Document [original]

Particular Timelike Flows in Global Lorentzian Geometry

vorgelegt von

Diplom-Physiker

Alexander Dirmeier

aus Lindau

Von der Fakultät II – Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

Dr. rer. nat.

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Wolfgang König

Berichter/Gutachter: Prof. Dr. Helga Baum

Prof. Dr. Mike Scherfner

Prof. Dr. John M. Sullivan

Tag der wissenschaftlichen Aussprache: 10.12.2012

Berlin 2013

D 83

Fakultät für Mathematik und Naturwissenschaften

der Technischen Universität Berlin

Zusammenfassung

Spezielle zeitartige Flüsse in der globalen Lorentzgeometrie

von Alexander Dirmeier

Diese Arbeit untersucht topologische und kausale Eigenschaften Lorentz’scher Mannigfaltig-

keiten (M, g), die als zusätzlich Struktur ein vollständiges, zeitartiges Einheitsvektorfeld

V, oder in anderen Worten einen globalen zeitartigen Fluss, aufweisen. Diese Lorentz’schen

Mannigfaltigkeiten sind in natürlicher Weise Raumzeiten. Es werden allgemeine geometrische

Bedingungen hergeleitet, die dazu führen, dass diese Raumzeiten eine Produktstruktur R×S

aufweisen, wobei das Vektorfeld Ventlang des Faktors Rzeigt und Sder Raum der Inte-

gralkurven von Vist. Die möglichen Kausalstufen für diese Produktraumzeiten werden

analysiert und eine vollständige kausale Klassifikation wird angegeben. Durch eine Klas-

sifikation bezüglich einer Zerlegung der kovarianten Ableitung ∇Vdes gegebenen zeitar-

tigen Vektorfelds können Unterklassen dieser Produktraumzeiten gewonnen werden. Die

speziellen Unterklassen der geblätterten Raumzeiten und der stationären Raumzeiten wer-

den hinsichtlich ihrer globalen Hyperbolizität analysiert und mehrere neue Beziehungen wer-

den gewonnen. Für stationäre und homothetische Raumzeiten wird eine neue Version der

Lorentz’schen Bochnertechnik hergeleitet. Schließlich werden konforme Lorentz’sche Sub-

mersionen und insbesondere Hubble-isotrope Raumzeiten analysiert und Bedingungen für

deren globale Hyperbolizität und geodätische Vollständigkeit werden gewonnen.

School for Mathematics and Natural Sciences

of Technische Universität Berlin

Abstract

Particular Timelike Flows in Global Lorentzian Geometry

by Alexander Dirmeier

This work investigates the topological and causal characteristics of Lorentzian manifolds

(M, g), which possess a complete and timelike unit vector field V, or in other words a global

timelike flow, as an additional structure. Naturally, these Lorentzian manifolds are space-

times. General geometric requirements for these spacetimes to split diffeomorphically as a

product R×S, with the vector field Valong the R-factor and Sthe space of flow lines of

V, are derived. The possible causality conditions for these splitting spacetimes are analyzed

and a complete causal classification is given. Sub-classes of these splitting spacetimes can

be obtained by a classification according to a decomposition of the covariant derivative ∇V

of the given timelike vector field. The specific sub-classes of sliced spacetimes and station-

ary spacetimes are analyzed with regard to global hyperbolicity and several new relations

are obtained. For stationary and homothetic spacetimes, a new version of the Lorentzian

Bochner technique is derived. Finally, conformal Lorentzian submersions, and particularly

Hubble-isotropic spacetimes, are analyzed and conditions for their global hyperbolicity and

geodesic completeness are obtained.

TABLE OF CONTENTS

Page

Chapter 1: Introduction ................................. 1

Chapter 2: Notation, Conventions and Preliminaries ................. 5

2.1 Semi-Riemannian Geometry ............................ 10

2.2 Global Randers Geometry ............................. 15

2.3 R-manifolds and Principal Bundles ........................ 19

2.4 Flows and Dynamical Systems ........................... 26

Chapter 3: Lorentzian Manifolds and Spacetimes ................... 32

3.1 Kinematical Quantities ............................... 37

3.2 Classification of Spacetimes by Kinematical Quantities ............. 41

3.3 Raychaudhuri Equations .............................. 43

3.4 The Causal Ladder ................................. 50

Chapter 4: Diffeomorphic Splitting of Lorentzian R-manifolds ............ 55

4.1 Topology of Kinematical Spacetimes ....................... 56

4.2 Causality of Splitting Spacetimes ......................... 69

4.3 The Low-dimensional Cases ............................ 78

Chapter 5: Sliced Spacetimes .............................. 85

5.1 Global Hyperbolicity of Sliced Spacetimes .................... 87

5.2 Cauchy Hypersurfaces in Stationary Spacetimes ................. 94

5.3 Lorentzian Bochner Technique ...........................108

Chapter 6: Conformal Lorentzian Submersions ....................125

6.1 Hubble-isotropic Spacetimes ............................128

6.2 Topological and Causal Properties ........................130

6.3 Completeness and Singularities ..........................142

Chapter 7: Outlook ...................................147

Bibliography .........................................152

ACKNOWLEDGMENTS

It is a pleasure for me to thank all people and institutions that were directly or indirectly

involved in supporting me materially and immaterially during the work on this thesis.

First of all I would like to express my gratitude to M. Scherfner for his supervision and

scientific guidance. Furthermore, I wish to thank my colleagues S. Born and M. Plaue for

many inspiring discussions and proofreading of this thesis, although all errors are my own.

Furthermore, I would like to acknowledge Technische Universität Berlin and, particularly, the

Department of Mathematics for providing me with a workplace to write this thesis and the

possibility to pursue teaching during the time of my graduation. I am particularly thankful

for the financial support provided by the NaFöG and the Elsa-Neumann-Scholarship of the

federal state of Berlin.

I also wish to express my sincere gratitude to H. Baum and J. M. Sullivan for reviewing my

thesis, and W. König for agreeing to officiate as chairman.

I would like to express my warmest gratitude to my partner Catharina for her constant

moral support and for her patience with me during the writing of this thesis.

I also wish to thank all the other people I have met, if only briefly, who have added to my

graduate experience through fruitful discussions, or in any other way.

iii

DEDICATION

For Catharina.

Chapter 1

INTRODUCTION

After the Golden Age of General Relativity in the 1960s and early 1970s (cf. [Tho03]),

when the theory and the mathematics of Lorentzian manifolds were developed intensively

by theoretical physicists, Lorentzian geometry took great leaps forward in recent years by

progress from a pure mathematical, though physically inspired, perspective. A selection

of these developments, which are relevant to this work, include: the Lorentzian splitting

theorem, which has started to emerge already in the 1980s, based on the simplified proof

of the Riemannian Cheeger–Gromoll splitting theorem by J.-H. Eschenburg and E. Heintze

(see [EH84]) and was then developed and constantly improved by J.K. Beem, P.E. Ehrlich,

J.-H. Eschenburg, E. Heintze, R. Bartnik, G.J. Galloway and others (see, e.g., [Gal89] and

the references therein for a reasonably strong version of the theorem), the recent progress in

the notion of global hyperbolicity of spacetimes, particularly in the work of A. Bernal and

M. Sánchez (see, e.g., [BS07] and [BS05] or [S´

11] for an overview), as well as the further

development of the causal ladder (see [MS08]), especially in the work of E. Minguzzi (e.g.,

[Min08c] [Min08d] [Min09a] [Min09b] [Min09c]), where a particular highlight is the connec-

tion of Lorentzian causality theory to order theory (see [Min10]). Other important, more

specific, recent developments in global Lorentzian geometry, on which this work is based, are:

the question of a diffeomorphic splitting of (conformally) stationary spacetimes analyzed by

M.A. Javaloyes and M. Sánchez (see [JS08]), the correspondence of global hyperbolicity to

the completeness of Finslerian metrics of Randers type for stationary spacetimes, devised in

the seminal work [CJM11] by E. Caponio, M.A. Javaloyes and A. Masiello and developed

further in [CJS11] by the first two authors and M. Sánchez, the concept of a Lorentzian

Bochner technique established by A. Romero and M. Sánchez (see [RS96] and [RS98]), as

well as the notion of regularly sliced spacetimes (see [CB09] for the introduction of this

terminology) developed in [CBC02] and [Cot04] by Y. Choquet-Bruhat and S. Cotsakis.

This work is concerned with global Lorentzian geometry, i.e., questions of the topological

and causal structure of manifolds admitting a Lorentzian metric. Local considerations,

particularly questions of curvature, will be of secondary importance, but will for example

come into play in section 5.3 in connection with the Lorentzian Bochner technique. We will

analyze Lorentzian manifolds which carry the additional structure of a global and timelike,

complete unit vector field or, in other words admit a global timelike flow. Naturally, these

Lorentzian manifolds are spacetimes. This will allow us to prove general theorems on the

topological and causal structure of these spacetimes in chapter 4and, subsequently, analyze

particular classes of these spacetimes in chapters 5and 6.

Particularly, this work is structured along the following train of thought. In chapter 2, we

first establish the basic mathematical concepts used in the later chapters, concerning the

differential geometry of semi-Riemannian manifolds, and fix the notational conventions we

will use. Then we provide an overview of the existing theory of R-actions on manifolds,

generalized principal R-bundles and dynamical systems, connect these concepts and cast

them into a form applicable to a global timelike flow on a Lorentzian manifold in the later

chapters. In chapter 3, we will establish the basic theory on specific Lorentzian geometry

used in this work. Particularly, we define in that chapter the basic geometric objects that

we analyze in this work, i.e., kinematical spacetimes (cf. Def. 3.15). Having at hand, not

only a Lorentzian metric, but also a specific vector field on a spacetime, allows us to define

some derived geometric quantities (called kinematical quantities; see section 3.1), give a

classification of spacetimes according to these quantities (section 3.2) and analyze their

evolution, which leads to the Raychaudhuri equations (section 3.3). The basic Lorentzian

causality theory will also be established in that chapter in section 3.4.

The main new results are contained in chapters 4,5and 6. In chapter 4, after an introduction

to different splitting philosophies, we establish in section 4.1 conditions for a kinematical

spacetime to split as a product manifold and derive various propositions on the geometric

structure of the resulting manifold. In section 4.2 we analyze the possible causality conditions

of kinematical spacetimes in the splitting case. A main result of those two sections can, for

example, be summarized as follows (cf. Thm. 4.13, Thm. 4.19, Thm. 4.29 and Prop. 4.31):

Main Result. Let (M, g, V )be a kinematical spacetime, i.e., a Lorentzian manifold (M, g)

with a timelike, complete unit vector field V. If the integral curves of Vare non–partially-

imprisoned and the space Sof integral curves of Vis a manifold (particularly, this is the case

if there is a Lorentzian metric ˜gin the conformal class of g, such that Vis ˜g-geodesic), then

Mis diffeomorphic to R×Sand Vis mapped to a vector field along the factor Rby a map

realizing this diffeomorphism. Furthermore, such a spacetime is causally continuous if it is

feebly distinguishing and if Sis compact, it is globally hyperbolic if it is non-imprisoning.

Besides the known splitting results and causality properties in the case of a Killing vector

field Vestablished in [JS08], which are included as special cases in the results of this work,

there are several other examples of related works with a similar aim. Classical articles

that deal with similar splitting questions in the (semi-)Riemannian case are for example

[Wad75], [Her60] or [PR93]. But these have always the aim to achieve “more” than only a

diffeomorphic splitting by establishing, for example, the structure of a twisted or warped

product manifold. An approach to the splitting of stably causal spacetimes was taken in

[GRK96], while paying attention to possible connections to timelike completeness. Ideas

most similar to the approach in this work were pursued in [GO03] and [GO09], although

only for a vector field Vwith an integrable vertical distribution. Various results in these two

references can be derived as special cases from the propositions established in this work.

In section 4.3, we will analyze the causality of splitting spacetimes in the particular case of

spacetime dimension smaller or equal to four. In this situation there exists a specifically close

connection between the topology of the spacetime and the causality conditions. Furthermore,

the propositions established in that section provide interesting (and maybe surprising) con-

nections between Lorentzian causality theory and Poincaré–Bendixson theory of dynamical

systems and between Lorentzian causality theory and contact geometry.

Chapter 5contains three sections, each of which deals with a different sub-class of splitting

spacetimes. We use the results about their topological and causal structure, derived in the

chapter before, as a basis to establish more specific results about (regularly) sliced spacetimes

in section 5.1, global hyperbolicity of stationary spacetimes in section 5.2 and the Lorentzian

Bochner technique in section 5.3. The notion of regularly sliced spacetimes was introduced

in [CBC02], to put bounds on the components of a Lorentzian metric of a spacetime and

derive simple conditions for global hyperbolicity in this case. In Thm. 5.11, we develop

this concept further to infer conditions for the existence of a regularly sliced metric in the

conformal class of a globally hyperbolic metric. Section 5.2 contains the following proposition

on the completeness of Riemannian metrics as a main result (see Thm. 5.22):

Main Result. The conformal transformation g

A2of a complete Riemannian metric gby a

positive function Aon a non-compact manifold Mis complete if and only if the function 1

is not an L1-function along any curve escaping to infinity on Mwith respect to g.

The remainder of that section deals with the application of this result to stationary space-

times in connection to the correspondence of global hyperbolicity of stationary spacetimes to

the completeness of Finslerian metrics of Randers type established in [CJS11]. We are able

to derive various propositions on the global hyperbolicity of stationary spacetimes based on

Riemannian completeness and the growth of specific functions (cf. Thm. 5.26, Prop. 5.28,

Prop. 5.29 and Thm. 5.31). In section 5.3, we first establish connections between the formu-

las of Lorentzian Bochner technique developed in [RS96] and [RS98] and the Raychaudhuri

equations from section 3.3, and then we are able to establish a new version of the Lorentzian

Bochner technique for particular non-compact stationary spacetimes, by regarding them as

Lorentzian submersions. Furthermore, we are able to extend this technique to the case of

a homothetic vector field instead of a Killing vector field. The main Bochner-like results of

this section are contained in the Thms. 5.39,5.45 and 5.50. The first theorem deals with the

stationary case and a compact base manifold of the submersion, the second theorem deals

with the stationary case and a base manifold of the submersion with certain conditions at

infinity, and the third theorem includes the homothetic case.

In chapter 6, we identify kinematical spacetimes obeying a certain condition on the kinemat-

ical quantities (namely vanishing shear; cf. section 3.1) with conformal Lorentzian submer-

sions. This gives the possibility to describe large classes of spacetimes, which are inspired

by constraints on the kinematical quantities from physics, by purely geometric conditions.

Hence, the remainder of that chapter deals with the global structure and the causality condi-

tions of kinematical spacetimes which are, at the same time, particular conformal Lorentzian

submersions with a Riemannian base manifold and totally geodesic fibers. It turns out that

these are the so-called Hubble-isotropic spacetimes (cf. e.g. [HP99]; the idea for the analysis

of these metrics is also inspired by [GPS+10]). We introduce these spacetimes in section

6.1 and analyze their global properties, particularly their metric structure with respect to

a diffeomorphic splitting (see Thm. 6.8), and conditions for their global hyperbolicity (see

Thm. 6.18 and Thm. 6.20) in section 6.2. Section 6.3 contains two propositions on the time-

and spacelike completeness of Hubble-isotropic spacetimes.

Finally, in chapter 7, we will discuss some interesting open problems arising from the the-

orems and propositions established in this work. We will also give some preliminary ideas

for directions of further research about these problems and state a conjecture together with

ideas towards its proof.

Chapter 2

NOTATION, CONVENTIONS AND PRELIMINARIES

In this chapter, we fix some notation and conventions from differential geometry that will be

used throughout this work. In section 2.1, we give some definitions and well-known results on

semi-Riemannian manifolds, but will also introduce some concepts which are non-standard.

In section 2.2, we will establish some facts about Finslerian geometry, particularly about

Finslerian metrics of Randers type, which will be needed for the analysis in subsequent

chapters. In sections 2.3 and 2.4, we introduce some details about R-actions on manifolds,

i.e., global flows, which lead to a connection with the theory of dynamical systems. Most

results in these sections are well-known, but we give a different comprehensive viewpoint on

these matters that will be suitable for the purposes of this work in the following chapters.

The propositions and theorems established in those two sections are the groundwork for the

splitting results which will be derived in chapter 4. Standard references for this first chapter

are [Con01], [O’N83] and [KN63], additional references will be cited where applicable.

Basic Essentials

The subset relation ⊂is assumed to be reflexive, i.e., A⊂Afor all sets A. Let A⊂T,

where Tis some topological space. We will denote the interior of Aby ˚

A, its closure by

Aand its complement with respect to Tby T\A. For subsets of the real numbers, we

use R≥a= [a, ∞),R≤a= (∞, a]and R>a = (a, ∞),R<a = (∞, a)for all fixed a∈Ras

abbreviations.

All (topological) manifolds Mof dimension dim(M) = nare assumed to be connected,

second countable, locally homeomorphic to Rnand Hausdorff. Occasionally, we will indicate

the dimension of a manifold by a superscript, i.e., Mnis the manifold Mwith dimension n

indicated. This definition also implies that all manifolds have empty boundary. Manifolds

with boundary can be included in this definition by considering them locally homeomorphic

to the half-space Hn={(x1, . . . , xn)∈Rn|x1≥0}(cf. [Con01, Sec. 1.6]). A space ˜

fulfilling all conditions of a topological manifold except for being Hausdorff, but is a T1-

space only instead, will be called a near-manifold. Then one can readily define differentiable

structures on near-manifolds in the same way as on ordinary manifolds (see e.g. [Hic65]).

As for manifolds, we will assume that they carry a fixed differential structure of degree Cr

for some r∈N∪{∞}. We will take smooth and diffeomorphic to mean of the same class of

differentiability as the underlying differentiable structure of the manifold. If not indicated

otherwise, all geometric objects (curves, maps, fields, etc.) are considered smooth, and

we will usually assume that the manifolds we consider are sufficiently smooth, i.e., for most

applications C3is enough. In some places, where we consider manifolds or geometric objects

of a lower differentiability class—or even continuous objects or topological manifolds—this

will be stated explicitly. The group of diffeomorphisms of a manifold Mof class r > 0to

itself will be denoted by Diffr(M)and the class of Cs-functions M→Rwill be denoted

by Cs(M)for some 0≤s≤r. Usually, we will consider smooth functions Cr(M)for a

Cr-manifold M.

We call a function d:T×T→R≥0, with Tsome topological space, a generalized distance if

it satisfies the conditions d(x, y)=0⇔x=y,d(x, y)≤d(x, z) + d(z, y)for all x, y, z ∈T.

A function d:T×T→R≥0which is a generalized distance and additionally fulfills d(x, y) =

d(y, x)for all x, y ∈Tis called a distance. We reserve the term metric for metric tensors on

manifolds. Nevertheless, we call a topological space Ttogether with a (generalized) distance

d, which is compatible with the topology, a (generalized) metric space (T, d). Generalized

metric spaces will arise in the context of Finslerian metrics (cf. [BCS00, Sec. 6.2] for details).

A metric space (T, d)will be called complete if every Cauchy sequence converges, i.e., for all

sequences {xk}x∈N⊂Tobeying d(xk, xl)→0as k, l → ∞, there is some x∞∈Tsuch that

d(xk, x∞)→0as k→ ∞.

We refer to [Rov98] for the following

Definition 2.1. Let Jbe some index set. A family {σi}i∈Jof connected subsets of a manifold

Mnof class Cr(r≥1) is called a k-dimensional foliation or codimension n−kfoliation of

class Cs(0≤s≤r) if

(i) Si∈Jσi=M,

(ii) i6=j⇒σi∩σj=∅,

(iii) for all x∈Mthere exists a Cs-chart map φx:Ux→Rn, with Uxsome open neigh-

borhood about x, such that the connected components of φx(Ux∩σi)are given by the

following parts of parallel affine subspaces

Ac1,...,cn−k={(x1, . . . , xn)∈φx(Ux)|xk+1 =c1, . . . , xn=cn−k},

for some constants cl∈R,l= 1, . . . , n −k, whenever Ux∩σiis non-empty.

