from the einstein{hilbert action to an action principle

University of Hamburg Physics Department

From the Einstein–Hilbert action to anaction principle for Finsler gravity

Christian Pfeifer

II. Institute for Theoretical Physics

Physics Department

MIN Faculty, University of Hamburg

May 03, 2010

Gutachter:Dr. Mattias WohlfarthII. Institute for Theoretical physicsUniversity of Hamburg

Zweitgutachter:Prof. Dr. Klaus FredenhagenII. Institute for Theoretical physicsUniversity of Hamburg

2

Abstract

In this thesis we will present a generalization of the Einstein–Hilbert action by chang-ing the underlying mathematical structures from semi-Riemannian geometry to Finslergeometry.

In the first part we give an introduction of the necessary mathematical concepts atthe basis of Finsler geometry. In particular we deal with general connections on fibrebundles, the curvature of such connections, and especially with a description of thetangent bundle. The latter has some important special features: the almost productstructure, the dynamical covariant derivative, and special vector fields, so-called sprays,are of major technical importance and will be explained. For physical reasons we reviewautoparallels on the tangent bundle and the Jacobi equation. The physical interpretationof these mathematical concepts connects gravity and point particles if we use certain.Moreover we describe Finsler geometry itself.

The second part of this thesis contains the development of the structures we need togeneralize the Einstein–Hilbert action. With the help of the mathematical concepts fromthe first part the generalization of the action will be done explicitly. Furthermore weshow how general relativity can be deduced in the generalized language we developed.Finally we deduce in a short-hand notation new extended gravity equations from ourgeneralization of the Einstein–Hilbert action.

Kurzfassung

Die vorliegende Arbeit befasst sich mit einer Verallgemeinerung der Einstein–HilbertWirkung durch Verallgemeinerung der zugrundeliegenden mathematischen Konzepte.Anstelle der semi-Riemannschen Geometrie benutzen wir die Finslergeometrie, um eineWirkung fur eine verallgemeinerte Gravitationstheorie zu formulieren.

Im ersten Teil geben wir eine Einfuhrung in die mathematischen Konzepte, die fur dieVerallgemeinerung der Wirkung notwendig sind. Wir beschreiben allgemeine Faserbundel,Zusammenhange auf Faserbundeln und als wichtigen Spezialfall das Tangentialbundel.Da Letzteres eine herausragende Rolle in der Finslergeometrie spielt, werden wir weitereStrukturen auf dem Tangentialbundel einfuhren, wie zum Beispiel die sogenannte almostproduct structure, die dynamische kovariante Ableitung und spezielle Vektorfelder, soge-nannte sprays. Um die Physik aus Sicht des Tangentialbundels zu beschreiben, werdenwir Autoparallelen aus dieser Sichtweise heraus definieren und eine Jakobi Gleichungfur selbige herleiten sowie deren physikalische Bedeutung erklaren. Eine kompakteEinfuhrung in die Finslergeometrie gegeben wir am Ende des ersten Teils.

Der zweite Teil dieser Arbeit beschaftigt sich mit der Verallgemeinerung der Einstein–Hilbert Wirkung zu einer Finsler-Gravitation sowie den Problemen die dabei auftretenund deren Losung. Außerdem werden wir die Einsteinschen Gleichungen in dem entwick-elten Formalismus herleiten, bevor wir die verallgemeinerte neue Gravitationsgleichungin kompakter Schreibweise darstellen.

3

Contents

Introduction 7

I. Mathematical background:from fibre bundles to Finsler geometry 11

1. Fibre bundles and connections 151.1. Basic definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2. Connections on fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . 171.3. Curvature of a connection . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2. The tangent bundle of a manifold 212.1. Induced coordinates and distinguished tensor fields . . . . . . . . . . . . 222.2. Equivalent structures to the connection . . . . . . . . . . . . . . . . . . . 252.3. Autoparallels and the dynamical covariant derivative . . . . . . . . . . . 262.4. Semisprays and Sprays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5. Jacobi equation of autoparallels . . . . . . . . . . . . . . . . . . . . . . . 31

3. Finsler geometry 333.1. The Finsler function and the Finsler metric . . . . . . . . . . . . . . . . 343.2. Geometrical objects of Finsler geometry . . . . . . . . . . . . . . . . . . 363.3. Finsler geodesics and Cartan nonlinear connection . . . . . . . . . . . . . 373.4. Curvature tensors in Finsler geometry . . . . . . . . . . . . . . . . . . . . 40

II. Finsler gravity from an action principle 41

4. New perspective on the Einstein–Hilbert action 45

5. The sphere bundle 475.1. Idea and definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2. Finsler spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 485.3. Transformation of the coordinate basis . . . . . . . . . . . . . . . . . . . 505.4. Integration over the sphere bundle . . . . . . . . . . . . . . . . . . . . . . 52

6. Finsler gravity and the calculus of variation 556.1. Finsler gravity action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5

6.2. Equivalence of Finsler equations . . . . . . . . . . . . . . . . . . . . . . . 566.3. Calculus of variation on the sphere bundle . . . . . . . . . . . . . . . . . 58

7. Einstein’s equations from a Finsler point of view 617.1. Vacuum equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.2. Complete gravity equations including matter . . . . . . . . . . . . . . . . 63

8. Finsler gravity equations 678.1. Organising the variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 678.2. Variation of the Finsler gravity action . . . . . . . . . . . . . . . . . . . . 68

Conclusion and outlook 71

Appendices 73

A. Additional calculations for Part I 75A.1. The horizontal bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.2. The geodesic equation of a Finsler space . . . . . . . . . . . . . . . . . . 76

B. Detailed derivations for Part II 77B.1. Pullback of the Sasaki metric to the sphere bundle . . . . . . . . . . . . . 77B.2. Diagonalisation of the pullback of the Sasaki metric . . . . . . . . . . . . 78B.3. Variation of the Finsler metric with respect to the Finsler function . . . . 81B.4. Rewriting the variation from the Finsler metric to the Finsler function . 81B.5. Details of the derivation of the Einstein equations in Finsler geometry . . 82

B.5.1. Variation of the fibre metric h . . . . . . . . . . . . . . . . . . . . 83B.5.2. Derivation of the vacuum equations . . . . . . . . . . . . . . . . . 83B.5.3. Derivation of the complete gravity equation

including matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 85B.6. Variation of the derivatives of components of the Finsler metric . . . . . 86B.7. Variation of the Finsler gravity action . . . . . . . . . . . . . . . . . . . . 87

6

Introduction

Almost a hundred nears after its formulation Einstein’s Theory of general relativity isstill a physicists accepted answer to questions concerning gravity. No other theory de-scribes the dynamics of the solar system, the cosmological evolution of the universe,the deflection of light by massive objects, the paths of space vehicles through the solarsystem, and much more, consistently to such high accuracy, and is at the same timebased on so few assumptions. Nevertheless there exist phenomena which seem to re-quire assumptions beyond general relativity, like the path of the Pioneer probe throughspace after leaving the solar system [1] or the accelerated expansion of the universe incosmological models [2, 3]. Furthermore it was not possible to find a quantized theoryof gravity yet, which could complete the quantized description of all fundamental forcesof physics.

Nowadays for the pioneer anomaly there exist a number of explanation attempts,some of them and further informations can be found in [4]. The acceleration of theuniverse can be solved most simply by the introduction of a cosmological constant inthe Einstein equations, done in standard textbooks like [5]. A modern interpretationof the cosmological constant is dark energy. The problem of a quantized theory isencountered in String Theory [6] and Loop Quantum Gravity [7]. However it appearsthat none of these explanations account the problems mentioned above fully satisfactoryyet.

This is the reason for the existence of a number of historical attempts to formulatealternative theories of gravity, like for example the one of Brans-Dicke [8]. Unfortunatelythese theories seem not to agree with experiments as well as general relativity [9], orthey offer little predictiveness beyond.

In this work we will develop a new generalization of general relativity through ageneralization of the underlying mathematics, and so we pave the way towards newequations which govern the dynamics of gravity. Since the mathematical extensionwe provide looks so natural we hope that in these new equations of gravity lies theexplanation of at least some of the not yet understood phenomena mentioned above.

From semi-Riemann to Finsler geometry

The mathematical foundation of general relativity is given by semi-Riemannian geome-try, in which the geometrical properties of a manifold M are defined through a metricg, which is a non-degenerate bilinear form on the tangent spaces of M , and a covariantderivative ∇ on M induced by a linear connection on the tangent bundle of the manifold.These two objects lead to the notion of distances, lengths, angles, curvature and parallel

7

Introduction

transport on the manifold. For example the length of a curve γ on M is explicitly givenby

L1[γ] =

∫dτ√g(γ)abγaγb

This mathematical setting is used to describe physics when we say that our physicalspacetime is a four dimensional manifold with a Lorentz metric and its correspondingLevi-Civita connection defines the curvature of spacetime. In the absence of any otherforce than gravity, geodesics on this spacetime are interpreted as worldlines of physicalobservers, particles and light.

The starting point of the extension of semi-Riemannian geometry to Finsler geometryis, that the length of a curve γ on our spacetime M is not longer given by the expressionabove but by the generalized expression

L2[γ] =

∫dτF (γ, γ),

where F is a function on the tangent bundle TM of M . In order to make this defi-nition well-defined F has to be homogeneous of degree one with respect to its secondargument, since otherwise the integral would not be reparametrization-invariant. Wecall F the Finsler function of the manifold M . It is immediately clear that semi-Riemannian geometry is a special case of Finslerian geometry with Finsler functionF =

√g(γ)ab(p)γaγb, p ∈M .

We will see that it is possible in Finsler geometry to define a non-degenerate bilinearform gF out of F , which we call the Finsler metric. The difference to a semi-Riemannianmetric is that its components gFab are not longer functions of the manifold, but functionsof its tangent bundle. Also, given a Finsler function F , we can construct a connectionon the tangent bundle which leads to the notion of curvature, parallelism and geodesicdeviation on the manifold. In contrast to semi-Riemannian geometry the tensor fieldsthat describe these structures on the manifold will be special tensor fields on the tangentbundle instead of tensors fields on the manifold. Hence we will recover all geometricalobjects that we know from semi-Riemannian geometry from Finsler geometry, but withstructural modifications.

To use Finsler geometry to extend general relativity is of physical interest since everygravity theory must describe, at least in some limiting process, how to determine curvesγ that are worldlines of point particles on the spacetime manifold. Technically thesecurves should be governed by a point particle action S[γ] that looks like an integral ofthe form of L2[γ]. Therefore physics always connects to some Finsler geometry. Anexample of an alternative gravity theory where a point particle action like L2[γ] appearsis area metric gravity [10].

Finsler geometry gravity theories

There exist several approaches to a Finsler formulation of a gravity theory in the liter-ature, but all of them differ from the one we present in this thesis. One attempt, done

8

Introduction

in [11], is to generate equations which formally look like the Einstein equations on thetangent bundle. Another is to study special tensors which fulfil certain Bianchi identitiesand use these tensors to write down equations that describe gravity as it is done in [12]and in [13]. The authors of [12] also show in [14] how Finsler geometry can be used towrite down a gravity theory that contains modified Newtonian dynamics [15] in a weakfield limit and how this could be a possible alternative to the dark matter hypothesis.

Our aim in this thesis will be a Lagrangian-based Finsler gravity theory, with anaction that reduces straightforwardly to the Einstein–Hilbert action known from generalrelativity in the special case of a Finsler geometry that reduces to semi-Riemanniangeometry. Our central results include how we rewrite and extend the Einstein–Hilbertaction in the Finsler geometry setting.

Structure of this thesis

The presented work is mainly divided in two parts.In Part I we will review all mathematical concepts needed to understand Finsler

geometry. We start with the introduction of the concepts of fibre bundles, connectionson fibre bundles and curvature in chapter 1. In chapter 2 we proceed with the descriptionof a very specific fibre bundle, the tangent bundle of a manifold M . We finally give anintroduction to Finsler geometry with focus on the concepts we need to generalize theEinstein–Hilbert action in chapter 3. References where most of the content of part I canbe found are [16, 17, 18], and [11].

Part II begins with the introduction of a rewriting procedure of the standard Einstein–Hilbert action on the semi-Riemann geometry level in chapter 4. In chapter 5 we developthe mathematical concepts of in integral over the manifold and its tangent space direc-tions in a Finsler geometry setting and use these developments to construct the intendedgeneralization of the Einstein–Hilbert action. How calculus of variations has to be donefor an action that is based on Finsler geometry will be described in chapter 6, before weshow in chapter 7 how it is possible to understand general relativity as a theory basedon Finsler geometry. We deduce the Einstein equations with the help of our new con-cepts explicitly. Finally in chapter 8 we vary the generalized Einstein–Hilbert action todeduce new gravity equations and express them in a preliminary compact form. Nearlyall results we present in the second part of this thesis are based on long calculations thatcan be found in appendix B

9

Part I.

Mathematical background:from fibre bundles to Finsler geometry

The first part of this thesis aims to review the necessary mathematical backgroundon which our development of Finsler gravity in the second part is based.

First, in chapter 1, we will review the basic notion of fibre bundles, connections on fibrebundles and how these terms lead to the concept of curvature. Second we will apply thesegeneral concepts to a very specific fibre bundle, namely the tangent bundle of a manifold(chapter 2). Finally we describe the foundations of Finsler geometry in chapter 3 whichlives on the tangent bundle and is a natural extension of semi-Riemannian geometrybased on a general reparametrization-invariant measure of length.

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1. Fibre bundles and connections

This chapter will give a short review about general fibre bundles, connections on fibrebundles and the corresponding curvature. These concepts are essential ingredients inthis thesis. More specifically, Finsler geometry takes place on the tangent bundle ofa manifold and our Finsler gravity action will be formulated on a subbundle of thetangent bundle which we will call the sphere bundle. All concepts explained throughoutthis chapter can be found in detail in [16, 17, 18].

1.1. Basic definition

We start right away with the definition of fibre bundles.

Definition 1.1 Locally trivial fibre bundleLet E,M ,F be manifolds and π be a projection π : E →M such that π−1(p) = F, ∀p ∈M ,then (E

π−→ M,F ) is called a locally trivial fibre bundle, if there exists an open cover(Ui)i∈I of M and for every open set of this cover Ui ⊂M there exists a diffeomorphismψi : π−1(Ui)→ Ui × F such that π ψ−1

i = pr1.

This definition can be combined into the following commuting diagram:

π−1(Ui)

π

ψi // Ui × F

pr1yysssssssssss

Ui

The manifold E is called the total space, M the base space, F the fibre and π theprojection of the bundle.

The notational convention we use in the following will be that small Latin indicesa, b, ... run from 1 to the dimension of the base manifold M , that small Greek indicesα, β, ... run from 1 to the dimension of the fibre manifold F , and that contracted indicesobey the Einstein summation convention.

Because of the existence of the diffeomorphisms ψi, coordinates φ = (x1, ..., xa) on Mand coordinates Φ = (y1, ..., yα) on F induce local coordinates on E via ψi(u) = (x, y).The ψi are called the trivialization of E. In the following we refer to a locally trivialfibre bundle simply as fibre bundle and name it after its total space.

An example for a fibre bundle is the tangent bundle TM of a manifold. The totalspace E in this case is given by E =

⋃p∈M

TpM , which is itself a manifold, as is known

15


Figure 1.1.: Illustration of a fibre bundle

from differential geometry. The fibre F is simply isomorphic to RdimM and the basemanifold is clearly M itself. In local coordinates φ = (x1, ..., xdimM) on M a point u ∈ Eis given by a vector in the tangent space at a point p ∈ M with component functionsu(p)a, namely u = u(p)a ∂

∂xa |p. The projection π maps this vector on its base point p so

we have π(u) = p, which holds for all vectors of TpM . Staying in the coordinates on Mand taking the canonical identification of TpM with RdimM the trivializations ψi leadto ψi(u) = (x1(p), .., xdimM(p), u(p)1, ..., u(p)dimM). A much more detailed description ofthe tangent bundle will be given in chapter 2, since this will be the bundle of interestfor Finsler geometry.

A common picture for a fibre bundle is a manifoldM with another manifold F attachedto each point as displayed in figure 1.1. In the case of the tangent bundle the manifoldgets an RdimM attached to every point which is nothing else but the manifold with itstangent spaces as mentioned above.

Since the fibre bundle E is a manifold in its own right, we may now consider theassociated tangent bundle TE. The projection π : E → M provides the bundle witha natural map between TE and TM through its differential dπ. Regarding the kernelker(dπ|u) of the differential at a point u ∈ E, it is easy to realize that this is precisely thepart of tangent space TuE in the directions of the fibre F . This is true since a generalvector X ∈ TuE can be expressed in the coordinate basis induced by coordinates fromthe base manifold M and the fibre F , as X = Xa(u) ∂

∂xa |u + X(u)α ∂∂yα |u

. Therefore

dπ|u : TuE → Tπ(u)M dπ(X) = Xa(u)∂

∂xa |π(u), (1.1)

and hence ker(dπ|u) = span( ∂∂yα |u

) with dim(ker(dπ|u)) = dimF , as claimed above. We

call ker(dπ|u) the vertical tangent space VuE at u ∈ E and

∂∂yα |u

the vertical basis.

This works at every u ∈ E and since dπ has constant rank all over E, V E :=⋃u∈E VuE

is a subbundle of TE, called the vertical bundle. Every fibre bundle is equipped withthis natural structure via its projection.

In the next section we will see how, by defining a connection, we can consider thetangent space TuE of a fibre bundle E at a point u ∈ E as a direct sum of the vertical

16


tangent space VuE and a complement HuE called the horizontal tangent space.

1.2. Connections on fibre bundles

While the vertical tangent space to a fibre bundle E is canonically defined as above,there is a choice involved in the definition of an appropriate complement. This choice isencoded into a connection one-form ω on E. We will see that ω allows us to write thetangent space of E at every point u ∈ E as a direct sum TuE = VuE ⊕HuE.

Definition 1.2 ConnectionA connection one-form, or short connection, ω on the fibre bundle (E

π−→M,F ) is a V Evalued one-form on E such that ω ω = ω with Im(ω) = V E. Hence ω is a projectionon the vertical bundle.

Expressed in the induced coordinate basis on E, the connection one-form ω can beexpressed as

ω = (dyα +N(x, y)αb dxb)⊗ ∂

∂yα, (1.2)

which we will show in appendix A.1. The connection components N(x, y)αb are functionson E with a very specific transformation behaviour under a change of coordinates onE. The transformation can be derived from the fact that ω is a (1, 1)-tensor field. Wewill provide more details only for the case of the tangent bundle in chapter 2, sincethe tangent bundle plays a central role in Finsler geometry and therefore in our gravitytheory.

In the case that the fibre F of the bundle E is a vector space, the connection com-ponents can be linear, homogeneous or nonlinear with respect to their fibre argument.These three cases will play a major role from chapter 2 onwards. To indicate the re-spective additional property of the connection components we will call the connection ωitself either linear, homogeneous, or nonlinear. The natural connection we will encounterin Finsler geometry will be of the homogeneous type, as we will see in chapter 3.

From the definition of the connection one-form ω we may determine its kernel as

ker(ω)|u = span

(δ

δxa |u

),

δ

δxa |u=

∂

∂xa |u−Nα

a (u)∂

∂yα |u, (1.3)

omitting some details presented in appendix A.1. The dimension of the kernel satisfiesdim(kerω|u) = dimM . It is now possible to decompose the tangent space at every pointu ∈ E in TuE = ker(ω)|u ⊕ VuE, which is clear from the coordinate expressions inequation (1.3). This leads to the definition of the horizontal tangent space HuE and thehorizontal bundle HE,

HuE := ker(ω)|u HE :=⋃u∈E

HuE . (1.4)

17


The properties of the connection one-form ensure that HE is really a well-definedsubbundle of TE. To have a picture in mind we can think about HuE as the part of thetangent space TuE along the base manifold with some fibre direction modifications.

