of orbits of the Lie group action (21).

The Liouville vector ﬁeld Cis tangent to the ﬁbers (

+)−1[(x,˙x)] (i.e., it is

+-vertical), which, taking

into account that these ﬁbers are 1-dimensional, means that Cactually generates the tangent spaces to these

ﬁbers.

In its turn, PT M +is a ﬁbered manifold over M.More precisely, we have the following result.

Proposition 14 (Structure of the bundle (PTM+,

M,M,Sn−1))The triple (PT M+,

M,M), where

PTM+→M,[(x,˙x)] 7→ x, is a ﬁbered manifold with ﬁbers diffeomorphic to Euclidean spheres.

Proof. The projection

Mis obviously a surjective submersion, meaning that (PT M+,

M,M)is, indeed, a

ﬁbered manifold. Its ﬁbers

−1

M(x) = {[(x,˙x)] |˙x∈TxM}are orientation coverings of the projective tangent

spaces PTxM≃PRn; but, the orientation covering of the projective space PRnis nothing but the round

sphere Sn−1.

Moreover, PTM+is a natural bundle over M, meaning that it is obtained from Mvia a covariant functor

(see the Appendix for more details on natural bundles); the natural lift to PTM+of a diffeomorphism

Mathematical foundations for ﬁeld theories on Finsler spacetimes 15

f:M→Mis given by [(x,˙x)] 7→ [( f(x),d fx(˙x))], which is well deﬁned by virtue of the linearity of d fx.

On natural bundles, one can speak about general covariance of Lagrangians, which is essential in ensuring

the existence of a well deﬁned notion of energy-momentum tensor.

Note: As already stated above, the bundle PT M+is better known in the literature on positive deﬁnite

Finsler spaces under the name of projective sphere bundle over M, see, e.g.,4; though the name is very well

justiﬁed by the above Proposition, we preferred to avoid this terminology, in order to avoid any confusions

with the indicatrix bundle L =1. This distinction is necessary since, for positive deﬁnite Finsler structures,

the ﬁbers Ix=L−1(1)of the indicatrix bundle are diffeomorphic to Euclidean spheres (i.e., diffeomorphic

to the ﬁbers of PT M +), while in Lorentzian signature, this is no longer the case; actually, in the latter

case, we have already seen that the ﬁbers of the indicatrix bundle are generally disconnected, containing the

observer spaces Oxas connected components. Moreover, it is essential for our later considerations to stress

that the construction of PT M+is completely independent of any pseudo-Finslerian (or pseudo-Riemannian)

structure whatsoever.

2. From PTM+to T M and back

Local computations on PT M+are much simpliﬁed if one uses local homogeneous coordinates instead of

the usual local coordinates (xi,u

)deﬁning the manifold structure, in the same fashion as on PT M, see65 .

For a given equivalence class [(x,˙x)], local homogeneous coordinates2are deﬁned as the coordinates

(xi,˙xi)in the corresponding chart on T M of an arbitrarily chosen representative of the class [(x,˙x)]; i.e.,

homogeneous coordinates are only unique up to multiplication by a positive scalar of the ˙x-coordinates.

In these coordinates, local computations on PT M+will become identical to those on T M,just with due

care that the involved expressions in xi,˙xi- which formally correspond to geometric objects on T M -

should really deﬁne objects on PT M+. A necessary (but not always sufﬁcient) condition is that these for-

mally deﬁned geometric objects on T M should be positively 0-homogeneous in ˙x, i.e., invariant under the

ﬂow of C.Here we list the most frequently encountered examples:

•Functions. A function f:◦

TM →R,f=f(x,˙x)can be identiﬁed with a function f+on PT M +if and

only if it is positively 0-homogeneous in ˙x; more precisely, f+:PT M +→Ris deﬁned by:

f+[(x,˙x)] = f(x,˙x),

i.e., f:=f+◦

+; in homogeneous coordinates, f+and fhave identical coordinate expressions. The

function f+is differentiable at [(x,˙x)] if and only if fis differentiable at one representative (x,˙x);

•Vector ﬁelds. For a vector ﬁeld X=Xi

∂

i+˜

Xi˙

∂

i∈X(◦

T M), the projection

X+:= (

+)∗X

is a well deﬁned vector ﬁeld on PT M +if and only if Xis positively 0-hom ogeneous in ˙x, i.e., LCX=0.

This is justiﬁed as follows. Having in view that

+is surjective, the necessary and sufﬁcient condition

for X+to be a well deﬁned vector ﬁeld on PT M+is that the mapping [(x,˙x)] 7→ X+

[(x,˙x)] = (

+)∗X(x,˙x)

is independent on the choice of (x,˙x)in the class [(x,˙x)] ; but this means precisely 0-homogeneity of X.

2Here, the word "local" is just meant to stress that these coordinates do not make sense globally, but only over a coordinate neigh-

borhood. (Local) homogeneous coordinates are, obviously, not local coordinates as typically deﬁned in differential geometry, since

their number is equal to 2dim(M), whereas the dimension of PT M+is 2dim(M)−1 .

Mathematical foundations for ﬁeld theories on Finsler spacetimes 16

In coordinates, this boils down to the fact that the functions Xiare positively 0-homogeneous, while

Xiare 1-homogeneous in ˙x.

An interesting remark is that the correspondence X7→ X+is surjective, but not injective, as all vector

ﬁelds of the form X+fCon T M , where fis a 0-homogeneous function, descend onto the same vector

ﬁeld X+∈X(PT M+).

•Differential forms. For differential forms

on ◦

T M, 0-homogeneity is necessary, but not sufﬁcient in

order to be identiﬁed with differential forms on PT M+.The following result (derived in43) is just a

coordinate-free restatement of a similar result on PT M,see65:

Proposition 15 Let

∈Ω(TM)be deﬁned on a conic subbundle of T M.Then, there exists a unique

differential form

+∈Ω(PTM+)such that

= (

+)∗

+if and only if the following conditions are

fulﬁlled:

is 0-homogeneous in ˙x,i.e.,

=0; (28)

+-horizontal, i.e.,

=0.(29)

Remark 16 43

1. The projection

+is locally represented in homogeneous coordinates as the identity:

+:(xi,˙xi)7→

(xi,˙xi). The latter relation tells us that the coordinate expressions of geometric objec ts on T M (e.g.,

f,X,

) that can be identiﬁed with geometric objects f +,X+,

+etc. on PT M+,will be identical to

the expressions of the latter in homogeneous coordinates.

2. Exterior differentiation of forms

+∈Ω(PT M+)can be carried out identically to exterior differenti-

ation of the corresponding form

∈Ω(◦

TM),since:

=d(

+)∗

+=

+∗d

In p articular, differentiation of functions on PT M +is carried out identically to the one o n T M .

B. Over Finsler spacetimes (M,L)

After having introduced PTM+in the previous section, we now demonstrate that the geometry of a Finsler

spacetime can be understood in terms of geometric objects on PT M+. This eventually enables us to write

down the desired action integrals for ﬁeld theories in a mathematically precise way.

1. Finsler Geometry on PTM+and volume forms

On a Finsler spacetime, we deﬁned the conic subbundles A,A0,T,N⊂◦

T M, and the observer space

O, see Section II A. We will denote by a plus sign, e.g., T+=

+(T),A+=

+(A)etc., their images

through

+:◦

T M →PT M+; also, we will always use local homogeneous coordinates on PT M+.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 17

Canonical nonlinear connection.

The canonical nonlinear connection Non A, see equation (14) can be transplanted to A+in a natural

way, as follows. Start with an arbitrary vector X+∈TA+. As we have seen above, it always corresponds to

a positively 0-homogeneous vector X∈TA(which is unique up to a multiple of C). Then, Xis decomposed

into its N-horizontal and vertical components hX=Xi

iand vX=˙

Xi˙

∂

i; both components are positively 0-

homogeneous, due to the homogeneity of N, hence they descend back onto vectors hX+,vX+on TA+.

Moreover, hX+,vX+are uniquely deﬁned by X+, as the possible multiple of Cappearing in the procedure

will be projected back to PTM+into the zero vector. This naturally gives rise to a splitting

X+=hX++vX+,

i.e., to a connection N+on A+:=

+(A):

TA+=HA+⊕VA+.(30)

The vectors hX+= (

+)∗(hX)and vX+:= (

+)∗(vX)are expressed in homogeneous coordinates as:

hX+=Xi

i,vX+=˙

Xi˙

∂

Similarly, the Chern-Rund connection Dgives rise to a linear connection D+on A+,having the same local

expression of covariant derivatives as D.

Contact structure and volume form for the set of non-null admissible directions.

In the following, we will identify a canonical volume form on the set of admissible non-null directions

A+

+(A0). The Hilbert form

=F·idxi, deﬁned on A0obeys the conditions:

=0,LC

=diC

+iCd

=0,

which means that it can be identiﬁed with a differential form

+on A+

0⊂PTM+,such that (

+)∗

;

in homogeneous coordinates:

+=F·idxi(31)

and

+=1

gi j −F·iF·j)

˙xj∧dxi.(32)

A direct calculation, see,65, shows that, for dim M=4,

+∧d

+=3!det g

L2iC(d4x∧d4˙x) = 3! detg

L2Vol0,(33)

where

Vol0=iC(d4x∧d4˙x),(34)

is always nonzero. In other words, the Hilbert form

+deﬁnes a contact structure on PT M +.

In contact geometry, the Reeb vector ﬁeld ℓ+∈X(A+

0)corresponding to the contact structure

+is

uniquely deﬁned by the conditions

iℓ+(

+) = 1,iℓ+d

+=0.(35)

In our case, this gives:

Mathematical foundations for ﬁeld theories on Finsler spacetimes 18

Proposition 17 The Reeb vector ﬁeld ℓ+corresponding to the contact structure

+on A+

0is expressed in

local homogeneous coordinates as:

ℓ+=li

i,li=˙xi

√F.

Proof. We have iℓ+

+=1 and

iℓ+d

+=F−1(

gi j −F·iF·j)iℓ+(

˙xi∧dxj) = F−1(

gi j −F·iF·j)lj

˙xi=0,(36)

where for the last equallity we used that F·jlj=1 and

gi jlj=F·i.

The importance of the Reeb vector ﬁeld is given by the following result.

Proposition 18 Let c :[a,b]→M,s7→ x(s)be a non-lightlike admissible curve parametrized by arc length

and C :[a,b]→A+

0,s7→ [(x(s),˙x(s))],its canonical lift. Then, C is an integral curve of ℓ+if and only if c

is an arc-length parametrized geodesic of (M,L).

Proof. In homogeneous coordinates, ˙

C=˙xi(s)

s˙xi(s)˙

∂

i, where

s˙xi(s) = ¨xi(s) + 2Gi(xj(s),˙xj(s)); that

is, Cis an integral curve of ℓ+is and only if:

˙xi(s) = li,

s˙xi(s) = 0.

The ﬁrst condition above is trivially satisﬁed by any curve parametrized by arc length, since L(x,˙x(s)) =

sign(L); taking into account the properties of the canonical nonlinear connection, the second condition is

equivalent to the fact that cis a arc-length parametrized geodesic of (M,L), see (12).