The pairs (Ux, φx)are called foliated charts. The elements σiof a foliation are called leaves.

A vector field X:M→TM is called transversal to the (differentiable) foliation {σi}i∈Jif

for any foliated chart

h∂

∂xi

, φx∗Xi 6= 0,

for some i∈ {k+ 1, . . . , n}, i.e., the vector field Xis nowhere tangential to the leaves.

Hence, naturally, we can have foliations of a lower differentiability class than the ambient

manifold. For example, continuous foliations in differentiable manifolds will occur in the

following chapters.

Vector and Tensor Fields

We denote vector fields on a manifold M(of class Cr) by capital letters and as sections

in the tangent bundle, e.g., X:M→T M or X∈Γ(TM)is a (global) vector field on M.

Single vectors in some tangent space TyM,y∈Mwill be denoted by a subscript Xyor by

lower case letters, e.g., v=Xy∈TyM. We denote differential forms and covariant vector

fields by lower case letters and as sections in the co-tangent bundle or some tensor bundle,

respectively, e.g., a one-form u:M→T∗Mor u∈Γ(T∗M). But at some places we will

also deviate from these general rules and for example denote special vector fields by lower

case or Greek letters. Furthermore, we denote the covariant tensor bundle of k-th order

(k∈N0) over Mby TkM, with the natural identifications T0M=Cr(M),T1M=T∗M,

T2M=T∗M⊗T∗M, and so forth, with ⊗denoting the tensor product. The symmetric and

textcolorblueantisymmetric tensor bundles of k-th order (k∈N0) over a manifold Mnwill be

denoted by ΣkMand ΛkM, respectively. In the usual way, we have Σ0M= Λ0M=Cr(M),

Σ1M= Λ1M=T∗Mand Λj>nM=∅. Hence, for example, we have a differential k-form

was a map w:M→ΛkMor as a section w∈Γ(ΛkM). In the same way Λ−1Mcan be

defined to be the set of constant functions over Mif necessary.

For vector fields X, Y , we denote the Lie bracket and the Lie derivative by [X, Y ] = £XY.

We denote the symmetrized and antisymmetrized tensor product by ∨and ∧, respectively.

The inner product of a vector field Xwith a covariant tensor field wwill be denoted by

Xcw. We adopt the following convention of normalization of these products:

For a k-form uand an l-form w, a (k+l)-form (u∧w)is given by

(u∧w)(X1, . . . , X(k+l)) = 1

(k+l)! X

τ∈Pk+l

sgn(τ)u(Xτ(1), . . . , Xτ(k))w(Xτ(k+1), . . . , Xτ(k+l)),

for k+lvector fields X1, . . . , X(k+l), and with Pk+lbeing the group of all permutations of

k+lnumbers. Moreover, u∧w= (−1)kl(w∧u)holds. Similarly, for two totally symmetric

tensor fields sand tof order kand lrespectively, we have a totally symmetric tensor field

(s∨t)of order (k+l)given by

(s∨t)(X1, . . . , X(k+l)) = 1

(k+l)! X

τ∈Pk+l

s(Xτ(1), . . . , Xτ(k))t(Xτ(k+1), . . . , Xτ(k+l)).

We adopt the inner product notation cfor all tensor fields, not only for differential forms.

Hence for k∈N0

(Xct)(Y1, . . . , Yk) := (k+ 1) ·t(X, Y1, . . . , Yk),

for all vector fields X, Y1, . . . , Yk:M→TM and all tensor fields t∈Γ(Tk+1M)and specif-

ically for t∈Γ(Λk+1M)or t∈Γ(Σk+1M). Note that in this convention the inner product

picks up a factor equal to the order of the tensor field on which it operates. This is necessary

to obtain consistency with the factors in the antisymmetrized product defined above and

the exterior derivative defined below.

The inner product and the wedge product obey the following distributive law for differential

forms. Let ube a k-form and wan l-form, then

Xc(u∧w)=(Xcu)∧w+ (−1)ku∧(Xcw),

for all X:M→TM.

Maps Between Manifolds

Let f:Mm→Nnbe a smooth mapping from a manifold Mof dimension mto a manifold

Nof dimension n. Then we denote by dfx:TxM→Tf(x)Nthe differential map of fat

x∈M. This yields a bundle map df:T M →TN, which is, in general, neither injective nor

surjective. Therefore, one usually distinguishes three important special cases: If m=nand

fis a diffeomorphism, then df:TM →TN is a diffeomorphism and dfmaps vector fields on

Mto vector fields on N. If m < n and dfxis injective for all x∈M, the mapping fis called

an immersion. If fis additionally injective, it is called an embedding and, in this case, df

maps vector fields on Mto vector fields tangential to the image of Munder f. Specifically,

for an embedding f,dfassigns a subbundle imdf(T M)⊂TN along imf(M)⊂N. This

operation on vector fields X:M→T M is called push-forward, written f∗X, from Mto N

and is defined by setting

df(X) = f∗X: imf(M)→imdf(TM),(f∗X)f(x):= dfx(Xx).

If m>nand the mapping fisasubmersion, i.e., if dfxis surjective for all x∈M, given a

vector field X∈Γ(T M), its image df(X)is generally not a well-defined vector field on any

subset of N. But one can now define an operation for one-forms defined on imf(M)⊂N,

which values in the set of one-forms on M. For a one-form u: imf(M)→T∗(imf(M)) this

operation is called pull-back, written f∗u, and is defined by setting

f∗u:M→T∗M, (f∗u)x(Yx) := uf(x)(dfx(Yx)),

for any vector field Yon M. Moreover, a smooth mapping f:M→Nbetween two mani-

folds Mand Nwill be called a surjective submersion if it is a submersion and additionally

imf(M) = Nholds.

If m=nand f:Mn→Nnis a diffeomorphism, the pull-back and push-forward operations

can be defined for fin the same way.

Subsequently, the definitions above carry over to the pull-back and push-forward of tensor

fields of any type. Moreover, in the special case of fbeing a diffeomorphism one can define a

push-forward for differential forms and a pull-back for vector fields in the obvious way using

the inverse mapping f−1:N→M. Specifically, we can now define the differential dfof any

smooth function f:M→R, which need not be a submersion. The mapping x7→ dfx(Xx)

for any vector field Xon Mvalues in Tf(x)R≃Rfor all x∈M. Hence, df(X)∈Cr(M),

df∈Γ(T∗M)and df(X)has the obvious interpretation of a directional derivative of f

along X, which we will occasionally also denote by Xf := df(X).

Definition 2.2. Let Mnbe a manifold (of class Cr) with n≥2. A subset S⊂Mis called

asubmanifold of dimension k < n and class Cs,s≤rif there is an embedding j:P→M

of class Csfrom a k-dimensional manifold P(necessarily of class Cs), such that j(P) = S.

Furthermore, if jis only a topological embedding, i.e., a homeomorphism onto its image, S

is called topological submanifold and if k=n−1,Sis called a hypersurface. Furthermore,

if jis only an immersion, Sis called an immersed submanifold.

Naturally, the leaves of a foliation are submanifolds.

Curves

A (parametrized) curve is a continuous map γ: [a, b]→Mwith [a, b]⊂R,−∞ <a<b<∞

an interval of the real numbers and Ma manifold. If γ(t) = γ(a)for all t∈[a, b], the curve

will be called constant. The points γ(a)and γ(b)will be called end-points of γ. Usually,

we will require curves to be smooth, i.e., γ|(a,b)is smooth and all necessary left- and right-

sided derivatives at the endpoints exist, but we will also allow for curves defined on non-

compact subsets of the real numbers and piecewise smooth curves. Thus, we naturally have

curves with one end-point (γ: [a, b)→Mwith −∞ < a < b ≤ ∞ or γ: (a, b]→Mwith

−∞ ≤ a < b < ∞) or no end-point (γ: (a, b)→Mwith −∞ ≤ a < b ≤ ∞). A continuous

curve γ: [a, b]→Mis called piecewise smooth if there is an n∈N, such that a partition

of [a, b]by nsub-intervals a=t0< t1<··· < tn−1< tn=bexists, with γ|[a,b]\Tnfor

Tn={ti}i=0,...,n being smooth and all necessary left- and right-sided derivatives at points in

Tnexist. Subsequently, a curve will be called inextendible if there is no continuous extension

of γsuch that its domain is compact in R={−∞}∪R∪{∞}. Moreover, we call a curve

γ:R≥a→Mwith x=γ(a)aray starting at x∈M.

Let (s, x)∈R×Mbe points in a product manifold. We call ∂tthe canonical vector field

along Ron R×Mgiven by ∂t=∂

∂s (s, x) = (1,0) ∈Γ(T(R×M)). Then the tangent

vector field or the velocity of a curve γ: [a, b]→Mwill be denoted ˙γ: [a, b]→TM with

˙γ(t0) = γ∗(∂t|t=t0)∈Tγ(t0)M, in the sense of a left- or right-sided derivative at the end-

points. If γ(a) = γ(b), the curve will be called a closed curve.

We use the designation curve here explicitly for a map γ: [a, b]→M, i.e, for a fixed domain

[a, b]of the curve parameter. Certainly, the image imγ([a, b]) ⊂Mof a curve can be obtained

by different maps possessing different domains, related to [a, b]by reparametrizations. Hence,

in situations where we are interested in the image of a curve only, we will refer to that image

as an unparametrized curve.

Derivatives

The exterior derivative d: ΛkM→Λk+1Mwill be defined as follows. Let ube a k-form,

then the (k+ 1)-form duis given for any k+ 1 vector fields X0, . . . , Xk∈Γ(T M)by

du(X0, . . . , Xk) = 1

(k+ 1)! X

(−1)iXiu(X0,..., ˆ

Xi, . . . , Xk)+

(k+ 1)! X

i<j

(−1)i+ju([Xi, Xj], X0,..., ˆ

Xi,..., ˆ

Xj, . . . , Xk),

where the hat ˆstands for an omitted vector. For one-forms, we have

du(X, Y ) = 1

2(Xu(Y)−Y u(X)−u([Y, Y ])) ,

for all vector fields X, Y ∈Γ(TM). This definition assures the following properties. Assume

there is a torsion-free connection with covariant derivative ∇on a manifold M. Let ube a

one-form on M, then duis the antisymmetric part of the tensor ∇u. Specifically,

(∇u)(X, Y )=du(X, Y ) + sym(∇u)(X, Y ),

for all vector fields X, Y on M, with sym(∇u)(X, Y ) = 1

2((∇u)(X, Y )+(∇u)(Y, X)). Fur-

thermore, naturally identifying Λ0Mwith the smooth functions on M, the exterior derivative

coincides with the differential on functions, i.e., 0-forms. The exterior derivative obeys the

following law of compatibility with the wedge product. Let ube a k-form and wan l-form,

then

d(u∧w) = (du)∧w+ (−1)ku∧(dw).

Now Cartan’s magic formula can be used to define a Lie derivative for differential forms.

Let ωbe a differential form, then

£Xω=Xcdω+ d(Xcω),

for all X∈Γ(TM). Hence, we have a Lie derivative for all differential forms, particularly

also for functions, i.e., 0-forms and

£Xf=Xf = df(X)

holds for all X∈Γ(TM)and all smooth functions f:M→R. Moreover, this yields a Lie

derivative for tensor fields of any kind, using the chain rule in the obvious way. The Lie

derivative commutes with the exterior derivative: £X(du) = d(£Xu).

2.1 Semi-Riemannian Geometry

Let Mnbe a manifold. A tensor field t:M→T∗M⊗T∗Mis called non-degenerate if

tx(v, w) = 0 for all x∈Mand all w∈TxMimplies v= 0. Any non-degenerate, covariant

2-tensor field t∈Γ(Σ2M)yields a non-degenerate scalar product tx:TxM×TxM→Rin

every tangent space. As any vector space allows for an orthonormal basis, there is a basis

{e1, . . . , en} ⊂ TxMin every tangent space, such that

tx(ei, ej) = εjδij, εj:= tx(ej, ej) = ±1.

Then s=Pn

i=1 εiis called signature of tat x∈M. The signature is independent of the

chosen basis. Furthermore, as we assume all manifolds to be connected, the signature of a

non-degenerate, symmetric, covariant 2-tensor field is constant over M.

Let x∈Mand denote by Ux⊂Man open neighborhood about x. If Uxis sufficiently

small, given a basis {e1, . . . , en} ⊂ TxM, we can extended this basis to a set of vector fields

{E1, . . . , En} ⊂ Γ(TUx), such that (Ei)x=eiand {(Ei)y}i=1,...,n is a basis of TyMfor

all y∈Ux, which will be called a local frame. A locally finite covering {Ua⊂M|a∈

A, Uaopen}of Mwith Asome index set (i.e. M=Sa∈A Ua=M), together with local

frames {E1, . . . , En}a⊂Γ(TUa)for all a∈ A will be called a frame on M. Using a frame

we can define global quantities, which are independent of the choice of the covering and the

local frames. Hence, in the remainder of this work, we will denote a frame on a manifold

Mnjust by {E1, . . . , En}.

Definition 2.3. A non-degenerate tensor field g:M→Σ2Mis called semi-Riemannian

metric on Mn. If the signature of gis n, it is called a Riemannian metric and if the

signature is n−2, it is called a Lorentzian metric. The ordered pair (M, g)is called a

Riemannian, Lorentzian or semi-Riemannian manifold accordingly.

A frame {E1, . . . , En}on Mfor which g(Ei, Ej) = εjδij holds, will be called a pseudo-

orthonormal frame or a g-orthonormal frame and just an orthonormal frame in the special

case of a Riemannian metric g. It is a standard result that g-orthonormal frames exist with

respect to any semi-Riemannian metric g. With the use of a frame, it is now possible to

define the trace of a tensor field with respect to a semi-Riemannian metric. Let w∈Γ(T2M),

then Tr(w): M→Ris given by

Tr(w) := X

εiw(Ei, Ei).

The summation runs over all indices that label the elements of a (pseudo-)orthonormal frame

and it can easily be shown that this definition is independent of the choice of the frame.

We will often denote the norm of a vector field Xon a Riemannian manifold (M, g)by

vertical double bars, i.e.,

kXkg:= pg(X, X)

is a function M→R. Particularly, for some point x∈Mand v∈TxMwe will write

kvkg

x:= pgx(v, v). The same notation will be used for the norm of one-forms uon Mby

defining

kukg

x:= sup

v∈TxM\{0}

|bx(v)|

pgx(v, v),

which equivalent to kukg=pPiu(Ei)2when using an orthonormal frame {Ei}i=1,...,n on

M. This definition can be naturally extended to covariant tensor fields of any rank. For

some particular quantities defined in section 3.1, we will use single bars |·| to indicate the

norm. We will omit the supersrcipt indicating the metric if it is no source of confusion.

Remark 2.4. These definitions of a norm above can be extended to semi-Riemannian man-

ifolds. If (M, g)is a semi-Riemannian manifold and X:M→TM a vector field, we can

set

kXkg=(pg(X, X) if g(X, X)≥0

−p|g(X, X)|if g(X, X)<0.

And in the same way, for a one-form u:M→T∗Musing a pseudo-orthonormal frame

{Ei}i=1,...,n, we can also set kukg=±p|Piεiu(Ei)2|, depending on the sign of Piεiu(Ei)2

as above.

We will denote the spaces of Riemannian and Lorentzian metrics on a manifold Mby

R(M) = {g∈Γ(Σ2M)|gRiemannian}and Lor(M) = {g∈Γ(Σ2M)|gLorentzian}.

Particularly, Lor(M)can be empty.

Let (M, g)be a semi-Riemannian manifold, then a vector 06=v∈TxM, such that gx(v, v) =

0is called isotropic at x∈M, and a vector field K:M→TM is called isotropic vector

field if Kxis isotropic for all x∈M.

For any semi-Riemannian manifold (M, g)there is a unique torsion-free connection with

covariant derivative ∇associated to g, such that ∇g= 0. This connection is called Levi-

Civita connection associated to g. In this case we have

£XY= [X, Y ] = ∇XY−∇YX,

for all vector fields X, Y :M→T M. The curvature associated to ∇will be introduced in

section 3.3.

The divergence of a vector field X:M→TM on semi-Riemannian manifold (M, g)is

defined by

divg(X) = X

εig(∇EiX, Ei),

for any pseudo-orthonormal frame {E1, . . . , En}on Mand with ∇the Levi-Civita connection

associated to g. Normally, we will omit the subscript gto the divergence if it is clear with

respect to which metric the divergence is computed.

Now let (M, g)be a Riemannian manifold. In this case we can assign a positive number

l(γ), called length, to any piecewise smooth curve γ: [a, b]→Mby setting

l(γ) := Zb

ak˙γ(t)kgdt:=

n−1

i=0 Zti+1

tiqgγ(t)(˙γ(t),˙γ(t))dt,

where {ti}i=0,...,n are the (finitely many) values in [a, b]where γis only continuous but not

differentiable. Certainly, this definition is independent of the parametrization chosen for the

curve. Hence, defining the space of piecewise smooth curves connecting two points x, y ∈M

Ω(x, y) := {γ: [0,1] →M|γis piecewise smooth, γ(0) = x, γ(1) = y}

leads to the definition of the distance

dg(x, y) := inf

γ∈Ω(x,y)l(γ)

associated to the Riemannian metric gon M. This definition makes (M, dg)a metric space

and the topology associated to dgcoincides with the manifold topology. We will say that a

Riemannian manifold (M, g)is complete or gisacomplete Riemannian metric on Mif the

the metric space (M, dg)is complete. The following lemma is well known.

Lemma 2.5. If the manifold Mis compact, any Riemannian metric gon Mis complete.

Proof. See, e.g., [Pet98, Cor. 3.5, p. 116].

A curve γ: [a, b]→Mon a Riemannian manifold (M, g)is called a geodesic if ∇˙γ˙γ= 0

holds. At least, in some appropriate sense, short geodesics are minimizers of the length

of all piecewise smooth curves connecting two points, such that dg(γ(a), γ(b)) = b−a

holds. As ususal, the geodesic condition fixes the parameter of a geodesic curve up to affine

transformations, which also yields g(˙γ, ˙γ) = const along the curve. A geodesic is called

complete if it can be extended to a geodesic defined on all of R. The extension is unique

up to affine reparametrization in this case. Then we call a Riemannian manifold (M, g)

geodesically complete if all geodesics are complete. Then the famous theorem by Hopf and

Rinow holds:

Theorem 2.6. Let (M, g)be a Riemannian manifold. The following is equivalent:

(i) (M, g)is complete.

(ii) (M, g)is geodesically complete.

(iii) Every closed and dg-bounded subset of Mis compact.

Proof. See, e.g., [Pet98, Thm. 7.1, p. 125].

The notion of geodesic completeness carries over to semi-Riemannian manifolds without

substantial adjustments. Although there is no Hopf–Rinow type theorem in the semi-

Riemannian case, as there is no associated metric space in general.

We will also need the following

Lemma 2.7. Let (M, h)be a complete Riemannian manifold, ηany Riemannian metric on

Mand f:M→R>0, such that infx∈Mf(x)>0. Then g=fh+ηis a complete Riemannian

metric on M.

Proof. See [FM78, Lem. 2].

Let Mnbe a manifold. A distribution ∆⊂TM in the tangent bundle of a manifold, is

a smooth assignment x7→ ∆x⊂TxMof a subspace ∆xto all points in M, such that

dim(∆x) = m≤nis constant over M. Then we write ∆ = Sx∈M∆xand call mthe

dimension of ∆. Subsequently, a vector field Xis in ∆, i.e., we write X:M→∆or

X∈Γ(∆) if Xx∈∆xfor all x∈M. Any non-vanishing one-form u:M→T∗Mfosters a

distribution ∆uassociated to it by

∆u

x:= {v∈TxM|ux(v)=0},

which is of dimension n−1.