It is now evident that there is some freedom in choosing the complement to VuE inTuE. The freedom gets fixed by specifying functions Nα

b on E with correct transforma-tion behaviour as components of the connection one-form, or by giving the connectionone-form as projector. If the Nα

b are not vanishing, then HuE is not spanned by the

induced coordinate basis of M but by the so-called horizontal basis

δδxa |u

. In contrast

to the vertical basis the horizontal basis is not a coordinate basis.After decomposing TE into the horizontal and vertical bundle and since the connection

one form ω is the projector onto V E we find the projector h onto HE

h : TE → HE , h = idTE − ω. (1.5)

This projector will be one of the key ingredients in defining the curvature of a connectionin the following section.

The whole construction of vertical and horizontal bundles can as well be performed onthe cotangent bundle T ∗E of a fibre bundle. In similar fashion this leads to the verticaland the horizontal cotangent bundle V ∗E and H∗E. These are given by

V ∗E := span(dxa) H∗E := span(δyα = dyα +Nαb dx

b) (1.6)

In order to avoid confusion, note that the vertical cotangent bundle is dual to thehorizontal tangent bundle and vice versa.

1.3. Curvature of a connection

Curvature can be understood as the failure of the horizontal bundle HE to be thetangent bundle of the base manifold M . How this is encoded in a tensor becomes clearby virtue of the Frobenius theorem.

Consider a subset Dp in the tangent space TpM of a manifold at all points p ∈ M .We call the union of all these subsets D =

⋃p∈M

Dp distribution. Therefore D is a subset

of the tangent bundle of M . A distribution is called integrable if it is itself the tangentbundle of some submanifold N ⊂ M , i.e. D = TN . The Frobenius theorem tells uswhat has to hold for a distribution D to be integrable. A distribution D is integrable ifand only if for any two X, Y ∈ D follows [X, Y ] ∈ D. This fact can be reformulated inthe sense that if D gets spanned by basis vector fields ea then D is integrable if andonly if [ea, eb] ∈ D ∀ a, b.

From its definition it is clear that the horizontal bundle is a distribution that getsspanned by δ

δxa (see equation (1.3)). The curvature of a connection is nothing but the

failure of the horizontal bundle to be integrable, so the failure of [ δδxa, δδxb

] to lie in HE.

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Definition 1.3 CurvatureLet X, Y be vector fields on E and h be the horizontal projector given by equation 1.5.The curvature R of a connection ω on a fibre bundle (E

π−→M,F ) is a (1,2)-tensor fielddefined by the following expression

R(X, Y ) = [h(X), h(Y )]− h[X, Y ]− h([X, h(Y )])− h([h(X), Y ]) . (1.7)

The curvature components Rαab are easily calculated in the horizontal-vertical basis

of TE, and essentially contain the commutator of two horizontal basis vector fields,

R

(δ

δxa,δ

δxb

)=

[δ

δxa,δ

δxb

]=

[∂

∂xa−Nα

a (x, y)∂

∂yα,∂

∂xb−Nβ

b (x, y)∂

∂yβ

]=

(δNα

a

δxb− δNα

b

δxa

)∂

∂yα

=: Rαab

∂

∂yα. (1.8)

In the horizontal-vertical basis of TE we can write the curvature to be

R =1

2Rα

abdxb ∧ dxa ⊗ ∂

∂yα. (1.9)

According to Frobenius theorem, HE is integrable if and only if the commutator ofits basis vector fields is again a vector field in HE, as explained in the beginning ofthis section. From the calculation of the curvature components we see that this canonly be the case if they vanish. Therefore the curvature measures the deviation of theintegrability of the horizontal bundle.

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2. The tangent bundle of a manifold

We will now specialize the general fibre bundle concepts explained in the previous chapterto the tangent bundle of a manifold. This is a very specific fibre bundle that will play acentral role in Finsler geometry.

What makes the tangent bundle so special is that its fibre is a vector space of thesame dimension as the underlying manifold, so that also the horizontal and the verticalbundle have the same dimension. Furthermore it is possible to use coordinates on thefibres which are induced through coordinates of the base manifold M . On the tangentbundle itself this guarantees coordinates which are also solely determined by those onthe base space. This fact causes some implications we will work out. One is the existenceof tensors on the tangent bundle whose components transform as if they were tensorson the manifold, not on the bundle.

Another speciality of the tangent bundle we come across is that due to its close relationto the base manifold there exist much more canonical structures than on a general fibrebundle. We will describe the structures we need for Finsler geometry and our gravitytheory, and how they are connected to a connection on the bundle. Our focus lies onthe so-called tangent structure, the adjoint structure and the almost product structure.

Moreover it will be possible to study curves γ on the manifold M from the tangentbundle point of view, i.e., to look a the curve and its tangent vector field γ together. Theset (γ, γ) ∈ TM is the natural lift γ of the curve γ to the tangent bundle. A connectionω on the tangent bundle then leads to the notion of autoparallels on M and how theycan be deduced from a so-called dynamical covariant derivative.

We also present a special type of vector fields, so-called semisprays or sprays, on thetangent bundle which allow us to find a geometry adapted to systems of differentialequations, in such a way that the solutions to the system of differential equations areautoparallels of a certain connection.

In the end of this chapter will derive a deviation or Jacobi equation for autoparallelsthat links this kind of general geometry on the tangent bundle to gravity.

Since we introduced the general concept of a connection on fibre bundles in the previ-ous chapter we will compare all structures that depend on a general connection on thetangent bundle to the case if the connection is linear, since this is the well known case ifwe consider basic differential geometry on a manifold. We will find that all generalizedobjects reduce to familiar ones.

A detailed introduction to all topics and the detailed proofs to the theorems mentionedcan be found in [11].

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2.1. Induced coordinates and distinguished tensor fields

Since the tangent bundle of a manifold is closely connected to the manifold itself thereexist special coordinates on the bundle induced by the ones from the manifold. There-fore a change of coordinates on the manifold induces a change of coordinates on thebundle. An important side effect of the existence of these coordinates is the possibilityto construct tensors on the bundle which transform under induced coordinate changesas if they were tensors on the manifold. They are called d-tensors.

The formal definition of the tangent bundle is as follows:

Definition 2.1 Tangent bundleThe unification of all tangent spaces of a n ∈ N dimensional manifold M is called thetangent bundle TM of a manifold,

TM =⋃p∈M

TpM . (2.1)

It is a fibre bundle with fibre F = Rn and base manifold M .

If we choose coordinates (U, φ = (x1, ..., xn)) on U ⊂ M and a point p ∈ U such that

φ(p) = 0, then a natural basis of TpM is given by

∂∂xa |p

. A vector X in TpM can

be expressed as equivalence class of curves [γ = (x(τ)1, ..., x(τ)n)] on the manifold andis given by X = xa(0) ∂

∂xa |p. Since TpM is isomorphic to Rn, X can be rewritten as

X = yαeα = xα(0)eα where eα is the canonical basis of Rn. The (yα) are the manifoldinduced coordinates on the fibre and lead to coordinates (xa, yα) = (xa(0), xα(0)) onπ−1(U) ⊂ TM .

In contrast to a general fibre bundle, the fibre F of the tangent bundle has the samedimension as the base manifold M , so a and α both run from 1 to n = dimM = dimF .Furthermore coordinate transformations on M will induce a special kind of coordinatetransformations on TM , which will be described now. Consider two intersecting coordi-nate patches (U, φ = (x1, ..., xn)) and (V,Φ = (x1, ..., xn)) on M . Let p ∈ M be a pointin the intersection of U and V . The two different coordinates of p are related by ancoordinate transformation xi = xi(x) which induces a change of coordinates on the fibre

yα =∂xα

∂xβyβ , (2.2)

and consequently on the tangent bundle

(xa, yα) 7→(xa(x), yα(x, y) =

∂xα

∂xβyβ). (2.3)

Some trouble with the index notation seems to appear here since suddenly there arecoordinates x labelled with an α. In the case of the tangent bundle this is no mistakesince the dimensions of the fibre and the manifold are the same. Throughout this chapter

22


this kind of index confusion will appear more often but for the lack of a better notationwe will use this one and we will not mention it explicitly any more.

As for every manifold there is an induced coordinate basis on the tangent space TuTM

at u ∈ TM given by

( ∂∂xa |u,

∂∂yα |u

)

. Their behaviour and the behaviour of their duals

under induced coordinate transformations is

∂

∂xa |u=

∂xb

∂xa∂

∂xb |u+∂yβ

∂xa∂

∂yβ |u

∂

∂yα |u=

∂xb

∂yα∂

∂xb |u+∂yβ

∂yα∂

∂yβ |u=∂xβ

∂xα∂

∂yβ |u

dxa|u =∂xa

∂xbdxb|u; dyα|u =

∂xα

∂xβdyβ|u +

∂yα

∂xbdxb|u . (2.4)

From the previous chapter we know that the vertical bundle V TM and the verticalcotangent bundle V ∗TM get spanned locally by ∂

∂yα respectively dxa. We remark

that these bases transform just as if they were basis vector or covector fields on themanifold.

The components of a connection Nαa on TM can be regarded as components of the

following (1, 1)-tensor on TM , see equation (1.2)

ω = (dyα +Nαa dx

a)⊗ ∂

∂yα= δαγ dy

γ ⊗ ∂

∂yα+Nα

a dxa ⊗ ∂

∂yα. (2.5)

A change of coordinates of the type given by equation (2.4) leads to the transformationbehaviour of the Nα

a

∂yβ

∂yαNαa =

∂yβ

∂xa+Nβ

b

∂xb

∂xa. (2.6)

We want to remark here that for a connection whose connection components N(x, y)αaare linear with respect to their fibre argument on TM we can write N(x, y)αb = Γ(x)αbcy

b.An investigation of the transformation behaviour of the Γαbc with the help of equation(2.6) leads to the result that the Γαbc transform like components of a standard covariantderivative ∇ on M , known from differential geometry. Hence we have that for every co-variant derivative on the manifold there exists a linear connection on TM and vice versa.In the case of semi-Riemannian geometry this is the Levi-Civita covariant derivative orLevi-Civita connection.

An investigation of the transformation behaviour of the basis vectors of the horizontal

bundle

δδxa |u

and basis covectors of the horizontal cotangent bundle δyα shows that

they transform under coordinate changes of the form (2.4) just if they were vector orcovector fields on the manifold, in the same way as this holds for the vertical bases.

The basis

δδxa, ∂∂yα

of TTM and its dual dxa, δyα which are adapted to the

horizontal-vertical structure of the bundle are called Berwald basis. As mentioned abovethey transform like vector fields, respectively one forms, on the manifold under inducedcoordinate changes and are extremely useful to construct higher order tensors whichhave the same behaviour.

23


Since tensors on TM which are adapted to the horizontal and vertical bundle structureof TTM transform like tensors on the manifold M , we distinguish them from the all othertensors. These kind of tensors are highly interesting since we will build the generalizedgravity action from them (chapter 6.1). Moreover, they have the property that they canbe used to define tensors on the manifold if their components are independent of thefibre coordinate.

Definition 2.2 D-tensorsLet X i; i = 1, ...s be vector fields, Ωj; j = 1, ...r be covector fields, qj be projectors on thehorizontal or vertical cotangent bundle and pi be projectors on the horizontal or verticaltangent bundle of TM .An (r, s)-tensor field T on TM is called d-tensor field (distinguished tensor field) if andonly if

T (Ω1, ...,Ωr, X1, ..., Xs) = T (q1(Ω1), ..., qr(Ωr), p1(X1), ..., pr(X

s)) . (2.7)

Their local component expression can always be written in the form

T = T a1a2...arb1b2...bs

δ

δxa1⊗..⊗ δ

δxai⊗ ∂

∂yai+1⊗..⊗ ∂

∂yaj⊗dxb1⊗..⊗dxbk⊗δybk+1⊗..⊗δybl . (2.8)

The number of δδxai

, ∂∂yaj

, dxbk and δybl depends on the nature of the projectors in

equation (2.7). These tensor fields behave under manifold induced coordinate transfor-mations (equation (2.4)) as if they were tensor fields on the base manifold. An examplefor a (1, 2)-d-tensor field on TM is the curvature R of a connection ω on TM , sinceit is given by equation (1.9) and so fulfils equation (2.7). Moreover its componentsRα

ab transform like components of a tensor field on the manifold under the consideredcoordinate changes.

Another feature of some d-tensors is that in some cases they define tensors on themanifold if the connection ω is linear. Again the curvature from equation (1.9) is suchan example. If N(x, y)αb = Γ(x)αbcy

b then

Rcab =

δN ca

δxb− δN c

b

δxa

=∂

∂xbΓcapy

p − Γqbpyp ∂

∂yqΓcary

r − ∂

∂xaΓcbpy

p + Γqapyp ∂

∂yqΓcbry

r

=

(∂

∂xbΓcap −

∂

∂xaΓcbp + ΓqapΓ

cbq − ΓqbpΓ

cap

)yp

= Rcpbay

p = −Rcpaby

p. (2.9)

So the curvature components reduce mainly to the components of the Riemann curva-ture tensor from standard differential geometry on a manifold, with covariant derivativecomponents Γ(x)αbc.

24


2.2. Equivalent structures to the connection

The tangent bundle offers a lot more canonical structure than a general fibre bundle.Part of this structure is the so-called tangent structure J , which is a map from thetangent bundle TTM of the bundle to its vertical bundle V TM . Details and proofs ofthe theorems presented below can be found in [11].

Definition 2.3 Tangent structureConsider the tangent bundle in the standard induced coordinates. The tangent structureJ of a tangent bundle TM is a map J : TTM → V TM defined by

J =∂

∂yα⊗ dxα . (2.10)

The tangent structure is a globally well-defined (1, 1)-tensor field on TM , which be-comes clear by applying the coordinate transformations given in equation (2.4). It hasthe property KerJ = ImJ = V TM hence J J = 0.

Further structures on the tangent bundle, which turn out to be equivalent to a con-nection one-form, can now be defined with the help of J . Most interesting for us is thealmost product structure P because it will lead us to the canonical connection of Finslergeometry in chapter 3. This connection is a generalization of the Levi-Civita connectionknown from semi-Riemannian geometry.

Definition 2.4 Almost product structureAn almost product structure P of a tangent bundle TM is a (1, 1)-tensor on TM definedby

P J = −J J P = J . (2.11)

The following theorem holds:

Theorem 2.1 An almost product structure P on TM is equivalent to a connection one-form ω on TM .

To understand this result, one expresses P in the Berwald basis as

P =δ

δxi⊗ dxi − ∂

∂yα⊗ δyα . (2.12)

In the context of theorem 2.1 this expression has to be understood in the followingway. If there is a connection ω that defines the horizontal and vertical bundle then P isgiven by the above expression, and the properties (2.11) hold. Conversely, if an almostproduct structure defined by definition 2.4 is given, then there exists a connection suchthat P can be written in the above form. Furthermore the expression makes clear howwe can determine explicitly the connection induced by the almost product structure.From the Berwald basis expression of P given by equation (2.12)) it is clear that thealmost product structure acts on ∂

∂xalike

P(∂

∂xa) =

δ

δxa−Nα

a

∂

∂yα=

∂

∂xa− 2Nα

a

∂

∂yα. (2.13)

25


Therefore we read off the connection components Nαa for any given almost product

structure through its action on the manifold part of the induced coordinate basis ofTTM .

Another structure of interest is the adjoint structure which will play a major role whenwe introduce the dynamical covariant derivative in the second half of section 2.3.

Definition 2.5 Adjoint structureAn adjoint structure Θ of a tangent bundle TM is a (1, 1)-tensor on TM defined by

Θ Θ = 0 , id = Θ J + J Θ . (2.14)

Alternatively one can view Θ as a map Θ : TTM → HTM . A similar theorem liketheorem 2.1 exists for the adjoint structure which is also equivalent to a connectionone-form. In the same way as explained for the almost product structure we can writeΘ explicitly in the Berwald basis

Θ =δ

δxi⊗ δyi . (2.15)

For completeness we mention the almost complex structure F which is also equivalentto a connection on TM , but it is of no further need to us yet.

Definition 2.6 Almost complex structureAn almost complex structure F of a tangent bundle TM is a (1, 1)-tensor on TM definedby

F F = −id id = F J + J F . (2.16)

Its Berwald basis expression is

F =δ

δxi⊗ δyi − ∂

∂yi⊗ dxi . (2.17)

2.3. Autoparallels and the dynamical covariantderivative

The physical motion of free point particles on a manifold must be modelled by a dis-tinguished class of curves on the spacetime manifold. In general relativity these curvesarise as the autoparallels of a specific linear connection. Indeed, the calculus of vari-ations applied to the metric action for a point particle leads exactly to the fact thatthe relevant connection is the Levi-Civita connection. As we will discuss in the nextchapter 3, Finsler geometry in general gives rise to a connection. This motivates us hereto study autoparallels for connections. In order to to this explicitly coordinate indepen-dently we will introduce the dynamical covariant derivative after we have establishedthe basic facts about autoparallels.

Recall from the beginning of this chapter that a curve γ on M has a natural lift γwhich is a curve on TM given by itself and its tangent vector field. To be precise, in

26


induced coordinates the natural lift of γ is given by γ = (γ, γ) = (x(t), x(t)). Such a liftof a curve from the manifold to the tangent bundle is called horizontal lift if the tangentvector field of the lifted curve is a horizontal vector field, i.e., ˙γ ∈ HE.

Definition 2.7 AutoparallelA smooth curve γ : I → M is called an autoparallel of the connection ω on TM if andonly if its natural lift γ : I → TM is a horizontal curve.

Expanding ˙γ and comparing it with a general horizontal vector field X = Xa δδxa

leadsto

˙γ = xa(τ)∂

∂xa+ xα(τ)

∂

∂yα!

= Xa ∂

∂xa−XaNα

a

∂

∂yα(2.18)

⇒ xa = Xa and xα = −Nαa x

a . (2.19)

Therefore to be an autoparallel γ has to fulfil the second condition in equation (2.19),called the autoparallel equation

xα +Nαa x

a = 0 . (2.20)

If N(x, y)αa is linear with respect to the fibre coordinates it can be written in the formN(x, y)αa = Γ(x)αbay

b (see section 2.1). The autoparallel equation reduces in this case tothe autoparallel equation for a covariant derivative ∇ on M with components Γαab

xα + Γαbaxbxa = ∇γ γ = 0 . (2.21)

In order to rewrite the generalized autoparallel equation (2.20) explicitly coordinateinvariant, like it can be done with a covariant derivative on the manifold in the linearcase, we will introduce the dynamical covariant derivative of a connection. We will showhow it acts on arbitrary d-tensors and define through this a metricity condition thatwill help us to find a unique connection in Finsler geometry. It can be thought of as anequivalent condition as the covariant constance of the metric in the Levi-Civita covariantderivative in semi-Riemannian geometry.

Definition 2.8 Dynamical covariant derivativeLet Nα

a be the components of a connection on TM , given in standard induced coordinatesand S be the vector field S = yα δ

δxα. A dynamical covariant derivative ∇ is a map

∇ : V TM → V TM defined by

∇(Xα ∂

∂yα

)=

(yα

δ

δxαXβ +Nβ

aXa

)∂

∂yβ=(S(Xβ) +Nβ

aXa) ∂

∂yβ. (2.22)

For functions f on TM the action of ∇ is given by

∇(f) = S(f) . (2.23)

27


Properties of the dynamical covariant derivative are that it is linear and that it fulfilsa Leibniz rule. For f ∈ F(TM) and X ∈ V E we have that

∇(fX) = S(f)X + f∇X. (2.24)

The vector field S = yα δδxα

is globally defined on TM . Since the fibre coordinates yα

have a nice transformation behaviour (equation (2.2)) and the dimension of the fibre isequal to the dimension of the manifold, it is possible to make this definition. On generalbundles this would fail, which emphasizes once more the speciality of the tangent bundle.