The contact structure

+, now enables us to identify a canonical volume form on A+

Deﬁnition 19 (Canonical volume form) Let (M,L)be a Finsler spacetime, A+

0⊂PTM+, the set of its

admissible, non-null directions and

+,the Hilbert form on A+

0. Then

dΣ+:=sign(detg)

+∧(d

+)3=|detg|

L2Vol0,(37)

where Vol0is as in (34), is called the canonical volume form on A+

Note that, on A+

0,gis nondegenerate, so, dΣ+is well deﬁned.

With respect to this volume form, the divergence of horizontal and vertical vector ﬁelds, X=Xi

iand

Y=Yi˙

∂

i, on A+

0, is43:

div(X) = (Xi|i−PiXi),(38)

div(Y) = (Yi·i+2CiYi−4

LYi˙xi),(39)

where Piis the trace of the Landsberg tensor (17) and Ciis the trace of the Cartan tensor (9). For any

f:A+

0→R, the above equations imply

∇f=div(fℓ) = div(f li

i).(40)

Mathematical foundations for ﬁeld theories on Finsler spacetimes 19

2. Integration on PTM+and integration on observer space

In positive deﬁnite Finsler spaces, the unit sphere bundle L−1(1)is globally diffeomorphic to PT M+,4.

But, passing to Finsler spacetimes, this is no longer true; this is easy to see since the ﬁbers Ix=L−1(1)∩TxM

are non-compact (they are, even in the simplest case of Lorentzian metrics, hyperboloids), while the ﬁbers

of PTM+are compact. Still, we will be able to establish a correspondence between the observer space O

and the set of future pointing timelike directions T+:=

+(T).A preliminary result, proven in43, refers to

compact subsets of T.

Proposition 20 , see43:

1. For any admissible compact, connected subset D ⊂L−1(1),the projection

+:D7→

+(D)⊂

+(L−1(1)) is a diffeomorphism.

2. For any connected, admissible and non-null compact subset D+⊂

+(A0)and any differential form

+on PT M +:

D+

+=Z

,(41)

where

= (

+)∗

+is a differential form on ◦

T M and D := (

+)−1(D+)∩L−1(1).

The above result will be mostly applied to pieces D ⊂O⊂L−1(1), where, by a piece D⊂X, we will

understand,71, a compact n-dimensional submanifold of Xwith boundary. Yet, it can be extended to the

whole observer space, as long as we integrate compactly supported differential forms.

Proposition 21 In any Finsler spacetime:

1. The mapping

+:O→T+is a diffeomorphism;

2. for any compactly supported 7-form

+on T+:

T+

+=Z

,(42)

where

= (

+)∗

Proof.

1. Injectivity: Assume

+(x,˙x) =

+(x′,˙x′)for some (x,˙x),(x′,˙x′)∈O.It follows that [(x,˙x)] = [(x′,˙x′)],

i.e., x=x′and there exists an

>0 such that ˙x′=

˙x.Applying Lto both hand sides, we ﬁnd L(x,˙x′) =

2L(x,˙x); but, on O,L=1,which means that

2=1.Since

>0,it follows that (x,˙x′) = (x,˙x).

Surjectivity: Consider an arbitrary [(x,˙x)] ∈T+.It means that (x,˙x)∈T; as Tis a conic subbundle of

TM,the vector (x,

˙x),with

:=L(x,˙x)−1/2,also belongs to T.But L(x,

˙x) = 1,hence (x,

˙x)∈O.

Since

+(x,˙x) =

+(x,

˙x) = [(x,˙x)],it follows that [(x,˙x)] ∈

+(O).

Smoothness: of

+and of its inverse follow immediately, working in homogeneous coordinates, in

which

+is represented as the identity.

2. follows immediately from

= (

+)∗

+and point 1.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 20

In particular, the above result shows that:

O+=T+.(43)

With this section we have established that integration of differential forms on the observer space of Finsler

spacetimes can be understood as integration of differential forms on (subsets of) PT M+.

IV. FIBERED MANIFOLDS AND FIELDS OVER A FINSLER SPACETIME

Having understood how integrals of homogeneous functions on Finsler spacetimes can be constructed,

the next step in constructing action integrals is to understand ﬁelds (and their derivatives) as sections

into

ﬁbered manifolds Yover PT M+. But, with this aim, we need to understand the structure of such ﬁbered

manifolds.

For an improved readability of the article, we give a detailed summary of jet bundles over ﬁbered mani-

folds and coordinate free calculus of variations in Appendix A.

A. Fibered manifolds over PTM+

Consider a Finsler spacetime (M,L)and denote by (Y,Π,PTM +)an arbitrary ﬁbered manifold of dimen-

sion 7 +m.Then, Ywill acquire a double ﬁbered manifold structure:

YΠ

−→ PT M+

−→ M.(44)

As a consequence, Ywill admit an atlas consisting of ﬁbered charts (V,

= (xi,u

),i=0,..., 3,

=0,..., 2,

=1,..., mon Y,that are adapted to both ﬁbrations, i.e. , the two projections will be represented

in these charts as:

Π:(xi,u

)7→ xi,u

,

M:(xi,u

)7→ xi.

Further, corresponding to any induced local chart (Π(V),

= (xi,u

)on PTM+,we can introduce the

homogeneous coordinates (xi,˙xi), which we will sometimes denote collectively as (xA).This way, we obtain

on V=Π−1(U+)the coordinate functions

:= (xi,˙xi,y

) = (xA,y

)

on V, which we will call ﬁbered homogeneous coordinates. The corresponding ﬁber coordinate y

is typi-

cally not unique, its relation to the original coordinates (xi,u

)may depend on the choice of representa-

tive (x,˙x)of [(x,˙x)] ∈PT M+. The precise relation will be discussed in the applications.

In ﬁbered ho mogeneous coordinates, local sections (physical ﬁelds)

:W+→Y,[(x,˙x)] 7→

[(x,˙x)] (where

W+⊂PTM+is open) are represented as:

:(xi,˙xi)7→ (xi,˙xi,y

(xi,˙xi)).(45)

The set of all such sections is denoted by Γ(Y).

Mathematical foundations for ﬁeld theories on Finsler spacetimes 21

Remark 22 Generically, the physical ﬁelds we are considering are k-homogeneous with respect to ˙x. Hence

their coordinate representation in ﬁbered h omogeneous coordinates satisﬁes

(x,

˙xi) = (xi,

˙xi,y

(xi,

˙xi)) =

(xi,

˙xi,

(xi,˙xi)).

Alternatively we could represent them in the original coordinates on Y as

(xi,u

) = (xi,u

(xi,u

)),

where z

(xi,u

(x,˙x)) =:z

(xi,˙xi)is a zero-homogeneous in ˙x when expressed in terms of ˙x, i.e. does not

depend on the representative of [(x,˙x)] ∈PT M +.

On the jet bundle JrY,ﬁbered charts (V,˜

)induce the ﬁbered charts 3(Vr,˜

r), with:

r= (xi,˙xi,y

,i,y

·i,..., y

·i1·i2...·ir),

where, for k=1, ..., rand

∈Γ(Y)locally represented as in (45),

,i1...·ik(xj,˙xj) =

∂

xi1...

∂

˙xik(y

(xj,˙xj))

are all partial x,˙x-derivatives up to the total order k. The canonical projections Πr,s:JrY→JsY,Jr

(x,˙x)

7→

(x,˙x)

(with r>s), are then represented as:

Πr,s:(xi,˙xi,y

,i1,..., y

·i1·i2...·ir)7→ (xi,˙xi,y

,i1,..., y

·i1·i2...·is),

accordingly,

Πr:JrY→PT M+,(xi,˙xi,y

,i1,..., y

·i1·i2...·ir)7→ (xi,˙xi).

In the calculus of variations, we will need two classes of differential forms on JrY,namely, horizontal

forms and contact forms, see Appendix A 3 for more details.

1. Πr-horizontal forms

∈Ωk(JrY)are deﬁned as forms that vanish whenever contracted with a Πr-

vertical vector ﬁeld. In the natural local basis (dxi,d˙xi,dy

,...dy

·i1...·ir),they are expressed as:

i1i2...ikdxi1∧dxi2∧... ∧d˙xik,(46)

where

i1i2...ikare smooth functions of the coordinates on JrY. In particular, Lagrangians will be

characterized as Πr-horizontal 7-forms

=ΛVol0on JrY,where Vol0is as in (34).

Similarly, Πr,s-horizontal forms, 0 ≤s≤rare locally generated by wedge products of dxi,d˙xi,dy

,dy

,i...,dy

·i1...·is.

2. Contact forms on JrYare, by deﬁnition, forms

∈Ωk(JrY)that vanish along prolonged sections, i.e.,

∗

=0,∀

∈Γ(Y).For instance,

θσ

=dy

−y

,idxi−y

·id˙xi,

θσ

,i=dy

,i−y

,i,jdx j−y

,i·jd˙xjetc.(47)

are contact forms, composing the so-called contact basis {dxi,d˙xi,

θσ

,i,

θσ

·i,...

θσ

·i1...·ir−1,dy

,i1...,ir,...dy

·i1...·ir}

of Ω(JrY).

3We note that, since we are using homogeneous coordinates over each chart domain, the number of coordinate functions (y

,i,y

i)is

8m, not 7mas one would expect taking into account the dimension of the ﬁbers of J1Y→Y.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 22

An important class of contact forms are source forms (or dynamical forms),

∈Ω8(JrY)that can be

expressed, corresponding to any ﬁbered chart, as:

ρσθσ

∧Vol0(48)

(see the Appendix for a coordinate-free deﬁnition); Euler-Lagrange forms of Lagrangians fall into this

class.

Raising to Jr+1Y,any differential form

∈Ωk(JrY)can be uniquely decomposed as:

Πr+1,r∗

where h

is horizontal and p

is contact. The horizontal component h

is what survives of

when pulled

back by prolonged sections Jr

(where

∈Γ(Y),) i.e.,

∗

=Jr+1

∗(h

).(49)

The mapping h:Ω(JrY)→Ω(Jr+1Y)is a morphism of exterior algebras, called horizontalization. On the

natural basis 1-forms, it acts as:

hdxi:=dxi,hd ˙xi=d˙xi,hdy

,idxi+y

·id˙xietc.(50)

Accordingly, for any smooth function fon JrY, we obtain:

hd f =dAf d xA=dif dxi+˙

dif d ˙xi,(51)

where difand ˙

difrepresent total xi- and, accordingly, total ˙xi-derivatives (of order r+1). Using (49) for

=d f ,we ﬁnd:

∂

i(f◦Jr

) = dif◦Jr+1

,˙

∂

i(f◦Jr

) = ˙

dif◦Jr+1

.(52)

Alternatively, one may use a nonlinear connection on A+⊂PT M+(e.g., the canonical one), to introduce

the total adapted derivative operators

δi:=di−Gj

i˙

dj,(53)

which help constructing manifestly covariant expressions. More precisely, using these operators, we can

write (51) as

hd f = (δif)dxi+ ( ˙

dif)

˙xi.(54)

If fis a coordinate invariant scalar function, then

ifand ˙

difare d-tensor components.