Definition 2.8. Let ∆⊂TM be a distribution and h:M→Σ2M. The tensor field his

called a semi-Riemannian metric on ∆if for any x∈Mit holds that hx(v, w)=0for all

w∈∆ximplies either v= 0 or v /∈∆x.

A distribution ∆⊂TM is a subbundle of the tangent bundle in the natural way. If there

is a semi-Riemannian metric gon M, there is naturally also a perpendicular distribution

∆⊥⊂TM given by

∆⊥

x:= {v∈TxM|gx(v, w) = 0 for all w∈∆x},

for all x∈M. Note that ∆⊥

x∩∆x={0}for all x∈M. This gives rise to a dual distribution

∆∗⊂T∗Mby setting u∈Γ(∆∗)if u∈Γ(T∗M)and u(X)=0for all vector fields

X∈Γ(∆⊥). Extending the sections over the distributions by tensor products we naturally

also get tensor fields and tensor bundles over distributions.

Remark 2.9. Note that the above definition of dual distributions is ambiguous in the case

of isotropic planes ∆xat some x∈Mfor a distribution ∆⊂TM in a semi-Riemannian

manifold (M, g). The reason is that isotropic vectors are perpendicular to one another if and

only if they are parallel. In this case one is forced to consider lightlike geometry (cf. [DS10]).

But in this work we will exclusively consider non-isotropic distributions and subbundles.

Proposition 2.10. Let (M, g)be a semi-Riemannian manifold and X:M→TM a nowhere

vanishing vector field that is nowhere isotropic. Then the one-form u=g(X, ·)metrically

associated to Vgives rise to a distribution ∆u. The tensor field

h:= g−u⊗u

g(X, X)

is a semi-Riemannian metric on ∆u. Particularly, if (M, g)is Lorentzian and g(X, X)<0,

then his Riemannian.

Proof. Obviously, his symmetric. Let x∈Mbe an arbitrary fixed point and assume

hx(v, w)=0for all w∈∆u

x. Then we have

gx(v, w) = ux(v)ux(w)

gx(X, X).

Furthermore, vcan be g-orthogonally decomposed by v=λXx+˜v, where ˜v∈∆u

xand λ∈R.

As ux(Xx) = gx(X, X), we get

λgx(Xx, w) + gx(˜v, w) = λux(w)⇒gx(˜v, w) = 0.

Hence, ˜v= 0 and, therefore, either v= 0 or v /∈∆u

x. Now let (M, g)be Lorentzian,

g(X, X)<0and v∈∆u

x. Then

gx(v, v) = hx(v, v),

and it follows that gx(v, Xx) = 0, thus hx(v, v) = gx(v, v)>0.

Remark 2.11. The notion of a semi-Riemannian metric on a distribution bears resemblance

to sub-Riemannian geometry, and in fact it is a generalization of this concept. In sub-

Riemannian geometry (see e.g. [Str86]) one considers Riemannian manifolds (M, g)together

with a completely non-integrable distribution H⊂TM, such that g|His a Riemannian metric

on Hin the sense of Def. 2.8. Here the complete non-integrability means that any vector field

Y:M→TM is a finite linear combination of vector fields X1,[X1, X2],[X1,[X2, X3]], . . .

for X1, . . . , Xm:M→Hwith some m∈N. Then (M, H, g)is called a sub-Riemannian

manifold.

Contact geometry formalizes the concept of complete non-integrability of a distribution in

terms of a differential form. We give the following

Definition 2.12. Let M2n+1 be a manifold, n≥1and u∈Γ(Λ1M). Then (M, u)is called

contact manifold if u∧(du)n6= 0 everywhere, where (du)n=du ∧···∧du (n-times).

We will also need the following concept on a contact manifold.

Definition 2.13. Let (M, u)be a contact manifold. A vector field R∈Γ(TM)is called

Reeb vector field if u(R)=1and Rcdu = 0.

It is not difficult to see that every contact manifold admits a unique associated Reeb vector

field (see, e.g., [Gei08]). There is an interesting conjecture—formulated by A. Weinstein in

1979 (see [Wei79])—in contact geometry, that we will apply to causality theory of Lorentzian

manifolds in section 4.3 below.

Weinstein Conjecture. Let (M, u)be a closed and oriented contact manifold. Then the

Reeb vector field has at least one closed integral curve.

Actually, this conjecture has only recently been proven in dimension three by C.H. Taubes

in [Tau07]. All higher dimensions remain open in the general case.

Theorem 2.14. Let (M, u)be an closed and oriented contact manifold with dim(M) = 3.

Then the Reeb vector field has at least on closed integral curve.

Proof. See [Tau07] or [Hut10] for a review.

2.2 Global Randers Geometry

In this section, we briefly recall the notions of Finslerian geometry. We follow the conventions

and notation in [BCS00] and [CJS11], which are the standard references for this section. For

theorems on global hyperbolicity of stationary and related classes of spacetimes, we will only

need a special case of Finslerian metrics, namely the so called Randers-type metrics. These

metrics arise from a given Riemannian metric and a one-form on a manifold.

Definition 2.15. AFinslerian metric on a manifold Sis a function F:TS →R≥0, which

has the following properties for all (x, y)∈TS with x∈Sand y∈TxS:

(i) Fis continuous on TS, smooth on TS \{(x, 0)}and F(x, y) = 0 only for y= 0.

(ii) Fis fiberwise positively homogeneous of degree one, i.e., F(x, λy) = λF(x, y), for all

(x, y)∈TS and λ > 0.

(iii) h1

∂2(F2)

∂yi∂yj(x, y)iis a positive-definite matrix for any (x, y)∈T S \(x, 0).

In this case we call (S, F)aFinslerian manifold.

Since Fis only positive homogeneous of degree 1for a piecewise smooth curve γ: [a, b]→S,

the Finslerian length

lF(γ) = Zγ

F(γ, ˙γ) = Zb

F(γ(t),˙γ(t))dt

depends on the orientation of the curve. As a consequence, the Finslerian distance between

two points p, q ∈S:

dF(p, q) = inf

γ∈Ω(p,q)lF(γ),

where Ω(p, q)is the set of all piecewise smooth curves γ: [a, b]→Swith γ(a) = pand

γ(b) = qis not symmetric in pand q, in general. Thus, two non equivalent notions of

completeness make sense (cf. [BCS00, Sec. 6.2]).

Definition 2.16. A sequence {xn}n∈N⊂Sis called forward (resp. backward)Cauchy se-

quence if for all ε > 0there is a ν(ε)∈N, such that for all i, j with ν(ε)≤i≤j,d(xi, xj)≤ε

(resp. d(xj, xi)≤ε) holds. Furthermore, (S, F)is called forward complete (resp. backward

complete) if all forward (resp. backward) Cauchy sequences are convergent in the generalized

metric space (S, dF).

Introducing forward balls B+(x, r) := {y∈S|dF(x, y)< r}and backward balls B−(x, r) :=

{y∈S|dF(y, x)< r}for 0< r < ∞in a Finslerian manifold (S, F), there is a Finslerian

version of the Hopf–Rinow theorem. A subset of the generalized metric space (S, dF)is

called forward (backward) bounded if it is contained in some forward (backward) ball.

Theorem 2.17. Let (S, F)be a Finslerian manifold, then the following is equivalent.

(i) The Finslerian metric Fis forward (backward) complete.

(ii) Every closed and forward (backward) bounded subset in the generalized metric space

(S, dF)is compact.

Proof. See, e.g., [BCS00, Thm. 6.6.1].

Remark 2.18. By an appropriate definition one can include forward and backward Finsle-

rian geodesics in this Hopf–Rinow theorem, but we will not need that concept in this work.

A special class of the Finslerian metrics are those which are of Randers type. These metrics

consist of a given Riemannian metric gand a one-form bon a manifold S, with the g-norm

of bpointwise bounded by 1:

kbkg

x<1,

for all x∈S. Then for all (x, y)∈TS, the Finslerian metric of Randers type is given by

F(x, y) = pgx(y, y) + bx(y).

The completeness of Randers-type metrics depend on the completeness of the underlying

Riemannian metric and the norm of the one-form with respect to this metric. The following

proposition is known and appeared, e.g., in [CJM11, Rem. 4.1]. We give a full proof for the

sake of completeness. Also the proof demonstrates in detail a standard technique that will

be used later on in this work and we will refer to this proof for details if necessary.

Proposition 2.19. A Randers-type metric F=√g+bon a manifold Sis forward and

backward complete if the Riemannian metric gis complete and the g-norm of bis uniformly

bounded by 1, i.e.,

|||b|||g:= sup

x∈Skbkg

x<1.

Proof. In the following we will omit the superscript indicating the metric g. We denote by

B:S→TS the vector field metrically associated to the one-form b, such that b=g(B, ·)

and the norm at x∈Sis given by kbkx=pgx(B, B). Let X∈Γ(TS)be a vector field

obeying g(X, X) = 1. Then the angle αbetween Band Xis given by

cos(α) = g(B, X)

pg(B, B).

For the norm kbkxwe get

sup

y∈TxS\0

|gx(Bx, y)|

pgx(y, y)= sup

kykx=1 |gx(Bx, y)|=

= sup

α∈[−π,π]|cos(α)|pgx(Bx, Bx) = pgx(Bx, Bx) = kbkx.

Let ˙γbe the tangential vector field at curves γin the set Ω(p, q)of piecewise smooth curves

connecting pand q, with curve parameter s∈[0,1]. The Riemannian distance between p

and qis, hence, given by

dg(p, q) = inf

γ∈Ω(p,q)Z1

0qgγ(s)(˙γ(s),˙γ(s)) ds

and is obviously symmetric in pand q. Because gis complete, every Cauchy sequence

{xn}n∈N⊂Sin the Riemannian distance dg(xi, xj)converges. The Finslerian distance

between two points xiand xjis given by

dF(xi, xj) = inf

γ∈Ω(xi,xj)Zγ

F(γ, ˙γ) = inf

γ∈Ω(xi,xj)Zγ

(pg(˙γ, ˙γ) + g(Bγ,˙γ)).

Thus, for a forward Cauchy sequence {xn}n∈Nin (S, d), the Finslerian distance obeys

dF(xi, xj) = inf

γ∈Ω(xi,xj)Zγ

[(1 + g(Bγ,˙γ)

pg(˙γ, ˙γ))pg( ˙γ, ˙γ)] < ε,

for all ε > 0and for ν(ε)< i ≤j. Due to the assumption

|||b||| = sup

x∈S

sup

v∈TxS\0

|gx(Bx, v)|

pgx(v, v)<1,

it follows that

ε>dF(xi, xj)>(1 −|||b|||) inf

γ∈Ω(xi,xj)Zγpg(˙γ, ˙γ) = (1 −|||b|||)dg(xi, xj)

⇒ε0:= ε

1−|||b||| > dg(xi, xj).

Hence, every forward Cauchy sequence in (S, dF)is also a Cauchy sequence in (S, dg), which

converges to some limit point, due to the completeness of g. Now, we can make use of the fact

that the topology induced by the open balls of a Riemannian metric, the topology induced

by the forward (and backward) open balls of a Finslerian metric and the manifold topology

coincide. See [ST51, p. 44] for the Riemannian case and [BCS00, Sec. 6.1] for the Finslerian

case. Hence, a sequence converging in the distance induced by the the Riemannian metric,

converges in the manifold topology and also converges forward in the distance induced by

the Randers-type metric. An analogue proof holds for the backward case.

On the other hand we can now show that the forward or backward completeness of a Randers-

type metric implies the completeness of the underlying Riemannian metric. Compare, e.g.,

[CJM11] and [CJS11].

Lemma 2.20. If a Randers-type metric F=√g+bon a manifold Sis forward or backward

complete, then the Riemannian metric gis complete.

Proof. We prove this lemma by contradiction. Assume that gis not complete. Let {xn}n∈N

be a g-Cauchy sequence, which does not converge in (S, dg). Let i≤j∈Nand thus we have

similar to the proof of Prop. 2.19

dF(xi, xj) = inf

γ∈Ω(xi,xj)Zγpg(˙γ, ˙γ) + b( ˙γ) = inf

γ∈Ω(xi,xj)Zγ 1 + b( ˙γ)

pg(˙γ, ˙γ)!pg( ˙γ, ˙γ)≤

≤2 inf

γ∈Ω(xi,xj)Zγpg(˙γ, ˙γ)=2dg(xi, xj).

The same holds for dF(xj, xi). Hence, whenever dg(xi, xj)< ε, we also have dF(xi, xj)<2ε

and dF(xj, xi)<2ε, such that {xn}n∈Nis a forward and backward Cauchy sequence with

respect to F. But following the same reasoning as in the proof of Prop. 2.19, the sequence

{xn}n∈Ncannot converge in the metric space (S, dF), either, as this would imply a limit

point with respect to the manifold topology and subsequently in the metric space (S, dg).

Hence, Fis forward and backward incomplete, which is the desired contradiction.

The Prop. 2.19 and Lem. 2.20 above give only very rough estimates on completeness rela-

tions between Riemannian and Randers-type metrics. When employing these completeness

considerations to Lorentzian causality theory, we will reveal much more refined connections

between different completeness conditions involving growth conditions of the norm of the

constituting one-form and other particular functions. See section 5.2 for this analysis.

2.3 R-manifolds and Principal Bundles

Standard references for this section are [Pal61] and [KN63]. The results presented are mostly

standard and are contained, maybe using a different language, in other references, such as

[tD87]. In [Pal61] R. Palais investigated the action of non-compact Lie groups on topological

spaces, and examined, particularly, the question of the existence of slices of the action. We

will adapt the basic techniques, definitions and theorems established in [Pal61] to the action

of (R,+) on a Lorentzian manifold. Therefore, we specialize the notions to the Lie group

(R,+) and add results on differentiability. Essentially, what we end up with in this process

is a dynamical system. So, we prepare the ground for connecting the concepts and notions in

this section to the conceptions used in the topological theory of dynamical systems, which we

will introduce in the following section. As we will use results from the differentiable version of

Palais’ non-compact Lie group theory developed in this section and from dynamical systems

in our further analysis, we will prove several results which connect the concepts from both

fields. There are often notions in these two areas that look different at first sight, but are

essentially the same. Moreover, we recall in this section some basic results on principal

bundles—with and without a Hausdorff base space—and narrow them down to principal

bundles with a one-dimensional structure group, particularly to R-principal bundles.

Definition 2.21. Let (R,+) be the real line furnished with the structure of an additive

group. By an R-action on a manifold M(of class Cr,r > 0), we mean a homomorphism

fof (R,+) into Diffr(M), such that the map R×M→Mwith (t, p)7→ f(t)(p) =: t◦pis

Cr-smooth. Such a manifold M, together with a fixed R-action f, is called R-manifold.

In the following, we will usually omit the homomorphism f: (R,+) →Diffr(M)and stick

to the notation t◦p, for t∈Rand p∈M. We use the following standard characterization

of the group action.

Definition 2.22. Let Mbe an R-manifold. The R-action is called free on Mif t6= 0 implies

t◦p6=pfor all p∈M. The R-action is called locally free if there is a neighborhood I⊂R

with 0∈I, such that t∈I\{0}implies t◦p6=pfor all p∈M.

Definition 2.23. The set Rp={t◦p|t∈R}is called orbit of p∈Mand for any subset

S⊂Mwe define the set RS={t◦p|t∈R, p ∈S}, which is called the saturation of S.

The set of orbits of an R-manifold will be denoted by M/Rwith the canonical projection

πM:p7→ Rp. The quotient space M/Rcan be furnished with the quotient topology inherited

from the manifold topology of M. But it is known that this topology may lack some of the

separation properties of the manifold topology, depending on the structure of the R-action.

For example, in general, the quotient topology of M/Ris not Hausdorff (see e.g. [vQ76]) and

there are cases where it is not even T1. We will consider subsets of R, emerging from the

R-action on subsets S, T ⊂M, which are of the following form:

((S, T)) := {t∈R|(t◦S)∩T6=∅}.

Here we denote t◦S:= {t◦p∈M|p∈S}.

Definition 2.24. Let S, T ⊂Mbe two subsets of an R-manifold M. Then we call Sthin

relatively to Tif ((S, T)) is bounded in R. If Sis thin relative to itself, it is called thin.

In [Pal61], some basic features of thin sets are listed. The most relevant attribute which we

will use in this work is shown in the following

Lemma 2.25. Let S, T ⊂Mbe two subsets of an R-manifold M. If Sand Tare compact,

then ((S, T)) is closed in R. If additionally Sand Tare relatively thin, then ((S, T)) is

compact in R.

Proof. Let S, T ⊂Mbe compact subsets and assume ((S, T)) is not closed. Then there

is a sequence {tn}n∈N⊂((S, T)) such that tn→t0/∈((S, T)) as n→ ∞. Hence, for all

n∈Nwe have (tn◦S)∩T6=∅or there is qn∈Ssuch that tn◦qn∈T. By passing to a

subsequence of {qn}n∈N⊂S, we have that qn→q0∈Sas n→ ∞ and tn◦qn→r0∈Tas

n→ ∞, because Sand Tare compact. But as the R-action (t, p)7→ t◦pis smooth, we have

r0=t0◦p0, and thus (t0◦S)∩T6=∅and t0∈((S, T)) is the desired contradiction. Now, if

Sand Tare relatively thin, then ((S, T)) is closed and bounded, hence compact.

The following definition differs slightly from the convention used in [Pal61, Def. 1.1.2], but

will prove very well adapted to the theory developed in this work. Recalling that a near-

manifold is a topological space which fulfills all conditions of a manifold except for being

Hausdorff but is a T1-space only, we give the following

Definition 2.26. An R-manifold Mnwill be called a Cartan R-manifold if the following

two conditions hold:

(i) Each orbit is closed (as a point set) in Mand M/Ris an (n−1)-dimensional near-

manifold.

(ii) The map t7→ t◦pfrom Ronto Rpis a diffeomorphism for all p∈M.

Then we have the following

Lemma 2.27. If Mis a Cartan R-manifold, the R-action is free on M.

Proof. Assume there is p∈Mand t∈R, such that t◦p=p= 0 ◦p. Obviously, in this

case the map t7→ t◦pis not injective, hence not a diffeomorphism.

As we will see below, the idea of a Cartan R-manifold is to have the manifold Mas a

generalized R-principal bundle over M/R. The terminology Cartan R-manifold is justified

by the following considerations. The free (R,+) action on Mcan be characterized as follows:

let R⊂M×Mbe the set of pairs (p, q)∈M×Msuch that pand qbelong to the same

orbit, then there is a unique element tp,q ∈R, such that q=tp,q ◦p. Based on this

characterization we can introduce the notion of a Cartan principal bundle, such that the

map R3(p, q)7→ tp,q ∈Ris smooth, similar to the definition given by H. Cartan in [Car49].

It can be shown (see, e.g., [Pal61] for the continuous case) that such a Cartan principal

bundle fulfills in fact all conditions of a Cartan R-manifold.

The following is the standard example of an R-manifold, which is not a Cartan R-manifold.

Example 2.28. Let the two-torus T2be given as the coordinate patch (x, y)∈[0,1] ×

[0,1] ⊂R2modulo the identifications x∼x+ 1 and y∼y+ 1. Let (R,+) act on T2by

t◦(x, y) = (x+t, y +at)with a∈R\Qan irrational constant. If we had a∈Q, each

orbit would be an embedded circle. Now, it is an application of L. Kronecker’s diophantine

approximation theorem (see [Kro68]) to conclude that each orbit is dense in T2in this case.

So, there can be no orbit which is a closed set in this case, and the orbit space T2/Ris not

even a T1-space.

It is important to notice that under very mild assumptions, the R-action on an R-manifold

gives rise to a one-dimensional foliation of Mby the equivalence classes, that is the orbits,

of the action. This is shown in the following

Lemma 2.29. Let Mnbe an R-manifold. If the R-action is locally free, Mis foliated by

the family of orbits F:= {π−1

M(ξ)}ξ∈M/Ras one-dimensional leaves.