With the help of the adjoint structure (definition 2.5) it is possible to extend the actionof the dynamical covariant derivative ∇ to arbitrary vector fields on TM . Every vectorfield X ∈ TTM can be written as X = h(X) + v(X) if h is the horizontal projectordefined in equation (1.5) and v the vertical projector, that is the connection one-form ω(see definition 1.2). The extended dynamical covariant derivative of X ∈ TTM is thendefined by the expression

∇(X) = Θ∇(J(h(X)) +∇(v(X)) , (2.25)

into which the tangent structure J and the adjoint structure Θ enter. Acting with thedynamical covariant derivative ∇ on arbitrary d-tensors is now straightforward. We use∇ as tensor derivative and so it acts on higher order d-tensors according to the tensorderivative Leibniz rule. For an (r, s)-d-tensor field T the components of the dynamicalcovariant derivative are

∇T a1...arb1...bs

= S(T a1...arb1...bs

) + Na1m T

m...arb1...bs

+ ...+Narm T

a1...mb1...bs

− Nmb1T a1...arm...bs

− ...−Nmbs T

a1...arb1...m

. (2.26)

To say a (0, 2)-d-tensor field g is dynamical covariant constant therefore means

∇gab = S(gab)−Nma gmb −Nm

b gma = 0 . (2.27)

This fact will be one ingredient when we determine the unique connection of Finslergeometry.

We just saw that the dynamical covariant derivative has a number of properties thatare very close to those of a covariant derivative on the base manifold M . Indeed, in thecase of a linear connection on the tangent bundle they are corresponding structures inthe following sense. Since the connection components of a linear connection N(x, y)αainduce components Γ(x)αab of a covariant derivative ∇ on M , via N(x, y)αa = Γ(x)αbay

b

we can compare the covariant derivative ∇ on M and the dynamical covariant derivative∇ on TM .

Consider a vector field V and a linear connection with components Nαa = Γαbay

b onTM . The dynamical covariant derivative of V then is

∇V i = S(V i) +N imV

m = yc∂V i

∂xc− ycyqΓpcq

∂V i

∂yp+ ΓimcV

myc

= yc(∂V i

∂xc+ ΓimcV

m)− ycyqΓpcq∂V i

∂yp. (2.28)

28


If we now introduce a vector field V on M by V = V (x)a ∂∂xa

, and its naive vertical lift to

the tangent bundle˜V = V (x)a ∂

∂ya, then we can rewrite the components of the dynamical

covariant derivative ∇ of˜V into components of a covariant derivative ∇ of V evaluated

at a point u = (x, y) in TM to be

∇ ˜V i|u = yc∇ ∂

∂xcV i . (2.29)

Therefore the components of the covariant derivative of some base manifold vector field Vwith respect to another base manifold vector field Y = ya ∂

∂xaare equal to the components

of the dynamical covariant derivative of the induced linear connection on the tangentbundle of the lift of V to the tangent bundle evaluated at u = (x, y). Choosing y tobe ya = V (x)a the dynamical covariant derivative is the covariant derivative of V alongitself.

Our goal to rewrite the autoparallel equation (2.20) for a curve γ = (xa(τ)) ∈ Mexplicitly coordinate invariant is achieved by writing

∇(xα(τ)

∂

∂yα

)=(S(xα(τ)) + xβ(τ)Nα

β

) ∂

∂yα= 0 . (2.30)

The equivalence of this equation with equation (2.20) can be seen by setting yα = ∂∂txα

and so recognizing that for autoparallels

d

dt=∂xa

∂t

∂

∂xa+∂yα

∂t

∂

∂yα=∂xa

∂t

∂

∂xa−Nα

a

∂xa

∂t

∂

∂yα= S . (2.31)

Now it is clear that the bracket in (2.30) is precisely equal to the autoparallel equa-tion (2.20).

2.4. Semisprays and Sprays

In the last section we used a connection on the tangent bundle and the notion of adynamical covariant derivative to deduce an autoparallel. In other words we deduced asystem of second order differential equations with the help of a connection that definesparallelism.

Another possibility to reach a notion of parallelism in correspondence with a systemof second order differential equations is to go the other way around. In this chapter westart with differential equations expressed in tangent bundle coordinates and functions.It can be shown that, if these equations fulfill some nice properties, we can construct aconnection on the tangent bundle in such a way that the differential equations we startedfrom are the autoparallel equations of this connection. An important tool during thisconstruction will be a special vector field S on TM called spray or semispray. It will turnout that the vector field we encountered by defining the dynamical covariant derivative(definition 2.8) is a spray.

29


For us this construction of a connection-based geometry adapted to a system of differ-ential equations will be the key to the natural connection in Finsler geometry in section3.3, and so the key to the whole geometry we will use to write down our new approachto Finsler gravity in part II of this thesis.

On a local induced coordinate chart of TM consider the system of second order dif-ferential equations of the type

d2

dt2xα + 2Gα

(x,dx

dt

)= 0 . (2.32)

This equation is well-defined on all of TM if on intersections of coordinate patches undercoordinate transformations given by equation (2.4) the functions Gα transform like

Gα =∂yα

∂yβGβ − 1

2

∂yα

∂xβyβ . (2.33)

Our autoparallel equation (2.20) is exactly of this type with 2Gα = dxβ

dtNαβ .

Definition 2.9 SemisprayLet J be the tangent structure of TM and C be the vector field C = yα ∂

∂yαon TM called

the Liouville vector field. A vector field S on TM is called a semispray if and only ifJ(S) = C.

In local coordinates this leads to:

Proposition 2.1 Let Gα be functions on TM fulfilling equation (2.33). A vector fieldS on TM is a semispray if and only if

S = ya∂

∂xa− 2Gα ∂

∂yα. (2.34)

All integral curves of a semispray S solve the differential equation (2.32) the semispraybelongs to. If the components of a semispray S are homogeneous with respect to theiry coordinates S is called a spray.

Definition 2.10 SprayLet S be a semispray, Gα as in equation 2.32 then S is called a spray if and only if Gα

is homogeneous of degree two with respect to the fibre coordinates, i.e., ∂Ga

∂ybyb = 2Ga.

Referring again to the autoparallel equation (2.20) it is clear that the associated vectorfield is a spray if and only if the connection that corresponds to the autoparallel equationis homogeneous. We mention here again that we will encounter this case when we talkabout Finsler geometry in chapter 3. For our purposes the most important property ofa semispray and a spray is the following:

Theorem 2.2 Let S be a semispray and J be the tangent structure on TM , then theLie derivative of J with respect to S is an almost product structure P = −LSJ on thetangent bundle.

30


According to theorem 2.1 the almost product structure P leads to a connection ω onTM determined through the action of P on ∂

∂xa(see equation (2.13)). Theorem 2.2 states

that −LSJ is an almost product structure. By acting with −LSJ on ∂∂xa

and comparingthe result with the above result for a general almost product structure (equation (2.13))we can read off the connection componentsNα

a induced by the semispray to beNαa = ∂Gα

∂ya.

The autoparallel equation of this connection is the system of second order differentialequations the semispray or spray belongs to.

In this section we showed that we can associate a connection to a system of secondorder differential equations in such a way that this system of differential equations arethe autoparallel equations of the associated connection. We constructed a geometryadapted to the equations of choice. When we introduce Finsler geometry in chapter 3we will use this procedure to find a connection that makes a special set of curves, thatwe want to interpret as worldlines of particles, autoparallels.

As a remark we wish to point out that nothing else is done in semi-Riemanniangeometry. When one considers geodesics, curves that minimize the length integral, onefinds that the Levi-Civita connection is the connection of choice to make these curvesautoparallels. So in fact everything we described in this chapter is just a generalization ofa well known procedure. But sometimes this point of view of the Levi-Civita connectionis not pointed out in semi-Riemannian geometry.

2.5. Jacobi equation of autoparallels

We will derive a Jacobi equation that connects the deviation of autoparallel curves withthe components of the curvature of a connection defined in equation (1.8). The Jacobiequation tells us that the dynamics of the deviation vector field, describing the deviationof two autoparallels, is governed by the curvature. This is interpreted as how two testparticles get affected by gravity differently, and so links gravity to curvature.

Consider a smooth variation of autoparallels Γ. That means a smooth mapping

Γ : I × (−ε, ε) → M (2.35)

(t, s) 7→ Γ(t, s) . (2.36)

For every fixed s, Γ is an autoparallel fulfilling equation (2.20) and we call ∂Γ∂s

thevariational or deviation vector field of Γ which satisfies

[∂Γ∂s

∂Γ∂t

]= 0. Take a local co-

ordinate patch (U, φ = (x1, ..., xn)) on M such that the autoparallel Γ(t, 0) is given byΓ(t, 0) = (xa(t)). For small parameter |ε|, Γ(t, ε) = (xa(t) + ε∂Γ

∂s(t, 0)) is an autoparallel

close to Γ(t, 0). If we have a look at the autoparallel equation (2.20) for Γ(t, ε) to firstorder in ε we get the Jacobi equation for a deviation vector field. For simplicity werename ∂Γ

∂s(t, 0) in V and get

V α +

(∂Nα

b

∂xaxb +Nα

a

)V a +

∂Nαb

∂xaxbV a = 0 . (2.37)

An equivalent form of equation (2.37) where we see the connection to the curvature

31


explicitly is

∇∇V α +

(∂Nα

b

∂xaxb −Nα

a

)∇V a +Rα

abxbV a = 0 . (2.38)

Equation (2.37) is calculated out of equation (2.20) simply by Taylor expandingNαa (Γ(t, ε), Γ(t, ε)) around the canonical lift of Γ(t, 0) = (xa(t)) to TM . To get the

equivalent form displayed in equation (2.38) recall that S = ddt

for autoparallels (shownin equation (2.31)).

If one considers a linear connection on the tangent bundle, it is possible to rewrite theconnection components N(x, y)αa as N(x, y)αa = Γαaby

a, as mentioned before. Thereforethe Jacobi equation reduces in the linear connection case to

∇∇V α +Rαabx

bV a = 0 . (2.39)

With this conclusion we see that the basic geometric objects known from differentialgeometry with a covariant derivative on a manifold are just special cases of a moregeneral setting we explained in this and the previous chapter.

In the standard program of differential geometry the next step would be the introduc-tion of a semi-Riemannian metric. This concept will be generalized in the next chapter.

32

3. Finsler geometry

Finsler geometry is a straightforward generalization of semi-Riemannian geometry basedon works by Finsler from the beginning of the 20th century [19]. Its heart is the gen-eralization of the lengths of curves which plays the role of an action for point particlesand observers.

In semi-Riemannian geometry the length of a curve γ on a manifold M in local coor-dinates is given by

LSR[γ] =

∫ t1

t0

dt√gab(x(t))x(t)ax(t)b . (3.1)

This measure of length gets generalized by introducing an arbitrary tangent bundlefunction F that is homogeneous of degree one with respect to the fibre coordinates inorder to make the following integral parametrization invariant

L[γ] =

∫ t1

t0

dt F (x(t), x(t)) . (3.2)

The function F is called Finsler function and will be described in detail in the firstsection of this chapter. From this starting point all geometric objects describing amanifold are generalized. The geometrical tensors corresponding to those known fromsemi-Riemannian geometry will all become d-tensors on the tangent bundle, and so willbe fibre coordinate dependent. One example we already encountered is the nonlinearcurvature R given by equation (1.9).

The letter F gets used from here on for a manifold, the fibre of some bundles, and forthe Finsler function. It should always be clear from the context what is meant.

An overview of Finsler geometry and its development can be found in [20] and [11].Physical applications, for instance involve the description of anisotropic optical media.In this chapter we will review Finsler geometry bearing in mind that we want to applyit to formulate generalized gravitational physics. We begin with the basic definitionsof Finsler Geometry including some geometrical objects which will be important lateron. Going on with the study of Finsler geodesics resulting from the generalized lengthof curves, it will be possible to define a unique natural connection such that Finslergeodesics are the autoparallels of this connection. Finally we will see that the Jacobiequation connects Finsler geodesics with the curvature of the canonical connection. Thisis the crucial link between Finsler Geometry and gravity.

33

3. Finsler geometry

3.1. The Finsler function and the Finsler metric

Finsler geometry starts with the definition of a Finsler function F on the tangent bundlewhich is necessary to define the generalized length integral presented in equation (3.2).This leads to a Finsler metric g equal to a semi-Riemannian metric under special cir-cumstances we will mention. To be able to formulate the action of a physical theory, wehave to define an integral over the space where our physical fields live. In order to doso, we need a canonical or physical motivated volume form on the space of interest. Forthe tangent bundle the so-called Sasaki lift of the Finsler metric will help us to define anatural volume form.Definition 3.1 Finsler functionConsider the tangent bundle in induced coordinates u = (x, y). A function F : TM → Ris called a Finsler function if it satisfies globally

1. F is C∞ on TM := TM \ (x, 0) and continuous on TM ,

2. F is homogeneous of degree one with respect to y, i.e. F (x, λy) = λF (x, y),

3. the Hessian of its square, in the fibre coordinates ∂2F 2

∂yα∂yβ, has constant rank and is

non-degenerat for every u ∈ TM .

In the developements below, it will be of major importance that Euler’s theorem holdsfor homogeneous functions:

Theorem 3.1 Let h : TM → R be a function that is homogeneous of degree n withrespect to the y coordinates, then the following formula holds

yα∂h

∂yα= nh . (3.3)

A proof can be found in [20]. From the third requirement in definition 3.1 we cometo the Finsler metric:

Definition 3.2 Finsler metricLet F be a Finsler function on TM and consider TM in the standard induced coordinates.A Finsler metric is the symmetric (0,2) d-tensor g(x, y) with components

gαβ(x, y) =1

2

∂2F 2(x, y)

∂yα∂yβ. (3.4)

Since F is homogeneous of degree one with respect to y, the Finsler metric g ishomogeneous of degree zero, thus by Euler’s formula (3.3)

yα∂

∂yαgβγ(x, y) = 0 and gαβ(x, y)yαyβ = F 2(x, y) . (3.5)

34

3. Finsler geometry

With the help of these homogeneity properties we can rewrite the generalized lengthintegral in equation (3.2) as

L[γ] =

∫ t1

t0

dt√gαβ(x(t), x(t))x(t)αx(t)β . (3.6)

Comparing the last expression to the semi-Riemannian length of a curve γ ∈M (equation(3.1)) it is clear that the difference between Finsler geometry and semi-Riemanniangeometry lies in the fibre coordinate dependence of the metric components gab. So tosome sense semi-Riemannian geometry is Finsler geometry with an imposed symmetry,namely invariance of the metric components along fibre coordinates.

To summarize, a manifold M with Finsler function F is called a Finsler space (M,F ),and g the Finsler metric of the Finsler space. If the Finsler metric does not depend on they coordinates, it defines a usual semi-Riemannian metric on (M,F ). In local coordinatessimply take g = gab(x)dxa ⊗ dxb, where the dxa are not the vertical covector basis ofV ∗TM but the covector basis on the manifold.

We want to remark here, that a general Finsler metric g is technically not a metricon M in the usual semi-Riemannian sense, since it is a d-tensor on TM . Hence itmaps vector fields on TM to functions on TM and not vector fields on M to functionson M . By adding a small technical detail we can well define how the Finsler metricacts on vector fields on M . Since g is a d-tensor it can be considered as metric thatmeasures only vertical or horizontal vectors on TM if we define it to be g = gabdx

a⊗dxbor g = gabδy

a ⊗ δyb. Therefore its action on a vector field X = Xa ∂∂xa

on M can beexplained by introducing the vertical and horizontal lift of X to TM . Its vertical liftis given by Xv = Xa ∂

∂yawhile its horizontal lift is Xh = Xa δ

δxa. The action of g on

these vector fields is well-defined if we interpret g as corresponding metric along HTMor V TM respectively. Hence we can measure the lifts of a vector field on the manifoldwith the Finsler metric and interpret this number as measure of the vector field itself.

With the help of the Finsler metric g, it is possible to define a canonical metric G onTM that makes the horizontal and vertical bundle orthogonal complements

G = gab(u)dxa ⊗ dxb + gαβ(u)δyα ⊗ δyβ. (3.7)

This kind of lift of the Finsler metric is called Sasaki lift or Sasaki metric. The naturalvolume form $G induced through the Sasaki metric is given by

$G =√

detG dx1 ∧ dx2 ∧ ... ∧ dxn ∧ δy1 ∧ δy2 ∧ ... ∧ δyn (3.8)

= det g dx1 ∧ dx2 ∧ ... ∧ dxn ∧ dy1 ∧ dy2 ∧ ... ∧ dyn . (3.9)

The last equality is a consequence of the properties of the wedge product and theform of the horizontal cotangent basis δyα defined in equation (1.6). Hence a naturalwell-defined integral over a tangent bundle function h in Finsler geometry looks like∫

$G h(x, y) =

∫dnxdny detg h(x, y) . (3.10)

The Finsler metric is only one of a few more d-tensors that can be defined via theFinsler function F , as we will see in the next chapter.

35

3. Finsler geometry

3.2. Geometrical objects of Finsler geometry

A Finsler function F on a manifold M provides us with several natural d-tensors definedthrough the functions n-th derivative with respect to the fibre coordinate. Among otherswe will present the Cartan tensor C which measures the deviation from semi-Riemanniangeometry and the Cartan two form Ω which is a canonical symplectic structure on TM .Moreover all of these structures and the Finsler metric itself satisfy some importantrelations because of their homogeneity. Since these relations are crucial for most of thecalculations to come, we will present them here.

The first derivative of F 2 gives rise to a well-defined d-covector field on TM .

Definition 3.3 Natural Finsler d-covector fieldThe natural Finsler covector field p is defined by its components

pα =1

2

∂F 2

∂yα. (3.11)

Because of the homogeneity of F the pα fulfil yαpα = F 2 and they are nothing morethan

pβ = yα∂

∂yα1

2

∂F 2

∂yβ= gαβy

α = yβ . (3.12)

Second derivatives of the Finsler function with respect to the fibre coordinates, givethe Finsler metric we studied in section 3.1. We move on to the third derivative ofthe Finsler function and the corresponding totally symmetric (0, 3)-d tensor, called theCartan tensor.

Definition 3.4 Cartan tensorThe Cartan tensor C of a Finsler space (M,F ) is a well-defined totally symmetric (0, 3)-

d tensor field on TM with components

Cαβγ =1

4

∂3F 2

∂yα∂yβ∂yγ=

1

2

∂gαβ∂yγ

. (3.13)

By the definition it is clear, that in case of a semi-Riemannian structure, instead ofa Finsler structure the Cartan tensor vanishes. Another relation that holds obviously,having equation (3.5) in mind, is yαCαβγ = 0.

Definition 3.5 Cartan one-formThe Cartan one-form θ is a well-defined one-form on TM given by

θ = pαdxα . (3.14)

We only mention it here because it leads directly to the natural symplectic structureon TM , the Cartan two form Ω.

Definition 3.6 Cartan two formThe Cartan two form Ω is the exterior derivative of the Cartan one-form.

Ω = dθ. (3.15)

36

3. Finsler geometry

Performing the derivative, leads to the coordinate expression of Ω

Ω = −gαβdxα ∧ dyβ +1

4

(∂2F 2

∂yα∂xβ− ∂2F 2

∂yβ∂xα

)dxα ∧ dxβ . (3.16)

The importance of this two form will become clear at the moment we determine theunique canonical connection of Finsler geometry in section 3.3. Before we are able todo that, we have to study Finsler geodesics and aim for the geodesic equation of Finslergeometry.

3.3. Finsler geodesics and Cartan nonlinear connection

As explained in the first section and in the beginning of this chapter, Finsler geometryis based on a generalized length functional that can be used as point particle actionin physical theories. A natural question that arises from both the mathematical andphysical point of view is which curves on the manifold extremize the length integral: whatare the equations of motion for a point particle? Both can be answered by the calculusof variations, and the answer is given by the Euler-Lagrange equations. Furthermore wewant to find a connection on TM whose autoparallel equations are the Euler-Lagrangeequations obtained by extremizing the length integral.

Varying the generalized Finsler length functional given by equation (3.2) and requiringthe variation to vanish leads to the Euler-Lagrange equations

∂

∂xaF (x, x)− d

dt

∂

∂xaF (x, x) = 0 . (3.17)

As in semi-Riemannian geometry the curves γ on M that solve this equations are calledgeodesics.

Definition 3.7 Finsler geodesicLet (M ;F ) be a Finsler space and γ be a curve on M . The curve γ is called a Finslergeodesic if it extremizes the length functional given by equation (3.2). This is equivalentto say that γ is a solution of equation (3.17).