B. Fibered automorphisms

Variations of sections and, accordingly, of actions, are given by 1-parameter groups of ﬁbered automor-

phisms of Y. But, in the case of Finsler spacetimes, these will also have to take into account the doubly

ﬁbered structure of the conﬁguration bundle Y. This is why we introduce:

Mathematical foundations for ﬁeld theories on Finsler spacetimes 23

Deﬁnition 23 (Automorphisms of Y)An automorphism of a ﬁbered manifold (Y,Π,PT M+)is a diffeomor-

phism Φ:Y→Y such that there exists a ﬁbered automorphism

of (PT M+,

M,M)with Π◦Φ=

◦Π.

In particular, this means that there exists a diffeomorphism

0:M→Mwhich makes the following

diagram commute:

YΦ//



PTM+



PTM+



0//M

(55)

Locally, a ﬁbered automorphism of Yis represented as:

˜xi=˜xi(xj),·

˜xi=·

˜xi(xj,˙xj),˜y

=˜y

(xi,˙xi,y

An automorphism of Yis called strict if it covers the identity of PT M+, i.e.,

=idPTM+.

Generators of 1-parameter groups {Φ

}of automorphisms of Yare vector ﬁelds Ξ∈X(Y)that are

projectable with respect to both projections Πand

M; in ﬁbered homogeneous coordinates, this is expressed

as:

Ξ=

i(xj)

∂

i+˙

i(xj,˙xj)˙

∂

i+Ξ

(xj,˙xj,y

)

∂

.(56)

In particular, strict automorphisms are generated by Π-vertical vector ﬁelds Ξ=Ξ

(xj,˙xj,y

)

∂

Given such a 1-parameter group {Φ

},any section

∈Γ(Y)is deformed into the section

γε

:=Φ

◦

−1

In ﬁrst approximation, if

is locally represented as:

:xi,˙xi7→ xi,˙xi,y

xi,˙xi,then:

γε

:xi,˙xi7→ xi,˙xi,y

xi,˙xi+

(˜

◦J1

)|(xi,˙xi)+O(

2),

where

:= (Ξ

−

,i−˙

·i).(57)

The functions ˜

Ξ◦J1

, deﬁned on each local chart in the domain of

, are commonly (though in a somewhat

sloppy manner) denoted by

The automorphisms Φ

:Y→Yare prolonged into automorphisms JrΦ

of JrYby the rule:

JrΦ

(Jr

(x,˙x)

):=Jr

(x,˙x)

γε

The generator of the 1-parameter group {JrΦ

}is called the r-th prolongation of the vector ﬁeld Ξand

denoted by JrΞ(see the Appendix for the precise coordinate formula for J1Ξ).

Mathematical foundations for ﬁeld theories on Finsler spacetimes 24

C. Homogeneous geometric objects on TM as sections

In order to apply the apparatus of calculus of variations with Finslerian geometric objects (e.g., Finsler

function L,metric tensor g,nonlinear/linear connection, homogeneous d-tensors) as d ynamical variables, we

will describe these geometric objects as sections of ﬁber bundles (Y,Π,PTM+).A priori,k-homogeneous

Finslerian geometric objects are (locally deﬁned) sections f:◦

T M →◦

Yinto some ﬁber bundle ◦

Ysitting

on ◦

TM (e.g, a bundle of tensors, or a bundle of connections over ◦

T M etc.). The key idea allowing us to

reinterpret them as sections of a bundle Ysitting over PT M+, is that k-homogeneity can be interpreted as

equivariance, with respect to the action of R∗

+,·on the ﬁber bundle ◦

Yand, respectively, on the principal

bundle (◦

T M,

+,PT M+,R∗

+).

The construction of the conﬁguration bundle (Y,Π,PTM+)follows essentially the same line of reasoning

as the one made in66 (Sec 5.4), in the case of principal connections and relies on factoring out the action of

R∗

+, from both the total space and the base of the original bundle ◦

Consider a ﬁber bundle ◦

Y◦

→◦

TM,with typical ﬁber Z.Noticing that ◦

Ycan be identiﬁed with the ﬁbered

product ◦

T M ×◦

◦

Y(via the isomorphism (◦

Π,id◦

Y)covering the identity of ◦

T M), it will be convenient to

abuse the notation by explicitly mentioning the base point of any element in ◦

Y. This means that we will

identify elements y∈◦

Yas triples (x,˙x,y),where (x,˙x) = ◦

Π(y). We assume that (R∗

+,·)acts on ◦

Yby ﬁbered

automorphisms:

H:R∗

+×◦

Y→◦

Y,H(

,·) = H

∈Aut(◦

Y)(58)

as:

(x,˙x,y) = x,

˙x,

ky,(59)

for some ﬁxed k∈R.

In particular, the above rule means that:

• Each automorphism H

∈Aut(◦

Y)covers the homothety

χα

:◦

TM →◦

T M, deﬁned in Deﬁnition 5.

• The action is free and proper (properness is proven by verifying that the mapping f:R∗

+×◦

Y→◦

Y×◦

(

,x,˙x,y)7→ (x,

˙x,

ky,x,˙x,y)is proper; the latter holds as the projection of a compact subset of a

Cartesian product onto each factor is compact).

Note. In the following, we do not assume a speciﬁc form of the ﬁber Zof ◦

Y,we just assume that, for

a given k,rescaling of ﬁber elements yby the power

k,∀

>0,makes sense; e.g., in the case of vector

bundles over ◦

TM,this makes sense for any k∈R,whereas for bundles whose ﬁbers do not admit a rescaling

of elements, one is forced to choose k=0. An important example of bundles ◦

Yare the pullback bundles

∗

TM|D(Tp

qM), whose sections are the anisotropic tensors introduced in Deﬁnition 9.

This way (see70, Ch. 21), the space of orbits of the action H, i.e., the set:

Y=◦

Y/∼,(60)

Mathematical foundations for ﬁeld theories on Finsler spacetimes 25

where the equivalence relation ∼is given by:

(x,˙x,y)∼(x′,˙x′,y′)⇔ ∃

>0 : (x′,˙x′,y′) = H

(x,˙x,y),

is a smooth manifold. Moreover, ◦

Ybecomes a principal bundle over Y,with ﬁber R∗

+and projection projY:

◦

Y→Y,(x,˙x,y)7→ [x,˙x,y].

Theorem 24 1. The manifold Y =◦

Y/∼is a ﬁber bundle over PT M+,with typical ﬁber Z.

2. k-homogeneous sections f :Q→◦

Y,where Q⊂◦

T M is a con ic subbundle, are in a one-to-one corre-

spondence with local sections

:Q+→Y,where Q+=

+(Q)⊂PTM+.

Proof.

1. First, let us deﬁne the projection:

Π:Y→PTM+,[(x,˙x,y)] 7→ [(x,˙x)].

This mapping is independent of the choice of representatives in the class [(x,˙x,y)],as Π[x,

˙x,

ky] =

[(x,

˙x)] = [(x,˙x)] = Π[(x,˙x,y)] and surjective.

A local trivialization of Ycan be obtained using the principal bundle structures of both ◦

Yover Yand

of ◦

T M over PT M+.More precisely, start with ◦

V=◦

Π−1(U)⊂◦

Y,where U∈ {Ui,Ui′} ⊂ ◦

T M is

a (small enough) coordinate neighborhood on which, say, ˙x3keeps a constant sign (as introduced in

Section III); then, ◦

Vis diffeomorphic to U×Z.

But, on the one hand, using the principal bundle structure of (◦

T M,

+,PTM+,R∗

+),the coordinate

neighborhood Uis diffeomorphic to U+×R∗

+,where U+=

+(U)and, on the other hand, using the

principal bundle structure of ◦

Yover Y, the coordinate neighborhood ◦

Vis in its turn, diffeomorphic to

V×R∗

+,where V:=projY(◦

Y).

This way, a trivialization of ◦

Ycan be written as follows:

V×R∗

+//

◦



(U+×R∗

+)×Z

proj1

ww♦

♦

U+×R∗

A system of R∗

+-adapted ﬁbered coordinate functions on ◦

Yis of the form xi,u

,˙x3,z

,where

xi,u

,with u

=˙x

˙x3,

=0,1,2 are coordinate functions on U+⊂PTM+.

A trivialization of Yis then obtained by discarding the R∗

+factor in the above diagram:

V//



U+×Z

proj1

{{✇

✇

U+

Mathematical foundations for ﬁeld theories on Finsler spacetimes 26

It remains to show that V=Π−1(U+):

⊂: If [x,˙x,y]∈V=projY(◦

V),then there exists a representative (x,˙x,y)∈◦

V=◦

Π−1(U)of the class

[x,˙x,y].But then, Π0(x,˙x,y) = (x,˙x)∈U,which means [(x,˙x)] = Π[x,˙x,y]∈U+,i.e., [x,˙x,y]∈

Π−1(U+).

⊃: Starting with [x,˙x,y]∈Π−1(U+),we ﬁnd that [(x,˙x)] ∈U+; picking a representative (x,˙x)∈Uof

the class [(x,˙x)],the corresponding representative (x,˙x,y)of the class [(x,˙x,y)] belongs to ◦

Π−1(U) =

◦

V.That is, the class [(x,˙x,y)] is in projY(◦

V) = V.

Using the above trivialization, the corresponding ﬁbered coordinates on V⊂Yare then obtained by

discarding the ˙x3coordinate from the coordinates (xi,u

,˙x3,z

)on ◦

Yi.e.,

= (xi,u

2. Let f:Q→◦

Y,(x,˙x)7→ f(x,˙x)∈◦

Y(x,˙x)be a k-homogeneous section, i.e., f(x,

˙x) =

kf(x,˙x),

∀

>0 and deﬁne:

:Q+→Y,

[(x,˙x)] = [x,˙x,f(x,˙x)].

The mapping

is independent of the choice of representatives (x,˙x)∈[(x,˙x)] by virtue of the k-

homogeneity of f.Moreover, (Π◦

)[(x,˙x)] = [(x,˙x)] for all (x,˙x)∈Q,which makes

a well deﬁned

local section of Y.

The correspondence f7→

is obviously injective. To prove surjectivity, pick an arbitrary

∈Γ(Y)and

deﬁne f(x,˙x),for every representative (x,˙x)∈[(x,˙x)],as the third component yof the representative

(x,˙x,y)∈

[(x,˙x)]; then, f(x,

˙x) =

kyby the deﬁnition of equivalence classes in Y,which means

that fis a k-homogeneous section of ◦

Note: Coming back to our discussion before Remark 22. Since we ﬁxed the group action of R∗

+, deﬁned

in (58), (59), on Y, on each ﬁbered chart domain V=Π−1(U+),we can explicitly introduce homogeneous

ﬁbered coordinates as the local coordinates

xi,˙xi,y

:= (xi,˙x3u0,˙x3u1,˙x3u2,˙x3;(˙x3)kz

)

of an arbitrarily chosen representative of the class [x,˙x,y]where u0=˙x0

˙x3etc. These are, obviously unique up

to positive rescaling, i.e., xi,˙xi,y

and (xi,

˙xi,

)will represent the same class.

1. Finsler functions L:A→R.In this case, ◦

Y=◦

TM ×Ris a trivial line bundle, which means the

conﬁguration bundle Y=◦

Y/∼is the space of orbits of the Lie group action H:R∗

+×◦

Y→◦

Ygiven by

the ﬁbered automorphisms:

:◦

Y→◦

Y,H

(x,˙x,y) = (x,

˙x,

2y),∀

>0.