Proof. All points p∈Mare elements of precisely one orbit. To see this, assume there is a

p∈Msuch that p∈Rqand p∈Rq0for Rq6=Rq0. By the definition of the orbits there are

two numbers t, t0∈Rsuch that p=t◦qand p=t0◦q0. But then also q=t−1◦t0◦q0= (t0−t)◦q0

and thus Rq=Rq0. Moreover, for all p∈Mthe R-action produces an orbit Rpwith p∈Rp.

This assures that

M=G

ξ∈M/R

π−1

M(ξ).

The foliated charts are constructed in the following way. Take any manifold chart (U, ϕ)

about p∈U⊂M. Maybe by making Usmaller, the locally free R-action on Mconstrains

to a free R-action on U, such that the orbits in Uare given by Rq∩Ufor all q∈U. This

is always the case as any locally free homomorphism of (R,+) into Diffr(M)yields only

0∈Ras a trivial stabilizer. By the map ϕ, these orbits of the R-action on Uare carried

to one-dimensional smooth submanifolds of ϕ(U)⊂Rn. So, there is a diffeomorphism

ψ:ϕ(U)→R×Rn−1which identifies these submanifolds ϕ(Rq∩U)with affine subspaces

(R~c)∩(ψ◦ϕ)(U)⊂R×Rn−1for ~c = (0, c1, . . . , cn−1)∈R×Rn−1∩(ψ◦ϕ)(U). Thus

(U, ψ ◦ϕ)is the desired foliated chart.

The following lemma connects the notion of thin sets to Cartan R-manifolds. The proof of

this lemma can essentially already be found in [Pal61] for the continuous case and general

group actions. We reproduce the proof here along the ideas in [Pal61], but we fill the gaps

to differentiability and specialize to an R-action.

Lemma 2.30. Let Mnbe an R-manifold. If all p∈Mpossess a thin neighborhood, then

Mis a Cartan R-manifold.

Proof. (i) Assume there is a sequence {tn}n∈N⊂R, such that the sequence {tn◦p}n∈N⊂Rp

for a fixed p∈Mconverges to a point tn◦p→q /∈Rpas n→ ∞. As qadmits a thin

neighborhood U, there is an N0∈N, such that tn◦p∈Ufor all n > N0. Fixing N1> N0

we have (tn−tN1)◦p0=tn◦pfor some p0=tN1◦p∈Usuch that tn−tN1∈((U, U)) for

n > N0. But as ((U, U)) is bounded, we can pass to a sub-sequence of tn−tN1in ((U, U))

that converges to some number τ∈((U, U)). Hence (tn−tN1)◦p0→q=τ◦p0∈Rp, which

yields the desired contradiction. Now that M/Ris a T1-space is a standard result in topology

for spaces admitting closed orbits (see [vQ76]). In the same way, the connectedness of Mand

the second countability of the topology are passed on to M/R. Let (U, ψ ◦ϕ)be a foliated

chart about p∈Mas in Lem. 2.29 and V=πM(U)an open neighborhood of ξ=πM(p)in

M/R. Then by π−1

Mthe points η∈Vare homeomorphically identified with orbit segments

Rq:= Rq∩U⊂U, such that η=πM(q). Thus, by pr2◦ψ◦φ:U→Rn−1the orbit

segments f

Rqare homeomorphically carried to the open set pr2(ψ◦φ(U)) ⊂Rn−1. Hence,

M/Ris locally homeomorphic to Euclidean space, and thus it is an (n−1)-dimensional

near-manifold.

(ii) In [Pal61] it was shown that Ris homeomorphic to Rpfor all p∈M, such that there is a

topological embedding R→Rpfor all p∈M. By (i) each orbit is closed in Mand for each

p∈Rpthere is a smooth coordinate chart (U, ϕ)about p∈Msuch that ϕ(U∩Rp)⊂ϕ(U)

is the image of the section of the orbit through p, which is a submanifold of ϕ(U). Thus the

topological embedding R→Rpis also a smooth immersion.

However, we will see in the following section that admitting a thin neighborhood about any

point is a much stronger condition than being a Cartan R-manifold. This will become clear

by using the topological theory of dynamical systems.

Given a manifold Mand the action of the one-dimensional Lie group (R,+) on M, we call

the triple (M, M/R, π)an R-principal bundle if the following conditions hold: Racts freely

on M, the canonical projection π:M→M/Ris a surjective submersion onto a manifold

M/Rand Mis locally trivial, i.e., for all xin M/Rthere is a neighborhood U, such that

there is a bundle isomorphism between π−1(U)⊂Mand the trivial bundle (R×U, U, pr2).

As (R,+) is Abelian, there is no need to distinguish between left and right actions.

Obviously, the definition above can be relaxed to include the quotient space to be a near-

manifold. We will call the bundle a generalized R-principal bundle in this case. Occasionally,

we will be also interested in the compact structure group R/Z=U(1). Particularly, this will

occur in the case of a non-free R-action with isotropy subgroup Z.

The following proposition connects R-principal bundles to Cartan R-manifolds.

Proposition 2.31. An R-manifold Mnis a generalized R-principal bundle if and only if

it is a Cartan R-manifold. Furthermore, every Cartan R-manifold is locally an R-principal

bundle.

Proof. Firstly, we show that if Mis a Cartan R-manifold, then it is a generalized R-principal

bundle. We have to prove that (i) the projection πM:M→M/Ris a surjective submersion

and (ii) for any ξ∈M/Rthere is local trivialization t: (π−1(U), U, πM)→(R×U, U, pr2)

for some neighborhood Uof ξ.

(i) We note that imπM(M) = M/Rby definition. It remains to show that dπMhas maximal

rank. Let x∈Mand (U, φ)a foliated chart about x, given by Lem. 2.29. Suppose that

φ(U)⊂R×Rn−1is open and φ(x) = (0,~

0) ∈R×Rn−1. Now we define two vectors

a, y ∈TxMby a=φ∗(1,~

0) and y=φ∗(0, ~x)for some ~x ∈Rn−1. Then we have dπM(a) =

ϕ∗(~

0) and dπM(y) = ϕ∗(~x), with the isomorphism ϕ∗:Rn−1→TπM(x)(M/R)induced by the

properties of the foliated chart. As this holds for all x∈Mand all ~x ∈Rn−1, we conclude

that dπMhas rank n−1.

(ii) Take any ξ∈M/Rand any x∈π−1

M(ξ)with a chart neighborhood W⊂M. Suppose

again that (W, φ)is a foliated chart about x∈M, with φ(W)⊂R×Rn−1being open and

φ(x) = (0,~

0) ∈R×Rn−1. Let the (n−1)-disk B⊂Rn−1be defined by B={(0, ~x)∈φ(W)}.

Then ˜

B=φ−1(B)⊂Mis an (n−1)-dimensional submanifold with boundary in M, such

that x∈˜

Band all orbits of the R-action that intersect W, intersect ˜

Bonly a countable

number of times. Even more so, as the R-action is free and any orbit is closed as a point

set, we conclude that all orbits of the R-action that intersect W, intersect ˜

Bonly a finite

number of times. Hence, by making Wsmaller we can achieve that all orbits that intersect

W, intersect ˜

Bexactly once. Thus, the neighborhood U=πM(W)about ξin M/Rhas a

chart mapping ϕ:U→Rn−1with varphi(U) = Band ϕ(ξ) = ~

0. The saturation R˜

Bin

Mis now diffeomorphic to π−1

M(U), so that there is a diffeomorphism t:π−1

M(U)→R×U,

as we can identify a point p∈R˜

Bby ϕ(Rp)∈Band by a unique number t∈Rsuch that

t◦p∈˜

Secondly, suppose that (M, M/R, π)is a generalized R-principal bundle. Then the closed set

property of any orbit is a straightforward consequence of the local trivialization condition.

For similar reasons, every orbit is diffeomorphic to R.

Now consider any ξ∈M/Rfor a generalized R-principal bundle (M, M/R, π). Then certainly

there is a neighborhood Uof ξ, such that there is a local trivialization (R×U, U, pr2)of

the bundle over Uand Uis homeomorphic to Rn−1in the sense of M/Rbeing a locally

Euclidean near-manifold. Then (R×U, U, pr2)is a generalized R-principal bundle in its

own right, but it is even an R-principal bundle, because Uis homeomorphic to a Euclidean

space and fulfills, therefore, all separation properties of the Euclidean space. Hence, Uis a

manifold.

Principal fiber bundles have the advantage to admit connections. We will work with the

following notion of connections, which makes sense because the Lie algebra of the Lie group

(R,+) is just the one-dimensional Euclidean vector space R.

Definition 2.32. Let (M, M/R, π)be a principal fiber bundle. Let Φ: R×M→Mbe

the R-action on M, such that Φt∈Diffr(M)for all t∈R. Furthermore, we denote by

Aa∈Γ(TM)the fundamental vector field belonging to the value ain (the Lie algebra) R.

Then a one-form u∈Γ(Λ1M)is called connection in (M, M/R, π)if

(i) u(Aa) = afor all a∈Rand

(ii) Φ∗

tu=ufor all t∈R.

Any principal bundle admits a connection. That is a standard result (cf. Thm. 2.1, p. 67

in [KN63]). The crucial condition in the proof of this fact is that the quotient M/Ris

indeed a manifold, particularly paracompact. Moreover, it is also a standard result that

an R-principal bundle is globally trivial or trivializable, i.e., there is a principal bundle

isomorphism (M, M/R, πM)→(R×Q, Q =M/R,pr2)(see, e.g., Thm. 5.7, p. 58 in [KN63]

or Prop. 16.14.5 in [Die69]), the proof of which requires Qto be a manifold. Thus, a Cartan

R-manifold is trivializable or admits a connection if M/Ris Hausdorff.

Connections allow for the definition of unique horizontal lifts of vector fields and curves.

For an R-principal fiber bundle (M, M/R, π)with connection u∈Γ(Λ1M)and a vector

field X∈Γ(T(M/R)), there is a unique vector field X∗∈Γ(TM), called the horizontal

lift of X, obeying π∗(X∗

p) = Xπ(p)for all p∈Mand u(X∗)=0. Particularly, given a

curve γ: [a, b]→M/Rwith tangential vector field ˙γ, there is a unique horizontal lift ˙γ∗on

π−1(γ)⊂M. Hence, fixing some p∈π−1(γ(t0)) for a fixed t0∈[a, b]yields a unique lifted

curve γ∗: [a, b]→Mwith γ(t0) = p. These are standard results (cf. [KN63]).

Moreover, the notion of a connection can be extended to Cartan R-manifolds or generalized

principal bundles, by extending the definition of a fundamental vector field in a natural

way. For example, let Φ: R×M→Mbe the R-action on a Cartan R-manifold. Then

every ain the Lie algebra Rinduces an invariant vector field ˜

Aaon the Lie group (R,+)

by the exponential map, hence ˜

Aa∈Γ(TR). Now extend ˜

Aato a vector field (˜

Aa,0) ∈

Γ(T(R×M)) ≃Γ(TR×TM)on R×M. Then the fundamental vector field Aaon Mis

given by the push-forward Aa= Φ∗(˜

Aa,0). Thus, we can prove the following

Proposition 2.33. Let Mbe a Cartan R-manifold with associated generalized principal

bundle (M, M/R, π). If (M, M/R, π)admits a (global) connection, then M/Ris Hausdorff.

Proof. Let γ1, γ2: [0,1] →M/Rbe two curves contained in M/R. As (M, M/R, π)admits

a connection, given p1∈π−1(γ1(0)) and p2∈π−1(γ2(0)), there are unique horizontal lifts

γ∗

1, γ∗

2: [0,1] →Mof γ1, γ2, such that γ∗

i(0) = piand γ∗

i(1) ∈π−1(γ1(1)) for i= 1,2. Now,

assume that M/Ris not Hausdorff. Then there are two points x, y ∈M/Rthat cannot be

separated by disjoint open neighborhoods. Assume Uxand Uyare open neighborhoods of x

and yrespectively, such that W:= Ux\{x}=Uy\{y}. Assume that the curve γ1is closed

at x, i.e., γ1(0) = γ1(1) = xand γ1(0,1) ⊂W. Furthermore, assume that γ2connects xand

yand coincides with γ1in W. That is γ2(0) = x,γ2(1) = yand γ2(0,1) = γ1(0,1). As the

horizontal lift is unique for all given p∈π−1(x), we also have γ∗

2(0,1) = γ∗

1(0,1) in M. By

continuity, it also follows that γ∗

1(1) = γ∗

2(1). Thus, γ∗

2(1) ∈π−1(x)and as this holds for

all given p∈π−1(x), we have that the fibers over xand ycoincide in contradiction to the

non-Hausdorff condition.

Now, we are ready to assess the question when a Cartan R-manifold Mallows for a Haus-

dorff quotient M/Rand is, therefore, a (globally trivial) R-principal bundle, from a purely

topological perspective. To this end we state the following

Definition 2.34. Let Mbe an R-manifold. A subset S⊂Mis called small if each point

p∈Mhas a neighborhood which is thin relative to S. And an R-manifold is called proper if

each point p∈Mhas a small neighborhood.

The following theorem assembles important equivalent conditions for an R-manifold to be

proper.

Theorem 2.35. Let Mnbe an R-manifold. Then the following items are equivalent.

(i) For all p, q ∈Mthere exist relatively thin neighborhoods S, T ⊂Mwith p∈Sand

q∈T.

(ii) Mis a proper R-manifold.

(iii) Mis a Cartan R-manifold and M/Ris an (n−1)-dimensional manifold.

(iv) If K⊂Mis compact, ((K, K)) ⊂Ris compact.

(v) The projection πMis closed.

Furthermore, every Cartan R-manifold is locally proper.

Proof. We refer to [Pal61] for the proof of the equivalence of items (i),(ii) and (iv). Therein

it was also shown that for a proper R-manifold M, the quotient M/Ris Hausdorff, i.e.,

that (ii) implies (iii). Item (v) implies (iii) because Mis a manifold and hence a normal

space. Thus, a closed projection implies that M/Ris a normal space, too, and in particular

Hausdorff.

(iii)⇒(v): Suppose that A⊂Mis closed, but B=πM(A)⊂M/Ris not closed. So, there

is x∈B\B, such that every open neighborhood Uabout xhas a non-empty intersection

with B. Hence, because M/Ris a T1-space, we can assume without loss of generality that

Ais compact. Let {xn}n∈N⊂Bbe a sequence with xn→xas n→ ∞. As Ais closed, we

certainly have A∩π−1

M(x) = ∅. Hence, we can lift xnto a sequence convergent in A. This

is always possible because xninduces a sequence in any local section σ:B∩U→Mthat

is contained in A. The local sections exist because of the local R-principal bundle structure

of the Cartan R-manifold. Thus the sequence pn=σ(xn)converges to some p∈A. But

this implies that πM(p)6=xin M/Rand πM(p)and xcannot be separated by disjoint open

neighborhoods contradicting the Hausdorff property.

(iii)⇒(i): Suppose (M, M/R, πM)is an R-principal bundle, which is equivalent to item

(iii) (cf. Prop. 2.31). Suppose first that there are two points p, q ∈Msuch that p /∈Rq.

Then x=πM(p)and y=πM(q)can be separated by disjoint open neighborhoods Ux

and Uyin M/R. Hence, π−1

M(Ux)and π−1

M(Uy)are trivializable, open and disjoint in M.

Hence, any two neighborhoods Wpand Wqof pand qare relatively thin as long as they

are contained in π−1

M(Ux)or π−1

M(Uy), respectively, as we have ((Wp, Wq)) = ∅. Suppose

now that p∈Rq, i.e., there is t∈Rsuch that p=t◦q. Now, take a neighborhood U

about x=πM(p) = πM(q)in M/R, such that there is a trivialization R×Uof π−1

M(U)

obeying p= (0, x)and q= (t, x). Then there is a τ∈R, such that Wp= (−τ, τ)×Uand

Wq= (t−τ, t +τ)×Uare open neighborhoods of pand qin R×Urespectively. Then

it is easily checked that ((Wp, Wq)) ⊂[t−2τ, t + 2τ], which implies that Wpand Wqare

relatively thin.

It is now a straightforward application of Prop. 2.31 to show that every Cartan R-manifold

is locally proper.

2.4 Flows and Dynamical Systems

We aim to apply the differentiable version of Palais’ theory of R-manifolds from the previous

section to analyze the global splitting properties of Lorentzian manifolds. Therefore, we are

particularly interested in R-actions on manifolds which are generated by a given vector field

on M. To connect vector fields to R-actions as introduced above, we define in the following

the notion of the flow of a vector field. The main reference for the treatment of flows is

[Con01, Ch. 4].

Definition 2.36. Aglobal flow on a manifold Mis a map

Φ: R×M→M,

such that for all t∈R, the stage Φt(·) := Φ(t, ·)is a diffeomorphism M→Mand

(i) Φ0= idM,

(ii) Φt+s= Φt◦Φsfor all t, s ∈R.

Definition 2.37. Let p∈Mand Φa global flow on M. The curve γp:R→Mdefined by

γp(t) := Φt(p),

such that γp(0) = pis called a flow line of Φthrough p.

Definition 2.38. Let Φbe a global flow on M. The vector field X:M→TM defined

pointwise by

Xp= ˙γp(0) ∈TpM

is called the generator of Φ. Conversely, any given vector field Y:M→TM is called

complete if it is the generator of some global flow on M.

It is a standard result (see [Con01]) that the vector field as in Def. 2.38 is indeed globally

well-defined and smooth, and that we have the following

Lemma 2.39. On a compact manifold M, every vector field X:M→TM is complete.

Theorem 2.40. Let Mbe a manifold of class Cr,r > 0. There is a one-to-one correspon-

dence between a global flow Φon a manifold Mand an R-action on M. The orbits of the

R-action correspond to the flow lines of Φ.

Proof. Let Φbe a global flow on M. Consider the map f:t7→ Φt(·). Then fis the

homomorphism R→Diffr(M)constituting the R-action, because due to the definition of

the global flow each f(t)=Φt(·)∈Diffr(M)and f(0) = idM, as well as f(t+s) = f(t)◦f(s)

for all s, t ∈R. Conversely, let there be an R-action on Mgenerated by a homomorphism

f:R→Diffr(M). Now define Φ: R×M→Mby Φt(p) := f(t)(p). By the definition of the

R-action, this map is smooth. And as fis a homomorphism, we also have Φ0= idMand

Φt+s= Φt◦Φsfor all t, s ∈R. Now due to

γp(t)=Φt(p) = t◦p,

for all (t, p)∈R×M, we have γp(t) = Rpfor all p∈M.

Remark 2.41. Hence, obviously, a global flow coincides with the global R-action of a gen-

eralized R-principal bundle or a Cartan R-manifold, respectively. This also leads to the

conclusion that the vector field generating the global flow is a fundamental vector field of the

R-action.

Hence, if we want to use the theory of R-manifolds to analyze splitting and slicing results

with respect to flows generated by a vector field, this vector field has to be complete, so

that we have a global flow. Subsequently, we will call a global flow that leads to a Cartan

R-action on the manifold MaCartan flow. Moreover, we state the following

Definition 2.42. Let Mbe a manifold and V:M→TM a globally defined, nowhere

vanishing and complete vector field. Then the ordered pair (M, V )is called a kinematical

manifold. Furthermore, if Vgenerates a Cartan flow we call (M, V )aCartan kinematical

manifold.

Obviously, a kinematical manifold is an R-manifold with the R-action generated by the

global flow of V. The natural connection of kinematical manifolds to Lorentzian metrics

arise from the following considerations. The preconditions for the existence of a globally

defined and nowhere vanishing vector field and for a Lorentzian metric on a manifold Mare

the same, namely, either Mbeing non-compact or having Euler characteristic zero (see the

following chapter). This is caused by the signature of a Lorentzian metric, that distinguishes

a special direction in the tangent bundle at all points in M, which will be called timelike

direction. A vector field, basically, provides the same distinction by a decomposition of the

tangent bundle.