A slightly tedious calculation postponed to appendix A.2 lets the geodesic equationbecome

gαβ

(xα +

1

2gασ

(∂gσµ∂xν

+∂gσν∂xµ

− ∂gνµ∂xσ

)xµxν

)=dF

dt

∂F

∂xβ. (3.18)

If we perform a special reparametrization of the curve γ in equation (3.2) from t tos(t) it is possible to define a canonical parameter of the curve called its arclength. Thisreparametrization is given by

s(t) =

∫ t

t0

dτ F

(x(τ),

dx

dτ

); t0, t ∈ I . (3.19)

37

3. Finsler geometry

The usual properties of arclength parametrization follow immediately

ds

dt= F

(x(t),

dx

dt

)⇒ dt =

ds

F(x(t), dx

dt

) ⇒ L[γ] =

∫ s(t1)

s(t0)

ds . (3.20)

In the notion of Finsler geometry arclength parametrization fulfils F(x(s), dx

ds

)= 1.

In arclength parametrization we recover the form of the geodesic equation, we know fromsemi-Riemannian geometry including the appearance of formal Christoffel symbols Γ

xα +1

2gασ

(∂gσµ∂xν

+∂gσν∂xµ

− ∂gνµ∂xσ

)xµxν = xα + Γαµν x

µxν = 0 . (3.21)

The only difference between the semi-Riemannian equation and the Finsler geometryequation is whether the metric used to write it down is velocity dependent or not. Againit is obvious that the Finsler geometry case and the semi-Riemannian case coincide ifand only if the Cartan tensor vanishes, hence the semi-Riemannian setting is a specialcase of Finsler geometry. Rewriting equation (3.21) in the form of equation (2.32)prepares us for finding the natural associated connection whose autoparallels are theFinsler geodesics. The rewritten equation is given by

xα + 2Gα = 0 Gα =1

2Γαµν x

µxν . (3.22)

Our final task will be to find a unique connection associated to equation (3.21), onwhich we base all further proceedings. According to section 2.4 we can associate aspecial vector field S, a so-called semispray or spray, to the geodesic equation of Finslergeometry (equation (3.21)). This can be done most easily if we consider the rewrittenform in equation (3.22)

S = ya∂

∂xa− 2Gα ∂

∂yα, Gα =

1

2Γαµνy

µyν . (3.23)

Actually we are talking about a spray since its components Gα are homogeneous ofdegree two. The homogeneity of Gα is easy to see if we realise that the homogeneityof Γαµν is zero since it is built of Finsler metrics only. Because the integral curves of Sare the Finsler geodesics (section 2.4) we call S the geodesic spray of the Finsler space(F,M).

From theorem 2.1 we know that the geodesic spray S induces a connection ω viathe almost product structure P = −LSJ with components Nα

b = ∂Gα

∂yb. The connection

induced by the geodesic spray is the unique connection of Finsler geometry we are lookingfor.

Definition 3.8 Cartan nonlinear connectionLet (M ;F ) be a Finsler space and S be the geodesic spray of the Finsler space. TheCartan nonlinear connection is the connection ω on TM that is induced by S ,withcomponents

Nαb = Γαbcy

c − gαpCpbdΓdmnymyn . (3.24)

38

3. Finsler geometry

Homogeneity properties of the components Nαb are

ybNαb = 2Gα; and yc

Nαb

∂yc= Nα

b . (3.25)

By construction the autoparallels of the Cartan nonlinear connection, coincide withthe Finsler geodesics (see section 2.4). Further special characteristics of the Cartannonlinear connection are that its dynamical covariant derivative (defined in section 2.3)is metric, with respect to the Finsler metric of (M,F ), and that the Cartan two form Ω(definition 3.6) vanishes on its horizontal bundle HTM ⊂ TTM , i.e.

∇gab = 0, Ω(h(X), h(Y )) = 0;X, Y ∈ TTM , (3.26)

which leads to the following theorem.

Theorem 3.2 Let (M,F ) be a Finsler space. The Cartan nonlinear connection is theunique connection on TM fulfilling both parts of equation (3.26).

For further details and a proof of this theorem we refer to [11]. We showed that it ispossible to find the canonical connection of Finsler geometry in two ways. One is toconsider the explicit form of the geodesic spray of a Finsler space (M,F ) directly, whichwe did first. The other is to ask for the unique connection fulfilling the conditions ofequation (3.26), mentioned in theorem 3.2.

Since the Cartan nonlinear connection is homogeneous of degree one with respect tothe fibre coordinates y (equation (3.25)), the Jacobi equation, we studied in section 2.5,becomes

∇∇V α +Rαabx

bV a = 0 . (3.27)

It links the deviation of Finsler geodesics to the curvature of the Cartan nonlinearconnection. In a physical interpretation this tells us that the deviation of test particlesmoving on different Finsler geodesics is caused by the curvature of the Finsler space theymove in. We identify this fact with how gravity acts differently on the test particles ondifferent geodesics. An example could be asteroids getting attracted by the sun or theplanets in the solar system. In the latter case one might think about a deviation vectorfield that connects Earth and Mars.

A final remark in this section is that in the case of a vanishing Cartan tensor C(definition 3.4), the components of the Cartan nonlinear connection reduce to Na

b =Γabcy

c. Obviously it becomes a linear connection and the Γabc become the Christoffelsymbols, which are the well known components of the Levi-Civita covariant derivativeon the base manifold. Thereby the Cartan nonlinear connection is the generalization ofthe Levi-Civita connection on the tangent bundle level.

In order to write down an action for a theory of gravity we have to find curvaturescalars, like the Ricci scalar in Semi-Riemannian geometry. These objects did not crossour way yet. They will do so in the next section of this chapter.

39

3. Finsler geometry

3.4. Curvature tensors in Finsler geometry

The Curvature associated to a connection was encountered already in section 1.3. Nowwe will explain how the Ricci and scalar curvature are constructed in the Finsler geom-etry setting. We always assume here to have an n-dimensional Finsler space (M,F ) asunderlying structure.

Recalling the curvature and its reduction to the Riemannian curvature, when there isa linear connection on TM (section 1.3), we consider the generalization

Rbc =1

2

∂

∂yb∂

∂yc(Rp

pdyd) =

1

2

∂

∂yb∂

∂yc

((δNp

p

δxd− δNp

d

δxp

)yd). (3.28)

Under coordinate transformations given by equation (2.4), Rbc transforms like a tensoron the manifold. It is a well-defined (0, 2) d-tensor on TM with the property that it ishomogeneous of degree zero with respect to the fibre coordinates. It was first consideredin [21] and is called Ricci tensor of Finsler geometry. The name has its full right sincein the case of vanishing Cartan tensor, it reduces, up to a sign, to the standard Riccitensor of semi-Riemannian geometry. The calculations are easy to be done with havingin mind that for a vanishing Cartan tensor the Cartan nonlinear connection becomeslinear (see section 3.3) and furthermore that for a linear connection the components ofthe curvature tensor become R(x, y)ppd = −R(x)pqpdy

q (see end of section 2.1).In the literature, for example in [20] and [22], the following expression is called Ricci

scalar of Finsler geometryRic = Rp

pdyd. (3.29)

It is indeed a well-defined scalar on TM but in the case of vanishing Cartan tensor itdoes not reduce to the Ricci scalar familiar to us from semi-Riemannian geometry, butto Ric→ −Raby

ayb.A quantity that does reduce up to a sign to the Riemannian Ricci scalar is the integral

over all fibre directions over Ric, if the Finsler metric is a Riemannian metric:

4vol(Sn)R = −∫SnRic = −

∫SnRp

pdyd = −

∫SnRp

bdpybyd =

∫SnRp

bpdybyd . (3.30)

The Riemannian metric g is necessary to define all fibre directions at a point p of themanifold to be Snp := y ∈ TpM | g(y, y) = 1. The proof of the result above is postponedto the next chapter since it will be the starting point of a velocity dependent viewpointof general relativity.

A d-tensor R that is a scalar and reduces to the Ricci scalar in the case of a semi-Riemannian structure can be constructed with the Finser metric and the generalizedRicci tensor Rbc from above

R = gbcRbc = gbc1

2

∂

∂yb∂

∂yc(Rp

pdyd) . (3.31)

It will have no further appearence during this theses since we do not need it for thegeneralization of the Einstein–Hilbert action we present.

40

Part II.

Finsler gravity from an action principle

As discussed in the introduction there exist various approaches to a Finsler formulationof gravity theory in the literature, but none of these seems to be based on an actionprinciple. The new approach we work out in this part of the thesis will differ in thisrespect. We aim at an action-based generalization of general relativity to a Finslergeometry setting. Our Finsler-geometric extension of the Einstein–Hilbert action willbe constructed in such a way that it reduces back, if we consider a Finsler metric withoutvelocity dependence, i.e., a semi-Riemannian metric.

We begin this part of the thesis by rewriting the Einstein–Hilbert action as an averageover tangent space directions in chapter 4. At this point the action will no longer be anintegral over the manifold itself but over a seven dimensional subbundle of the tangentbundle. In chapter 5 we will extend this construction to Finsler geometry; the resultingsubbundle will be called the sphere bundle on which we also define a natural integral.

This enables us immediately to formulate our Finsler-geometric extension of generalrelativity. As usual in physics we aim at equations of motion for a dynamical variableby performing the calculus of variations with respect to the dynamical variable in theaction integral. In the Finsler geometry setting the fundamental dynamical variable isno longer the metric, as in semi-Riemannian geometry, but the Finsler function. Anywaywe can show in chapter 6 that the Finsler function and the Finsler metric are informationequivalent and it will turn out that the equations we get by varying with respect to theFinsler function are equivalent to the equations we get by varying with respect to theFinsler metric. Furthermore we explain some general subtleties concerning the calculusof variations.

As a test case for our variational formalism on the sphere bundle we show in chap-ter 7 how the Einstein equations can be derived from the rewritten Einstein–Hilbertaction. Finally we use our Finsler gravity action to deduce extended gravity equationsin chapter 8.

Throughout this part we always consider a four dimensional Finsler space (M,F )as the basic structure upon which everything else is built. We will make use of thenatural Cartan nonlinear connection defined in section 3.3, as well as of the correspondingcurvature defined in section 1.3. Furthermore the standard induced coordinates on thetangent bundle explained in chapter 2 are used.

43

4. New perspective on theEinstein–Hilbert action

Finsler geometry takes place on the tangent bundle and we want to use it to describephysics. One reason is the possibility to rewrite the Einstein–Hilbert action as integralover a seven dimensional subbundle of the tangent bundle instead of an integral overthe manifold. Actually this can only be done in the case of a Riemannian metric on themanifold. as we will proof in this chapter. The restriction to a Riemannian metric doesnot cause any problems in our further proceeding since we will show in chapter 7 thatwe can deduce the Einstein equations from this point of view and afterwards search forthe physical solutions to the equations with Lorentz signature.

One part in this program will be the verification of equation (3.30) that is still missing.Consider a Riemannian manifold (M, g) and at every point p ∈ M the three sphereS3p := X = ya ∂

∂xa |p ∈ TpM |√g|p(X,X) = 1 as submanifold in the tangent space TpM

with volume form $S3p. At every point of the manifold we can consider the average over

all directions of the manifold over some function f on TM that depends not only on apoint p ∈M but also on the coordinates y of TpM∫

S3p

$S3pf(p, y)|S3

p. (4.1)

In the following we will be interested in f = Rabyayb .If we now consider an orthonormal frame ea and its dual ea around p we can

rewrite the metric around p as g = δabea⊗eb. Therefore we look at the tangent space TpM

in the basis ea|p and have a scalar product on TpM given by the standard euclideanscalar product δ. From the coordinates (y1, ..., y4) on TpM regarded as a vector spacewith basis ea|p we perform a coordinate transformation to standard four dimensionalspherical coordinates (θ1, θ2, θ3, r) via

y1 = r cos θ1 sin θ2 sin θ3; y2 = r sin θ1 sin θ2 sin θ3;

y3 = r cos θ2 sin θ3; y4 = r cos θ3;

r ∈ (0,∞); θ1 ∈ (0, 2π); θ2, θ3 ∈ (0, π) . (4.2)

Setting r = 1 leads to coordinates on S3p and gives us the possibility to perform a

pull-back of the scalar product δ that we can also interpret as constant metric on TpMto S3

p . The natural volume form that results is

$S3p

= d3θ sin θ2 sin2 θ3 . (4.3)

45

4. New perspective on the Einstein–Hilbert action

So choosing an orthonormal frame ea it is easy to see that

R =4

vol(S3p)

∫S3p

d3θ sin θ2 sin2 θ3 Rbdyb(θ)yd(θ) . (4.4)

Simply pull Rab out of the integral, which is possible since it does not depend on the ycoordinates of TpM , and use the coordinate transformation given by equation (4.2) toperform the integral. For every index combination a 6= b the integral vanishes, but fora = b it returns four times the volume of the three sphere. So the result of the integralcontracted with the Ricci tensor is nothing else but the Ricci scalar times the volume ofthe three sphere.

Now this is proven we can rewrite the Einstein–Hilbert action as

SEH [g] =

∫M

d4x√gR =

∫M

∫S3p

d4xd3θ sin θ2 sin2 θ3 √g Rabya(θ)yb(θ) . (4.5)

Hence we can view the Einstein–Hilbert action as average over all directions of themanifold at every point over the function f = Raby

ayb. As explained in section 3.4 wecan rewrite f as f = −Rp

pbyb with Rp

pb seen as the components of the curvature of alinear connection on TM induced by the Levi-Civita covariant derivative on M . If wefurthermore introduce hµν to be the pullback of the scalar product from TpM to Sp3 asexplained above we can express the rewritten Einstein–Hilbert action by

SEH [g] =

∫M

∫S3p

d4xd3θ√h√g (−Rp

pbyb) . (4.6)

We begin with the generalization of general relativity exactly here. We want to look atthe Einstein–Hilbert action from a Finsler geometry point of view. The basic underlyingspacetime will not longer be a manifold M with a metric g but a Finsler space (M,F )that naturally contains a Finsler metric gF (see section 3.1). In order to generalize theexpression above we have to clarify what it means to average over all directions of amanifold in a Finsler geometry setting, when we do not have a metric on the manifold,and hence no scalar product on its tangent spaces to define S3

p . Furthermore it will nolonger be possible to perform the integration over θ explicitly since the curvature partof the scalar integrand will be directional dependent. It will be the curvature of thenatural Cartan nonlinear connection of Finsler geometry (section 3.3).

Our aim is to construct the seven dimensional integral of the rewritten Einstein–Hilbert action in a Finsler geometrical setting and to perform the calculus of variationsin order to find an equation that determines a Finsler function or Finsler metric thatextremizes the action. Furthermore if we assume a Finsler function whose Finsler metricdefines a metric on M , i.e., a Finsler metric without fibre coordinate dependence, theconstruction should reduce to the one explained in this chapter. We reach this goalin section 6.1 and it will have exactly the same form as the Einstein–Hilbert action inequation (4.6), only the underlying mathematics will be changed.

46

5. The sphere bundle

To be able to generalize the Einstein–Hilbert action we have to construct a well-definedintegral in the language of Finsler geometry over some seven dimensional manifold thatencodes the integral over the base manifold and all of its directions. It will turn out thatthis seven dimensional manifold is a subbundle of the tangent bundle, we call spherebundle. Furthermore we show that there exist coordinates on the tangent bundle whichare perfectly adapted to the sphere bundle and make it possible to perform calculationsin a manageable way. These coordinates are spherical coordinates for the tangent bundlein the notion of Finsler geometry.

Moreover there is more than one natural way of constructing an integral over the spherebundle but in the end only one that is useful for our generalization of the Einstein–Hilbertaction that we will write down in the next chapter.

5.1. Idea and definition

One way to define an integral over the manifold and all possible directions is to takean integral over the whole tangent bundle and integrate out the absolute value of thevelocities. In the fibre bundle language we integrate out one degree of freedom along thefibre. We would like to integrate out the absolute value of all vectors and just be leftwith the integral over all unit vectors, everything encoded in the natural coordinates ofthe tangent bundle. The natural volume form on TM we know from chapter 3.1, is thevolume form induced by the Sasaki metric.

Since the functions we are interested in to construct the generalized Einstein–Hilbertaction, are all homogeneous of degree n ∈ N with respect to the fibre coordinates thefollowing holds. If f is such a function on TM then f(n + 1) = ∂

∂yb(fyb). Hence an

integral over f , with volume form absorbed into f , can be written as∫d4xd4y f =

∫d4xd4y

∂

∂yb(fyb)

1

n+ 1. (5.1)

Using Stoke’s theorem we can rewrite the integral as integral over the boundary ofTM , but since the boundary of TM lies at infinity the yb in the integration would be-come infinite. The easiest way this can be seen is to introduce spherical coordinates(r, θ1, θ2, θ3) on the fibre and perform the r integration. In those coordinates the ho-mogeneity in y translates in a homogeneity in r. We come back to this argument afterthe introduction of Finsler spherical coordinates and their implications for homogeneousfunctions on the tangent bundle in equation (5.11).

47


As long as we do not have good arguments why the fibre coordinate should be boundedthis is not a practical way to generalize the Einstein–Hilbert action. We remark herethat the manifold coordinates do not have these problems since we always assume thatphysical fields appearing in f , like the curvature, vanish at the boundary of the manifold.So the homogeneity properties of the curvature scalars in question are the reason whywe want to restrict our integration from beginning on to a seven dimensional manifoldwe call sphere bundle. This will lead to a well-defined integral we can use for thegeneralization of the Einstein–Hilbert action. The sphere bundle appears also in booksabout Finsler geometry [20], but mostly for different reasons than here.

Definition 5.1 Sphere BundleThe sphere bundle Σ is a submanifold of the tangent bundle TM of a Finsler space(M,F ) with Finsler metric g of definite signature, defined by

Σ := (x, y) ∈ TM |F (x, y) = 1 = (x, y) ∈ TM |√g(x, y)abyayb = 1 . (5.2)

The sphere bundle is a fibre bundle over a Finsler space (M,F ) with fibre S3Fp over

every fixed point p ∈M called Finsler sphere

S3Fp := X =

(ya

∂

∂xa

)|p∈ TpM |

√gab(x, y)yayb = 1 . (5.3)

The fibre S3Fp appears also in [20] but is called indicatrix.

There are some details in the definition of S3Fp that have to be mentioned, since g

is not a metric on M but a d-tensor field on TM . In a coordinate basis on M aroundp we can express X ∈ TpM as X = ya ∂

∂xa |p. Since the Finsler metric is a d-tensor

field it maps elements of the horizontal bundle HTM or the vertical bundle V TM toR. In order to have a well-defined fibre in the sphere bundle we have to explain whatthe expression

√gab(x, y)yayb means. Let u = (x, y) be a point in the tangent bundle.

Every vector X = ya ∂∂xa |π(u)

∈ Tπ(u)M can be lifted horizontally to HuTM or vertically

to VuTM . Its vertical lift is given by Xv = ya ∂∂ya

while its horizontal lift is Xh = ya δδxa

.

Now it is clear how to understand√gab(x, y)yayb. It is the Finsler metric at u acting

on the vertical or horizontal lift of the vector X depending on whether we see g asg = gabdx

a ⊗ dxb, g = gabdxa ⊗ δyb or g = gabδy

a ⊗ δyb. In any case S3Fp is well-defined.

If g is not dependent on the y coordinates S3Fp reduces to S3

p ⊂ TpM we encountered inthe previous chapter.

To perform an explicit integral over the sphere bundle it is necessary to have a nat-ural volume form in a coordinate basis on Σ. Therefore we introduce Finsler sphericalcoordinates.