This way, 2-homogeneous Finsler functions are identiﬁed with local sections

L7→

∈Γ(Y),

[(x,˙x)] = [x,˙x,L(x,˙x)].

In homogeneous ﬁbered coordinates, the class [x,˙x,L(x,˙x)] is represented as (xi,˙xi,L(x,˙x)).

Mathematical foundations for ﬁeld theories on Finsler spacetimes 27

2. The 1-particle distribution function of a kinetic gas, see31,72, can be understood as a 0-homogeneous

mapping

:Q→R,deﬁned on some conic subbundle Q⊂◦

T M.Again ◦

Y=◦

T M ×Rand the con-

ﬁguration bundle is Y= ( ◦

T M ×R)/∼,i.e. the space of orbits of the Lie group action H:R∗

+×◦

Y→◦

is given by

:◦

Y→◦

Y,H

(x,˙x,y) = (x,

˙x,y),∀

>0.

The corresponding section

:Q+→Y,

[(x,˙x)] = [x,˙x,

(x,˙x)] is represented in ﬁbered homogeneous

coordinates as:

:xi,˙xi7→ (xi,˙xi,

(x,˙x)).

3. 0-homogeneous metric tensors g:A→T0

2(◦

T M),g(x,˙x)=gi j (x,˙x)dxi⊗dx j(which are thus treated

as tensors of type (0,2) on ◦

T M,operating on horizontal vector ﬁelds X∈Γ(HT M ),see Section II C),

are obtained as sections

of the bundle Y=◦

Y/∼,where ◦

Y=T0

2(◦

T M). The Lie group action H:

R∗

+×◦

Y→◦

Yis given by

:◦

Y→◦

Y,H

(x,˙x,y) = (x,

˙x,y),∀

>0.

In ﬁbered homogeneous coordinates (which are naturally induced by the coordinates xion M), these

sections are represented as

:xi,˙xi7→ (xi,˙xi,gi j (x,˙x)).

Homogeneous d-tensors of any rank and any homogeneity degree can be treated similarly.

V. FINSLER FIELD LAGRANGIANS, ACTION, EXTREMALS

Finally, we are in the position to explicitly construct action based ﬁeld theories on Finsler spacetimes. The

Finsler-related geometric notions have been introduced in Section II. Afterwards, we discussed the proper

base manifold, PT M+,for action integrals having homogeneous ﬁelds as dynamical variables in Section III,

and we demonstrated that these homogeneous ﬁelds can be understood as sections into ﬁber bundles over

PTM+in Section IV.

A. Actions for ﬁelds as sections of PT M+

We now display all necessary deﬁnitions needed for well deﬁned action based ﬁeld theories on Finsler

spacetimes.

Deﬁnition 25 (Fields) A homogeneous ﬁeld on a Finsler spacetime (M,L)is a local section

of a ﬁbered

manifold (Y,Π,PT M+)over the positive projective tangent bundle PTM+.

Deﬁnition 26 (Lagrangians) On a conﬁguration bundle (Y,Π,PT M+)over a Finsler spacetime (M,L), a

Finsler ﬁeld Lagrangian of order r is a Πr-horizontal 7-form

+∈Ω7(JrY).

This deﬁnition is a particular instance of the general deﬁnition of Lagrangians given in Appendix A 3.

In homogeneous ﬁbered coordinates, any Lagrangian on Ycan be expressed as:

+=ΛdΣ+=LVol0.(61)

Mathematical foundations for ﬁeld theories on Finsler spacetimes 28

where Λ=Λ(xi,˙xi,y

,i,..., y

·i1...·ir)is the Lagrange function and dΣ+is an invariant volume form on an

appropriately chosen open subset Q+⊂PT M +; for instance, one can choose the canonical volume form

(37) on the set A+

0⊂PT M+of non-null admissible directions over a Finsler spacetime; in this case, we

obtain the Lagrange density

L=Λ|detg|

L2.(62)

Note. The pulled back form Jr

∗

+(where

∈Γ(Y)) is a differential form on PT M +,hence, it must be

invariant under positive rescaling in ˙x. In coordinates, this becomes equivalent to the result below.

Proposition 27 In local homogeneous coordinates corresponding to any ﬁbered chart (Vr,

r)on JrY , any

Finsler ﬁeld Lagrangian function Λ:Vr→Rmust obey:

˙xi˙

diΛ=0.(63)

Proof. Pick an arbitrary section of Π,say,

:U→Y,where U⊂PT M+is a local chart domain. The

function Λ◦Jr

is then deﬁned on a subset of PT M+,hence, it must be 0-homogeneous in ˙x; that is,

˙xi˙

∂

i(Λ◦Jr

) = 0.

But, from (52), ˙

∂

i(Λ◦Jr

) = ( ˙

diΛ)◦Jr+1

.Substituting into the above relation and taking into account the

arbitrariness of

, we get the result.

The action attached to the Lagrangian (61) and to a piece D+⊂PT M+is the function SD+:Γ(Y)→R,

given by:

SD+(

) = Z

D+

∗

By Proposition 20, such action integrals on timelike domains D+can equivalently be understood as inte-

grals over pieces D⊂O, i.e. as actions formulated on the observer space. The advantage in the represen-

tation of the action as integrals on PT M +, is that the domain of the integral does not depend on the Finsler

Lagrangian.

The preparation from the previous sections, in particular, the formulation of ﬁelds as sections of a conﬁg-

uration bundle (Y,Π,PT M+), allows us now to straightforwardly apply the coordinate-free formulation of

the calculus of variations for Finsler ﬁeld Lagrangians.

The variation of the action under the ﬂow {Φ

}of a doubly projectable vector ﬁeld Ξ∈X(Y)is given

by the Lie derivative, see Appendix A:

ΞSD+(

) = Z

D+

∗LJrΞ

+.(64)

A ﬁeld

∈Γ(Y),[(x,˙x)] 7→

[(x,˙x)] on a Finsler spacetime is a critical section for S,if for any piece

D+⊂PT M+and for any Π-vertical vector ﬁeld Ξsuch that supp(Ξ◦

)⊂D+:

ΞSD(

) = 0.

For any Lagrangian

+∈Ω7(JrY),there exists (see71, or Appendix A) a unique source form E

+∈

Ω8(JsY)with s≤2r,called the Euler-Lagrange form of

+,such that:

∗(LJrΞ

+) = Js

∗iJsΞE

+−d(Js

∗JΞ),(65)

Mathematical foundations for ﬁeld theories on Finsler spacetimes 29

for some JΞ∈Ω6(JsY).The 6-form JΞ(which is interpreted as a Noether current), is only unique up to

a total derivative; in integral form, the above relation reads:

D+

∗(LJrΞ

+) = Z

D+

∗iJsΞE(

+)−Z

∂

D+

∗JΞ.(66)

In a local contact basis,(47), E

+is thus given as:

+=E

σθσ

∧Vol0.

The precise meaning of the requirement that E

+is a source form is that the interior product

iJsΞE

+= ( ˜

)Vol0,

only depends on the functions ˜

=Ξ

−

,i−˙

·i, not on higher order components of Ξ.

In order to identify the Euler-Lagrange form, Π-vertical variation vector ﬁelds Ξ=Ξ

σ∂σ

are sufﬁcient.

More general transformations will, yet, be used when determining energy-momentum tensors.

The ﬁeld equations of

+are then given by

◦Js

=0.

B. Finsler gravity sourced by a kinetic gas

As an example for a ﬁeld theory on Finsler spacetimes we discuss in the jet bundle language the dynamics

of a Finsler spacetime sourced by a kinetic gas - a theory which is considered as an extension of general

relativity30,31,43,72.

We ﬁst discuss the purely geometric (vacuum) ﬁeld theory, where the Finsler function Litself is the

dynamical ﬁeld, and then add a matter Lagrangian as source of these dynamics.

1. Finsler gravity Lagrangian

We have shown above that, for theories using the 2-homogeneous Finsler function L:A→Ras the

dynamical variable, the appropriate conﬁguration bundle is (60), with ﬁber R; we will re-denote it here as

(Yg,Πg,PT M+)and the homogeneous coordinates corresponding to a ﬁbered chart on Ygby (xi,˙xi,ˆ

L), i.e.,

comparing to the notations in Section IV,y1=ˆ

L. The hat is meant to distinguish the last coordinate function

on Ygfrom mappings L:A→R,L=L(x,˙x),i.e., from components of sections

of the conﬁguration

bundle; more precisely, L=ˆ

L◦

Brieﬂy, we have:

Yg:= ( ◦

T M ×R)/∼,Πg:[(x,˙x,ˆ

L)] 7→ [(x,˙x)].

As already said above, we identify 2-homogeneous functions L:A→R(where A⊂T M is a conic sub-

bundle) with sections

∈Γ(Yg),

L7→

:A+→Yg,

[(x,˙x)] = [x,˙x,L(x,˙x)] .

Locally,

is described as: (xi,˙xi)7→ (xi,˙xi,L(x,˙x)).

Mathematical foundations for ﬁeld theories on Finsler spacetimes 30

On Yg,a Lagrangian of order ris expressed in ﬁbered homogeneous coordinates as

+=ΛdΣ+,where

dΣ+is the canonical volume form (37) and Λ=Λ(xi,˙xi,ˆ

L,ˆ

L,i,ˆ

L·i,... ˆ

L·i1...·ir)becomes a 0-homogeneous

function of ˙xwhenever evaluated along sections

∈Γ(Y). The local contact basis of Ω(JrY)is then denoted

by {dxi,d˙xi,

,i,

·i,...,

·i1...·ir−1,dˆ

L,i1...,ir,...dˆ

L·i1....·ir}, where the (unique) ﬁrst order contact form is:

=dˆ

L−ˆ

L,idxi−ˆ

L·id˙xi.(67)

On Yg,it is convenient to use formal adapted derivatives:

δi:=di−Gj

i˙

dj,

where the word "formal" means that Gj

i∈F(Yg)are considered as functions on a chart of the jet bundle

JrY- constructed by the usual formula from the coordinate functions ˆ

L;ˆ

L,i;...;ˆ

L·i·j; i.e., only when evaluated

along sections

,they become the usual canonical nonlinear connection coefﬁcients, deﬁned on charts of

T M. In particular, we get:

δiˆ

L=0.(68)

Using the latter relation and (54), the contact form

can be written in a manifestly covariant form:

=dˆ

L−ˆ

L·i

˙xi.(69)

Source forms on JrYgare locally expressed as:

∧dΣ+,

where f=f(xi,˙xi,ˆ

L,ˆ

L,i,ˆ

L·i,... ˆ

L·i1...·ir).

On the bundle Yg,the following Lagrangian is a natural (generally covariant) one:

g=R0dΣ+,

where, again, R0is constructed by means of the usual formula, in terms of the the coordinate functions ˆ

L,ˆ

L,i

etc.; using (13), (14) and (15), we ﬁnd that

gcontains fourth order derivatives of L, i.e.,

g∈Ω7(J4Yg).

Naturality of this Lagrangian follows taking into account that, along any section, both R0(which is an

invariant scalar, since it is constructed using only operations with d-tensors) and dΣ+are invariant under

coordinate changes on T M induced by arbitrary coordinate changes on M, see also the Appendix A 3 for a

discussion of natural Lagrangians.