The main reference for the treatment of dynamical systems is [NS60], but other references

will also be used and cited where applicable. What we will be concerned with here, is the

so called general theory of dynamical systems or the question of the topological structure

of a manifold on which a dynamical system is defined. This differs from more specific

investigations of dynamical systems defined by given differential equations. In these cases

one is particularly interested in stability properties of specific orbits or the whole system, or

in fixed points and bifurcations. These arise for example by closed integral curves or zeros

of a vector field, which defines a dynamical system on a manifold. This is not the theory

fitting the applications in this work, as we will be concerned with kinematical manifolds and,

subsequently, timelike vector fields on Lorentzian manifolds. Hence, the important part of

the dynamical systems theory for the following chapters is about unstable or parallelizable

dynamical systems (see [Mar69]).

We start out with a definition of dynamical systems adapted to our purposes.

Definition 2.43. Let Mbe a non-compact manifold. A dynamical system on Mis a global

flow according to Def. 2.36. We will denote this by the ordered pair (M, Φ).

Also for dynamical systems, we will use the terms flow line and orbit interchangeably.

Definition 2.44. Let γ:R→Mbe any orbit of a dynamical system (M, Φ). Then the

(possibly empty) sets

ω(γ) = \

t∈R

γ(R≥t)and α(γ) = \

t∈R

γ(R≤t)

are called ω-limit set and α-limit set of the orbit γ, respectively. We also set Ω(γ) = α(γ)∪

ω(γ)for all orbits γ. In the same way, the limit sets

ω(Φ) = [

ω(γ)and α(Φ) = [

α(γ),

as well as Ω(Φ) = ω(Φ) ∪α(Φ) are defined, where the union is understood over all orbits.

Obviously, Ω(Φ) is a closed set for any dynamical system (M, Φ), but further conclusions

cannot be drawn about Ω(Φ) in the generic case.

Remark 2.45. Note that the notion of a limit set already makes sense for individual curves

defined over the real numbers. We will use this in causality theory of Lorentzian manifolds

in the sections 3.4 and 4.2.

In the usual way, we can now define a periodic orbit of a dynamical system (M, Φ), as a flow

line of Φwhich is diffeomorphic to an embedding of S1into M. Similarly, we call a point

x0∈Mafixed point if γx0(t) = x0for all t∈R. This is the case if and only if the generating

vector field of the flow possesses a zero at x0. Obviously, periodic orbits and fixed points

are contained in Ω(Φ) and even in ω(Φ) and α(Φ).

It turns out that the emptiness of the limit sets of a dynamical system is the condition

corresponding precisely to a Cartan R-action in the differentiable version of Palais’ theory

developed in the previous section. Hence, we have the following

Proposition 2.46. The dynamical system (M, Φ) fulfills Ω(Φ) = ∅if and only if Mis a

Cartan R-manifold with R-action corresponding to the global flow Φ.

Proof. By Thm. 2.40 the global flow Φcorresponds to an R-action.

“⇒”: Ω(Φ) = ∅implies that ω(γ) = ∅and α(γ) = ∅for all orbits γof the dynamical system.

Particularly, this implies that every orbit is closed as a point set. This can be seen as follows.

Assume there is a point x∈Mand a flow line γxthrough x, which is not closed as a point

set. Thus, there is a sequence {tn}n∈N⊂Rwith |tn| → ∞ and γx(tn)converging to some

point y /∈γx. But, hence, y∈Ω(γx), which is a contradiction. Furthermore, for all x∈M

the map Φ(·, x): R→Mis a diffeomorphism of Ronto the orbit γxthrough x. So, it

remains to show that the quotient M/Ris a near-manifold. But this follows in exactly the

same way as in the proof of Lem. 2.30, as the orbits being closed implies that M/Ris a T1-

space, and the remaining conditions follow from investigating foliated charts.

“⇐”: If Mis a Cartan R-manifold we conclude by a similar argument as above that Ω(γ) = ∅

for every orbit γ. Assume x∈γfor any orbit γin an R-manifold or a dynamical system.

Then there is a sequence {tn}n∈N⊂Rsuch that γ(tn)→xas n→ ∞. Then either tn

admits a subsequence which converges to some τ∈R, in which case x∈γ, or a subsequence

obeying |tn|→∞as n→ ∞, in which case x∈Ω(γ). This shows that γ=γ∪Ω(γ), which

proves that Ω(Φ) is empty if all orbits are closed as a point set.

The following two notions are standard definitions in the general theory of dynamical sys-

tems.

Definition 2.47. Let (M, Φ) be a dynamical system. A point x0∈Mis called wandering if

there exists a neighborhood Uof x0and a number T∈R, such that Φ(t, U)∩U=∅for all

|t|> T, with Φ(t, U) = Sx∈Uγx(t). All other points are called non-wandering. Furthermore,

the dynamical system is said to have an improper saddle point (also called a saddle at infinity)

if there are two sequences {τn}n∈N,{tn}n∈N⊂Rwith 0< τn< tnfor all n∈Nand τn→ ∞

as n→ ∞, as well as two points x, y ∈Mand a sequence {xn}n∈N⊂Mwith xn→xand

Φ(tn, xn)→yas n→ ∞, but the sequence {Φ(τn, xn)}n∈Nhas no limit point.

The following lemma extends the correspondence between R-manifolds and dynamical sys-

tems even further.

Lemma 2.48. A point xin a dynamical system (M, Φ) is wandering if and only if it has a

thin neighborhood with respect to the R-action corresponding to the global flow Φ.

Proof. Just compare Def. 2.47 with Def. 2.24.

We will see below that the notion of the non-existence of an improper saddle point for a

dynamical system is a relatively strong condition. Although improper saddle points do not

really arrange properly according to the limit point scheme. See item (ii) in Lem. 2.50 below.

Following [Mar69] we give the subsequent definitions.

Definition 2.49. A dynamical system (M, Φ) is called

(i) unstable if no compact set K⊂Mcontains an entire half-orbit γx(R>0)or γx(R<0)

for some x∈M,

(ii) completely unstable if all points are wandering and

(iii) parallelizable if it is unstable and has no improper saddle point.

Item (i) in the following lemma is a standard result in the topological theory of dynamical

systems. We can prove it now by highlighting the equivalence of certain conditions to the

theory of R-actions. Item (ii) is a new implication that will be useful for the analysis

conducted in this work.

Lemma 2.50. For a dynamical system (M, Φ) we have the following implications:

(i) completely unstable =⇒Ω(Φ) = ∅=⇒unstable,

(ii) no improper saddle point =⇒Ω(Φ) = ∅or every half-orbit γx(R>0)(or γx(R<0)) with

ω(γx)6=∅(or α(γx)6=∅) for x∈Mis entirely contained in some compact set K⊂M.

Proof. (i): By Lem. 2.48 and Prop. 2.46 the first implication is equivalent to Lem. 2.30. For

the second implication assume that there is a point x∈Msuch that the half-orbit γx(R>0)

is entirely contained in a compact set K. Then for any sequence {tn}n∈N⊂Rwith tn>0

and tn→ ∞ as n→ ∞ the sequence γx(tn)⊂Khas a subsequence in K, which converges to

some y∈K. Hence, y∈ω(γx)and an equivalent reasoning applies if the half-orbit γx(R<0)

is contained in some compact set. Thus we have non-empty Ω(Φ) in these cases.

(ii): We will show the following: if Ω(Φ) is not empty and there is a half-orbit γx(R>0)

with ω(γx)6=∅, which is not entirely contained in a compact set K⊂M, then there is an

improper saddle point. Again, the case for the half-orbit γx(R<0)is completely analogous

and will be omitted. Let x∈Mbe a point such that ω(γx)6=∅. Then the condition

that γx(R>0)is not entirely contained in a compact set is characterized in the following

way. Take any compact exhaustion of Mbased at x, i.e., a family of non-empty, compact

sets {Ki}i∈N0⊂ P(M)(the power set of M), such that K0={x},x∈Kifor all i∈N,

Ki⊂˚

Ki+1 and Si∈NKi=M. Then there is a sequence {ti}i∈N⊂R, such that ti>0for

all i∈Nand γx(ti)∈Ki+1 \Ki. Hence, Φ(ti, x)has no limit point in Mand ti→ ∞ as

i→ ∞. But as ω(γx)6=∅, there is also a sequence {τi}i∈N⊂Rwith τi>0for all i∈Nand

τi→ ∞ for i→ ∞, such that γx(τi)→y∈ω(γx)as i→ ∞. Thus we have Φ(τi, x)→y

as i→ ∞. By switching to subsequences if necessary we can achieve that ti< τi, and by

regarding xas a constant sequence, we conclude that there is an improper saddle point.

Remark 2.51. All the implications in Lem. 2.50 do not generally hold in the opposite di-

rection (cf. [NS60]). Finding examples for dynamical systems which are unstable but not

completely unstable is particularly involved. In fact, it can be shown that in R2every un-

stable dynamical system is completely unstable (see chapter V. Thm. 10.02 in [NS60]). Ex-

amples can be constructed in R3and we refer to chapter V. Ex. 10.03 in [NS60] for such a

construction.

Moreover, we have the following characterization of parallelizability. This characterization

has already been shown in [Mar69], although therein, parallelizability was given as a dy-

namical system which is completely unstable and has no saddle at infinity. It was only in

[BF72] that the possibility of relaxing this assumption was shown. We give here a completely

different proof. To this end we first establish the following

Lemma 2.52. If a dynamical system (M, Φ) is unstable and has no improper saddle point

and if there are sequences {xn}n∈N⊂Mand {tn}n∈N⊂Rsuch that xn→xas n→ ∞ and

Φ(tn, xn)converges to some y∈Mas n→ ∞, then {tn}n∈Nis bounded.

Proof. See chapter V. Lem. 10.06 in [NS60].

Proposition 2.53. A dynamical system is parallelizable if and only if the R-action corre-

sponding to the global flow Φon Mis proper if and only if it corresponds to an R-principal

bundle.

Proof. By Lem. 2.50, parallelizability of a dynamical system (M, Φ) implies Ω(Φ) = ∅.

Hence, by Prop. 2.46 and Prop. 2.31 we conclude that Mis a generalized R-principal bundle.

So, it remains to show that no improper saddle point implies that M/Ris Hausdorff. We

prove this by showing that the non-Hausdorffness of the quotient space implies the existence

of an improper saddle point. Assume a, b ∈M/Rare two points which cannot be separated

by disjoint open sets. Let Uaand Ubbe two neighborhoods of aand brespectively, such

that W=Ua\{a}=Ub\{b}, and such that the generalized bundle (M, M/R, πM)allows

for a local trivialization over Uaand Ubrespectively. Let {ξn}n∈N⊂Wbe a sequence that

converges to the double point a, b in M/R. Then assume two local sections σa:Ua→M

and σb:Ub→Min the local trivializations over Uaand Ubrespectively. We denote by

xn=σa(ξn)and yn=σb(ξn)two sequences in Mand by p=σa(a),q=σb(b)two points in

the fibers over aand b, which are lifted by the local sections. Then obviously xn→pand

yn→qas n→ ∞. Furthermore, as πM(xn) = πM(yn) = ξn, we conclude that the elements

xnand ynof the respective sequences lie in the same fiber over ξn. So for all n∈N, there

is a tn∈Rsuch that Φ(tn, xn) = yn, and we can assume tn>0without loss of generality.

Moreover, we must have that tn→ ∞ as n→ ∞. To see this we suppose that {tn}n∈N, or a

subsequence of {tn}n∈N, had a limit point t0∈R. But then we would have Φ(t0, p) = qand

hence qwould be an element of the fiber over pand M/Rwas Hausdorff. Now the existence

of an improper saddle point follows from Lem. 2.52.

Prop. 2.53 will be the foundation for certain diffeomorphic splitting results of Lorentzian

manifolds, which will be derived in the subsequent chapters. At last, we will need the

concept of invariant and minimal sets for dynamical systems, given in the following

Definition 2.54. A subset A⊂Min a dynamical system (M, Φ) is called invariant if

Φ(t, x)∈Afor all x∈Rand all x∈A. Furthermore, Ais called minimal if it is a

non-empty, closed and invariant set and it does not contain an invariant proper subset.

Remark 2.55. Obviously, every union of orbits of a dynamical system is invariant and

every individual orbit is minimal, if it is closed as a point set. Furthermore, the limit sets

ω(Φ) and α(Φ) are closed, invariant sets for any dynamical system Φand they each contain

at least one minimal set (these are standard results, cf. [NS60, Ch. 5]).

Chapter 3

LORENTZIAN MANIFOLDS AND SPACETIMES

In this chapter, we will establish some basic definitions and properties about the main geo-

metric objects of this work, and we will also prove some propositions about them. Standard

references that will be used throughout this chapter are [BEE96], [O’N83] and [HE73]. We

define Lorentzian manifolds, spacetimes and Lorentzian manifolds with a fixed complete

timelike vector field, i.e., a global timelike flow. We will also define and establish some

specific terminology that only occurs in Lorentzian geometry. This will provide the basis

for the subsequent analysis of these geometric objects, and particular cases of them, in the

following chapters. The sections 3.1,3.2 and 3.3 will then be concerned with geometric

quantities constructed from the covariant derivative of the timelike vector field, the so-called

kinematical quantities, and their relation to the geometry, topology, and partly also to the

curvature of the Lorentzian manifold. In section 3.4 we establish some basic notions about

Lorentzian causality theory, which will be needed in the following chapters.

Based on the definition of semi-Riemannian manifolds in chapter 2, we give now the following

more detailed definition of a Lorentzian manifold, which will be used in the subsequent

analysis.

Theorem & Definition 3.1. Let Mn+1 be a manifold with n≥1, either non-compact or

with Euler characteristic zero. Then Madmits a non-degenerate metric g:M→Σ2Mof

signature n−1, i.e., a Lorentzian metric. An ordered pair (M, g), consisting of a manifold

Mand a Lorentzian metric g, is called a Lorentzian manifold.

Proof. See e.g. [MS08, Thm. 2.4].

We use standard causal classification of tangent vectors by the unique bilinear form on each

tangent space induced by the metric g. For all p∈M, the Lorentzian metric gpconstrained

to TpMis a non-degenerate scalar product gp:TpM⊗TpM→Ron TpM, such that TpM

admits a pseudo-orthonormal basis, cf. section 2.1.

Definition 3.2. For all p∈M, a tangent vector v∈TpMis classified as:

(i) timelike if gp(v, v)<0,

(ii) lightlike if gp(v, v) = 0 and v6= 0,

(iii) causal if vis timelike or lightlike, or

(iv) spacelike if gp(v, v)>0or v= 0.

The same classification holds for vector fields and curves. A C1-curve (resp. a vector field)

on Mis called {timelike, lightlike, causal, spacelike}if all its tangent vectors (resp. values)

are {timelike, lightlike, causal, spacelike}.

At first sight it seems as if the causal classification is constrained to differentiable curves.

But in the subsequent analysis, we also need the notion of a continuous causal curve. Such

a notion will be given if the Lorentzian manifold is a spacetime (see Def. 3.7 and 3.8 below).

Furthermore, the causal classification of vectors yields a causal classification of hypersurfaces

(i.e., submanifolds of dimension nin a spacetime (Mn+1, g)).

Definition 3.3. Let Sn⊂Mn+1 be a hypersurface in the spacetime (M, g). If there is a

uniquely defined vector field ν:S→NS obeying |g(ν, ν)|= 1 (where we denote the normal

bundle of Sby NS), then Sis called a non-degenerate hypersurface (i.e., the metric induced

on Sby gis nowhere degenerate) and Sis spacelike if g(ν, ν) = −1and timelike if g(ν, ν) = 1

everywhere on S.

Remark 3.4. Let (M, g)be a Lorentzian manifold. A pointwise conformal transformation

of (M, g)is a change of the Lorentzian metric g7→ g0by a function φ:M→Rsuch that

g0=eφg. A conformal transformation of (M, g)is a map ϕ:M→M, such that there

is a pointwise conformal transformation g0=eφgobeying ϕ∗g=g0. Obviously any vector

v∈TpMthat is {timelike, lightlike, causal, spacelike}with respect to a Lorentzian metric

gon M, is also {timelike, lightlike, causal, spacelike}with respect to any pointwise confor-

mally transformed metric g0. Thus the causal classification of tangent vectors is conformally

invariant.

We name the collection of causal or lightlike tangent vectors at a point in the following way.

The subset

Lp={v∈TpM|vlightlike} ⊂ TpM

is called light cone at p, the subset

Cp={v∈TpM|vcausal} ⊂ TpM

is called causal cone and the subset

Tp={v∈TpM|vtimelike}=Cp\Lp⊂TpM

is called time cone at p.

Lemma 3.5. For all p∈M,Tphas two connected components.

Proof. Let Bp={e0, . . . , en}be any pseudo-orthonormal basis of TpMwith e0∈Tp. Then

any vector v∈TpMcan be expanded with respect to Bpby v=vaea=v0e0+Piviei. If

v∈Tp, we have gp(v, v) = −(v0)2+Pi(vi)2<0and necessarily |v0|>0. Thus, all vectors

v∈Tpnaturally come in two classes with respect to e0∈Bp, namely T↑

e0={v∈Tp|v0>0}

and T↓

e0={v∈Tp|v0<0}with T↑

e0∩T↓

e0=∅. Thus, it remains to show that T↑

e0and

T↓

e0are open sets. Consider v, w ∈T↑

e0with (v0)2>Pi(vi)2,(w0)2>Pi(wi)2and v0>0,

w0>0. Hence, we have by the Cauchy-Schwarz inequality

gp(v, w)≤ −v0w0+|X

viwi|≤−v0w0+sX

(vi)2sX

(wi)2<−v0w0+|v0w0|= 0.

Thus, particularly, gp(v+w, v +w) = gp(v, v)+2gp(v, w) + gp(w, w)<0for all v, w ∈T↑

e0,

which makes T↑

e0an open set. An analogue reasoning holds for T↓

e0. As obviously e0∈T↑

e0,

we can choose any other basis ˜

Bp={˜e0,...,˜en}as long as ˜e0∈T↑

e0, which makes the notion

of the two connected components independent of the basis.

By extension, and because the vector v= 0 in TpMis spacelike, then also each causal cone

and each light cone has two connected components. The concept of future and past is the

unique assignment of one of the connected components of the time cone in TpMto each point

pin a Lorentzian manifold. Such an assignment of past and future to each point p∈Mis

called time orientation on (M, g)and following [Car71] it is a map τ(p): Rn+1 →TpM, for

all p∈M, which maps the Euclidean solid half cone

x0>

X

1≤i≤n

(xi)2

,

with x0, . . . , xnCartesian coordinates in Rn+1, into TpMin a non-degenerate, homogeneous

and linear way. Moreover, it is required to change continuously with p. The image τ(p)will

be called future cone at p∈M. A Lorentzian manifold which admits a time orientation is

called time-oriented.

Remark 3.6. Time orientability—its dual, space orientability—as well as orientability are

logically independent concepts for a Lorentzian manifold (M, g). Generally, these can be

assessed in terms of Stiefel–Whitney classes associated to a decomposition TM =ξ⊕νof

the tangent bundle (cf. [Hus94, Ch. 17]). This is understood as the direct g-orthogonal sum of

a maximal timelike subbundle ξand a maximal spacelike subbundle ν. Such a decomposition

exists for every Lorentzian manifold due to the vanishing Euler characteristic. Now, one

can see that (M, g)is time-orientable if ξis orientable, i.e., if the first Stiefel–Whitney

class of ξvanishes, it is space-orientable if νis orientable, i.e, the first Stiefel–Whitney

class of νvanishes and it is orientable if TM is orientable, i.e., the first Stiefel–Whitney

class of TM vanishes. Furthermore, one can conclude from this that, firstly, any Lorentzian

manifold admits a time-orientable double cover (see [Pen72]) and, secondly, if any two kinds

of orientability hold, the third one is implied. In fact, this is a simple corollary of the product

property of Stiefel–Whitney classes: w1(TM) = w1(ξ) + w1(ν).

Definition 3.7. Aspacetime is a Lorentzian manifold together with a fixed time-orientation

τon M.