5.2. Finsler spherical coordinates

Comparable to standard spherical coordinates given by equation (4.2) which lead tocoordinates on the three sphere S3

p we will now construct coordinates, so-called Finsler

48


spherical coordinates, for the tangent bundle which lead to coordinates on the Finslersphere S3F

p .Consider a coordinate chart (U, φ = (x1, ..., x4)) on a Finsler space (M,F ) and the in-

duced coordinate chart (π−1(U),Φ = (x1, ..., x4, y1, ..., y4)) (see chapter 2) on the tangentbundle TM . These standard coordinates are related to the Finsler spherical coordinates(x1, ..., x4, θ1, θ2, θ3, r) by

xa(x1, ..., x4, θ1, θ2, θ3, r) = xa; a = 1, 2, 3, 4 (5.4)

ya(x1, ..., x4, θ1, θ2, θ3, r) =yar

F (x, y); a = 1, 2, 3, 4 . (5.5)

The θ dependence of these coordinates is hidden in the ya, a = 1, ..., 4. They are standardspherical coordinates

y1 = cos θ1 sin θ2 sin θ3; y2 = sin θ1 sin θ2 sin θ3;

y3 = cos θ2 sin θ3; y4 = cos θ3;

θ1 ∈ (0, 2π); θ2, θ3 ∈ (0, π). (5.6)

An investigation of F (x, y), under use of its homogeneity properties, leads to the follow-ing remarkable insight

F (x, y) = F

(x,

yr

F (x, y)

)=

r

F (x, y)F (x, y) = r . (5.7)

Furthermore it is clear that the inverse coordinate transformation for the manifoldcoordinates xa is the identity and that the θµ, µ = 1, 2, 3, coordinates are related tothe y coordinates through the y via

ya√δpqypyq

=yar

F (x, y)

√F 2(x, y)

r2δpqyqyp

= ya . (5.8)

The step to come from the y to the θµ, µ = 1, 2, 3, is given by the inverse coordinatetransformation of standard spherical coordinates from equation (5.6).

Our further calculations will become much easier if we use all homogeneity properties.Equation (5.8) makes it easy to demonstrate that y is homogeneous of degree zero withrespect to y, since

λya√δpqλypλyq

=ya√δpqypyq

⇒ yb∂ ya

∂yb= 0 . (5.9)

For the θµ a similar condition holds

yb∂θµ

∂yb= yb

∂θµ

∂ ya∂ ya

∂yb= 0 . (5.10)

We presented the complete coordinate transformation on the tangent bundle, startingfrom standard manifold induced coordinates to Finsler spherical coordinates and back.

49


It is obvious that if we set r = 1 the fibre part of the Finsler spherical coordinatesare local coordinates on S3F

p . As for standard spherical coordinates one has to takeout the north or south pole of the sphere to get a coordinate patch. It is possible tothink about the sphere bundle locally as parametrized submanifold of the tangent bundlewith parametrization given by (xm, θµ), m = 1, 2, 3, 4; µ = 1, 2, 3 with x ∈ (−∞,∞),θ1 ∈ (0, 2π) and θ2, θ3 ∈ (0, π) defined by the relations (5.4) and (5.5).

Transforming from standard induced coordinates on TM to the Finsler spherical co-ordinates gives us the possibility to restrict every function on TM to Σ in a very easyway, by just setting r equal to 1. Since all functions on TM we are interested in havehomogeneity properties they all can be regarded as functions on Σ and then be extendedback to all of TM via their homogeneity if it is necessary. Let f be function on TMthat is homogeneous of degree n. Expressed in Finsler spherical coordinates this leadsto

f(x, y) = f(x, θ1, θ2, θ3, r)

= f(x,yr

F (x, y)) = rnf(x,

y

F (x, y)) = rnf(x, θ1, θ2, θ3, 1) = rnf|Σ . (5.11)

The components of the Finsler metric g for example are homogeneous of degree zero,hence have no dependence on r. So if they are known on Σ they are known on all of TMwithout any rescaling. Functions of homogeneity different then zero have to be rescaledto be extended from Σ to TM , as seen in equation (5.11).

In the beginning of section 5.1 we argued that it does not work to construct an integralover Σ by integrating out one degree of freedom in the fibres of the tangent bundle. Thereason why this integral will always be infinite can be seen here in the Finsler sphericalcoordinates. The absolute value integration we wanted to perform translates in the rintegration and for positive homogeneous functions of degree n we see from equation(5.11) that their r dependence is given by the n-th power of r. Bearing in mind thatTM is not bounded the r integration will always diverge.

Transforming the derivatives, the coordinate basis of TTM and the coordinate basisof T ∗TM from the standard induced coordinates to the Finsler spherical coordinateson TM gives us an insight about how some tensor structures can be mapped from TMto Σ.

5.3. Transformation of the coordinate basis

The components of the Finsler metric g are given by g(x, y)ab = 12∂2F 2

∂ya∂yb(see section

3.1). In order to rewrite it in Finsler spherical coordinates we have to transform thederivatives with respect to the fibre coordinates into derivatives with respect to thenew coordinates. Furthermore a canonical volume form on Σ is needed to write downa well-defined integral. Such a volume form can be constructed through the pull-backof the Sasaki metric (see section 3.1) from TM to Σ. Therefore we have to study thetransformation of the standard coordinate basis one-forms dxa, dyb and the standardcoordinate basis vector fields ∂

∂xa, ∂∂yb to Finsler spherical coordinates.

50


Standard coordinates (zA, A = 1, ..., 8) on the tangent bundle are given by (zA) =(xa, ya); a, a = 1, 2, 3, 4. This kind of index notation is confusing in the beginning but itmakes the 4 + 4 split of the tangent bundle into manifold and fibre clear. New coordi-nates on TM , the Finsler spherical coordinates, get labelled by (zM) = (xm, θµ, r);m =1, 2, 3, 4;µ = 1, 2, 3. As for every coordinate transformation on TM we have

dzA =∂zA

∂zMdzM = P−1A

MdzM and

∂

∂zA=∂zM

∂zA∂

∂zM= PM

A

∂

∂zM. (5.12)

More explicitly for the transformation from standard coordinates to Finsler sphericalcoordinates defined in equation (5.4)

zA(z) = (xa(z), ya(z)) =

(xa,

ya(θ)r

F (x, y(θ))

). (5.13)

The transformation matrices P defined in equation (5.12) results in

P−1AM =

(δam 0 0∂ya

∂xm∂ya

∂θµ∂ya

∂r

)=

(δam 0 0P a

m P aµ P a

r

)=

(δam 0P a

m P am

)(5.14)

PMA =

δma 00 ∂θµ

∂ya∂r∂xa

∂r∂ya

=

δma 00 P µ

a

P ra P r

a.

=

(δma 0P m

a P ma

). (5.15)

The index m combines the new fibre coordinates θ and r to one block. To be preciseP ma = ∂ym

∂xaand P m

a = ∂ym

∂ya, where ym = θµ; m = 1, 2, 3 and ym = r; m = 4. In some

circumstances this allows us a more compact notation. Remarkable is the vanishing of∂θµ

∂xm, which is perhaps not immediately clear. A look at equation (5.8) should remove

all, since θµ is only dependent on y and y only on y and not on x coordinates.The upper case indices matrices from equations (5.14) and (5.15) will not appear

as whole matrices in our calculations, only their blocks with lower case indices are offurther interest. Therefore it is worth it to study the implications of the fact that theyare inverse of each other on their blocks

P−1AMP

MC =

(δamδ

mc 0

∂ya

∂xmδmc + ∂ya

∂r∂r∂xc

∂ya

∂θµ∂θµ

∂yc+ ∂ya

∂r∂r∂yc

)!

=

(δac 00 δac

)(5.16)

PMCP−1C

N =

(δmc δ

cn 0

∂ym

∂xcδcn + ∂ym

∂yc∂yc

∂xn∂ym

∂yc∂yc

∂yn

)!

=

(δmn 00 δmn

). (5.17)

Hence the lower case indices P -blocks defined in equations (5.14) and (5.15) satisfy

P amδ

mc = −P a

r Prc ; P a

µPµc + P a

r Prc = δac

P mc δ

cn = −P m

c Pcn; P m

c Pcn = δmn . (5.18)

Further useful Properties fulfilled by the different parts of the coordinate transforma-tions are

yaP µa = 0; yaP

aµ = 0; yaP r

a = r; r∂P r

a

∂r= 0;

P µc P

cµ = 3; P r

c Pcr = 1; P r

a =yar

; P ar =

ya

r. (5.19)

51


These are calculated with help of equations (5.7), (3.5) and (3.3). We mention allthese properties here because they will simplify the calculation to come enormously.For example we can rewrite the components of the Finsler metric in Finsler sphericalcoordinates. Remembering F (x, y)2 = r2 the components of g become

g(x, y)ab =1

2

∂2F 2

∂ya∂yb=

1

2

∂2r2

∂ya∂yb=

∂

∂yarP r

b

= P raP

rb + r

∂P rb

∂ya= P r

aPrb + rP µ

a

∂P rb

∂θµ+ rP r

a

∂P rb

∂r

= P raP

rb + rP µ

a

∂P rb

∂θµ. (5.20)

Since we know that g(x, y) is homogeneous of degree zero with respect to y, hence withrespect to r in the Finsler spherical coordinates (see equation (5.11)), and moreover thesame holds for the P r

a it follows from the above equation (5.20) that the P µa have to be

homogeneous of degree −1 with respect to r.In our index notation on the tangent bundle, the components of the Finsler metric

have indices from the beginning of the alphabet with bar. A problem that will come upsoon is that the Finsler metric will appear in several calculations contracted with variousindices since it rises, lowers and contracts them. So for the Finsler metric we will breakthe index convention: it will appear with arbitrary indices. The objects where it is reallyimportant which kind of index labels them are the blocks of the basis transformations P(equations (5.14) and (5.15)). These will always be determined by an indices is writtenwith bar or without or from the beginning or middle of the alphabet.

After having realized how to transform the coordinate bases on TM we can focus onthe integration over the sphere bundle Σ.

5.4. Integration over the sphere bundle

From the previous two sections we know how to find suitable coordinates on the spherebundle in which we could carry out an integration. One last missing thing is a naturalvolume form to integrate over. On the tangent bundle we encountered the Sasaki metricin section 3.1 and the corresponding volume form on TM . If we perform a pullback ofthe Sasaki metric to Σ along the natural embedding of Σ into TM we get a metric anda volume form on the sphere bundle.

It will turn out that for a vanishing Cartan tensor (definiton 3.4) an action basedon this construction reduces to the Einstein–Hilbert action shown in equation (4.5), asit should. Furthermore we will find out that on the sphere bundle exists a horizontal-vertical bundle structure which is inherited from the Cartan nonlinear connection (chap-ter 3.3) on the tangent bundle.

Consider the mapping Ψ that maps Σ into TM , given in the Finsler spherical coordi-nates from section 5.2

Ψ : Σ→ TM Ψ(xm, θµ) =

(xm,

ya(θ)

F (x, y(θ))

). (5.21)

52


The Sasaki metric given by equation (3.7) is a metric on the tangent bundle that canbe pulled back via Ψ to the sphere bundle Σ. As for every manifold the determinant ofa metric can be used to define a volume form in the coordinate basis of the manifold.Hence on the sphere bundle the pull-back of the Sasaki metric defines a natural volumeform. Before we can perform the pull-back we need to transform the Sasaki metric into acoordinate basis of TM since it is defined in the horizontal-vertical basis (see chapter 1)and our mapping Ψ is not. For the sake of clarity we will write these calculations inmatrix notation. Indices a, b, ... run from 1 to 4 and indices α, β, ... run from 1 to 3. Theblock structure of the matrices should become clear from the indices.

Regard the Sasaki metric G as an eight by eight matrix in the horizontal-vertical basisof TM defined by the Cartan nonlinear connection (see definition 3.8)

G =

(gab 00 gab

). (5.22)

A change of basis on TTM from the horizontal-vertical basis to the standard coordinatebasis is given by the matrix S which has to act on G in the following way, known fromlinear algebra

S =

(δbc 0N bc δbc

)G = ST ·G · S . (5.23)

The form of S can easily be read off by expanding equation (3.7) in the induced coordi-nate basis of TM . Let X, Y be vector fields on Σ and Ψ∗G be the pullback of G in thecoordinate basis of the sphere bundle then

Ψ∗G(X, Y ) = G(dΨ(X), dΨ(Y )) . (5.24)

Expressed by matrix multiplication in the coordinate basis of Σ and TM , Ψ∗G is givenby

Ψ∗G = dΨT · G · dΨ = dΨT · ST ·G · S · dΨ . (5.25)

Ψ∗G really is a seven by seven matrix since dΨ is an eight by seven matrix that lookslike

dΨ =

(δcm 0∂yc

∂xm∂yc

∂θµ

). (5.26)

Doing all the calculation leads to the result for the pullback of the Sasaki metric inthe coordinate basis on the sphere bundle (details see appendix B.1)

Ψ∗G =

gmn + P rmP

rn + gabN

amN

bn +N b

ngbcPcm +N b

mgbcPcn gdbN

bn∂yd

∂θµ

gdbNbm∂yd

∂θνgabP

aµP

bν

. (5.27)

Performing a coordinate transformation T on the sphere bundle we can diagonalizeΨ∗G which makes it easier to calculate its determinant. The coordinate transformationneeded has the property that its determinant is one and so the coordinate transformedmetric G has the same determinant as Ψ∗G

T =

(δmc 0lµc δµγ

)detT = 1⇒ det(Ψ∗G) = det(T T · G · T ) = det(G) . (5.28)

53


Take

lµc = P µaN

ac ; hµν = gabP

aµP

bν (5.29)

then the resulting diagonal metric G and its determinant are simply given by

G =

(gmn 00 hµν

); det G = det(gmn) det(hµν) . (5.30)

A detailed calculation can be found in appendix B.2. The basis that diagonalizes Ψ∗Gis the horizontal-vertical basis of the sphere bundle defined by connection componentsNµc = lµc = P µ

aNac inherited from the Cartan nonlinear connection components N a

c onTM

δ

δxm=

∂

∂xm−Nµ

m

∂

∂θµ;

∂

∂θµ

. (5.31)

The dual horizontal-vertical basis is constructed as explained in equation (1.6). Hencea natural well-defined metric on the sphere bundle is given by

G = gmndxn ⊗ dxm + hµνδθ

µ ⊗ δθν . (5.32)

What happened by going from the tangent bundle to the sphere bundle is that themetric along the horizontal bundle, gets not altered and along the vertical bundle themetric gets pulled back to the exchanged fibre. Since hµν can be regarded as metric onS3Fp for a fixed p ∈M it has an inverse hµν we want to mention here explicitly

hµν = gabP µa P

νb ; hµαhαν = δµν . (5.33)

It will play an important role in the calculus of variation we will perform to get newequations of gravity. To proof the inverse equation above we simply use the relationsgiven by equations (5.18) and (5.19),

hµαhαν = gabP µa P

αb gpqP

pνP

qα = gabgpqP

µa P

pν (δq

b− P q

r Prb ) (5.34)

= P µp P

pν − gabgpqP

µa P

pν

yq

r

ybr

= P µp P

pν = δµν . (5.35)

Calculations of this kind will appear over and over again in our further proceeding.Most of the time they will be abandoned to the appendix because of their lengthy nature.The proof of the inverse relation is presented as a model calculation so the reader getsa feeling about how a lot of calculations are done in principal.

A well-defined integral over a sphere bundle function f(x, θ) finally looks like∫Σ

d4xd3θ√G f(x, θ) =

∫Σ

d4xd3θ√g√h f(x, θ); h = det(hµν); g = det(gmn) . (5.36)

This integral is the key to write down a new generalization of the Einstein–Hilbert action.

54

6. Finsler gravity and the calculus ofvariation

With the sphere bundle technology we are now in the position to present one of the keyresults of this theses, a generalized action SG[F ] for Finsler gravity.

Afterwards we want to deduce dynamical equations which determine a Finsler func-tion F that defines the properties of spacetime. As usual for an action based theoryin physics we require this Finsler function to extremize the action. The answer to thequestion how to find this Finsler function is to perform calculus of variation on theaction and solve the resulting equations. Here two important problems arise from theconcepts we used to construct the action. These will be answered in this chapter, beforewe derive the Einstein equations from the Einstein–Hilbert action using the formalismwe developed in the next chapter.

First from Finsler geometry we have a choice of what we regard as fundamental dy-namical variable, the Finsler function or the Finsler metric since they are equivalent.Therefore if a certain Finsler function F extremizes the action its corresponding Finslermetric g should do so also, and vice versa. In the second section of this chapter we willshow that this is the case: a dynamical equation for the Finsler function that determinesspacetime is equivalent to dynamical equations for the components of the Finsler metricgab.

The second problem arises from the fact that our action is an integral over the spherebundle. In the generalized action integrand there are derivatives with respect to thestandard induced coordinates on the tangent bundle acting on the dynamical variable ofchoice. In order to read off dynamical equations at the end of the variation we have tocommute the variation with the derivatives. But since we are on the sphere bundle, andthe sphere bundle is defined with the help of our dynamical variables as explained inchapter 5, the variation does not commute with derivatives with respect to the standardinduced coordinates on the tangent bundle, it only commutes with the sphere bundlecoordinates. The solution to this problem will be presented in the final section of thischapter.

6.1. Finsler gravity action

We showed how our construction of an integral over the sphere bundle works and thatintegrals over the sphere bundle encode integrals over the base manifold and all ofits tangent space directions. Recall that for the definition of the sphere bundle theFinsler metric must be of definite signature. The new generalization of the Einstein–

55

6. Finsler gravity and the calculus of variation

Hilbert action SG we will present now is the integral over the sphere bundle functionf = (Rp

pbyb)|Σ

SG[F ] =

∫Σ

d4xd3θ√g√h (Rp

pbyb)|Σ h = det(hµν); g = det(gmn) . (6.1)

The function f we encountered already in section 3.4 where we called it Ric andshowed that it reduces to −Raby

ayb in the case of a linear connection on the tangentbundle. Since for an underlying metric structure the Cartan nonlinear connection isa linear connection as mentioned in section 3.3, the action reduces to the rewrittenEinstein–Hilbert action we presented in chapter 4, if the Finsler function leads to aFinsler metric that defines a real Riemannian metric on M .

Remarkably our generalization of the Einstein–Hilbert action and the Einstein–Hilbertaction itself have exactly the same form, up to a sign that does not matter. The exten-sion we present really lies in the extension of the underlying mathematical concepts weintroduced. An advantage of our way of extending the gravity action is that it alwaysyields back the original gravity action as mentioned above and that we can see in allsteps of all upcoming calculations were we get corrections to the original theory, and howeverything reduces back to general relativity if we impose the symmetry of a vanishingCartan tensor what is equal to the independence of the Finsler metric from the tangentspace coordinates.

6.2. Equivalence of Finsler equations

The equivalence of Finsler function F and Finsler metric g is clear since if we start froma Finsler function the corresponding Finsler metric is given by gab = 1

2∂2F 2

∂ya∂yband if we

start from a given Finsler metric gab the Finsler function it belongs to is F =√gabyayb.

Therefore we can always construct one out of the other in a one to one correspondence.Hence we can view our action SG as depending on F , SG[F ], or depending on g, SG[g].It is important that we clarify that the dynamical equation obtained from the calculusof variations with respect to the Finsler function is equivalent to the equations we get byvariation with respect to the Finsler metric, since we could do both on the same action,depending on whether we regard the action as depending on F or as depending on g.

In order to prove that the equations for F and g are equivalent we have to rewritethe variation with respect to F into a variation with respect to gab and vice versa. Webegin with the deduction of the explicit dependence of δF on δgab.

δF (x, θ) = δF (x, y(θ)) = δ

√gaby

ayb =1

2

δgabF

yayb = F (x, θ)1

2δgaby

ayb. (6.2)

An important remark about the notation has to be made here. Wherever throughoutthis thesis a δF or δF

Fappears it is always meant to be evaluated at the sphere bundle

coordinates, i.e., δF = δF (x, θ) = δF (x, y(θ)) and δFF

= δF (x,y(θ))F (x,y(θ))

. Otherwise, if any-

where appears a F (x, y) in a variation we will always write it as r since F (x, y) = r. Forthe definition of the different coordinates see section 5.2.

56


In equation (5.20) we expressed the Finsler metric in Finsler spherical coordinates

g(x, θ)ab = P raP

rb + P µ

a

∂P rb

∂θµ. (6.3)

Varying the components of the Finsler metric with respect to the Finsler function Fleads to

δg(x, θ)ab = 2gabδF

F+

(3P r

(aPµ

b)+ Pα

(a

∂P µ

b)

∂θα

)∂

∂θµδF

F+ P µ

a Pνb

∂2

∂θµ∂θνδF

F=: Dab(

δF

F) .