The Euler-Lagrange form of

gis E

+=E

∧dΣ+, where,43:

E=1

2gi j(LR0)·i·j−3R0−gLi j (Pi|j−PiPj+ (∇Pi)·j),(70)

and thus the ﬁeld equation, which determines the extremal points of the action is: E=0.

2. Kinetic gas Lagrangian

It turned out that there exists a physical ﬁeld which naturally couples to Finsler geometry and can act as

source for the dynamics of a Finsler spacetime. This ﬁeld is the 1-particle distribution function (1PDF)

a kinetic gas, which describes the dynamics of a kinetic gas on the tangent bundle of spacetime73–75.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 31

Usually, the gravitational ﬁeld of a kinetic gas is described in terms of the Einstein-Vlasov equations76,

which, however, only take the averaged kinetic energy of the particles constituting the gas into account. By

coupling the 1PDF of the kinetic gas directly to the Finslerian geometry of spacetime, this averaging can be

omitted; thus, the velocity distribution of the gas particles contributing to the gravitational ﬁeld can be fully

taken into account31,72.

A kinetic gas is deﬁned as a collection of a large number of particles, whose properties are encoded into

1PDF, i.e. a function

:O→R,

(x,˙x).

Its interpretation is the following. The number of particles crossing a given (6-dimensional) hypersurface

⊂Ois

] = Z

vol,(71)

where vol =1

3!d

∧d

is the canonical invariant volume form on

, determined by the Lorentzian

(or pseudo-Finslerian) structure on spacetime. This volume form induces a coupling between the geometry

of spacetime and the 1PDF. Prolonging

to TM by 0-homogeneity as discussed in Section IV C, we can

equivalently regard

as a function deﬁned on O+=

+(O)⊂PTM+. The partial functions

(x,·)

are all assumed to be compactly supported (which is physically interpreted as the fact that the speeds of the

particles composing the gas have an upper bound lower than the speed of light).

The Lagrangian deﬁning the dynamics of the kinetic gas on a Finsler spacetime is, see31,

m=m

dΣ+=m

|det(g)|

L2Vol0=LmVol0(72)

where the Lagrange density Lm=m|det(g)|

depends on xi,˙xi,ˆ

L,..., ˆ

L·i·jand mis the mass parameter of the

gas particles, here assumed all of the same mass for simplicity. The ˆ

L,..., ˆ

L·i·jdependence in the Lagrange

density appears due to the dependence of the volume form on the Finsler Lagrangian and on the Finsler

metric tensor g. Yet, for the sake of uniformness (since we will couple it to

g,which lives on J4Yg), we

will regard Lmas a function on J4Yg,rather than on J2Yg.

Consider on J4Ygthe Lagrangian

+=1

then, calculation of the Euler-Lagrange form by variation with respect to Lleads,31,72, to the Finsler gravity

equations sourced by a kinetic gas:

2gi j(LR0)·i·j−3R0−gLi j (Pi|j−PiPj+ (∇Pi)·j) = −

,(73)

where

is the gravitational coupling constant.

We note that the above equation determines nonzero values of L; accordingly, in the construction of the

actions correspond ing to

gand

m, one must only consider non-lightlike domains for L. This is a difference

from actions of metric ﬁeld theories built directly over the spacetime manifold M, which do not distinguish

between possible causal properties of vectors.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 32

VI. ENERGY-MOMENTUM DISTRIBUTION TENSOR

An important concept in physics, which is derived from the action of a ﬁeld theory, is the the energy-

momentum tensor. One way to interpret the energy momentum tensor mathematically is that it measures77,

“the response of the matter Lagrangian to compactly supported diffeomorphisms of spacetime”. This in-

terpretation will be preserved in Finslerian ﬁeld theory. In other words, naturality (general covariance, or

general invariance78) of Lagrangians will still be understood as invariance under (lifted) diffeomorphisms

of spacetime - though, in this case, the base of our conﬁguration bundle is not spacetime, but its positively

projectivized tangent bundle. This will require an extension of the technique presented in79 and will result

in a “weaker” (averaged) energy-momentum conservation law.

A. Generally covariant Lagrangians

To identify the energy-momentum tensor in our construction of ﬁeld theories on Finsler spacetimes, we

need some preparations:

1. Lifts of diffeomorphisms

0of Minto doubly ﬁbered automorphisms of Y, that cover the natural

lifts4of

0to PTM+, see the diagram (55), since a priori diffeomorphisms of Mdo not act on Y.

2. A splitting of the total Lagrangian

+of the theory into a background (vacuum) Lagrangian and

a matter one and, accordingly, of the variables of the theory into background and dynamical ones.

The background Lagrangian (which we denote by

g) will only depend on the background variables

(e.g., metric components, Finsler function etc), whereas the matter Lagrangian

mwill depend on

all the variables, see66. Roughly, denoting the background coordinates by y

Band non-background or

dynamical variables y

D, we have:

+(y

B,..., y

B,i...·j,y

D,...y

D,i...·j) =

g(y

B,..., y

B,i...·j) +

m(y

B,..., y

B,i...·j,y

D,...y

D,i...·j).

For instance, in general relativity, one has y

B=gi j, whereas y

Dcan be, e.g., the electromagnetic 4-

potential. The names "background" vs. "dynamical" come from the fact that the Lagrangian can be

split into a part

g, which only contains the background variables, and a part

mwhich contains all

information about the dynamical variables and their coupling to the background variables. Hence,

even if one leaves

gaside and ﬁxes a value of the background ﬁelds, one can study the dynamics of

the dynamical ﬁelds coupled to a ﬁxed background.

Then, under the assumption that the matter Lagrangian

mis generally covariant (see again the end

of Appendix A 3), it will be invariant under any one-parameter group of canonical lifts of diffeo-

morphisms of the spacetime manifold M, thus giving rise to conserved Noether currents JΞ(where

Ξ=F(

0), is the canonical lift to Yof a diffeomorphism generating vector ﬁeld

0from M). Roughly

speaking, the energy-momentum tensor will be given by the correspondence

07→ JΞ.

In the case of Finsler spacetimes, the fundamental background variable is the Finsler Lagrangian Litself.

Yet, the whole construction can be done in a completely similar manner, e.g., for the Finsler metric tensor

components gi j ,as background variables.

4Such lifts exist, e.g. when Yis a bundle of k-homogeneous d-tensors, which is the R∗

+-orbit space of a bundle ◦

Yof d-tensors on ◦

TM.

Diffeomorphisms

0of Mare naturally lifted into ﬁbered automorphisms d

0of TM and further, by tensor lifting to ◦

Mathematical foundations for ﬁeld theories on Finsler spacetimes 33

Consider a ﬁbered product

Y:=Yg×PTM+Ym

over PT M+, where Yg= ( ◦

T M ×R)/∼was constructed in Section V B 1 and Ymis both a a ﬁber bundle over

PTM+and a natural ﬁber bundle over M. In particular, Yhas a double ﬁbered manifold structure:

YΠ

−→ PT M +

−→ M

We denote the homogeneous coordinates corresponding to a doubly ﬁbered chart on Yby (xi,˙xi,ˆ

L,y

D),

where yB=ˆ

Lis the coordinate on the ﬁber of Ygand y

Dare local coordinates on the ﬁber of Ym.

As both Ymand Ygare natural bundles over M it follows that any vector ﬁeld

0∈X(M)admits a

canonical lift Ξ∈X(Y).

Consider a generally covariant Lagrangian

m∈Ω(JrY):

m=Lm(xi,˙xi,ˆ

L,ˆ

L,i,ˆ

L·i,..., ˆ

L·i1...ir,y

D,..., y

D·i1...ir)Vol0,

which will be interpreted as the matter Lagrangian (as already mentioned above, the total Lagrangian of the

theory will be obtained as

+:=

m). Since

mis generally covariant, for any compactly supported

vector ﬁeld

0∈X(M),

mis invariant under the ﬂow of the r-th jet prolongation of the canonical lift

Ξ:=F(

0),i.e.,

LJrΞ

m=0.(74)

In the following, we will explore in detail the consequences of this invariance of

B. The energy momentum distribution tensor and the energy momentum density

We will ﬁrst give the technical precise deﬁnition of the energy-momentum distribution tensor, and demon-

strate the concept on the example of the kinetic gas at the end of this subsection.

Assume {

}is a 1-parameter group of compactly supported diffeomorphisms of M,generated by

0∈

X(M),

∂

i.Then:

1. Each

is ﬁrst naturally lifted to T M ,as

φε

:=d

.The generator of {

φε

}is the complete lift

∈X(◦

T M)of

∂

i+˙

i˙

∂

i,˙

,j˙xj.(75)

Since the canonical lift

is 0-homogeneous, we can identify it with a vector ﬁeld on PT M+(more

precisely, with its pushforward by

+), see Section III A 2.

2. Further, taking into account that Yg= ( ◦

T M ×R)/∼is obtained as a quotient space of the trivial bundle

◦

T M ×R,the canonical lift,66 Φg,

:Yg→Ygof

φε

is also a trivial one i.e., it acts on the ﬁber variable

Las the identity:

Φg,

[(x,˙x,ˆ

L)] = [(

φε

(x,˙x),ˆ

L)];

Mathematical foundations for ﬁeld theories on Finsler spacetimes 34

The above mapping is well deﬁned (i.e., independent on the choice of the representative of the class

[(x,˙x,ˆ

L)]), due to the linearity of

φε

in ˙x.As the lifted diffeomorphisms act trivially on ˆ

L, the generator

is canonically lifted into a vector ﬁeld Ξg∈X(Yg),with vanishing

∂

Lcomponent, i.e.:

Ξg=

∂

i+˙

i˙

∂

i+0

∂

3. According to our ﬁrst assumption at the beginning of Section VI A, there exists a canonical lift

to Ym,

into some vector ﬁeld Ξmof the form Ξ=

∂

i+˙

i˙

∂

i+Ξ

σ∂

∂

D.All in all, we obtain that the canonical

lift of

0∈X(M)to the ﬁbered product Y=Yg×PTM+Ymis expressed in a ﬁbered chart by adding to

the contributions describing the transformation of each of the ﬁber variables

Ξ=

∂

i+˙

i˙

∂

i+0

∂

L+Ξ

σ∂

∂

D,(76)

where, see77,Ξ

are functions of the coordinates xi,˙xi,y

D,..., y

D·i1...irand of a ﬁnite number of the

derivatives of

First variation formula.

Accordingly, the Euler-Lagrange form E(

m)will be split into a Ygand a Ym-component as

m) = Eg(

m) + Em(

m),

where:

Eg(

m) =

∧Vol0,Em(

m) =

θσ

D∧Vol0,(77)

and

θσ

D=dy

D−y

D,idxi−y

D·id˙xi. Since hLJrΞ

m=0 (which follows from the invariance condition (74)),

this leads to:

hiJsΞEg(

m) + hiJsΞEm(

m)−hdJΞ=0.

But, on-shell for the variables y

D,i.e., along sections

:= (L,

m)such that the “matter ﬁeld”, i.e. the section

m:PTM+→Ym,xi,˙xi7→ xi,˙xi,y

Dxi,˙xi, is critical for

m,the Em-term above vanishes, i.e.:

hiJsΞEg(

m)−hdJΞ≃

m0,(78)

where ≃

mmeans equality on-shell for the matter ﬁeld

The energy-momentum distribution tensor.