The points p∈Mare also called events if (M, g)is a spacetime. In a spacetime, the notions

of future and past directedness for causal curves and vector fields can now be defined. We

say that a causal vector v∈TpMat some event pin a spacetime (M, g)is future-directed

(resp. past-directed) if it is an element of the future (resp. past) component of the causal cone

Cp, distinguished by the fixed time-orientation of (M, g). This definition naturally carries

over to causal vector fields and to velocities of causal curves, i.e., we can also distinguish

future- and past-directed causal curves.

Now we are in a position to extend the notion of causal curves to the continuous case. Only

for causal curves, this can be conducted in a unique and sensible way. For example, there

is no unique notion of continuous timelike curves (see [MS08, Rem. 3.17]). To this end we

need the notion of a causally convex set. An open set U⊂M, with (M, g)a spacetime, is

called causally convex if every future-directed causal curve with end-points in Uis entirely

contained in U. It is a standard result that every event in a spacetime has a causally convex

neighborhood. We can now use [MS08, Prop. 3.16] as a definition for continuous causal

curves.

Definition 3.8. Let (M, g)be a spacetime and I⊂Ra (bounded) interval. A continuous

curve γ:I→Mis future-directed causal if for each causally convex set U, given t1, t2∈I,

t1< t2with γ([t1, t2]) ⊂U, there is a differentiable, future-directed causal curve connecting

γ(t1)∈Uto γ(t2)∈U.

It is also possible to assess continuous causal curves from the perspective of Sobolev spaces

(cf. [MS08, Rem. 3.18] and [CS08]).

Definition 3.9. A globally defined timelike vector field V:M→TM on a Lorentzian

manifold (M, g)is called reference frame if it has unit length, i.e., g(V, V ) = −1.

Obviously, a time-orientation can be given by assigning a timelike vector Vpto every point

in the Lorentzian manifold, which varies smoothly from point to point. The future cone

is then given by choosing the cone of which Vpis an element. Moreover, we then have a

globally defined timelike vector field, which can be chosen to be a global reference frame.

But we emphasize the viewpoint of assigning a unique future cone to all points as opposed

to distinguishing a particular timelike vector at each point, because in a spacetime the

important feature is that such a distinguished direction is given but a timelike vector field

is not fixed. Any timelike unit vector field chosen such that all of its vectors are elements

of the future cone can be used to serve as a global reference frame. We emphasize that a

fixed global timelike vector field is an additional structure on the Lorentzian manifold, and

in this work, we will mostly be concerned with Lorentzian manifolds which carry this kind

of additional structure. This gives rise to the following definition.

Definition 3.10. Akinematical Lorentzian manifold (M, g, V )is a Lorentzian manifold

(M, g)together with a globally defined timelike vector field V:M→TM, which is complete.

Obviously, following definition 2.42, a kinematical Lorentzian manifold possesses both the

features of a kinematical manifold (M, V )and of a (time-orientable) Lorentzian manifold

(M, g). Moreover, the Lorentzian and kinematical structures are required to be compatible

in the sense that the complete vector field Vhas to be timelike everywhere.

Proposition 3.11. Every kinematical Lorentzian manifold is a spacetime.

Proof. As Vis timelike for all p∈M, the vector Vp∈TpMis an element of a specific

connected component of Tpat p, say Vp∈TV

p. Then set

T↑

p=τ(p) := TV

for all p∈M, which gives the time-orientation. The future cone changes continuously with

pas Vis smooth.

Proposition 3.12. Every spacetime can be made a kinematical Lorentzian manifold by an

appropriate choice of a vector field V:M→TM.

Proof. Let (M, g)be a spacetime. Choose any globally defined timelike vector field ˜

V:M→

TM, which exists due to the time orientation τon M. If ˜

Vis not complete, choose any

auxiliary complete Riemannian metric gRon M, which always exists (see, e.g., [NO61]).

Then set

V:= ˜

|gR(˜

V , ˜

V)|1

which is a complete timelike vector field on M.

Remark 3.13. Moreover, Prop. 3.12 shows that the completeness of the vector field V:M→

TM is independent of the Lorentzian metric gon M, which makes (M, g)a spacetime.

Example 3.14. Consider the two-dimensional Minkowski space with coordinates (t, x)∈R2

and the canonical flat metric η=−dt2+ dx2, which we denote as a Lorentzian manifold

(R2, η), and which is a spacetime with the usual time cone structure inherited from Minkowski

space. Now remove the origin of R2leading to ˚

R2:= R2\ {(0,0)}, and (˚

R2, η)is still

a spacetime. By setting V1=∂t, we get a spacetime (˚

R2, η, V1)with a fixed incomplete

reference frame V1. Now

g=dt2+ dx2

t2+x2

is a complete Riemannian metric on ˚

R2. Then we set

V2=V1

kV1kg= (t2+x2)1

2V1,

such that (˚

R2, η, V2)is obviously a kinematical Lorentzian manifold with complete timelike

vector field V2, but η(V2, V2)6=−1.

Definition 3.15. Akinematical spacetime (M, g, V )is a kinematical Lorentzian manifold,

with the globally defined and complete timelike vector field V:M→TM having unit length

g(V, V ) = −1, i.e., being a reference frame.

As Example 3.14 shows, not all spacetimes can be made into kinematical spacetimes, but

it is a proper additional requirement to the metric gand the vector field Von M. In

some sense the metric and the vector field have to be adjusted to the topological setting

of the underlying manifold. If we allow for conformal changes of the Lorentzian metric the

kinematical spacetime condition can be achieved.

Proposition 3.16. For all kinematical Lorentzian manifolds (M, g, V ), there is a conformal

factor eφ:M→R>0leading to a pointwise conformal Lorentzian metric g∗=eφg, such that

(M, g∗, V )is a kinematical spacetime.

Proof. We have g(V, V )<0and

−1 = g∗(V, V ) = eφg(V, V ),

such that the choice

φ=−ln(−g(V, V ))

gives the desired result.

Combining this result with Prop. 3.12 we can infer that any spacetime (M, g)can be made

into a kinematical spacetime by an appropriate choice of a complete timelike vector field

and a conformal transformation.

Example 3.17. Recall ˚

R2and V2from Example 3.14. If we set

g:= η

t2+x2,

then (˚

R2, g, V2)is a kinematical spacetime. Furthermore, if we consider the global flow of V2

in (˚

R2, g, V2)as an R-action on ˚

R2, this is an example of a Cartan R-manifold which is not

proper. This can be seen as follows. The set of flow lines of V2is given by the set of lines

and half-lines

R2/R={{(t, x0)|t∈R}|x0∈R\{0}}∪{(t+,0) |t+>0}∪{(t−,0) |t−<0}.

We furnish this set with the quotient topology. Let Obe open in ˚

R2/Rif π−1(O)is open

in ˚

R2. Thus any open set in ˚

R2/Rwhich contains (t+,0) also contains elements from a

neighborhood of (t−,0) and vice versa. Hence, (t+,0) and (t−,0) cannot be separated by

disjoint open neighborhoods and ˚

R2/Ris not Hausdorff. In fact, ˚

R2/Ris homeomorphic to

the line with two origins (see, e.g., [SS78]).

Proposition 3.18. Every time-oriented and compact Lorentzian manifold is a kinematical

Lorentzian manifold.

Proof. Choose a time orientation by a timelike vector field, then the assertion is a direct

consequence of Lem. 2.39.

Particularly, in chapter 4, we will assume a fixed complete timelike vector field, and will

allow the Lorentzian metric to vary, in order to ensure the vector field to have unit norm, if

necessary. For the topological and causal properties investigated in that chapter, considering

the conformal class of metrics is sufficient.

3.1 Kinematical Quantities

In this section, we develop the theory and basic equations for the invariant parts of the

covariant derivative of a reference frame in a kinematical spacetime. Given any kine-

matical spacetime (M, g, V ), it is naturally equipped with a Lorentzian metric gand its

Levi-Civita connection ∇, as well as the reference frame Vand its metric dual one-form

u=g(V, ·). Thus, in a next step, the quantities we would like to examine are the tensor

fields ∇u=g(∇·V, ·)∈Γ(T2M)and ∇V(∇u)∈Γ(T2M). By taking into account the de-

composition TpM≃R⊕V⊥

pof each tangent space with respect to the reference frame V,

we can extract the invariant parts of ∇u—called kinematical quantities here—and ∇V(∇u)

with respect to that decomposition. Note that the kinematical quantities are particularly

famous in the physically oriented literature on general relativity and are often referred to

as kinematical invariants, as the decomposition in the tangent space at each event p∈M

yields the irreducible parts of the matrix ((∇u)p)ij, i.e., the invariant parts in terms of linear

algebra (cf., e.g., [HE73, Sec. 4.1] or [Ehl93]). For ∇V(∇u)the decomposition leads to a

set of geometric constraint equations for the kinematical quantities known as Raychaudhuri

equations. These can be regarded as evolution equations for the kinematical quantities along

the flow lines of V, and they are fulfilled because of the identities of the curvature tensor on

the manifold.

Remark 3.19. In this section, as well as in the sections 3.2 and 3.3, we state all results

for kinematical spacetimes (M, g, V )to achieve consistency with the larger part of this work.

But as all results in these sections are local by nature, they do also hold for spacetimes (M, g)

together with a reference frame V, which is not necessarily complete. This will be needed at

some points in section 5.3 and chapter 6.

In the following, we will assume (Mn+1, g, V )to be a kinematical spacetime and we will use

a pseudo-orthonormal frame {E0, E1, . . . , En}with E0=V. Summation over vector fields

Eiin the frame will run from 0to nif not indicated otherwise.

The timelike vector field Vinduces an integrable distribution in the tangent bundle of

codimension n, the integral manifolds of which are the flow lines of Vaccording to section

2.3. Thus, each tangent space allows for a decomposition

TpM= span{Vp}⊕V⊥

p≃:VpM⊕HpM.

Hence, Vinduces a horizontal distribution HM ⊂TM and a vertical distribution V M ⊂

TM. By Prop. 2.10, the projection

h=g+u⊗u

is a Riemannian metric on HM. The maps V:TM →V M and H:TM →HM given by

V(X) = −u(X)V

and

H(X) = X+u(X)V,

for all X∈Γ(TM)will be called projection operators. We will use the same notation Hand

Vfor the projection of differential forms and tensors. Hence, we have H(g) = hby setting

H(g)(X, Y ) := g(H(X),H(Y)) = g(X+u(X)V, Y +u(Y)V) =

=g(X, Y ) + u(X)g(V, Y ) + u(Y)g(V, X)−u(X)u(Y) = g(X, Y ) + u(X)u(Y) = h(X, Y ),

for all X, Y :M→T M. This carries over to all tensors and differential forms, for example

for any one-form w∈Γ(T∗M), the one-form H(w)∈Γ(H∗M)is given by

H(w)(X) = w(H(X)),

for all vector fields Xon M. Now we are ready for the following

Definition 3.20. Let (Mn+1, g, V )be a kinematical spacetime. The expansion Θis the

function M→Rgiven by

Θ := div(V) = Tr(∇u) = X

g(∇EiV, Ei).

The acceleration ˙uis the one-form M→T∗Mgiven by

˙u:= g(∇VV, ·).

The shear σis the symmetric trace-free tensor field M→Σ2M

σ:= sym(∇u) + u∨˙u−Θ

nh.

The rotation ωis the two-form field M→Λ2Mgiven by

ω:= du+u∧˙u.

Proposition 3.21. For the kinematical quantities of a kinematical spacetime as in Def. 3.20,

the following holds:

(i) σ(V, ·) = 0,ω(V, ·) = 0 and ˙u(V) = 0, hence, σ∈Γ(H∗M∨H∗M),ω∈Γ(H∗M∧

H∗M)and ˙u∈Γ(H∗M),

(ii) £Vg= 2 sym(∇u)=2σ−2u∨˙u+2

nΘh, and

(iii) ∇u=Θ

nh+σ+ω−u⊗˙u.

Proof. We use g(V, V ) = −1and the metricity of the Levi-Civita connection ∇.

(i): From the definition of the shear, rotation and acceleration we get:

2 ˙u(V)=2g(∇VV, V ) = ∇V(g(V, V )) = 0,

2σ(V, ·) = ∇Vu+ (∇·u)(V) + u(V) ˙u+ ˙u(V)u= ˙u−∇·(u(V)) −˙u= 0

and

2ω(V, ·) = Vcdu+u(V) ˙u−˙u(V)u= ˙u+∇·(u(V)) −˙u= 0.

(ii): From the properties of the Lie derivative we get

(£Vg)(X, Y ) = g(∇XV, Y ) + g(∇YV, X),

for any X, Y :M→T M, so that £Vgis indeed a symmetric tensor field M→Σ2Mequal

to 2 sym(g(∇.V, ·)) = 2 sym(∇u), and the assertion follows from the definition of the shear.

(iii): From the definition of the kinematical quantities we get

nh+σ+ω−u⊗˙u= sym(∇u) + u∨˙u+ du+u∧˙u−u⊗˙u= sym(∇u)+du=∇u.

The kinematical quantities can be regarded as unique tensorial objects that can be con-

structed from a given Levi-Civita connection ∇and a vector field V. There are certain

specific functions which can be formed from the Lorentzian metric gand the kinematical

quantities in a kinematical spacetime.

Definition 3.22. Let (M, g, V )be a kinematical spacetime and Θ,˙u,σ,ωthe kinemati-

cal quantities as above. We call the following set of functions M→Rscalar kinematical

quantities.

(i) The expansion Θitself,

(ii) the acceleration scalar |˙u|2:= g(∇VV, ∇VV) = ˙u(∇VV),

(iii) the rotation scalar |ω|2:= Pi,j ω(Ei, Ej)ω(Ei, Ej),

(iv) the shear scalar |σ|2:= Pi,j σ(Ei, Ej)σ(Ei, Ej).

It is also useful to know how the kinematical quantities change under a pointwise conformal

transformation ˜g=e2φgof the metric gof a kinematical spacetime (M, g, V ). As we would

like to still have a reference frame after the conformal transformation, Vhas also to be

changed to ˜

V=e−φV. Although, the resulting spacetime is not necessarily a kinematical

one any more as ˜

Vmay be incomplete. Hence, in [Pla12] it was shown that the following

holds.

Proposition 3.23. Let (Mn+1, g, V )be a kinematical spacetime and φ:M→Rinduces a

conformal transformation ˜g= e2φg, as well as ˜

V= e−φV. Then the kinematical quantities

Θ,˜ω,˜σand ˜

˙uof (Mn+1,˜g, ˜

V)read as follows:

eφ˜

Θ = Θ + ndφ(V),

˙u= ˙u+H(dφ),

e−φ˜ω=ω,

e−φ˜σ=σ.

Proof. See [Pla12, Prop. 3.3.1].

3.2 Classification of Spacetimes by Kinematical Quantities

In this section and in the remainder of this work we will make use of the following symmetry

properties of spacetimes.

Definition 3.24. A spacetime (M, g)admitting a timelike conformal vector field K:M→

TM, i.e., there is a function ϕ:M→R, such that £Kg=ϕ2g, is called conformally

stationary. The timelike vector field Kis called homothetic if ϕis a constant. If (M, g)

admits a timelike Killing vector field K:M→TM, i.e., £Kg= 0, it is called stationary, and

it is called static if additionally the reference frame Vparallel to Khas vanishing rotation.

There is a standard result that links symmetry properties of a kinematical spacetime (M, g, V )

to the kinematical properties of the timelike unit vector field V:M→TM. These have

firstly been proven in articles by J. Ehlers, P. Geren and R. Sachs [EGS68], as well as by

D.R. Oliver and W.R. Davis [OD77] and have been widely examined in [HP88], [Per90],

[DS99] and [DPS08].

Theorem 3.25. Let (Mn+1, g, V )be a kinematical spacetime. There is a conformal vector

field ξparallel to V, i.e., there are functions f:M→R,ϕ:M→R, such that ξ=efVand

£ξg=ϕ2gif and only if σ= 0 and ˙u−Θ

nuis exact, such that ˙u−Θ

nu= df. Furthermore,

ξis homothetic if and only if additionally there is a constant c∈Rsuch that Θ = c2e−f,

it is Killing if and only if additionally Θ = 0 and it is geodesically Killing if and only if

additionally Θ = 0 and f= 0.

Proof. The first two assertions have been widely examined in the references above. So we

only sketch the proofs here for the sake of completeness. Examining

(£ξg)(V, V )=(£efVg)(V, V ) = −ϕ2,

yields after a short computation

ϕ2= 2df(V)ef.

If we combine this with Tr(£ξg) = Tr(ϕ2g), we arrive at

ϕ2=2

nefΘ.(∗)

Using this to manipulate the equation

£ξg=£efVg=ef£Vg+ 2efdf∨u=ϕ2g

and using item (ii) from Prop. 3.21 one gets after some tedious algebra the desired conditions

σ= 0 and ˙u−Θ

nu= df.

On the other hand, starting from these conditions and setting ϕ2:= 2

nefΘand ξ:= efV

one can calculate £ξg=ϕ2g, again from item (ii) in Prop. 3.21.

The vector field ξis homothetic if ϕ2is constant. For Θ = c2e−fwe obviously get from (∗)

that ϕ2=2c2

nand if ϕ2=ϕ2

0=const, we arrive at Θ = n

2ϕ2

0e−f.

From (∗) it is clear that ϕ2= 0 if and only if Θ=0, so ξis Killing only in this case.

For f= 0, we have obviously ξ=V, and hence g(ξ, ξ) = −1. Take an arbitrary vector field

X:M→TM such that g(ξ, X)=0everywhere. Then we get

0 = ∇ξ(g(ξ, X)) = g(∇ξξ, X) + g(ξ, ∇ξX) = g(∇ξξ, X) + g(£ξX+∇Xξ, ξ) =

=g(∇ξξ, X) + £ξ(g(ξ, X)) + 1

2∇X(g(ξ, ξ)) = g(∇ξξ, X).

As this holds for any orthogonal vector field X, we conclude that ∇ξξ= 0 and therefore

ξ=Vis geodesic.

Remark 3.26. Obviously, the theorem above yields that if the reference frame Vin a kine-

matical spacetime (M, g, V )is itself Killing, then it is geodesically Killing.

The following classification result of kinematical spacetimes is a simple consequence of

Thm. 3.25 above.

Corollary 3.27. Let (Mn+1, g, V )be a kinematical spacetime. The spacetime (M, g)is

conformally stationary if σ= 0 and ˙u−Θ

nuis exact, it is stationary if σ= 0 and ˙uis exact

and it is static if σ= 0,ω= 0 and ˙uis exact.

Subsequently, the kinematical spacetimes with vanishing shear σ= 0 will be of special

interest, because—as we will investigate in chapter 6—they possess the structure of conformal

pseudo-Riemannian submersions in many particular situations. The following definitions are

standard for those spacetimes (cf. [GRK96]).

Definition 3.28. A kinematical spacetime (M, g, V )is called spatially conformally station-

ary if there is a vector field ξ:M→TM parallel to V, a function f:M→Rsuch that

ξ=efVand a function ϕ:M→R, such that £ξh=ϕh holds. It is called spatially homoth-

etic if additionally H(dϕ)=0and spatially stationary if additionally ϕ= 0. The vector field

ξis then called spatially conformal,spatially homothetic or spatially Killing, respectively.

The reference frame Vis also called rigid if it is parallel to a spatial Killing field.

Remark 3.29. Certainly, one could also define a spacetime (M, g)to be spatially confor-

mally stationary, spatially homothetic or spatially stationary if there exists a timelike vector

field ξ:M→TM, such that for the tensor field η=g+g(ξ, ·)⊗g(ξ, ·),£ξη=ϕη holds with

some function ϕ:M→R(with H(dϕ) = 0 in the spatially homothetic case and ϕ= 0 in

the spatially stationary case). But we will use these notions only for kinematical spacetimes

or for spacetimes with a fixed reference frame in this work.

Lemma 3.30. A kinematical spacetime (Mn+1, g, V )is

(i) spatially conformally stationary if and only if σ= 0,

(ii) spatially homothetic if and only if σ= 0 and H(d(Θef)) = 0,

(iii) spatially stationary if and only if σ= 0 and Θ = 0.