(6.4)For everyone who is interested in the details of this calculation they are explained inappendix B.3. The other way around is much easier to calculate

Now we can compare variations of an action with respect to F and g. Consider thefollowing action depending on the Finsler metric

S[g] =

∫Σ

d4xd3θ√g√h Q(g) =

∫Σ

d4xd3θQ(g). (6.5)

We absorbed the volume form factors in the integrand since they do not play a specialrole in the argument to come. Calculus of variation with respect to the Finsler metricapplied to the action formally results in

δS[g] =

∫Σ

d4xd3θδQ

δgpqδgpq. (6.6)

Hence the equations of motion for a Finsler metric from this action are given by δQδgpq

= 0.

If we consider the same action dependent on F instead of depending on g, the variationof the action with respect to F becomes

δS[g(F )] =

∫Σ

d4xd3θδQ

δgpqδgpq =

∫Σ

d4xd3θδQ

δgpqDab(

δF

F) . (6.7)

Since Dab is a differential operator determined by equation (6.4) we have to do partialintegration to read off the equation which determines the Finsler function that extremizesthe action. It is given by Dab(

δQδgpq

) = 0, where Dab is the differential operator that results

from partial integration of Dab. At this point it is clear that if we have a Finsler metricthat fulfils δQ

δgpq= 0 also the equation for the corresponding Finsler function Dab(

δQδgpq

) = 0

is solved. But what about the other way around?Consider the same action as above to be F dependent

S[F ] =

∫Σ

d4xd3θQ(F ) = S[g(F )] =

∫Σ

d4xd3θQ(g(F )). (6.8)

Its the variation leads to

δS[F ] =

∫Σ

d4xd3θδQ

δFδF . (6.9)

57


The equations of motion are given by δQδF

= 0. Then switching the variable to thecorresponding Finsler metric g the variation becomes

δS[F (g)] =

∫Σ

d4xd3θδQ

δFδF =

∫Σ

d4xd3θδQ

δFF

1

2yaybδgab . (6.10)

There is no differential operator involved by varying F with respect to gab, which enablesus to read off the equations that have to be satisfied by gab to extremize the action S[F (g)]

to be δQδF

12yayb = 0. If we have a Finsler function that satisfies δQ

δF= 0 its corresponding

Finsler metric satisfies δQδF

12yayb = 0.

We showed that if a Finsler metric extremizes an action its corresponding Finslerfunction does and that the same holds vice versa. Since the actions we consideredare the same, only regarded as depending on different variables, it is also clear thatthe equations produced through the different variations are equivalent. To make thispoint clear we remark that S[F (g)] = S[g] as well as S[F ] = S[g(F )] and therefore thefollowing holds

δS[F (g)] = δS[g] and δS[F ] = δS[g(F )]. (6.11)

Therefore if one of the variations vanishes for a certain solution of the equations fromthe calculus of variations, the other does also. On the level of the action integral this isclear. On the level of equations this implies

δQ

δF

1

2yayb =

δQ

δgaband

δQ

δF= Dab(

δQ

δgab) . (6.12)

It is clear that these equations are not independent. If the first equation in (6.12) holdsthen the second follows by applying Dab, since Dab fulfils Dab(

12fyayb) = f what is shown

in appendix B.4.Let us one more time summarize the result of this section. If we have an action that

depends on a Finsler function as dynamical variable and if we want to extremize thisaction, it does not matter whether we vary it with respect to the Finsler function orwith respect to the corresponding Finsler metric. The equation obtained in the case ofthe variation with respect to the Finsler function is equivalent to the equations obtainedby varying with respect to the Finsler metric. Furthermore equations (6.12) tells us howto rewrite the scalar equation for the Finsler function into the tensor equation for theFinsler metric and vice versa. From the viewpoint of a gravity theory that means thedynamics of gravity can be described by one scalar equation for the Finsler functionor by a symmetric (0,2)-tensor equation with ten components. Both lead to a Finslerfunction, directly or indirectly, that extremizes the gravity action.

6.3. Calculus of variation on the sphere bundle

In the Einstein–Hilbert action there appear derivatives with respect to the standardinduced coordinates on the tangent bundle restricted to the sphere bundle. If we rewrite

58


them in Finsler spherical coordinates the blocks of the basis transformation P (equations(5.14) and (5.15)) will show up and they are dependent on the Finsler function F orthe corresponding Finsler metric g depending on the viewpoint. Since we view theFinsler function or Finsler metric as fundamental dynamical variable we have to includethe coordinate transformation in the calculus of variations. In other words we cannotchange the order of variation δ and derivatives with respect to the standard inducedcoordinates, since they are F respectively g dependent:

δ∂

∂ya |Σ6= ∂

∂ya |Σδ . (6.13)

Therefore we study the variation of the basis transformation matrices P . Rememberthat dependence on F is equivalent to an dependence on g and vice versa since F (x, y) =√g(x, y)abyayb and g(x, y)ab = 1

2∂2F 2

∂ya∂yb(x, y). All variations that appear take place on

the sphere bundle. The first object of interest is the y coordinate itself

δya|Σ = δya(θ)

F (x, y(θ))= − ya(θ)

F (x, y(θ))2δF (x, y(θ)) = −ya|Σ

δF (x, y(θ))

F (x, y(θ))= −ya|Σ

δF

F. (6.14)

Most of the time we have d-tensor or scalar expressions defined on the tangent bundlewhich we want to vary restricted to the sphere bundle. Therefore the transformationmatrix of interest is PM

A (equation (5.15)),

δPMA = −PM

C PNA δP

−1CN . (6.15)

We deduce the variation of PMA this way since it is less work to vary P−1C

N . The variationof P−1C

N (equation (5.14)) yields

δP−1cN = −P−1c

N

δF

F− yc ∂

∂zNδF

F. (6.16)

All other components have vanishing variation. Putting the last two results together,and remembering the m notation from equations (5.14) and (5.15) the variation of δPM

A

becomes

δP ma = −P m

c PNa δP

−1cN = P m

c PNa (P−1c

N︸︷︷︸=δca=0

δF

F+ yc

∂

∂zNδF

F) = P m

c yc ∂

∂xaδF

F. (6.17)

The zero in the middle of the calculation is true since c is a y and a a x index, sothey can never be equal. After the last equal sign only the ∂

∂xasurvive since δF

Fis not

r dependent. Their dependence on the coordinates can be seen in equation (6.14). Asimilar calculations leads to the missing parts of the variation

δP ma = P m

a

δF

F+ P m

c ycP µ

a

∂

∂θµδF

F. (6.18)

Summing all up we have the following four important variation formulas

δP ra =

∂

∂xaδF

F; δP µ

a = 0; δP ra = P r

a

δF

F+ P µ

a

∂

∂θµδF

F; δP µ

a = P µa

δF

F. (6.19)

We want to be sure that the variation procedure we developed so far is consistent withthe results from the Einstein–Hilbert action of general relativity. Therefore we derivethe Einstein equations using our formalism in the next chapter.

59

7. Einstein’s equations from a Finslerpoint of view

All we developed so far would be useless if it were not possible to generate the equationsof general relativity in the framework we explained in the previous chapters. Thereforewe take the rewritten Einstein–Hilbert action from chapter 4 and rewrite it further intothe form of the generalized Einstein–Hilbert action (section 6.1) with the help of a Finslerfunction with vanishing Cartan tensor (see definition 3.4). Then we perform calculus ofvariations to show that indeed we can deduce the Einstein equations.

The calculation we present in this chapter is a new way of deducing general relativityfrom an action principle. It shows that our sphere bundle construction, including thevariation procedure we described in chapter 6, consistently produce Einstein’s theory ofgravity. If inconsistencies would appear on this level, there would be less hope to get aconsistent generalized theory that reduces well to general relativity not only on the levelof the action, but also on the level of equations.

However, note that the calculations of this chapter will not clarify whether the fullgeneralized theory without any constraints on the Finsler function will be consistentwith general relativity. This question has to be investigated at the point when we havethe dynamical equations from the generalized Einstein–Hilbert action in chapter 8.

Furthermore we argued in chapter 6 that the equations gained by variation of a metricwith respect to the Finsler metric are equivalent to those gained by the variation withrespect to the corresponding Finsler function. Both variations will be done in thischapter and we will see that the equations are equivalent even if they have not the sameform.

During the first part of this chapter we will derive the equations without any matterpresent. The resulting equation will not directly be form-equivalent to the Einsteinequations but their implication will be. Therefore in the second part we will includematter that couples to the metric and see that in the end we gain equations which canbe manipulated to look exactly like the Einstein equations.

7.1. Vacuum equations

Consider a Finsler space (M,F ) with Cartan nonlinear connection induced curvatureRa

bc (chapter 1.3) and vanishing Cartan tensor C (definition 3.4). We formulate theEinstein–Hilbert action in the Finsler geometry language as integral over the sphere

61

7. Einstein’s equations from a Finsler point of view

bundle Σ (chapter 5) given by

SEH = −∫

Σ

d4xd3θ√g√h (Rp

pbyb)|Σ = −

∫Σ

d4xd3θ√g√h (Rp

abpybya)|Σ

=

∫Σ

d4xd3θ√g√h( Raby

bya)|Σ . (7.1)

There is no need to restrict√g and

√h to Σ since they naturally live there. Details

concerning the rewriting of the nonlinear curvature in case of a vanishing Cartan tensorare clear from the end of section 2.1.

All objects that appear in the action are dependent on F respectively g. First we varythe action with respect to F and will see that the equation of motion that describes thedynamics of F are clearly equivalent to the Einstein equations. During the upcomingcalculation we omit the subscript |Σ for the restriction of the functions to Σ, all functionsare meant to be evaluated there. Standard tricks of calculus of variation are assumed tobe known, for example the variation of a determinant etc..

The first step in the variation is

δSEH [F ] =

∫Σ

d4xd3θ√g√h(

1

2Raby

aybgpqδgpq +1

2Raby

aybhµνδhµν + δRabyayb − 2Raby

aybδF

F

). (7.2)

Evaluating hµνδhµν = (gmn − ymyn)δgmn − 6 δFF

(explicitly done in appendix B.5.1) thevariation becomes

δSEH [F ] =

∫Σ

d4xd3θ√g√h(

1

2Raby

aybgpqδqpq +1

2Raby

ayb(gpq − ypyq)δgpq

− 5Rabyayb

δF

F+ δRaby

ayb) . (7.3)

Since√g and Rab are not dependent on θ in the considered case we can perform the

θ integration in the term with the variation of the Ricci tensor itself. Going in anorthonormal frame basis ea the integral that is left is∫

S3Fp =S3

p

d3θ√hyayb =

vol(S3p)

4gab. (7.4)

Therefore, as known from the standard derivation of the Einstein equations the partof the variation with δRab vanishes as total divergence of the manifold coordinates.Everything left from the variation can be summarized in

δSEH [F ] =

∫Σ

d4xd3θ√g√h

(Raby

ayb(gpq − 1

2ypyq)δgpq − 5Raby

aybδF

F

). (7.5)

From equation (6.4) we know how δgab and δF are related. Inserting this in thevariation equation (7.5) and expanding everything including partial integrations, theresult is

δSEH [F ] =

∫Σ

d4xd3θ√g√h (−6Rab + 2Rgab) y

aybδF

F. (7.6)

62


This is a long calculation done in appendix B.5.2. Hence we read off the equation thatdetermines F in the vacuum,

(−6Rab + 2Rgab)yayb = 0⇔ (Rab −

1

3Rgab)y

ayb = 0 . (7.7)

This result is obviously not form-equivalent to the Einstein vacuum equations

Rab −1

2Rgab = 0 . (7.8)

Anyway both equation reduce to the fact that in the vacuum the Ricci-Tensor mustvanish Rab = 0.

If we consider the action with Finsler metric as fundamental variable the calculationwould not differ until we reach equation (7.5). Just exchange F in the argument of thefunctional against g. Since it is fairly easy to rewrite δF in terms of δg see equation(6.2) we rewrite the variation into

δSEH [g] =

∫Σ

d4xd3θ√g√h

(Raby

ayb(gpq − 1

2ypyq)δgpq −

5

2Raby

aybδgpqypyq)

=

∫Σ

d4xd3θ√g√h(Raby

aybgpq − 3Rabyaybypyq)δgpq . (7.9)

Since δgpq has no dependence on θ in the case we consider during this chapter we haveto read off∫

S3p

d3θ(√hRaby

aybgpq − 3Rabyaybypyq) =

1

4vol(S3

p)Rgpq − 3Rab

1

8vol(S3

p)g(abgpq) = 0 .

(7.10)The calculation of the integral over the yaybypyq term was done in the same way as itwas done in chapter 4 for the yayb term. Switch to an orthonormal frame, which ispossible since we have a metric on the manifold, and do the integration for all indexcombinations.

Simplifying equation 7.10 leads to

0 =1

4Rgpq − 3

8Rabg

(abgpq) =1

4Rgpq − 3

8(1

3Rgpq +

2

3Rpq) =

1

8Rgpq − 2

8Rpq . (7.11)

It is obvious that the just deduced equation is equivalent to the Einstein vacuumequations (7.8). To make sure that the equation we got for the variation with respectto the Finsler function (equation (7.7)) takes also the well known form, we now deducethe full gravity equations including matter.

7.2. Complete gravity equations including matter

We will show that if we perform calculus of variation on a matter action with respectto the Finsler function, that the full gravity equations look like Einstein’s equations.

63


We restrict our investigation here to the Finsler function as fundamental variable sincethe equivalence to the case of g as fundamental variable was shown in section 6.2. Inthe variation with respect to the metric the energy momentum tensor would result.The argument that clarifies the last statement is that we can again only read off theintegration over θ since the variation of g is not θ dependent, as explained in the previoussection.

As before in this chapter, we consider a Finsler space with vanishing Cartan tensor. Amatter action S is based on a Lagrangian density L that is a scalar on the Finsler space,so it depends only on the manifold coordinates. It describes as usual the dynamics of thematter field of interest. In the Finsler geometry setting with Finsler function F regardedas fundamental variable that determines spacetime the matter action for matter fieldsψi looks like

SM [F, ψi] =

∫Σ

d4xd3θ√g√h L(g, ψi) . (7.12)

Since L and g are not dependent on θ we could in principle perform the integration overthe fibre coordinates which would lead to the standard matter action on the manifoldM if we divide out the volume of the three sphere.

Calculus of variation with respect to the metric performed on a matter action leadsusually to the Energy-Momentum tensor of the considered matter. In the Finsler ge-ometry setting we will show that a variation with respect to the Finsler function leadsto an expression, where the Energy-Momentum tensor T ab and its trace T = T abgab areinvolved. When we combine the matter action (equation 7.12) with the gravity action(equation 7.1) this leads to equations identical to Einstein’s equations.

After a calculation comparable long as for the derivation of the varied gravity action(7.6) the variation of SM with respect to the Finsler function becomes

δSM [F, ψi] =

∫Σ

d4xd3θ√g√h(12T ab − 2Tgab)yayb

δF

F. (7.13)

Details are presented in the appendix B.5.3. We combine the matter and the Einstein–Hilbert action to a total action that couples gravity to matter

S[F, ψi] = κSEH [F ] + SM [F, ψi]. (7.14)

After performing the variation with respect to F ,

S[F, ψi] = δSEH [F ] + δSM [F, ψi]

=

∫Σ

d4xd3θ√g√h(κ(−6Rab + 2Rgab) + 12Tab − 2Tgab)y

aybδF

F(7.15)

this leads to the following equation that determines the Finsler function F which in turndetermines the structure of spacetime,

(κ(−6Rab + 2Rgab) + 12Tab − 2Tgab)yayb = 0 . (7.16)

Recall that we restricted ourselves to a Finsler space with vanishing Cartan tensor.Therefore none of the tensors in the bracket of equation (7.16) are y dependent. Taking

64


the second derivative with respect to y on this equation and contracting all indices witha metric results in

κ2R + 4T = 0⇔ T = −1

2κR . (7.17)

Inserting equation (7.17) into equation (7.16), dividing by 12 and setting κ = 2 c4

8πG

finally leads to an equation that already looks very similar to the Einstein equations

(Rab −1

2Rgab)y

ayb =8πG

c4Taby

ayb. (7.18)

Again, a second derivative with respect to the fibre coordinates leads exactly to theequations of general relativity.

Using the construction we developed on the basis of Finsler geometry we showed thatit is possible to derive an equation equivalent to the Einstein equations. Actually thederivation is based on a metric with definit signature, since this is one basic assumptionto construct the sphere bundle (see chapter 5). But, as known from general relativitywe search for solutions of the dynamical equations which lead to a metric with Lorentzsignature. In the case of the Einstein equations ist is clear that they exist.

In the next chapter we release the constraint of a vanishing Cartan tensor on theaction level and vary the generalized Einstein–Hilbert action presented in section 6.1.The resulting equations for the dynamics of gravity will be deduced under the assumptionof an underlying Finsler metric with definite signature, since the generalized Einstein–Hilbert action is formulated on the sphere bundle. But, as explained in this chapter,the generated equations are expected to be useful determining a Finsler function withFinsler metric of Lorentzian signature.

65

8. Finsler gravity equations

In the last chapter it turned out that our variation procedure is consistent with a deriva-tion of general relativity. Now we want to perform calculus of variation on the generalizedEinstein–Hilbert action presented in section 6.1, which will lead to an extended gravityequation. Recall that this action

SG[F ] =

∫Σ

d4xd3θ√g√hRic(x, y)|Σ =

∫Σ

d4xd3θ√g√h(Rp

pd(x, y)yd)|Σ . (8.1)

reduces up to a sign to the Einstein–Hilbert action we investigated during the lastchapter. First we present a way to organize the variation before we finally perform thevariation and writhe down our new gravity equations in a compact way.

As usual we consider a Finsler space (M,F ) with Cartan nonlinear connection de-fined through connection components Na

b (x, y) (chapter 3.3) and corresponding curva-ture scalar Ric (chapter 3.4). All of these objects get restricted to the sphere bundle Σequipped with the canonical metric G defined in chapter 5.

8.1. Organising the variation

During the derivation of Einstein’s equations from the Einstein–Hilbert action the varia-tion of the Ricci tensor itself vanishes, since it can be shown that it is a total divergence.The nonlinear curvature Ra

bc does not have this feature. Therefore we have to keep itsvariation. Expanding the curvature in formal Christoffel symbols Γabc built out of theFinsler metric, remembering from section 3.3 that Ga = 1

2Γabcy

byc and Nab = ∂Ga

∂yb, leads

to

Ric = (δNp

p

δxc− δNp

c

δxp)yc = yayb(

∂Γpap∂xb

− ∂Γpba∂xp

+ ΓqapΓpbq − ΓpabΓ

qpq)

+ yaybyc(∂Γqab∂yp

Γpcq −∂Γpbq∂yp

Γqac −∂Γqbq∂yp

Γpac)

+1

2yaybyc

∂2Γpab∂yp∂xc

+ yaybycyd(1

4

∂Γpab∂yq

∂Γqcd∂yp

− 1

2Γpab

∂2Γqcd∂yp∂yq

) . (8.2)

One more time it is easy to see the reduction of this expression to the integrand in theEinstein–Hilbert action (equation (4.5)). If the Finsler metric is not dependent on they coordinates ,everything but the first term vanishes, and the expression equals minusthe Ricci tensor contracted with y′s.

67


It becomes also clear that the variation of this object with respect to the Finsler metricor Finsler function is quite some work. As explained in section 6.3 the variation does notcommute with the partial derivatives in standard tangent bundle coordinates restrictedto the sphere bundler and therefore every part has to be varied carefully. The completevariation will be a very long calculation. Therefore we develop an organisation patternthat gives us a good overview.

We want to work out the dependence of Ric on the Finsler function in order to performthe calculus of variations. Obviously Ric in equation (8.2) has no explicit dependenceon F , only an indirect one through the y coordinates and the Christoffel symbols Γ. Tokeep as much structure as possible during the variation, we vary Ric with respect to yand the Christoffel symbols which we vary afterwards with respect to the Finsler metricor the Finsler function.