The surviving Euler-Lagrange component hiJsΞEg(

m)in (78) can again be split into a linear expression

iand a divergence expression; the latter will couple with hdJΞinto a boundary term and will provide

the building block of the energy-momentum distribution tensor Θ. More precisely,

Lemma 28 For any natural Finsler ﬁeld Lagrangian

m∈Ω7(JrY),there exist unique F(M)-linear map-

pings Θ:X(M)→Ω(JsY),B:X(M)→Ω(Js+1Y),with Πs(respectively, Πs+1)-horizontal values (where

s≤2r)such that, for any

0∈X(M):

hiJsΞEg(

m) = B(

0) + hdΘ(

0).(79)

Mathematical foundations for ﬁeld theories on Finsler spacetimes 35

Proof. We will ﬁrst construct Θand Bin a ﬁbered chart and then, show that the obtained expressions are

independent on the choice of this chart. In any ﬁbered chart, Egis expressed as:

Eg(

m) =

∧Vol0=:−1

2Tˆ

L−1

∧dΣ+,(80)

where Tis a 0-homogeneous scalar which acts as source term for Finsler gravity equations (70), and the

factor ˆ

L−1is introduced to ensure this degree of homogeneity (as both ˆ

L−1

and dΣ+are 0-homogeneous.)

The precise expression of Tdepends on the chosen volume form. For instance, if dΣ+is the canonical

volume form (37), then:

T=−2ˆ

|detg|

L.(81)

Since

+is a natural Lagrangian,

Lis a scalar density and, accordingly, Tis a scalar invariant. Then,

iJsΞEg(

m) = (−1

2Tˆ

L−1iJsΞ

)dΣ++1

2(Tˆ

L−1)

∧iJsΞdΣ+; since the last term is a multiple of

, it is a

contact form; the remaining component is thus the horizontal component hiJsΞEg(

m)and can be expressed

(up to a pullback by Πs+1,sof the right hand side) as:

hiJsΞEg(

m) = −1

2(Tˆ

L−1iJsΞ

)dΣ+.(82)

Further, using

=dˆ

L−ˆ

L,idxi−ˆ

L·id˙xi=dˆ

L−δiˆ

Ldxi−ˆ

L·i

˙xiand δiˆ

L=0,we ﬁnd:

=dˆ

L−ˆ

L·i

˙xi=dˆ

L−2˙xi(d˙xi+Gijdx j);

inserting into

the lift (76) of

0,this becomes:

iJrΞ

=−2˙xi(˙

i+Gij

j) = −2 ˙xi(

i,j˙xj+Gij

j) = −2 ˙xi∇

where in the second equality we used: ˙

i,j˙xj.We can thus rewrite (82) as:

hiJsΞEg(

m) = Tˆ

L−1˙xi∇

idΣ+.

Taking into account that ∇˙xi=0 and ∇ˆ

L=0, this can be uniquely split into a linear term in

iand the

divergence of a linear term in

hiJsΞEg(

m) = [∇(Tˆ

L−1˙xi

i)−

i˙xiˆ

L−1∇T]dΣ+.(83)

Then, using ˙xi

|j=0 and ∇=˙xjD

j,we can rearrange the divergence term as

∇(Tˆ

L−1˙xi

i) = (Tˆ

L−1˙xi˙xj

i)|j,(84)

which suggests the notation:

Θj

i:=Tˆ

L−1˙xj˙xi(85)

As Tis a scalar invariant, the functions Θj

i,deﬁned on the given ﬁbered chart, transform under induced

ﬁbered coordinate changes as the components of a tensor on M(equivalently, as d-tensor components on

Mathematical foundations for ﬁeld theories on Finsler spacetimes 36

T M). Also, noticing that the last term in (83) can be written as:

i˙xiˆ

L−1∇T=

i˙xiˆ

L−1˙xjT|j=Θj

i|j

i, this

suggests to introduce the mappings Θ:X(M)→Ω6(JsY),B:X(M)→Ω7(Js+1Y)given by

Θ(

0) = (Θj

i)i

jdΣ+,(86)

0) = −Θj

i|j

idΣ+,(87)

(where

∂

i). These mappings are well deﬁned, i.e., independent on the chosen coordinate charts;

moreover, they have Πs(respectively, Πs+1)-horizontal values, they are both linear in

and obey (79),

which completes the proof of the existence. Uniqueness of Band Θfollows from the uniqueness of the

splitting (83) and the arbitrariness of

Note. The proof of the above result is based on a similar idea to the one of Lemma 2 in79. The essential

difference, in the Finslerian case, is that naturality of Lagrangians is based on the group of diffeomorphisms

of M(and not of PT M+as one would have expected following79), i.e., naturality comes from a manifold of

lower dimension than the one of the base space of our conﬁguration manifold Y. This will result, as we will

see below, in a “weaker” (averaged) form of the energy-momentum balance law.

Actually, taking into account (85), in homogeneous ﬁbered coordinates, Θis expressed as:

Θ=Θijdx j⊗i

idΣ+=T(ˆ

F·jdx j)⊗ili

idΣ+,(88)

where ˆ

F=q|ˆ

L|; a quick computation shows that ˆ

L−1˙xi˙xj=ˆ

F·jli,regardless of the sign of ˆ

L. Equivalently,

in a coordinate-free writing:

Θ=T

+⊗iℓdΣ+,(89)

where we have identiﬁed, by abuse of notation, the Reeb vector ﬁeld ℓ=li

i∈X(A+

0)with the vector ﬁeld

on Js+1Yobtained by replacing

iwith the formal total adapted derivative δi,i.e., with: liδi∈X(Js+1Y). In

the same fashion, the values

[(x,˙x)] of the mapping

+:A+

0→Ω1(M)are identiﬁed with their pullbacks

to Js+1Y.

Deﬁnition 29 (Energy-momentum distribution tensor) The energy-momentum distribution tensor associ-

ated to a natural Lagrangian

mon a bundle Y =Yg×PTM+Ym, which is natural over a Finsler spacetime

M, is the F(M)-linear mapping Θ:X(M)7→ Ω6(JsY)deﬁned by (89).

Deﬁnition 30 (Energy-momentum scalar) We call the function T:A+

0→R, deﬁned by the relation (80),

and explicitly given by (81), the energy-momentum scalar.

We will call the F(M)-linear mapping B:X(M)7→ Ω7(Js+1Y)deﬁned by (87), the balance function,

as energy-momentum conservation (or energy-momentum balance) law is naturally characterized in terms

of B,as we will see below.

Averaged energy-momentum conservation law.

Consider, in the following, local sections

∈Γ(Y)such that supp(Js

∗

m)⊂T+. This way, it makes

sense to integrate the form Js

∗iJsΞEg(

m)on the entire set T+

x=O+

xof timelike directions at x.

Consider a piece D0⊂Mand denote by

T+(D0):=∪

x∈D0T+

x=∪

x∈D0O+

Mathematical foundations for ﬁeld theories on Finsler spacetimes 37

the set of all timelike (equivalently, of observer) directions corresponding to D0.Then, (79) becomes, with

:= (L,

m):

T+(D0)

∗iJsΞEg(

m) = Z

T+(D0)

Js+1

∗B(

0) + Z

∂

T+(D0)

∗Θ(

0).(90)

But, on-shell for

m,we have, according to (78): Js

∗iJsΞEg(

m)−Js

∗dJΞ≃

m0; substituting into the

above relation, this gives:

T+(D0)

Js+1

∗B(

0) + Z

∂

T+(D0)

∗(Θ(

0)−JΞ)≃

m0.(91)

We are now able to prove the following result.

Theorem 31 Consider a bundle Ymover PTM+,which is natural over M, and an arbitrary section

(L,

m)∈Γ(Yg×PT M+Ym)such that supp(Js

∗

m)⊂T+, then:

1. Averaged energy-momentum conservation law: At any x ∈M and in any corresponding ﬁbered chart:

T+

(Θj

i|j◦Js+1

)dΣ+

x=0,(92)

where dΣ+=:d4x∧dΣ+

2. Θ(

0)is a “corrected Noether current”, i.e., for any

0∈X(M)

∂

T+(D0)

∗Θ(

0) = Z

∂

T+(D0)

∗JΞ,(93)

where Ξdenotes the canonical lift of

0to Y.

Proof.

1. Fix x0∈M.Consider an arbitrary piece D0⊂Mcontaining x0as an interior point and an arbitrary

0∈X(M)with support contained in D0.

Now, let us have a look at the boundary term in (91). Since the support of the integrand, at every

x∈M, is strictly contained in T+

x, the only possible nonzero values are obtained at points [(x,˙x)] with

x∈

∂

D0.But, at these points,

0identically vanishes (hence also Ξ=0, as Ξis built from

and its

derivatives), which means that this boundary term is actually zero. It follows:

T+(D0)

Js+1

∗B(

0)≃

m0.(94)

In coordinates, this is:

T+(D0)

(Θj

i|j◦Js+1

)

idΣ+≃

m0.

Squeezing D0around x0such that D0is contained into a single chart domain, the above integral

can be written as an iterated integral R

i(R

T+

(Θj

i|j◦Js+1

)dΣ+

x)d4x,which, taking into account the

arbitrariness of

i, leads to the result.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 38

2. follows then immediately from (91) and 1.

Relation (93) says that, the energy-momentum tensor Θ(

0)is, at least up to a term which does not

contribute to the integral (93)), the conserved Noether current JΞ- i.e. (see also77), it gives the correct

notions of energy and momentum of the system under discussion.

Remark 32 Taking into account that O+

x=T+

x, the averaged conservation law can be rewritten as:

O+

(Θj

i|j◦Js+1

)dΣ+

x=0.(95)

It is worth noting that, due to the fact that naturality of Lagrangians comes from M, which is a space of

lower dimension than the one of the space PT M+on which the action integral is considered, in the above

relation, integration over O+

x(or, equivalently, T+

x) cannot be removed, i.e., we can typically only establish

an averaged conservation law. This is a distinctive feature of Finslerian ﬁeld theory.

Energy-momentum density on M

The mapping Θ:X(M)→Ω(JsY)gives rise to an energy-momentum tensor density on M, by averaging

over observer (or timelike) directions O+

+(Ox).Consider an arbitrary ﬁbered chart on Y;Θ(

) =

Θij

j⊗i

idΣ+. Then, for any section

∈Γ(Y)such that supp(Js

∗

m)⊂T+, set

j(x):=Z

O+

(Θij◦Js

)|(x,˙x)dΣ+

x,∀x∈M.(96)

Under the above assumption this integral is ﬁnite, so the result is well deﬁned. Moreover, given the expres-

sion of dΣ+

x,the functions Ti

j(x)represent the components of a tensor density on M.

Example: the energy momentum distribution tensor of a kinetic gas.

The kinetic gas example, which motivated the whole above construction, has been previously presented

from a somewhat pedestrian perspective. In31 (Eqs. (42)-(43) ), the maps Θand B, can be read off. We brieﬂy

identify these maps here from the more abstract and mathematically precise construction we presented.