Proof. Using the identities £Vu= ˙u,£ψV u=ψ£Vu−dψand £ψV g=ψ£Vg+ 2dψ∨u

for any function ψ:M→R, as well as (ii) from Prop. 3.21, we compute

£ξh=£ξ(g+u⊗u) = £efVg+ 2(£efVu)∨u=

=ef[£Vg+ 2df∨u+ 2( ˙u−df)∨u] = ef(2σ+2

nΘh).

Hence, there is a function ϕ:M→Rsuch that £ξh=ϕh holds if and only if σ= 0, which

implies item (i). As ϕ=2

nefΘin this case, items (ii) and (iii) follow.

Moreover, we have the following

Lemma 3.31. A kinematical spacetime (Mn+1, g, V )is spatially conformally stationary if

and only if the reference frame Vis spatially conformal with £Vh=2Θ

nhand it is spatially

stationary if and only if Vis spatially Killing. Furthermore, in the case of vanishing shear,

Vis spatially homothetic if and only if H(dΘ) = 0.

Proof. Examining the computations in Lem. 3.30 yields

£Vh= 2σ+2Θ

and the results follow.

3.3 Raychaudhuri Equations

Regarding the kinematical quantities in section 3.1, one can now ask the question how

they evolve along the flow lines of the vector field Vin a kinematical spacetime (M, g, V ).

This leads to the investigation of the quantity ∇V(∇u). As this expression involves second

derivatives of the metric, we expect curvature terms to become involved. Originally, the

evolution of the expansion Θwas investigated first in [Ray55] by A.K. Raychaudhuri himself

and it was the evolution equation of the expansion that was named after him in the first

place. The evolution equations for the remaining kinematical quantities are, subsequently,

summarized under the term Raychaudhuri equations as well (cf. [KS07]). Geometrically,

all Raychaudhuri equations are in fact curvature identities, namely some components of

the curvature tensor ordered conveniently. In this section, we will derive all Raychaudhuri

equations in their full generality.

Let (Mn+1, g, V )be a kinematical spacetime of class Cr,r≥3and {E0, . . . , En}a pseudo-

orthonormal frame for g, where we sum over i, j, . . . = 0, . . . , n and set E0=Vas usual.

We denote by

R: Γ(TM)×Γ(TM)×Γ(T M)→Γ(TM)

R(X, Y )Z=∇X∇YZ−∇Y∇XZ−∇[X,Y ]Z

the curvature tensor associated to the Lorentzian metric g, for all vector fields X, Y, Z ∈

Γ(TM), which satisfies the usual symmetries.

We denote the Ricci tensor associated to Rby Ric: Γ(TM)×Γ(TM)→Cr(M),Ric ∈

Γ(Σ2M), which is defined by

Ric(X, Y ) = Tr(R(·, X)Y) = X

g(R(Ei, X)Y, Ei),

for all X, Y ∈Γ(T M).

Now, the Raychaudhuri equations follow from the investigation of the expression

X, Y 7→ g(R(X, V )V, Y ),

for all X, Y ∈Γ(T M)and for the reference frame V. Using

∇V∇XV=∇∇VXV+ (∇V(∇V))(X)

and (∇˙u)(X, Y ) = g(∇X∇VV, Y ), as well as the usual R(X, V, V, Y ) := g(R(X, V )V, Y )

yields

(∇V(∇u))(X, Y )=(∇˙u)(X, Y )−(∇u)(∇XV, Y )−R(X, V, V, Y ),(∗)

for all X, Y :M→T M. And we will uncover the evolution equations of Θ,σ,ωand ˙u,

involving their time derivatives ˙

Θ = ∇VΘ = dΘ(V),˙σ=∇Vσ,˙ω=∇Vωand ¨u=∇V˙u=

g(∇V∇VV, ·), by examining the invariant components of equation (∗) with respect to V,H

and the trace, as well as its symmetric and antisymmetric parts.

The term (∇u)(∇XV, Y )in (∗) is of specific relevance for the pending derivation of the

Raychaudhuri identities. Therefore, we first prove the following

Lemma 3.32. The tensor (∇u)(∇·V, ·)∈Γ(T2M)does only depend on the kinematical

quantities and the one-form u. Particularly, we have

(∇u)(∇XV, Y ) = Θ2

n2h(X, Y ) + 2Θ

n(σ(X, Y ) + ω(X, Y )) + X

σ(Ei, X)σ(Ei, Y )+

−X

ω(Ei, X)ω(Ei, Y ) + X

σ(Ei, X)ω(Ei, Y )−X

ω(Ei, X)σ(Ei, Y )−

−Θ

nu(X) ˙u(Y)−u(X)σ(∇VV, Y )−u(X)ω(∇VV, Y ),

for all X, Y :M→TM.

Proof. We use ∇u=Θ

nh+σ+ω−u⊗˙ufrom Prop. 3.21, as well as

(∇u)(∇XV, Y ) = X

(∇Xu)(Ei)(∇Eiu)(Y),

for any vector fields X, Y :M→TM and for a g-orthonormal frame {E0, . . . , En}with

E0=V, i.e., we sum over i= 0, . . . , n. This yields

(∇u)(∇XV, Y ) = X

iΘ

nh(Ei, X) + σ(Ei, X) + ω(X, Ei)−u(X) ˙u(Ei)×

Θ

nh(Ei, Y ) + σ(Ei, Y ) + ω(Ei, Y )−u(Ei) ˙u(Y)=

=Θ2

n2h(X, Y ) + 2Θ

n(σ(X, Y ) + ω(X, Y )) + X

σ(Ei, X)σ(Ei, Y ) + X

ω(X, Ei)ω(Ei, Y )+

σ(Ei, X)ω(Ei, Y )−X

ω(Ei, X)σ(Ei, Y )−Θ

nu(X) ˙u(Y)−

−u(X) ˙u(Ei)(σ(Ei, Y ) + ω(Ei, Y ))

by using Prop. 3.21, and the result follows.

Now, we are ready to derive the Raychaudhuri equations. To this end, we introduce the

shorthand notation ˙

V=∇VV. Furthermore, we will denote by Hsym(∇˙u)the symmetric

and by ^

Hsym(∇˙u)the symmetric trace-free part of H(∇˙u). Moreover, we define ˜

∨to be the

trace-free symmetrized tensor product, given for horizontal one-forms u, w ∈Γ(H∗M)by

u˜

∨w=u∨w−Pn

i=1 u(Ei)w(Ei)

nh,

where we use a pseudo-orthonormal frame {V, E1, . . . , En}, such that {E1, . . . , En}yields a

basis of HpMfor all p∈M.

Proposition 3.33. Let (Mn+1, g, V )be a kinematical spacetime. The kinematical quantities

Θ,ω,σand ˙uand their time derivatives obey the following geometrical identities, called

Raychaudhuri equations:

(i) Raychaudhuri equation for the expansion:

Θ = −Θ2

n+|ω|2−|σ|2+ div( ˙

V)−Ric(V, V ).

(ii) Raychaudhuri equation for the acceleration:

¨u−£V˙u=|˙u|2u−Θ

n˙u+1

2˙

Vcω−1

2˙

Vcσ.

(iii) Raychaudhuri equation for the rotation:

˙ω=−2Θ

nω+1

(Eicω)∧(Eicσ) + H(d ˙u) + u∧(˙

Vcω).

(iv) Raychaudhuri equation for the shear:

˙σ=−2Θ

nσ−1

(Eicσ)˜

∨(Eicσ) + 1

(Eicω)˜

∨(Eicω)+

Hsym(∇˙u) + ˙u˜

∨˙u+1

2u∨(˙

Vcσ)−R(·, V, V, ·) + Ric(V, V )

nh.

Proof. (i) Due to equation (∗) on page 44, we have

Θ = X

(∇V(∇u))(Ei, Ei) = X

[(∇˙u)(Ei, Ei)−(∇u)(∇EiV, Ei)−R(Ei, V, V, Ei)] .

Using Lem. 3.32, as well as Pi(∇˙u)(Ei, Ei) = div( ˙

V),Pih(Ei, Ei) = nand the definition

of the Ricci curvature, we get

Θ = div( ˙

V)−Θ2

n+X

i,j

σ(Ei, Ej)σ(Ei, Ej)−X

i,j

ω(Ei, Ej)ω(Ei, Ej)−Ric(V, V ).

Using Def. 3.22 for |σ|2and |ω|2yields the desired equation.

(ii) The Raychaudhuri equation for the acceleration is an identity that stems from expanding

the Lie derivative £V˙uwith respect to the kinematical quantities and from the identity

|˙u|2=g(∇VV, ∇VV) = −g(∇V∇VV, V ) = −¨u(V),

which is just a consequence of Vbeing a unit vector field in a kinematical spacetime. Using

Cartan’s magic formula we compute for any vector field X∈Γ(TM)

−(£V˙u)(X) = −(Vcd ˙u)(X) = 2d ˙u(X, V )=(∇X˙u)(V)−(∇V˙u)(X) =

g(∇X∇VV, V )−¨u(X) = −g(∇VV, ∇XV)−¨u(X) = −(∇u)(X, ˙

V)−¨u(X) =

=−Θ

nh(X, ˙

V)−ω(X, ˙

V)−σ(X, ˙

V) + u(X) ˙u(˙

V)−¨u(X) =

=−Θ

n˙u(X) + 1

2(˙

Vcω)(X)−1

2(˙

Vcσ)(X) + |˙u|2u(X)−¨u(X)

and the result follows.

(iii) The Raychaudhuri equation for the rotation follows from the anti-symmetric part of

equation (∗) on page 44, which is just given by ∇V(du). As du=ω−u∧˙u, we have

∇V(du) = ∇Vω−∇V(u∧˙u) = ˙ω−u∧¨u.

As R(·, V, V, ·)is symmetric and the anti-symmetric part of ∇˙uis d ˙u, it follows, using

Lem. 3.32, that

˙ω=u∧¨u+ d ˙u−2Θ

nω+X

ω(Ei,·)σ(Ei,·)−X

σ(Ei,·)ω(Ei,·)+

+Θ

nu∧˙u+u∧σ(˙

V , ·) + u∧ω(˙

V , ·),

and hence

˙ω=u∧¨u+ d ˙u−2Θ

nω+1

(Eicω)∧(Eicσ) + Θ

nu∧˙u+1

2u∧(˙

Vcσ) + 1

2u∧(˙

Vcω).

Inserting the Raychaudhuri equation for the acceleration to replace ¨uyields

˙ω=−2Θ

nω+1

(Eicω)∧(Eicσ) + u∧(Vcd ˙u) + d ˙u+u∧(˙

Vcω),

when using £V˙u=Vcd ˙u. Noting that for all vector fields X, Y ∈Γ(TM)

(H(d ˙u))(X, Y ) = d ˙u(HX, HY) = d ˙u(X+u(X)V, Y +u(Y)V) =

= d ˙u(X, Y ) + u(Y)d ˙u(X, V ) + u(X)d ˙u(V, Y ) = d ˙u(X, Y )+(u∧(Vcd ˙u))(X, Y )

holds, yields H(d ˙u) = d ˙u+u∧(Vcd ˙u)and, therefore, the desired equation.

(iv) Similar reasoning leads to the Raychaudhuri equation for the shear. In this case we

have

σ= sym(∇u) + u∨˙u−Θ

nh,

hence using the symmetric part of (∗) on page 44 yields

˙σ=∇V(u∨˙u)−∇V(Θ

nh) + sym(∇˙u)−sym[(∇u)(∇·V, ·)] −sym[R(·, V, V, ·)].

Now, we compute

∇V(u∨˙u) = u∨¨u+ ˙u⊗˙u

and

∇V(Θh) = ˙

Θh+ Θ∇V(u⊗u) = ˙

Θh+ 2Θ(u∨˙u),

and use Lem. 3.32 to arrive at

˙σ=u∨¨u+ ˙u⊗˙u−˙

nh−2Θ

n(u∨˙u) + sym(∇˙u)−Θ2

n2h−2Θ

nσ−1

(Eicσ)⊗(Eicσ)+

(Eicω)⊗(Eicω) + Θ

nu∨˙u+1

2u∨(˙

Vcσ) + 1

2u∨(˙

Vcω)−R(·, V, V, ·).

Now, we use the Raychaudhuri equation for the expansion to replace ˙

Θand get

˙σ=u∨¨u−Θ

n˙u+1

2(˙

Vcσ) + 1

2(˙

Vcω)+ ˙u⊗˙u+ sym(∇˙u)−2Θ

nσ−div( ˙

nh−

−1

(Eicσ)˜

∨(Eicσ) + 1

(Eicω)˜

∨(Eicω)−R(·, V, V, ·) + Ric(V, V )

nh.

Considerations similar to the rotation case above lead to

Hsym(∇˙u) = sym(∇˙u) + u∨(2¨u−£V˙u)−|˙u|2u⊗u

by using (£V˙u)(X) = ¨u(X)−(∇X˙u)(V)for all vector fields Xon M. Using this to replace

sym(∇˙u), and employing the Raychudhuri equation for the acceleration to get rid of ¨u, yields

˙σ=u∨(˙

Vcσ) + ˙u⊗˙u+Hsym(∇˙u)−2Θ

nσ−div( ˙

nh−

−1

(Eicσ)˜

∨(Eicσ) + 1

(Eicω)˜

∨(Eicω)−R(·, V, V, ·) + Ric(V, V )

nh.

Now the desired equation follows from

Tr( ˙u⊗˙u) = X

[ ˙u(Ei)]2=|˙u|2

and

Tr(Hsym(∇˙u)) = Tr(sym(∇˙u)) + |˙u|2[u(V)]2−2u(V)¨u(V) = div( ˙

V)−|˙u|2,

which implies

Hsym(∇˙u) + ˙u˜

∨˙u=Hsym(∇˙u) + ˙u⊗˙u−div( ˙

nh.

Several remarks are in order. The vertical part of the Raychaudhuri equation for the rotation

yields just the identity

Vc˙ω+˙

Vcω= 0,

which forces the horizontal part to read

H( ˙ω) = −2Θ

nω+1

(Eicω)∧(Eicσ) + H(d ˙u).

The same holds for the shear equation, which yields

Vc˙σ+˙

Vcσ= 0

and the corresponding horizontal equation

H( ˙σ) = −2Θ

nσ−1

(Eicσ)˜

∨(Eicσ) + 1

(Eicω)˜

∨(Eicω)+

Hsym(∇˙u) + ˙u˜

∨˙u−R(·, V, V, ·) + Ric(V, V )

nh.

The equations for the expansion and the shear are the only ones to include curvature terms.

In the case of the expansion it is Ric(V, V ), the Ricci curvature along the reference frame

V, also called Raychaudhuri scalar, and in the case of the shear it is the traceless part

of R(·, V, V, ·), which can be expressed in terms of the Weyl curvature tensor Wand the

traceless part of the Ricci tensor in the following way (cf. [KS07]).

Remark 3.34. Denote by

Π = ^

Hsym(Ric)

the symmetric trace-free part of H(Ric). Then after some tedious, but straightforward algebra

one gets

Ric(V, V )

nh−R(·, V, V, ·) = Π

n−1−W(·, V, V, ·).

In four spacetime dimensions (i.e., n= 3), the symmetric tracefree tensor E, defined by

E(X, Y ) := W(X, V, V, Y )for all X, Y :M→TM is called electric part of the Weyl tensor.

Besides that, there is, in arbitrary spacetime dimension, a so called magnetic part Hof

the Weyl tensor which depends on the kinematical quantities and their derivatives only (see

formula (B12) in [HOW12, Appendix B]). Furthermore, in four spacetime dimensions, the

Weyl tensor Wis completely determined by Hand the electric part Edefined above. Hence,

the Weyl tensor in four spacetime dimensions can be computed from Πand the kinematical

quantities. It is worth noting that in higher spacetime dimensions (i.e., n+1 ≥5) a splitting

of the Weyl tensor into electric and magnetic parts is possible, too (cf. [HOW12]), but the

electric part is not only given by W(·, V, V, ·)occurring in the Raychaudhuri equation for

the shear. This amounts to the fact that in higher dimensions the Weyl curvature is not

completely determined by the kinematical quantities and Π.

The considerations about curvature terms above, underline the insight that the rotation is

an attribute of the reference frame V, which is independent of the curvature of g. Especially,

since a simple computation shows that

£Vω=H(d ˙u),

which makes the horizontal part of the Raychaudhuri equation for the rotation even in-

dependent of the acceleration. However, curvature properties enter the development of ω

indirectly via the expansion Θand its Raychaudhuri equation.

In their full generality, the Raychaudhuri equations seem to be too complicated to be useful.

Therefore, we will primarily use them in special cases in the following sections and chapters.

Particularly, the case of vanishing shear will be of interest. In this case the Raychaudhuri

equation for the shear transforms into a constraint equation, which connects rotation, ac-

celeration and Weyl curvature of the kinematical spacetime. Basically, this implies that

these three quantities cannot be chosen independently in a spatially conformally stationary

kinematical spacetime (cf. Lem. 3.30). The Raychaudhuri equation for the expansion will

be of interest in the case of vanishing shear and vanishing expansion, particularly in sta-

tionary spacetimes. This will enable us to derive connections between rotation, acceleration

and Ricci curvature by a Lorentzian Bochner technique in section 5.3. The Raychaudhuri

equation for the rotation will become important in shear-free cases with geodesic reference

frame V, where it just reads

˙ω=−2Θ

nω

in chapter 6.

3.4 The Causal Ladder

In this section, we introduce the basic notions of causality conditions for spacetimes. Besides

the standard references for this chapter, we will also base this section on [MS08]. It is

important to keep in mind that all causality conditions are conformally invariant. Although

we will define the specific causality conditions for a single spacetime (M, g)only, it does

always hold for the whole conformal class of spacetimes (M, [g]) with [g] = {˜g∈Lor(M)|˜g=

eϕg, ϕ:M→R}, too.

First, we fix some causal binary relations for events in a spacetime and some definitions for

causal sets.

Definition 3.35. Let (M, g)be a spacetime and p, q ∈M. Then we say:

(i) pis chronologically related to q, denoted pq, if there is a future-directed timelike

curve connecting pwith q.

(ii) pis strictly causally related to q, denoted p < q, if there is a future-directed causal

curve connecting pwith q.

(iii) pis causally related to q, denoted p≤q, if there is a future-directed causal or constant

curve connecting pwith q.

(iv) pis horismotically related to q, denoted p→q, if there is a future-directed causal or

constant but not timelike curve connecting pwith q.

(v) The chronological future of pis the set I+(p) := {q∈M|pq}.

(vi) The causal future of pis the set J+(p) := {q∈M|p≤q}.

(vii) The future horismos of pis the set E+(p) := {q∈M|p→q}.

The chronological and causal past I−(p),J−(p)and the past horismos E−(p)are defined

analogously. Furthermore, the binary relations ,≤,→can be regarded as subsets of M×M

by setting

(i) I+:= {(p, q)∈M×M|pq},

(ii) J+:= {(p, q)∈M×M|p≤q}and

(iii) E+:= {(p, q)∈M×M|p→q}.

Moreover, we call a set U⊂Machronal (acausal) if there are no two events p, q ∈U, such

that pq(p<q) holds.

Using the notion of continuous causal curves from the beginning of this chapter, it becomes

clear that the causal relation and the chronological relation are both transitive, i.e., let

p, q, r be events in some spacetime (M, g), then p≤qand q≤r(pqand qr) implies

p≤r(pr). This is obvious, as the causal or constant curve connecting pto rcan

be differentiable almost everywhere but only continuous at q, as long as the limits of the

tangent vectors—when approaching qfrom the future or the past, respectively—are both

future pointing. This carries over to timelike curves.

Definition 3.36. Let (M, g)be a spacetime and γ: [a, b]→Ma closed curve in M. Then

γis called a CTC (closed timelike curve) if it is timelike and a CCC (closed causal curve)

if it is causal.