The following notations allow us a short way of notation of the variation of Ric

δgabn1...nsm1...mr = δ∂s+rgab

∂xn1 ...∂xns∂ym1 ...∂yms, Rica =

∂Ric

∂ya, (8.3)

Ricabn1...nrm1...ms =∂Ric

∂Γijk,c1...cr d1...ds

∂Γijk,c1...cr d1...ds

∂ ∂s+rgab∂xn1 ...∂xnr∂ym1 ...∂yms

, (8.4)

Γabc,n1...nsm1...ms=

∂r+sΓabc∂xn1 ...∂xns∂ym1 ...∂yms

. (8.5)

To order the variation it turned out that it is useful to consider the F dependence inRic in terms of derivatives acting on the Finsler metric, and to consider this dependencethrough the formal Christoffel symbols. Counting the derivatives acting on the Finslermetric g the maximum number is three,

Ric = Ric

(y, gab,

∂gab∂xc

,∂gab∂yc

,∂2gab∂xc∂xd

,∂2gab∂xc∂yd

,∂2gab∂yc∂yd

,∂3gab

∂xc∂xd∂yf,

∂3gab∂xc∂yd∂yf

).

(8.6)A variation of Ric with respect to the Finsler function becomes

δRic = Ricaδya +Ricabδgab +Ricabcδgabc +Ricabcδgabc + ...+Ricabcdfδgabcdf . (8.7)

Hence the variation of Ric ends up to be a lot of partial derivatives with respect toChristoffel symbols as well as their derivatives and the variation of derivatives of thecomponents of the Finsler metric. After we explained the structure of the variation wepresent the calculation in the next section.

8.2. Variation of the Finsler gravity action

The variation of the action (8.1) will be organised as explained in chapter 8.1. We beginby writing

δSG[F ] =

∫Σ

d4xd3θ√g√h(

1

2gpgδgpqRic+

1

2hµνδhµνRic+ δRic) . (8.8)

68


We already encountered most of the terms in the above equations. In appendix B.5.1hµνδhµν can be found and in sections 6.2 and 6.3 we already showed how to vary y andgab with respect to the Finsler function. Hence the only missing part is the variation ofRic. It can be calculated with the help of the variation of the coordinate transformationto Finsler spherical coordinates given by equation (6.19). We list the results here andperform the calculation in appendix B.6,

δ∂gab∂xc

=∂δgab∂xc

; (8.9)

δ∂gab∂yc

=∂gab∂yc

δF

F+ P µ

c

∂δgab∂θµ

; (8.10)

δ∂2gab∂xd∂xc

=∂2δgab∂xd∂xc

; (8.11)

δ∂2gab

∂yd∂xc=

∂2gab

∂yd∂xcδF

F+ P µ

d

∂2δgab∂θµ∂xc

; (8.12)

δ∂2gab

∂yd∂yc= 2

∂2gab

∂yd∂ycδF

F+

(∂P µ

c

∂yd

)∂δgab∂θµ

+ P µc P

νd

∂δ2gab∂θµ∂θν

; (8.13)

δ∂3gab

∂yd∂xf∂xc= P µ

d

∂3gab∂xf∂xc∂θµ

δF

F+ P µ

d

∂3δgab∂xf∂xc∂θµ

; (8.14)

δ∂3gab

∂yd∂yc∂xf= 2

∂3gab

∂yd∂yc∂xfδF

F+

(∂P µ

c

∂yd

)∂2δgab∂θµ∂xf

+ P µc P

νd

∂3δgab∂θµ∂θν∂xf

. (8.15)

Putting all together and performing the necessary integration by parts, the variationbecomes (details appendix B.7)

δSG[F ] =

∫Σ

d4xd3θ

(δFQ

δF

F+ δgQabδgab

). (8.16)

δFQ are all parts of the variation that naturally come with δFF

, namely

δFQ√g√h

= − 3Ric−Ricaya +Ricabc∂gab∂yc

+Ricabcd∂2gab∂yc∂xd

+ 2Ricabcd∂2gab

∂yc∂yd

+ Ricabcdf∂3gab

∂yc∂xd∂xf+ 2Ricabcdf

∂3gab

∂yc∂yd∂xf. (8.17)

We remark that no partial integration had to be carried out in the calculation to getδFQ and that if we consider a Finsler function that produces a vanishing Cartan tensormost of this term does not contribute to the final gravity equation.

Let us now have a closer look on the part of the variation where the variation of theFinsler metric and its derivatives appear with all partial integration carried out. Define

Ricn1,...,nr,m1,...,ms

:= Ricn1,...,nr,m1,...,ms√g√h , (8.18)

69


then δgQab can be written as

δgQab = (gab − 1

2yayb)Ric+ Ric

ab− ∂

∂xcRic

abc− ∂

∂θµP µc Ric

abc+

∂2

∂xcxdRic

abcd

+∂2

∂θµ∂xcP µ

dRic

abcd+

∂

∂θν(P ν

d

∂

∂θµ(P µ

c Ricabcd

))− ∂3

∂θµ∂xc∂xdP µ

fRic

abcdf

− ∂2

∂θν∂xf(P ν

d

∂

∂θµ(P µ

c Ricabcdf

)) . (8.19)

From equation (8.16) and with the help of the variation of the metric with respectto the Finsler function (equation (6.4)) we can read off the form of the equation thatdetermines F and so spacetime for a vacuum,

δFQ+ 2δgQabgab −∂

∂θµ(δgQabgµab) +

∂2

∂θµ∂θν(δgQabgµνab ) = 0. (8.20)

The only missing ingredients to investigate the equation further are the Ricabn1...nsm1...mr .There is strong evidence that the new gravity equations (8.20) reduce to the Einsteinequations if one searches for solutions with a Finsler metric that is not velocity dependentbut it is not proven yet. It looks very promising that not only on the action level but alsoon the vacuum gravity equation level there are no contradictions between our extendedtheory of gravity and general relativity.

70

Conclusion and outlook

In this this thesis we aimed for a generalization of the Einstein–Hilbert action by extend-ing the underlying mathematical structure from semi-Riemannian geometry to Finslergeometry.

After the introduction of the necessary mathematical concepts we rewrote the Einstein–Hilbert action by taking a velocity dependent point of view. As long as the underlyingspacetime is a manifold with a metric structure this is a trivial procedure; but it be-comes a true generalization if we consider instead a more general Finsler geometry forspacetime.

To achive this generalization a large amount of mathematical detail had to be workedout. In particular we developed a well-defined notion of integration over a Finsler spaceand its tangent space directions as well as calculus of variation in this setting. Fur-thermore we showed that general relativity could be deduced as a special case in thelanguage of the structures we developed.

Finally we were able to to provide a calculational scheme to determine a new dynamicalequation for gravity. This equation describes a classical vacuum theory of Finsler gravitythat is expected to be consistent with general relativity. The extended action principalfor Finsler gravity and the proof that its variation can be performed are the centralresults of this thesis.

For a complete generalized theory of gravity there are still some ingredients missingwhich should be provided in the future. On the one hand, the gravity vacuum equationwe wrote down in the final chapter of this thesis has to be invastigated in more detail. Inthe optimal case it can be recast in a form where we can identify a generalized Einsteintensor written in terms of d-tensors from Finsler geometry and some additional termsthat vanish if the Finsler structure is reducible to a semi-Riemannian structure. Whilethis thesis was written down it could be shown that this is really the case and notonly a vague hope. These results will appear in a related Diploma thesis by NiklasHbel. On the other hand, the matter part of the generalized gravity equation hasnot yet been investigated at all. The same generalization we did for the Einstein–Hilbert action during this thesis has to be done with matter actions for various physicalfields. A generalization of the energy-momentum tensor has to be found and interpreted.Furthermore all technical subtleties that arise during such a generalization have to betaken care of. All these generalizations should have the feature that they reduce tothe standard objects known from the original physical theories if we impose a Finslerstructure simply induced by a semi-Riemannian metric.

Once all this work is done we can search for solutions to the new gravity equationwhich give corrections to or modifications of the predictions of general relativity. Thesesolutions have to be confronted with experimental data. One way could be to adapt

71

Conclusion and outlook

the parametrized-post-newton formalism (PPN) to the new gravity equation (see [9]) tocheck its consistency with solar system experiments. These are the issues that can bestudied further if we think about the gravity equation.

If we believe that the physical spacetime is not a semi-Riemannian manifold but aFinsler space more general questions have to be answered:

• How do we define physical observers in a Finsler geometry setting?

• What kind of symmetry transformation relates observers?

• What are symmetries of a Finsler space?

• How do we implement the concept of causality?

• What is the interpretation of the fibre coordinate dependence of the geometricaltensors?

• How does a field theory in a Finsler space work?

• Is it possible to quantize field theories in a Finsler space?

• Is it possible to quantize the Finsler gravity theory we presented?

We already started to collect ideas to answer some of these questions while the answersfor others lie in the far future. A key point in finding the answers will be that we requireall structures which get generalized to reduce to the well known original ones if theunderlying Finsler structure reduces to a semi-Riemannian one.

72

Appendices

A. Additional calculations for Part I

During our description of the mathematical background we stated relations that areimportant for our constructions in part I of this thesis. The coordinate expression of aconnection one-form on a fibre bundle and the rewriting of the Finsler geodesic equationare not presented in every detail in the literature we cited, therefore we want to proofthese statements explicitly here.

A.1. The horizontal bundle

During our description of connections on fibre bundles in section 1.2 we presented anexplicit coordinate expression of the connection one-form ω and the basis of its kernel.We proof the validity of these expressions in this section.

The most general form of a connection one-form ω on a fibre bundle E and a generalvector field X ∈ TE in local coordinates u = (x, y) are given by

ω = (M(u)αβdyβ +N(u)αb dx

b)⊗ ∂

∂yα, X = X(u)c

∂

∂xc+ X(u)δ

∂

∂yδ. (A.1)

We suppress the u dependence in the following.The conditions ω ω = ω and Im(ω) = V E lead to

ω ω(X) = (XbN δbM

αγ + XβMγ

βMαγ )

∂

∂yα

!= (XbNα

b + XβMαβ )

∂

∂yα= ω(X) (A.2)

⇒Mαγ = δαγ . (A.3)

This proves the coordinate basis expression of a connection one-form (1.2)

ω = (dyα +Nαa dx

a)⊗ ∂

∂yα. (A.4)

The kernel of ω is given by all X ∈ TE fulfilling

ω(X) = 0 ⇒ (XbNαb + Xα)

∂

∂yα= 0 (A.5)

⇒ Xα = −XbNαb . (A.6)

That is why the basis vector fields of HE look like in equation (1.3)

X = Xa δ

δxa= Xa(

∂

∂xa−Nα

a

∂

∂yα) . (A.7)

75

A. Additional calculations for Part I

A.2. The geodesic equation of a Finsler space

In section 3.3, where we derived the Finsler geodesics, we postponed the proof that thefollowing two equations are equivalent

∂

∂xaF (x, x)− d

dt

∂

∂xaF (x, x) = 0 (A.8)

gαβ

(xα +

1

2gασ

(∂gσµ∂xν

+∂gσν∂xµ

− ∂gνµ∂xσ

)xµxν

)=dF

dt

∂F

∂xβ. (A.9)

This will be shown now. Consider the following as starting point

∂

∂xaF 2 − d

dt

∂

∂xaF 2 = 2F

(∂

∂xaF − d

dt

∂

∂xaF

)− 2

dF

dt

∂F

∂xa. (A.10)

Using the relations F 2 = gabxaxb, 2gabx

a = xa ∂∂xa

∂F 2

∂xb= ∂F 2

∂xbknown from sections 3.1 and

3.2, and performing the derivative with respect to t, gives

∂

∂xaF 2 − d

dt

∂

∂xaF 2 =

∂

∂xa(gpqx

pxq)− ∂2F 2

∂xq∂xaxq − ∂2F 2

∂xq∂xaxq

=∂

∂xa(gpqx

pxq)− 2∂

∂xq(gapx

p)xq − 2gaqxq . (A.11)

Since x and x are independent coordinates this leads to

∂

∂xaF 2 − d

dt

∂

∂xaF 2 = −2gaq

(xq − 1

2gqb

∂grp∂xb

xpxr + gqb∂gbp∂xr

xpxr)

= −2gaq

(xq +

1

2gqb(∂grb∂xp

+∂grb∂xr− ∂grp∂xb

)xpxr

). (A.12)

Putting all together leads to the resulting equation which makes the equivalence of (A.8)and (A.9) obvious

− 2gaq

(xq +

1

2gqb(∂grb∂xp

+∂grb∂xr− ∂grp∂xb

)xpxr

)= 2F

(∂

∂xaF − d

dt

∂

∂xaF

)− 2

dF

dt

∂F

∂xa. (A.13)

76

B. Detailed derivations for Part II

In Part II of this thesis we constructed the sphere bundle to define a generalized Einstein–Hilbert action which leads to extended gravity equations. During this program we en-countered many results that are based on calculations which are too long to be presentedin the main part of this thesis. In this appendix they are performed in full detail.

The topics to which details are presented are the pullback to the sphere bundle andthe diagonalization of the Sasaki metric, the variation of the Finsler metric with respectto the Finsler function, details about the derivation of the Einstein equations in thesphere bundle formalism, the variation of the derivatives of the Finsler metric and thevariation of the generalized Einstein–Hilbert action.

B.1. Pullback of the Sasaki metric to the sphere bundle

In section 5.4 we constructed an integral over the sphere bundle with the help of thepull-back of the Sasaki metric (equation (3.7)) from the tangent bundle to the spherebundle. We diagonalized it, and with its help we could define a natural volume form.How the pullback is done in detail is the topic of this section.

The natural embedding of the sphere bundle Σ to TM given in equation (5.21) wasdefined by

Ψ : Σ→ TM Ψ(xm, θµ) =

(xm,

ya(θ)

F (x, y(θ))

). (B.1)

The Sasaki metric G in the horizontal-vertical basis of the tangent bundle, the basistransformation S to the standard coordinate basis on TM , and the differential dΨ of Ψare given by

S =

(δbc 0

N bc δbc

); G =

(gab 0

0 gab

); dΨ =

(δcm 0

∂yc

∂xm∂yc

∂θµ

). (B.2)

Identical to the convention in section 5.4, indices a, b, .. run from 1 to 4 and indicesα, β, ... run from 1 to 3. The calculation to be performed, to find the pull-back of theSasaki metric Ψ∗G, as explained in section 5.4 is

Ψ∗G = dΨT · ST ·G · S · dΨ . (B.3)

Performing matrix multiplication with matrices in block form and indices on theblocks, the upper index labels the row and the lower index the column. For a transposedmatrix this is switched. Furthermore if an index with bar hits an index without bar,

77


the index without is modified. In order to avoid confusion, we display the transpose ofS and dΨ explicitly

ST =

(δad Na

d

0 δad

); dΨT =

δdn ∂yd

∂xn

0 ∂yd

∂θν

. (B.4)

We organise the calculation the way that we first perform G = ST ·G·S and afterwardsconsider Ψ∗G = dΨT · G · dΨ. We start the calculation with the following identity

G = ST ·G · S = ST ·(

gac 0gabN

bc gac

)=

(gdc + gabN

adN

bc gacN

ad

gdbNbc gdc

). (B.5)

From here we go on to show the desired formula

Ψ∗G = dΨT · G · dΨ (B.6)

= dΨT ·

(gdm + gabN

adN

bm + gacN

ad∂yc

∂xmgacN

ad∂yc

∂θµ

gdbNbm + gdc

∂yc

∂xmgdc

∂yc

∂θµ

)

=

gnm + gabNanN

bm + gacN

an∂yc

∂xm+ gdbN

bm∂yd

∂xn+ gdc

∂yc

∂xm∂yd

∂xngacN

an∂yc

∂θµ+ gdc

∂yc

∂θµ∂yd

∂xn

gdbNbm∂yd

∂θν+ gdc

∂yc

∂xm∂yd

∂θνgdc

∂yc

∂θµ∂yd

∂θν

.

Having in mind that gabyayb = F 2 = r2, and using the properties of the blocks of the

basis transformation matrices from standard induced to Finsler spherical coordinatesfrom equations (5.19) and (5.18), we can deduce further relations that simplify Ψ∗G.They are given by

gdc∂yc

∂xm∂yd

∂θν= gdcP

cmP

dν = −gdcP d

ν PcrP

rm = −1

rgdcy

cP dν P

rm = 0 (B.7)

gdc∂yc

∂xm∂yd

∂xn= gdcP

cmP

dn = gdcP

crP

rmP

dr P

rn = P r

nPrm . (B.8)

And finally the pull-back of the Sasaki metric looks like claimed in section 5.4

Ψ∗G =

gmn + P rmP

rn + gabN

amN

bn +N b

ngbcPcm +N b

mgbcPcn gdbN

bn∂yd

∂θµ

gdbNbm∂yd

∂θνgabP

aµP

bν

. (B.9)

B.2. Diagonalisation of the pullback of the Sasaki metric

In order to calculate the determinant of the Sasaki metric pulled back to the spherebundle we block-diagonalize it. It will turn out that the determinant of the coordinatetransformation that diagonalizes the Sasaki metric will be one, and so the diagonal metricwill have the same determinant as the non-diagonal one. We perform the calculation

78


again in matrix notation with matrices in block form, where the size of the blocks isgiven by their labels with indices. Identical to the convention in section 5.4 indices a, b, ..run from 1 to 4 and indices α, β, ... run from 1 to 3.

Consider a seven by seven block diagonal matrix G in some basis on the sphere bundle,representing a metric on the sphere bundle and a basis transformation matrix T as wellas its transpose

G =

(Gmn 0

0 Gµν

); T =

(δmd 0lµd δµδ

)T T =

(δnc lνc0 δνγ

). (B.10)

Performing a basis transformation on G with T results in the representation G of G

G = T T · G · T =

(Gcd +Gµνl

µd lνc Gνδl

νc

Gµγlµd Gδγ

). (B.11)

Since the determinant of T is 1, the determinants of G and G coincide by the determinantproduct rule.

We can identify G (equation (B.11)) with the pull-back of the Sasaki metric Ψ∗G fromequation (5.27) (or equation (B.9)) by specifying lµc , Gmn and Gµν to

Gµν = gabPaµP

bν = hµν ; Gµν = gabP µ

a Pνb = hµν ; GµαGαν = δµν ; (B.12)

lµc = GµνP dνN

ac gad; (B.13)

Gmn = gmn + P rmP

rn + gabN

amN

bn +N b

ngbcPcm +N b

mgbcPcn − lµmlνnGµν . (B.14)

It is clear now that we can diagonalize Ψ∗G with a basis transformation on the spherebundle given by T−1 with lµc chosen as specified above. The resulting block diagonalmatrix is G with block entries Gmn and Gµν . Since Ψ∗G is given in a coordinate basisof the sphere bundle, the basis it is diagonal in is given by

δ

δxm=

∂

∂xm− lµm

∂

∂θµ;

∂

∂θµ

. (B.15)

There is still a discrepancy between the result we just presented here and the diago-nalized pull-back of the Sasaki metric in equation (5.30) from section 5.4. This is due tothe fact that lµc and Gmn can be enormously simplified even if it does not look like thatat this point. We will show now that lµc = P µ

aNac as well as Gmn = gmn holds. The first

claim is easy to proof with help of the inverse relations of the coordinate transformationmatrices (equation (5.18)) and their homogeneity properties explained in equation (5.19)

lµc = hµνP dνN

ac gad = gadg

pbP µp P

νb P

dνN

ac = gadg

pbP µp N

ac (δdb − P

rb P

dr )

= P µp N

pc − gadgpbP

µp N

ac

yd

r

ybr

= P µp N

pc . (B.16)

The proof of the second claim has to be done step by step. First we compute lµmlνnGµν

to be

lµmlνnGµν = P µ

p NpmP

νq N

qngabP

aµP

bν = N p

mNqngab(δ

ap − P r

pPar )(δbq − P r

q Pbr )

= N pmN

qn(gqq − 2

ypyqr2

+ypyqr2

) = N pmN

qngqq −N p

mNqn

ypyqr2

. (B.17)

79


Plugging this into equation (B.14) gives

Gmn = gmn + P rmP

rn +N b

ngbcPcm +N b

mgbcPcn +N p

mNqn

ypyqr2

. (B.18)

Using equation (5.18) to rewrite P cm = −P c

rPrm = −yc

rP rm we can simplify Gmn further.