In the case of a kinetic gas discussed in Section V B 2, the kinetic gas Lagrangian (72) is given by

m=m

dΣ+,where

is the 1-particle distribution function, reinterpreted as a function of x,˙x,Land its

derivatives,

(x,˙x) = f(x,˙x,L(x,˙x),..., L·i·j(x,˙x))

and dΣ+is chosen as the canonical volume form (37). Varying

m, we use (81) to obtain T:=1

;

accordingly, the energy-momentum tensor distribution Θhas the local components, compare to (85),

Θij=1

2Lm

˙xi˙xj.

For any kinetic gas, the averaged conservation law (95) holds.

In particular, for collisionless gases, it is known that

is subject to the Liouville equation ∇

=0,

equivalently:

Dℓ

=0.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 39

Taking into account that li

|j=0, we notice that the Liouville equation is nothing else than a pointwise

covariant conservation law of Θ:

iΘij=0.

Particular case: Lorentzian spaces.

On a Lorentzian manifold (M,a), the quantities

Tij=1

p|deta|Ti

j(97)

represent the components of a tensor of type (1,1) on M,and their Levi-Civita covariant derivatives are,80,

just the integrals of the Chern covariant derivatives of Θ:Ti

j;i= (p|deta|)−1R

O+

∗Θij|i(x,˙x)dΣx.Hence,

the energy-momentum conservation law (92) reads

Tij;i=0.

In the particular case of kinetic gases on a Lorentzian spacetime, our expression (96) of the energy-

momentum density agrees to the known one, see75.

It is important to note that, in general Finsler spacetimes, we have no metric tensor on M,hence (97)

makes no sense. All we can get is an energy-momentum tensor density on M,by averaging over observer

directions as in (96) and, accordingly, the conservation law (95) of the energy-momentum distribution Θ.

VII. SUMMARY AND OUTLOOK

In this article we have proposed a general framework for action based ﬁeld theories on Finsler spacetimes.

The starting point of our construction is the assumption that physical ﬁelds are homogeneous sections of

suitable bundles deﬁned over (conic subbundles of) the tangent bundle of a Finsler spacetime. Using the

assumption of homogeneity, we have constructed an equivalent description of ﬁelds as sections of bundles

over the positive projective tangent bundle PT M+instead. This step is crucial for a well-deﬁned application

of the variational principle, as it allows for variations with compact support within PT M+, which is not

possible in the aforementioned approach using homogeneous sections over the tangent bundle. Within this

framework, we studied the implications of general covariance, and derived the corresponding conserved

energy-momentum distribution. As a particular example, we studied the kinetic gas.

Since the framework we propose is kept very general, it can be applied to a wide range of conceivable

theories. The most natural class of ﬁelds to study, besides the kinetic gas, would be d-tensor ﬁelds. The

latter provide a simple generalization of tensor ﬁelds on the spacetime manifold, which attain a dependence

on directions in the tangent space, in addition to their dependence on spacetime. This additional dependence

could be employed to model a velocity-dependent interaction between such ﬁelds with observers or particles.

Such a dependence would be expected in an effective description of the quantum nature of spacetime, and

leads to a modiﬁed dispersion relation for highly energetic particles, which could possibly be detected in

observations. An ongoing effort is to extend our construction to a well deﬁned notion of spinors and spinor

ﬁeld theories on general Finsler spacetimes.

Another potential application of our proposed framework is to address the so far unexplained observa-

tions in cosmology. The well-known standard model of cosmology, coined ΛCDM model as it models

95% of the matter content of the universe as dark energy Λand cold dark matter (CDM), both of which

Mathematical foundations for ﬁeld theories on Finsler spacetimes 40

have so far eluded direct detection, is under growing tension due to discrepancies between the measured

values of the Hubble parameter in different observations. The correct interpretation of these observations

depends crucially on understanding the propagation of electromagnetic radiation (and, with the advent of

multi-messenger astronomy, also of gravitational waves, neutrinos and high-energetic cosmic particles). A

modiﬁed propagation law, as it could arise for a ﬁeld propagating on a Finsler spacetime background, could

therefore provide alternative explanations that might resolve the observed tension.

ACKNOWLEDGMENTS

C.P. was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project

Number 420243324. M.H. was supported by the Estonian Research Council grant PRG356 “Gauge Gravity”

and by the European Regional Development Fund through the Center of Excellence TK133 “The Dark Side

of the Universe”. The authors would like to acknowledge networking support by the COST Action QGMM

(CA18108), supported by COST (European Cooperation in Science and Technology). Also, they would like

to express their thanks to the anonymous JMP referee, for his/her useful comments and questions.

This article may be downloaded for personal use only. Any other use requires prior permission of the

authors and AIP Publishing. This article appeared in the Journal of Mathematical Physics and may be found

at https://aip.scitation.org/doi/10.1063/5.0065944.

DATA AVAILABILITY STATEMENT

The study presented in this article is of purely theoretical and mathematical nature. All results and all

sources on which these results are based are cited. Data sharing is not applicable to this article as no new

data were created or analyzed in this study

Appendix A: Jet bundles and the coordinate-free calculus of variations

In this appendix we brieﬂy present the jet bundle formalism, which allows for a coordinate-free description

of calculus of variations, in terms of differential forms; for more details, we mainly refer to the monograph71.

1. Fibered manifolds and their jet prolongation

Aﬁbered manifold is a triple (Y,

,X),where X,Yare smooth manifolds with dim X=n,dimY=n+m

and

:Y→Xis a surjective submersion. The level sets Yx=

−1(x)are called the ﬁbers of Y.

Any ﬁbered manifold admits an atlas consisting of ﬁbered charts. These are local charts (V,

(xA,y

)such that there exists a local chart (U,

= (xA)on X, with

(V) = U,in which

is represented

:(xA,y

)7→ (xA).

In particular, ﬁber bundles, as understood in81, are ﬁbered manifolds that are locally trivial, i.e., in the

above, each Vis homeomorphic to a Cartesian product U×Z,where Zis a manifold, called the typical ﬁber.

Assume, in the following, that (Y,

,X)is a ﬁbered manifold. Local sections

:U→Y(with U⊂Xopen)

are smooth maps such that

◦

=idX; in a ﬁbered chart, they are represented as:

:(xA)7→ (xA,y

(xA)).

Mathematical foundations for ﬁeld theories on Finsler spacetimes 41

We denote by Γ(Y)the set of sections of (Y,

,X).In the following, capital Latin indices A,B,C,... will run

from 0 to n−1 and Greek indices

,... will run from 1 to m.

Physical interpretation. In ﬁeld theory, these manifolds and quantities are interpreted as follows:

• The manifold Yis called the conﬁguration space.

• The base manifold Xis typically (but not always) interpreted as spacetime; a notable exception to this

rule is Finslerian ﬁeld theory, where X=PTM+is the positively projectivized tangent bundle of the

spacetime manifold M(and the naturality of Lagrangians will be discussed with respect to M). In the

following, we will reserve the notation Mfor manifolds to be interpreted as spacetimes and denote by

Xgeneric base manifolds.

• Sections

∈Γ(Y)are interpreted as ﬁelds.

The jet bundle JrY={Jr

∈Γ(Y),x∈X}is naturally equipped with an atlas consisting of ﬁbered

charts (Vr,

r),

r= (xA,y

C1,...,y

C1C2...Cr)on JrY,induced by ﬁbered charts (V,

),via

C1...Ck(Jr

) =

∂

xC1...

∂

xCk(xA).(A1)

Any section of Yis naturally prolonged into a section Jr

of JrY; in a chart (Vr,

r):

:(xA)7→ xA,y

(xA),

∂

xA(xB),....,

∂

xA1...

∂

(xB).

When referring to local expressions of geometric objects on JrY, we always understand their expressions in

ﬁbered charts (Vr,

r)as above.

JrYis a ﬁbered manifold over all lower order jet bundles JsY,0≤s<r(where J0Y:=Y), with canonical

projections

r,s:JrY→JsY,(xA,y

C1,...,y

C1C2...Cr)7→ (xA,y

C1,...,y

C1C2...Cs).

JrYis also a ﬁbered manifold over X,with projection

r:JrY→X,(xA,y

C1,...,y

C1C2...Cr)7→ (xA).

2. Horizontal and contact forms

Let us introduce the following sets on JrY:

1. Ωk(JrY), the set of differential k-forms deﬁned over open subsets Wr⊂JrY

2. Ω(JrY):=L

k∈N

Ωk(JrY)the set of all differential forms over open subsets Wr⊂JrY;

3. X(JrY):=Γ(T JrY)the module of vector ﬁelds on Wr⊂JrY;

4. F(JrY), the set of all smooth functions f:W→Rdeﬁned on open subsets W⊂JrY.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 42

A differential form

∈Ωk(JrY)is

r-horizontal, if iΞ

=0 whenever Ξ∈X(JrY)is

r-vertical (i.e.,

whenever d

r(Ξ) = 0). In a ﬁbered chart, any

r-horizontal form is expressed as:

A1A2...AkdxA1∧dxA2∧... ∧dxAk,(A2)

where

A1A2...Akare smooth functions of the coordinates xA,y

C1,...,y

C1C2...Cron JrY. Similarly,

r,s-

horizontal forms, 0 ≤s≤rare locally generated by dxA,dy

,..., dy

C1...Cs. A particular example of horizontal

forms are Lagrangians, which we deﬁne in the next subsection.

The horizontalization operator is the unique morphism of exterior algebras h:Ωr(Y)→Ωr+1(Y)such

that, for any f∈F(JrY)and any ﬁbered chart: h f =f◦

r+1,rand

hd f =dAf d xA,(A3)

where dAf:=

∂

Af+

∂

A+...

∂

C1...Cr

C1...CrAis the total derivative (of order r+1) with respect to xA.

On the natural basis 1-forms, it acts as:

hdxA:=dxA,hdy

AdxA, ...,hdy

C1...Ck=y

C1...CkAdxA,k=1,r.(A4)

A useful property is the following. For any f∈F(JrY),

∈Γ(Y):

∂

A(f◦Jr

) = Jr+1

∗dAf.(A5)

A differential form

∈Ω(JrY)is a contact form if Jr

∗

=0,∀

∈Γ(Y). For instance,

θσ

=dy

−y

CdxC,

θσ

A1=dy

A1−y

A1CdxC, ...,

θσ

A1A2...Ar−1=dy

A1A2...Ar−1−y

A1A2...Ar−1CdxC,(A6)

are contact forms on a given chart domain Vr⊂JrY,providing a local basis {dxA,

θσ

,....,

θσ

A1...Ar−1,dy

A1...Ar}

of the module Ω1(JrY),called the contact basis.

Raising to the next “ﬂoor” Jr+1Y,any differential form can be uniquely split as



r+1,r∗

where p

is contact. Intuitively, h

is what will survive of

when pulled back to Xby prolonged sections

Jr+1

,where

∈Γ(Y),while p

becomes invisible: Jr+1

∗(p

) = 0.

In particular, a k-form

∈Ω(JrY)is 1-contact if iΞ

is a

r-horizontal form whenever Ξ∈X(JrY)is

r-vertical; in coordinates, 1-contact forms

can be recognized by the fact that, in their expression in the

contact basis, each term contains exactly one of the contact basis 1-forms

θσ

,...,

θσ

A1...Ardeﬁned in (A6)).

r,0-horizontal, 1-contact (n+1)-form

∈Ωr

n+1Yis called a source form. Locally, a source form is

expressed as:

ησθσ

∧dnx,(A7)

where

ησ

(xA,y

,....y

A1...Ar).