It becomes

Gmn = gmn + P rmP

rn −N q

n

yqrP rm −N p

m

yprP rn +N p

mNqn

ypyqr2

= gmn + (N pm

ypr− P r

m)(N qn

yqr− P r

n) . (B.19)

Before we reach the desired result Gmn = gmn we must show the vanishing of (N pmypr−

P rm). This calculation is a bit tedious. Recall that for the Cartan nonlinear connection

(chapter 3.3) the connection components Nab are given by equation (3.24). Pulling one

index down with the Finsler metric gives [11]

Nab = −∂gap∂yb

Gp +1

2

∂gab∂xq

yq +1

4

(∂2F 2

∂ya∂xb− ∂2F 2

∂yb∂xa

); Gp =

1

2Γpcdy

cyd . (B.20)

And since N pmyp = Npmy

p, we can deduce

Npmyp =

(−∂gpq∂ym

Gq +1

2

∂gpm∂xq

yq +1

4

(∂2F 2

∂yp∂xm− ∂2F 2

∂ym∂xp

))yp

= 0 +1

4

∂2F 2

∂xq∂ymyq +

1

2

∂F 2

∂xm− 1

4

∂2F 2

∂ym∂xpyp

=1

2

∂F 2

∂xm= F

∂F

∂xm. (B.21)

In Finsler spherical coordinates F = r (see eqn 5.7) and therefore

Npmyp = rP r

m . (B.22)

Ultimately we showed that Gmn = gmn since equation B.22 leaves no doubt that(N p

mypr− P r

m) is zero. This finally proofs that the diagonalized pull-back of the Sasaki

metric to the sphere bundle Ψ∗G is given by

gmndxm ⊗ dxm + hµνδθ

µ ⊗ δθν . (B.23)

Furthermore a side effect of this proof is that we see that there exists a horizontal vertcialbundle structure on the sphere bundle which is inherited from the tangent bundle. Asalready mentioned above the adapted vector basis is given by

δ

δxm=

∂

∂xm− lµm

∂

∂θµ;

∂

∂θµ

. (B.24)

80


B.3. Variation of the Finsler metric with respect to theFinsler function

When we compared the variation of an action with respect to the Finsler function Fand its corresponding Finsler metric g we needed the variation of g with respect to Fand claimed the result in equation (6.4) to be


F+

(3P r

(aPµ

b)+ Pα

(a

∂P µ

b)

∂θα

)∂

∂θµδF

F+ P µ

a Pνb

∂2

∂θµ∂θνδF

F. (B.25)

We now present a proof of this formula. The components of the Finsler metric gabwere calculated in equation (5.20) and restricted to the sphere bundle

gab = g(ab) = P r(aP

rb) + P µ

(a

∂P rb)

∂θµ= P r

aPrb + P µ

(a

∂P rb)

∂θµ. (B.26)

We placed symmetry brackets everywhere where the symmetry is not obvious in ordernot to miss any symmetry factors. With the help of the variation formulas (6.19) we canstep by step calculate the variation of the Finsler metric g with respect to the Finslerfunction F .

δgab =1

2

(δP µ

a

∂P rb

∂θµ+ P µ

a

∂δP rb

∂θµ+ δP µ

b

∂P ra

∂θµ+ P µ

b

∂δP ra

∂θµ

)+ δP r

aPrb + P r

a δPrb

=1

2

(P µa

∂P rb

∂θµδF

F+ P µ

a

∂

∂θµ(P r

b

δF

F+ P ν

b

∂

∂θνδF

F)

)+

1

2

(P µ

b

∂P ra

∂θµδF

F+ P µ

b

∂

∂θµ(P r

a

δF

F+ P ν

a

∂

∂θνδF

F)

)+ (P r

a

δF

F+ P ν

a

∂

∂θνδF

F)P r

b + (P rb

δF

F+ P ν

b

∂

∂θνδF

F)P r

a

= 2gabδF

F+

(3P r

(aPµ

b)+ Pα

(a

∂P µ

b)

∂θα

)∂

∂θµδF

F+ P µ

a Pνb

∂2

∂θµ∂θνδF

F. (B.27)

In the last step we just collected all terms standing before a certain derivative of δFF

,which finishes the proof.

B.4. Rewriting the variation from the Finsler metric tothe Finsler function

In this section we provide the details about the differential operator that appears byrewriting the variation of an action with respect to the Finsler metric into a variationwith respect to the Finsler function, as it is done the first time in section 6.2. Let φ be

81


a function on the sphere bundle (see section 5) and consider the following integral, thatappears during the variations

I =

∫Σ

d4xd3θφ(x, θ)δgab . (B.28)

Rewriting the integral into a variation with respect to F yields by equation (6.4) andpartial integration

I =

∫Σ

d4xd3θ φ(2gabδF

F+

(3P r

(aPµ

b)+ Pα

(a

∂P µ

b)

∂θα

)∂

∂θµδF

F+ P µ

a Pνb

∂2

∂θµ∂θνδF

F)

=

∫Σ

d4xd3θ

(2gabφ−

∂

∂θµ((3P r

(aPµ

b)+ Pα

(a

∂P µ

b)

∂θα)φ) +

∂2

∂θµ∂θν(P µ

a Pνb φ)

)δF

F

=

∫Σ

d4xd3θDab(φ)δF

F(B.29)

Therefore we read of how the differential operator Dab acts on sphere bundle function φ

Dab(φ) =

(2gabφ−

∂

∂θµ((3P r

(aPµ

b)+ Pα

(a

∂P µ

b)

∂θα)φ) +

∂2

∂θµ∂θν(P µ

a Pνb φ)

). (B.30)

Furthermore it fulfils the following identity that is true by the homogeneity propertiesof the P µ

a

Dab(1

2φyayb) = φ . (B.31)

B.5. Details of the derivation of the Einstein equationsin Finsler geometry

During the derivation of the Einstein equations from the Einstein–Hilbert action by cal-culus of variation in chapter 7, most of the results were presented without calculations.Details were missing for the variation of the volume element of the action integral espe-cially for the variation of

√h. Moreover, the way how to rewrite terms that naturally

came with the variation of gab into terms with the variation of F were presented withoutdetailed calculation. All these details will be presented in this section.

To have a good overview we mention the relations needed for the calculation

hµν = gabP µa P

νb ; hµν = gabP

aµP

bν ; P m

c Pcn = δmn ; yaP µ

a = 0; P ra =

yar

δP cµ = −P a

µ

δF

F− ya ∂

∂θµδF

F; P a

µPµc + P a

r Prc = δac . (B.32)

They are described in detail in sections 5.3 and 5.4.

82


B.5.1. Variation of the fibre metric h

The first equation we want to prove is hµνδhµν = (gmn − ymyn)δgmn − 6 δFF

. Beforestarting the proof we show one further formula that will be needed

hµνP aµ = gdcP µ

dP νc P

aµ = gdcP ν

c (δad − Par P

rd ) = P ν

c (gac − P ayc

r) = P ν

c gac . (B.33)

Now we deduce

hµνδhµν = hµν(δgabPaµP

bν + 2gabP

aµδP

bν )

= δgabPbνP

νc g

ac − 2gabPνc g

ac(P bν

δF

F+ yb

∂

∂θνδF

F)

= δgabgac(δbc − P b

rPrc )− 6

δF

F= (gab − yayb)δgab − 6

δF

F. (B.34)

B.5.2. Derivation of the vacuum equations

Next we prove the most important formula of chapter 7, namely the equality betweenequation (7.5) and equation (7.6)

δS[F ] =

∫Σ

d4xd3θ√g√h

(Raby

ayb(gpq − 1

2ypyq)δgpq − 5Raby

aybδF

F

)(B.35)

!=

∫Σ

d4xd3θ√g√h (−6Rab + 2Rgab) y

aybδF

F. (B.36)

We remind the reader that the variation of the Finsler metric with respect to F(equation (6.4)) is given by


F+

(3P r

(aPµ

b)+ Pα

(a

∂P µ

b)

∂θα

)∂

∂θµδF

F+ P µ

a Pνb

∂2

∂θµ∂θνδF

F. (B.37)

For the sake of readability we now evaluate step by step what the multiplication of theterms of δgpq and (gpq − 1

2ypyq) yields,(

gpq − 1

2ypyp

)2gpq = 7; (B.38)(

gpq − 1

2ypyp

)(3P r

(pPµq) + Pα

(p

∂P µq)

∂θα

)= gpq

(3P r

(pPµq) + Pα

(p

∂P µq)

∂θα

)= gpqPα

p

∂P µq

∂θα;(B.39)(

gpq − 1

2ypyp

)P µp P

νq = gpqP µ

p Pνq = hµν . (B.40)

Inserting these relations into expression (B.35) results in

δS[F ] =

∫Σ

d4xd3θ√g√h(

2Rabyayb

δF

F+Raby

aybgpqPαp

∂P µq

∂θα∂

∂θµδF

F+Raby

aybhµν∂2

∂θµ∂θνδF

F

). (B.41)

83


Integrating by parts in order to take the derivatives away from the variation, (where wedo not have to take care about

√g since g is not dependent on any θ,) leads to

δS[F ] =

∫Σ

d4xd3θ√gδF

F(√h2Raby

ayb − ∂

∂θµ(√hRaby

aybgpqPαp

∂P µq

∂θα) +

∂2

∂θµ∂θν(√hRaby

aybhµν)

). (B.42)

Studying ∂2


aybhµν) results in

∂2


aybhµν) =∂

∂θν(∂

∂θµ(√hRaby

ayb)hµν +√hRaby

aybgpqPαp

∂P νq

∂θα

+√hRaby

aybgpqP νp

∂Pαq

∂θα) . (B.43)

Therefore the variation reduces to

δS[F ] =

∫Σ

d4xd3θ√gδF

F(√h2Raby

ayb +∂

∂θν(∂

∂θµ(√hRaby

ayb)hµν +√hRaby

aybgpqP νp

∂Pαq

∂θα)

). (B.44)

Since there are no further partial integration to be done we can investigate parts of theintegrand on their own. Hence we equate

∂

∂θµ(√hRaby

ayb)hµν = 2√hRabh

µνybP aµ +

1

2

√hRaby

aybhµνhαβ∂

∂θµhαβ

= 2√hRaby

bgacP νc +√hRaby

aybhµνP βq

∂P qβ

∂θµ

= 2√hRc

bybP ν

c −√hRaby

aybgpqP νp

∂P βq

∂θβ, (B.45)

and the variation shrinks one more time to

δS[F ] =

∫Σ

d4xd3θ√gδF

F

(√h2Raby

ayb +∂

∂θν(2√hRc

bybP ν

c )

). (B.46)

An important formula we used in the calculation is

hαβ∂

∂θµhαβ = 2P β

q

∂P qβ

∂θµ= 2P β

q

∂P qµ

∂θβ= −2

∂P βq

∂θβP qµ . (B.47)

84


Last but not least it remains to calculate ∂∂θν

(2√hRc

bybP ν

c ),

∂

∂θν(2√hRc

bybP ν

c )

= 2R√h− 2

√hRc

bybyc + 2

√hRc

byb∂P

νc

∂θν+√hRc

bybP ν

c 2P βq

∂P qβ

∂θν

= 2R√h− 2

√hRc

bybyc + 2

√hRc

byb(∂P ν

c

∂θν+ P ν

c Pβq

∂P qβ

∂θν)

= 2R√h− 2

√hRc

bybyc + 2

√hRc

byb(∂P ν

c

∂θν− P ν

c

∂P βq

∂θβP qν )

= 2R√h− 2

√hRc

bybyc + 2

√hRc

byb(∂P β

q

∂θβyqyc)

= 2R√h− 8

√hRbcy

byc . (B.48)

Hence finally we reach the desired form of the variation

δS[F ] =

∫Σ

d4xd3θ√g√hδF

F

(2R− 6Rbcy

byc). (B.49)

B.5.3. Derivation of the complete gravity equationincluding matter

In order to complete the derivation of the full gravity equation we need to prove thatthe variation of

SM [F, ψi] =

∫Σ

d4xd3θ√g√hL(g, ψi) (B.50)

is

δSM [F, ψi] =

∫Σ

d4xd3θ√g√h(12T ab − 2Tgab)yayb

δF

F. (B.51)

Varying the matter action in equation B.50 with respect to the Finsler function Fleads to

δSM [F, ψi] =

∫Σ

d4xd3θ√g√h(

1

2Lgabδgab +

1

2Lhµνδhµν +

∂L

∂gabδgab)

=

∫Σ

d4xd3θ√g√h

1

2(T abδgab + Lhµνδhµν)

=

∫Σ

d4xd3θ√g√h

1

2(T abδgab + L(gab − yayb)δgab − 6L

δF

F)

=

∫Σ

d4xd3θ√g√h

1

2(2T

δF

F+ T ab3P r

aPµ

b

∂

∂θµδF

F

+ (T ab + Lgab)Pαa

∂P µa

∂θα∂

∂θµδF

F+ (T ab + Lgab)P ν

a Pµa

∂

∂θν∂θµδF

F).(B.52)

In the last step we simply inserted the variation of the Finsler metric with respect tothe Finsler function (see equation (6.4) or appendix B.3). Doing integration by parts

85


and performing one of the derivatives in the second derivative part changes the variationto be

δSM [F, ψi] =

∫Σ

d4xd3θ√g

1

2

δF

F(√h2T − T ab ∂

∂θµ(3√hyaP

µ

b)

+ (T ab + Lgab)∂

∂θµ(1

2

√hP µ

a Pνb h

αβ ∂hαβ∂θν

+√hP µ

a

∂P νb

∂θν))

=

∫Σ

d4xd3θ√g

1

2

δF

F(√h2T − T ab ∂

∂θµ(6√hyaP

µ

b)) . (B.53)

Performing the last partial derivative and summing everything up finishes the proof.

B.6. Variation of the derivatives of components of theFinsler metric

In section 8.2 we started the derivation of the extended gravity equations in the or-ganisational pattern explained in section 8.1. For this it is necessary to show how thevariation of derivatives of the Finsler metric components works.

Proving equations (8.9) to (8.15) is simply done by expanding the derivatives withrespect to the standard induced coordinates on the tangent bundle into derivatives withrespect to the Finsler spherical coordinates, everything restricted to the sphere bundle.Since we regard derivatives of the Finsler metric g, the restriction to the sphere bundlecauses no problems. Because of its homogeneity properties g naturally lives on thesphere bundle and has no r dependence, hence ∂gab

∂r= 0.

We explained the transformation of the derivatives in section 5.3 and their behaviourunder variation in section 6.3. The proof of equation (8.9) becomes very easy

δ∂gab∂xc

= δ(∂gab∂xc

+ P rc

∂gab∂r

) =∂δgab∂xc

. (B.54)

If the function to vary is not dependent on r, the variation always commutes with thederivative with respect to the standard induced tangent bundle coordinate x restrictedto the sphere bundle. The y derivative is a bit more complicated

δ∂gab∂yc

= δ(P µc

∂gab∂θµ

) = P µc

∂gab∂θµ

δF

F+ P µ

c

∂δgab∂θµ

=∂gab∂yc

δF

F+ P µ

c

∂δgab∂θµ

=∂gab∂yc

δF

F+∂δgab∂yc

.

(B.55)In the first step we already used that g is not dependent on r. We see that the yderivative does not commute with the variation and that we have to take care of anextra factor by interchanging them.

As shown above, the x derivative commutes with the variation which makes the fol-lowing two formulas true by performing a x derivative on equations (B.54) and (B.55)

δ∂2gab

∂yd∂xc=

∂2gab

∂yd∂xcδF

F+ P µ

d


, δ∂2gab∂xd∂xc

=∂2δgab∂xd∂xc

. (B.56)

86


Finally we are left with the slightly more complicated formula involving the two yderivatives

δ∂2gab

∂yd∂yc=

∂2gab

∂yd∂ycδF

F+

∂

∂ydδ∂gab∂yc

= 2∂2gab

∂yd∂ycδF

F+

∂

∂yd(P µ

c

∂δgab∂θµ

)

= 2∂2gab

∂yd∂ycδF

F+∂2δgab

∂yd∂yc. (B.57)

As before the two missing equations are proven by performing x derivatives on the firstpart in equation (B.56) and on equation (B.57),

δ∂3gab

∂yd∂xf∂xc= P µ

d


δF

F+ P µ

d


; (B.58)

δ∂3gab

∂yd∂yc∂xf= 2

∂3gab

∂yd∂yc∂xfδF

F+

(∂P µ

c

∂yd


+ P µc P

νd


. (B.59)

Hence we proved all relations that are needed to write down the extended gravityequations in section 8.2.

B.7. Variation of the Finsler gravity action

In section 8.2 we presented the result of the variation of the generalized Einstein–Hilbertaction, but left out certain details. We present these details in this section. Starting fromequation (8.8) we want to explain how to reach the gravity equation (8.20). Equation(8.16) is given by

δSG[F ] =

∫Σ

d4xd3θ√g√h(

1

2gpgδgpqRic+

1

2hµνδhµνRic+ δRic) . (B.60)

If we plug in hµνδhµν = (gab − yayb)δgab − 6 δFF

and expression (8.7) from section 8.1for δRic we get

δSG[F ] =

∫Σ

d4xd3θ√g√h((gab − 1

2yayb)Ricδgab − 3Ric

δF

F+Ricaδy

a (B.61)

+Ricabδgab +Ricabcδgabc +Ricabcδgabc + ...+Ricabcdfδgabcdf ) . (B.62)

In section 6.3 the variation of y is calculated to be δy = −δy δFF

and the variation ofthe δgabn1...nsm1...mr are given in section 8.2 (with proof in appendix B.6). Combining

87


everything yields

δSG[F ] =

∫Σ

d4xd3θ√g√h[(gab − 1

2yayb)Ricδgab − 3Ric

δF

F−Ricaya

δF

F

+ Ricabδgab +Ricabc∂δgab∂xc

+Ricabc(∂gab∂yc

δF

F+ P µ

c

∂δgab∂θµ

)

+ Ricabcd∂2δgab∂xd∂xc

+Ricabcd(∂2gab

∂yd∂xcδF

F+ P µ

d


)

+ Ricabcd(2∂2gab

∂yd∂ycδF

F+∂P µ

c

∂yd∂δgab∂θµ

+ P µc P

νd

∂2δgab∂θµ∂θν

)

+ Ricabcdf (P µ

d


δF

F+ P µ

d


)

+ Ricabcdf (∂3gab

∂yd∂yc∂xfδF

F+

(∂P µ

c

∂yd


+ P µc P

νd


)].(B.63)

We expressed all derivatives acting on a variation through derivatives with respectto coordinates we can use to perform integration by parts. Expanding all terms andperforming integration by parts the variation takes the form we claimed in section 8.2,

δSG[F ] =

∫Σ

d4xd3θ

(δFQ

δF

F+ δgQabδgab

), (B.64)

with

δFQ√g√h

= − 3Ric−Ricaya +Ricabc∂gab∂yc

+Ricabcd∂2gab∂yc∂xd

+ 2Ricabcd∂2gab

∂yc∂yd

+ Ricabcdf∂3gab

∂yc∂xd∂xf+ 2Ricabcdf

∂3gab

∂yc∂yd∂xf(B.65)

and

δgQab = (gab − 1

2yayb)Ric+ Ric

ab− ∂

∂xcRic

abc− ∂

∂θµP µc Ric

abc+

∂2

∂xcxdRic

abcd

+∂2

∂θµ∂xcP µ

dRic

abcd+

∂

∂θν(P ν

d

∂

∂θµ(P µ

c Ricabcd

))− ∂3

∂θµ∂xc∂xdP µ

fRic

abcdf

− ∂2

∂θν∂xf(P ν

d

∂

∂θµ(P µ

c Ricabcdf

)) . (B.66)

The ∂Pµc∂yd

terms where absorbed using

∂

∂θν(P ν

d

∂

∂θµ(P µ

c Ricabcd

)) =∂2

∂θµ∂θν(P µ

c Pνd Ric

abcd)− ∂

∂θµ(∂P µ

c

∂ydRic

abcd). (B.67)

Therefore the variation in section 8.2 is correct and also the form of the gravityequation (8.20).

88

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90

Erklarung gemaß Diplomprufungsordung

Ich versichere, diese Arbeit selbststandig und nur unter Benutzung der angegebenenHilfsmittel und Quellen angefertigt zu haben. Ich gestatte die Veroffentlichung dieserArbeit.

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91

from the einstein{hilbert action to an action principle

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