Fibered morphisms.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 43

An automorphism of a ﬁbered manifold (Y,

,X)is,71, a diffeomorphism Φ:Y→Ysuch that exists a

mapping

∈Diff(X)with

◦Φ=

◦

, i.e., the following diagram is commutative:

YΦ//



//X

(A8)

In this case, Φis said to cover

.In coordinates, these must be of the form:

:(xA)7→ ˜xA(xB)(A9)

Φ:(xA,y

)7→ (˜xA(xB),˜y

(xB,y

)).(A10)

The automorphism Φis called strict if

=idX.

Any generator Ξof a 1-parameter group {Φ

}of automorphisms of Yis a

-projectable vector ﬁeld, i.e,

∗Ξis a well deﬁned vector ﬁeld on X; in a ﬁbered chart, projectable vector ﬁelds are represented as:

Ξ=

A(xB)

∂

A+Ξ

(xB,y

)

∂

.(A11)

In particular, 1-parameter groups of strict automorphisms are generated by

-vertical vector ﬁelds Ξ=

(xB,y

)

∂

Automorphisms Φ:Y→Yare prolonged into automorphisms of JrYas: JrΦ(Jr

):=Jr

(x)(Φ◦

◦

−1).

The generator of the 1-parameter group {JrΦ

},with Φ

as above, is called the r-th prolongation of the

vector ﬁeld Ξand denoted by JrΞ.In particular, for r=1,this is given by:

J1Ξ=

∂

A+Ξ

σ∂σ

+Ξ

∂

A,Ξ

A=dAΞ

−y

3. Lagrangians and ﬁrst variation formula

ALagrangian is deﬁned as a

r-horizontal form

∈Ωr

nYof degree n=dimX ; locally,

=Ldnx,L=L(xA,y

,..., y

A1...Ar),(A12)

where dnx:=dx1∧... ∧dxn.

By a piece D ⊂X,we understand,71, a compact n-dimensional submanifold with boundary of X. The

action attached to the Lagrangian (A12) and to a piece D⊂Xis the function SD:Γ(Y)→R,given by:

SD(

) = Z

∗

Consider an arbitrary 1-parameter group {Φ

}of automorphisms of Y,with (

-projectable) generator Ξ∈

X(Y).This will induce a deformation

7→

γε

:=Φ

◦

−1

of sections

∈Γ(Y):

YΦ

//Y

−1

γε

OO(A13)

Mathematical foundations for ﬁeld theories on Finsler spacetimes 44

The variation

ΞSD(

):=d

=0S

φε

(D)(

γε

)is then expressed as the Lie derivative:

ΞSD(

) = Z

∗LJrΞ

.(A14)

A section

∈Γ(Y)is a critical section for S,if for any compact D⊂Xand for any

-projectable Ξ∈X(Y)

such that supp(Ξ◦

)⊂D, there holds:

ΞSD(

) = 0.

For any Lagrangian

∈Ωn(JrY)and any Ξ∈X(Y),there holds the ﬁrst variation formula:

∗(LJrΞ

) = Js

∗iJsΞE

−Js

∗dJΞ,(A15)

where:

•E

∈Ωn+1(JsY)is a source form of order s≤2r,called the Euler-Lagrange form5; locally, if

Ldnx,then:

σθσ

∧dnx,

with:

∂

−dA

∂

+... + (−1)rdA1...dAr

∂

A1...Ar

.(A16)

The section

∈Γ(Y)is critical for

if and only if E

◦Js

=0.

• The (n−1)-form JΞ∈Ωn−1(JsY)is called the Noether current associated with

and to the vector

ﬁeld Ξ.If Ξis a symmetry generator for

,i.e., if LJrΞ

=0,then, Noether’s ﬁrst theorem states that

the Noether current is conserved along critical sections:

∗dJΞ≈0,(A17)

where ≈denotes equality on-shell, i.e., for critical sections

The Euler-Lagrange form of

is unique, while the Noether current JΞis only unique up to an exact

form d

In integral form, the ﬁrst variation formula reads:

∗(LJrΞ

) = Z

∗iJsΞE(

)−Z

∂

∗JΞ.(A18)

Remark 33 1. The fact that E

σθσ

∧dnx is a source form implies that locally, only the

∂

Aand

∂

components of JrΞwill contribute to iJsΞE

(i.e., higher order components of JrΞwill not contribute

to it):

iJsΞE

= ( ˜

)dnx,˜

=Ξ

−y

C.(A19)

The functions (˜

◦Js

):X→Rare commonly denoted in the literature by

5The coordinate-free deﬁnition of the Euler-Lagrange form associated to a Lagrangian

employs the notion of Lepage equivalent

,see71, p. 122 and 125.; yet, for our purposes, the precise expressions of Lepage equivalents of our Lagrangians will not be

necessary.

Mathematical foundations for ﬁeld theories on Finsler spacetimes 45

2. In order to identify the Euler-Lagrange form, it is sufﬁcient to use

-vertical variation vector ﬁelds

Ξ∈X(Y).Yet, general vector ﬁelds are needed in discussing general covariance and its consequence,

energy-momentum conservation.

Natural bundles and natural Lagrangians.

Let Mndenote the category of smooth n-dimensional manifolds, with smooth embeddings as morphisms

and F B,the category of smooth ﬁber bundles, whose morphisms are smooth ﬁbered morphisms.

Anatural bundle functor over n-manifolds is,81 , a functor F:Mn→F B,such that:

1. For each M∈Ob(Mn),F(M)is a ﬁber bundle over M;

2. For each embedding

0:M→M′∈Mor f (Mn),the ﬁbered manifold morphism F(

0):F(M)→

F(M′)covers

If Y=F(M),then any automorphism

of M∈Ob(Mn)admits a canonical (or natural) lift Φ:=F(

)

to Y. These natural lifts encode the transformations of ﬁelds - more precisely, their local expressions are

identical to transition functions on Y(see, e.g.,66,82). For instance, if Yis a bundle of tensors of over M,then

the canonical lift Φ=F(

)of

∈Di f f (M)is given by pullback/pushforward.

Passing to inﬁnitesimal generators, any vector ﬁeld

∈X(M)admits a canonical lift Ξ:=F(

)∈X(Y);

in a ﬁbered chart, the components Ξ

can always be expressed in terms of the components

iof

and a

ﬁnite number of partial derivatives thereof,77.

For example, in the case of the bundle of tensors Y=Tp

qMof type (p,q)over M,one obtains Ξ=

∂

i+Ξi1...ip

j1... jq

∂

yi1...ip

j1... jq

,where:

Ξi1...ip

j1... jq=

,hyhi2...ip

j1... jq+...

,hyi1...ip−1h

j1... jq−

,j1yi1...ip

h j2... jq−

,j1yi1...ip

j1... jq−1h.

A globally deﬁned Lagrangian

∈Ωn(JrF(M)) is called natural, or generally covariant, if it is invariant

under canonical lifts of arbitrary diffeomorphisms of spacetime, i.e., JrF(

)∗

for all

∈Di f f (M),82.

Using the formal similarity between lifts of active diffeomorphisms

∈Di f f (M)and (manifold-induced)

ﬁbered coordinate changes on F(M),naturality amounts to the fact that

must be invariant to any such co-

ordinate changes (deﬁned on any manifold F(M),where M∈Ob(Mn)). In terms of inﬁnitesimal generators,

this reads:

LJrF(

)

=0,(A20)

for all

∈X(M).General covariance gives rise to a notion of energy-momentum tensor,77.

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Apr 2020

It is well-known that static vacuum solutions of Einstein equations are analytic in suitable coordinates. We ask here for an extension of this result in the context of Finsler gravity. We consider Finsler spacetimes that retain several properties of static Lorentzian spacetimes, are Berwaldand have vanishing Ricci scalar.

Foundations of Finsler Spacetimes from the Observers’ Viewpoint

Article

Full-text available

Apr 2020

Physical foundations for relativistic spacetimes are revisited in order to check at what extent Finsler spacetimes lie in their framework. Arguments based on inertial observers (as in the foundations of special relativity and classical mechanics) are shown to correspond with a double linear approximation in the measurement of space and time. While general relativity appears by dropping the first linearization, Finsler spacetimes appear by dropping the second one. The classical Ehlers–Pirani–Schild approach is carefully discussed and shown to be compatible with the Lorentz–Finsler case. The precise mathematical definition of Finsler spacetime is discussed by using the space of observers. Special care is taken in some issues such as the fact that a Lorentz–Finsler metric would be physically measurable only on the causal directions for a cone structure, the implications for models of spacetimes of some apparently innocuous hypotheses on differentiability, or the possibilities of measurement of a varying speed of light.

Modified Gravity and Cosmology An Update by the CANTATA Network: An Update by the CANTATA Network

Book

Jan 2021

With a focus on modified gravity this book presents a review of the recent developments in the fields of gravity and cosmology, presenting the state of the art, high-lighting the open problems, and outlining the directions of future research. General Relativity and the ΛCDM framework are currently the standard lore and constitute the concordance paradigm of cosmology. Nevertheless, long-standing open theoretical issues, as well as possible new observational ones arising from the explosive development of cosmology in the last two decades, offer the motivation and lead a large amount of research to be devoted in constructing various extensions and modifications. In this review all extended theories and scenarios are first examined under the light of theoretical consistency, and are then applied in various geometrical backgrounds, such as the cosmological and the spherical symmetric ones. Their predictions at both the background and perturbation levels, and concerning cosmology at early, intermediate and late times, are then confronted with the huge amount of observational data that astrophysics and cosmology has been able to offer in the last two decades. Theories, scenarios and models that successfully and efficiently pass the above steps are classified as viable and are candidates for the description of Nature, allowing readers to get a clear overview of the state of the art and where the field of modified gravity is likely to go. This work was performed in the framework of the COST European Action “Cosmology and Astrophysics Network for Theoretical Advances and Training Actions” - CANTATA.

Randers pp-waves

Article

Jul 2021

In this work, we study Randers spacetimes of Berwald type and analyze Pfeifer and Wohlfarth’s vacuum field equation of Finsler gravity for this class. We show that, in this case, the field equation is equivalent to the vanishing of the Finsler Ricci tensor, analogously to Einstein gravity. This implies that the considered vacuum field equation and Rutz’s equation coincide in this scenario. We also construct all exact solutions of Berwald-Randers type to vacuum Finsler gravity, which turns out to be composed of a Ricci-flat, CCNV (covariantly constant null vector) Lorentzian spacetime, or Brinkmann space, and a 1-form defined by its CCNV. Since the pp-waves are the most well-known metric in this class, we refer to the found solutions as Randers pp-waves.

A Note on Clifford Bundles and Certain Finsler Type Spaces

Article

May 2021

Ricardo Gallego Torromé

We study a relation between certain extensions of the Clifford bundle and Finsler type structures that naturally generalize the standard Clifford relation between (pseudo)-Riemannian metric structures and Dirac matrices. We show for flat metrics that there is a triangle map between Finsler structures constructed from a (pseudo)-Riemannian metric and 1-forms on [Formula: see text], the extension of the Clifford bundle and relevant Dirac type operators.

Mathematical foundations for field theories on Finsler spacetimes

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