Please note that a CTC (or CCC) is not necessarily periodic as ˙γ(a)6= ˙γ(b)in general.

Definition 3.37. Let (M, g)be a spacetime and γ:R→Ma (possibly non-smooth) future-

directed causal and inextendible curve in Mand K⊂Mcompact.

(i) The curve γis called future (resp. past) imprisoned in Kif there is a T∈R, such that

γ([T, ∞)) ⊂K(resp. γ((−∞, T]) ⊂K).

(ii) The curve γis called partially future (resp. past) imprisoned in Kif there is a sequence

{tn}n∈N⊂Rwith tn→ ∞ as n→ ∞ (resp. tn→ −∞ as n→ ∞), such that γ(tn)∈K

for all n∈N.

(iii) We call a curve γnon–(partially)-imprisoned if it is neither (partially) future nor

(partially) past imprisoned in any compact set.

(iv) A spacetime (M, g)is called non-imprisoning if all causal curves are non-imprisoned.

(v) A spacetime (M, g)is called non–partially-future-(resp. past)-imprisoning if it does

not contain any partially future (resp. past) imprisoned causal curve. It is called non–

partially-imprisoning if it is non–partially-future-imprisoning and non–partially-past-

imprisoning.

Remark 3.38. J.K. Beem has shown that a spacetime is non–future-imprisoning if and only

if it is non–past-imprisoning (see [Bee76], where this is called condition N). Therefore, as

opposed to partial imprisonment, we do not have to to distinguish between the future and the

past cases.

In [Min08c], E. Minguzzi showed the connection between limit sets of a curve and its im-

prisonment by the following

Proposition 3.39. Let (M, g)be a spacetime and γ:R→Ma (maybe continuous) future-

directed causal and inextendible curve in M. The curve γis partially future (past) imprisoned

in some compact K⊂Mif and only if ω(γ)6=∅(α(γ)6=∅) and it is future (past) imprisoned

in some compact K⊂Mif and only if ω(γ)6=∅and ω(γ)is compact (α(γ)6=∅and α(γ)

is compact).

Proof. See [Min08c, Prop. 3.2].

Definition 3.40. A spacetime (M, g)is called totally vicious if for all p∈Mthere is a

CTC intersecting p.

Note that for this worst causal property of a spacetime, there does not exist a simple topo-

logical reason. See Example 4.6 below.

Now we are ready to give the definitions for the steps on the causal ladder which are most

important in this work. We mainly base the definitions on [MS08], but we will also use some

other references. There is one condition, related in some ways to the causality conditions

below, which stands, nevertheless, outside the causal ladder.

Definition 3.41. A spacetime (M, g)is called reflecting if I+(p)⊃I+(q)⇔I−(p)⊂I−(q)

for all p, q ∈M.

We will see below that reflectivity helps to jump up the causal ladder from distinction to

causal continuity.

Definition 3.42. A spacetime (M, g)is called chronological if there is no CTC in (M, g).

Combining chronology with reflectivity yields the existence of a so-called semi-time function,

i.e., a continuous function t:M→Rfor a spacetime (M, g), such that pqimplies

t(p)< t(q)for all events p, q ∈M. In other words, tis strictly increasing along any future-

directed timelike curve. Then the following holds.

Proposition 3.43. A spacetime (M, g)which is chronological and reflecting admits a semi-

time function.

Proof. See [Min10, Thm. 5].

Definition 3.44. A spacetime (M, g)is called causal if there is no CCC in (M, g).

Obviously, causality of a spacetime implies that the spacetime is also chronological. The

next stronger condition on the causal ladder is non-imprisonment, as defined in Def. 3.37.

In [Min08d] the following new step on the causal ladder was established.

Definition 3.45. A spacetime (M, g)is called feebly distinguishing if (p, q)∈J+,p∈I+(q)

and q∈I−(p)implies p=q.

Definition 3.46. A spacetime (M, g)is called distinguishing if I+(p) = I+(q)implies p=q

and I−(p) = I−(q)implies p=q.

In [Min08d] and [Min08c] it was shown that distinction of a spacetime implies feeble distinc-

tion and feeble distinction, in turn, implies non-imprisonment.

The next stronger condition on the causal ladder is non–partial-imprisonment, which was

shown to be stronger than distinction, but weaker than strong causality defined below, in

[Min08c].

For the next step on the causal ladder there are several equivalent formulations. We will

omit most of them here as this step of the causal ladder is scarcely needed in this work

and will only give the formulation which can be cast into the shortest form, although this is

non-standard.

Definition 3.47. A spacetime (M, g)is called strongly causal if the Alexandrov topology,

generated by the base sets Bp,q ={I+(p)∩I−(q)|p, q ∈M}, equals the manifold topology.

Furthermore, there are some more subtle steps on the causal ladder around distinction and

strong causality (see, e.g., [Min09b], [Min08a], [Min08b]), which will be omitted here, as they

are not relevant for the causality conditions of kinematical spacetimes analyzed in chapter

4. In [MS08], it was shown that the next condition on the causal ladder is stronger than

strong causality.

Theorem & Definition 3.48. A spacetime (M, g)is called stably causal if it fulfills one of

the following equivalent properties.

(i) There is a time function on (M, g), i.e., a continuous function t:M→R, which is

strictly increasing along any future-directed causal curve.

(ii) There is a temporal function on (M, g), i.e., a smooth function τ:M→Rwith past-

directed timelike gradient ∇τ.

In fact, the equivalence of items (i) and (ii) above has only been rigorously established a

few years ago (see, e.g., [BS05]). The following step of the causal ladder will be of special

importance in this work.

Definition 3.49. A spacetime (M, g)is called causally continuous if it is reflecting and

feebly distinguishing.

Usually (cf. [MS08]), causal continuity was defined as a spacetime being reflecting and distin-

guishing. In [Min08d], it was proven that the assumption can be relaxed to feeble distinction.

Causal continuity is stronger than stable causality, as shown in [MS08].

The following two last and strongest causality conditions are usually connected to the com-

pleteness of some properly constructed Riemannian or Finslerian metric on (hypersurfaces

in) the spacetime under consideration (see chapters 4and 5).

Definition 3.50. A spacetime (M, g)is called causally simple if it is causal and J+(p),

J−(p)are closed sets for all p∈M.

Definition 3.51. A spacetime (M, g)is called globally hyperbolic if it is causal and J+(p)∩

J−(q)are compact sets for all p, q ∈M.

We again refer to [MS08] for the implication global hyperbolicity ⇒causal simplicity ⇒

causal continuity. We will also need the following

Theorem 3.52. A spacetime (M, g)is globally hyperbolic if and only if it is causal and the

space C(p, q)of all continuous and future-directed causal curves connecting the events pand

qin Mis compact in the C0-topology.

Proof. See [MS08, Thm. 3.79].

The C0-topology is the simplest topology one can impose on the space C(p, q)and is given

by considering open neighborhoods of the curves in the spacetime.

Moreover, we will need the characterization of global hyperbolicity in terms of Cauchy

hypersurfaces. We have the following

Definition 3.53. ACauchy hypersurface Sin a spacetime (M, g)is an achronal, topological

submanifold S⊂M, such that every inextendible timelike curve in Mintersects S.

Often one is interested not in general Cauchy surfaces, but in smooth and spacelike ones,

which are then acausal smooth hypersurfaces.

Theorem 3.54. A spacetime (M, g)is globally hyperbolic if and only if there is a (smooth

spacelike) Cauchy hypersurface in (M, g).

Proof. This theorem has originally been proven by R. Geroch in [Ger70], without the

requirements of the Cauchy hypersurface to be smooth and spacelike, and only recently

the problems of smoothability and the existence of spacelike Cauchy hypersurfaces were

solved (see [MS08, Thm. 3.78] for an overview and the references therein for details).

We conclude this section with a diagram that summarizes the steps on the causal ladder

which are most relevant for this work:

globally hyperbolic



causally continuous



stably causal



non–partially-imprisoning



feebly distinguishing



reflectivity

non-imprisoning



causal



chronological



non–totally-vicious

Chapter 4

DIFFEOMORPHIC SPLITTING OF LORENTZIAN R-MANIFOLDS

In the first section 4.1 of this chapter, we will derive the main diffeomorphic splitting result

for kinematical spacetimes by the use of the theory of R-actions, principal bundles and flows

established in chapter 2. Hence, we investigate the conditions under which a kinematical

spacetime has the topology of R×S, i.e., it is diffeomorphic to R×S, with Sbeing the

space of flow lines of the reference frame. Subsequently, we can derive some interesting

propositions, for example, on the existence of Lorentzian metrics that fulfill the preconditions

for a splitting on particular manifolds, or on the question of the local and global existence

of a spacelike codimension one foliation of the kinematical manifold. This question provides

a smooth transition to section 4.2, where we will analyze the causality conditions of such

splitting kinematical spacetimes in greater detail. Finally, in section 4.3 we will consider the

causality conditions of some low dimensional splitting kinematical spacetimes, which possess

a particularly close connection between topology and causality if the space of flow lines Sis

compact.

The general problem of any splitting theorem can be stated as follows: When does a given

splitting TM =ξ⊕νof the tangent bundle TM of a manifold M, yield a splitting of the

manifold M=X×N(such that TX =ξand TN =ν) and when does this, subsequently,

also yield the splitting of a semi-Riemannian metric gon M, such that g=s+h, with s

and ha (family of) semi-Riemannian metrics on Xand N, respectively? The first question

only asks for a diffeomorphic splitting of the manifold, without requiring any conditions for

the metric. So it is conceivable that, in general, it should be possible to formulate conditions

for this type of splitting without referring to a metric at all, but specific properties of the

metric will turn out to be exactly the adequate conditions for the splitting in particular

cases. This is precisely what we will establish in Thm. 4.13, Prop. 4.14 and Thm. 4.19

below for the diffeomorphic splitting of kinematical spacetimes. Furthermore, the splitting

of Lorentzian manifolds as M=R×Sis a natural condition to ask for, as all Lorentzian

manifolds naturally come with a splitting of the tangent bundle T M =ξ⊕ν(cf. Rem. 3.6).

The second splitting question also asks for a global splitting of the metric, which amounts

to g=−A2dt2+htin the case of a Lorentzian manifold (R×S, g), with Asome positive

function on R×Sand hta family of Riemannian metrics on Svarying with t∈R. Usually

this question is asked in spacetimes, i.e., without fixing a specific reference frame, and is

then connected to conditions on causality or completeness. One example for such a splitting

is the requirement of global hyperbolicity as in Thm. 5.4. Global hyperbolicity or geodesic

completeness are exactly the conditions that are necessary in the Lorentzian version of the

Cheeger–Gromoll splitting theorem, together with positive timelike Ricci curvature and the

existence of a timelike line (see, e.g., [Gal89]). In this case one even gets a splitting of the

metric as g=−dt2+hon R×Swith a fixed Riemannian metric hon S. If one adopts

the viewpoint of a kinematical spacetime (M=R×S, g, V ), certainly a splitting of gas

g=−A2dt2+htrequires the reference frame Vto have vanishing rotation if Vis parallel

to ∂t. The analysis of a diffeomorphic splitting in the case of an irrotational vector field V

is indeed a special case included in the investigation in this work. This particular case was

intensively examined in [GO03] and [GO09].

Hence, we will assume a diffeomorphic splitting M=R×Salong the—generally rotating—

flow lines of a complete reference frame in a kinematical spacetime (M, g, V ), and in the first

step, we will not be interested in a splitting of the metric. This approach complies with the

one taken in [JS08] and [Har92] for spacetimes with a complete (conformal) Killing vector

field. But we will generalize it to arbitrary complete reference frames, i.e., kinematical

spacetimes and will not use any assumptions on the causality of the spacetime. This in

turn will allow us to analyze the causality of arbitrary splitting spacetimes in section 4.2

independently, without having to assume a specific causality condition in order to achieve

the splitting.

4.1 Topology of Kinematical Spacetimes

Theorem 4.1. Let (M, g, V )be a kinematical spacetime. Then Mis a Cartan R-manifold

with R-action induced by the flow of Vif and only if the flow lines of Vare non–partially-

imprisoned curves.

Proof. Applying Prop. 3.39 to the integral curves of Vyields the fact that every integral

curve has empty limit sets if and only if it is non–partially-imprisoned. Thus, regarding the

kinematical manifold (M, V )as a dynamical system with the global flow Φinduced by Von

Mimplies that Ω(Φ) = ∅if and only if all integral curves of Vare non–partially-imprisoned.

Now we can apply Prop. 2.46 to this dynamical system and deduce that Mis a Cartan

R-manifold if and only if all integral curves of Vare non–partially-imprisoned.

There are two interesting simple consequences of Thm. 4.1.

Corollary 4.2. Every kinematical spacetime (M, g, V )with Cartan flow of Vis non-compact.

Proof. If Mwas compact, every flow line of Vwould be imprisoned in the compact set

Corollary 4.3. Assume there is a kinematical scalar invariant Θ,|σ|2,|ω|2or |˙u|2, which

is unbounded and monotonic along all flow lines of Vin a kinematical spacetime (M, g, V ),

then the flow is Cartan.

Proof. Assume the flow of Vis not Cartan. Thus, there is at least one p∈Mand a flow

line γp(t)through p, which is partially future or past imprisoned in a compact set K. Let

F∈ {Θ,|σ|2,|ω|2,|˙u|2}be the scalar kinematical invariant that is unbounded and monotonic

along γp(t), that is the function F◦γp:R→Ris unbounded and monotonically increasing

or decreasing. But as there is a sequence {tn}n∈N, such that γp(tn)⊂K, the sequence

(F◦γp)(tn)is bounded, which is the desired contradiction.

Remark 4.4. Of course, the corollary above is true for any unbounded function along the

flow lines. But in a kinematical spacetime the complete vector field Vprovides us with a

natural set of functions {Θ,|σ|2,|ω|2,|˙u|2}along the flow lines to test for being unbounded.

Thus, kinematical spacetimes (M, g, V )with a Cartan flow induced by the vector field V

are exactly those kinematical spacetimes for which the flow lines of Vare non–partially-

imprisoned. Based on the results in chapter 2, we see that these spacetimes have a quite

regular structure. They are generalized R-principal bundles over a base M/R(which can be

non-Hausdorff), and thus, in particular, they are locally trivial. Non–partial-imprisonment

is a considerably weaker assumption than chronology, which is usually made whenever the

theory of R-manifolds is used to obtain Lorentzian splitting results (see [Har92] and [JS08]).

Of course, chronology ensures for any kinematical spacetime (M, g, V )to have Cartan flow.

But in this case one leaves unregarded a whole interesting class of kinematical spacetimes

which do possess CTCs, that are created by peculiarities of the Lorentzian metric gand

not by topological obstructions of the spacetime manifold M. Thus, kinematical spacetimes

admitting a Cartan flow are optimal to study CTCs that are brought about by metric

properties. We could even define a CTC to be non-trivial if it occurs in a kinematical

spacetime with Cartan flow. Note that this definition of non-trivial CTCs differs from the

classical one given by B. Carter in [Car68]. Therein, a CTC is defined to be non-trivial if it

is homotopic to a point and trivial if this is not the case. A trivial CTC, therefore, could be

dissolved in this definition by passing to the universal covering spacetime. As the example

below shows, our definition also includes CTCs as non-trivial that would be excluded by

B. Carter’s definition. In our setup of kinematical spacetimes this definition would be too

limited. We want to examine the topology of a kinematical spacetime, with fixed reference

frame field and not its universal covering manifold. We assume a fixed reference frame

field Vand a Lorentzian metric that can be altered—while keeping Va timelike reference

frame—to avoid or create CTCs. Hence, the CTCs in kinematical spacetimes occur due to

a specific choice of the Lorentzian metric. As the examples below show, there are indeed

CTCs in a kinematical spacetime with Cartan flow that are not homotopic to a point, but

which occur due to a specific choice of the Lorentzian metric. Furthermore, B. Carter’s

definition of non-trivial CTCs include for example CTCs occurring in a compact spacetime

(S3, g), as any closed curve on the 3-sphere is homotopic to a point. But from the viewpoint

of Lorentzian geometry these CTCs occur trivially as the spacetime manifold S3is compact.

Example 4.5. Consider R2with global coordinates (t, x)∈R2. We wind up R2to a cylinder

S1×Rby the identification (t, x)∼(t+ 1, x). Now we can make different choices of global

complete vector fields and associated Lorentzian metrics. Let us consider

(i) V1=∂tand g1=−dt2+ dx2,

(ii) V2=∂t+1

2∂x,g2a=4

3g1,g2b=−4dtdx+ 4dx2and g2c= 2dt2−10dtdx+ 8dx2as

well as

(iii) V3=∂t+x2∂xand g3=−(1 + x4)dt2+ dx2.

Obviously, the vector fields given are complete and are normed to −1with respect to all the

metrics associated to them. Thus, in all cases, we have a kinematical spacetime structure on

the cylinder.

In case (i), however, V1obviously has the circles S1as flow lines, which are CTCs and the

corresponding R-action on M=S1×Rhas the isotropy subgroup Z. In this case, there is

actually an action by the compact group R/Z=S1on Mand, of course, Mis not a Cartan

R-manifold. The flow lines themselves are compact and they are, therefore, naturally future

and past imprisoned.

Things are completely different if we keep the metric conformally fixed and tilt the vector

field a little, as it is the case in (ii) with the complete vector field V2and the first metric

g2a. Any integral curve of the vector field ∂tis still a CTC, but the vector field V2gives

rise to to Cartan R-action on M, as its flow lines wind up the cylinder towards ±x-infinity.

Thus, they leave any compact set and are future and past non-imprisoned. The same is

true for the metrics g2band g2c, but in the case of g2bthe vector field ∂tis lightlike, so the

resulting spacetime is chronological but non-causal. For the metric g2c, the vector field ∂t

becomes spacelike and the spacetime is causal. These three choices of the metric show that—

although the CTCs or CCCs, that may occur, are non-homotopic to a point and would in

fact disappear if we unwind the cylinder, which is equal in this case to pass to the universal

covering manifold—their appearance crucially depends on the choice of the Lorentzian metric

on S1×R, and it is not just a result from topological obstructions. Moreover, in these three

subcases of (ii), the R-action induced by V2is even proper and the space of flow lines is

homeomorphic to S1.

The case (iii) has the integral curve of V3|x=0 =∂t|x=0 as a flow line of the complete vector

field V3. This flow line equals the orbit Rp0for any p0= (t0,0) ∈S1×R, it is compact and,

therefore, future and past imprisoned. Hence, the flow of V3is not Cartan in this case. But

the flow lines corresponding to the orbits Rp−and Rp+for all p−= (t, −c2)and p+= (t, c2)

are either only future imprisoned—in the case of Rp−in any compact subset S1×[−ε2,0]—or

past imprisoned—in the case of Rp+in any compact subset S1×[0, ε2]. The orbit space is

not a T1-space in this case, as it can be easily seen that the orbit Rp0cannot be separated

from the neighborhoods of Rp−and Rp+.

In Thm. 4.32 below, we will give a complete causal classification of Lorentzian cylinder

spacetimes, which will include the cases in the example above.

Example 4.6. The standard example of a topologically trivial spacetime providing non-trivial

CTCs is the Gödel spacetime. Consider R3with global coordinates (t, x, y)and vector field

V=∂t. Then, obviously, ∂tis complete and its flow lines cause a proper R-action on R3,

with the space of orbits homeomorphic to R2. This holds for any Lorentzian metric on R3.

We consider the Gödel metric on R3given by

g¨o=−dt2+ 2exdydt+ dx2−1

2e2xdy2.

Thus, (R3, g¨o, ∂t)is a proper kinematical spacetime (see Def. 4.7 below). CTCs occur because

the geometry induced on the global t= const slices, homeomorphic to the orbit space, is not

Riemannian. Thus, these slices are not spacelike when regarded as embedded submanifolds

in (R3, g¨o, ∂t)and CTCs occur.

As Example 3.17 shows, the non–partial-imprisonment of the flow lines of the vector field

Vin a kinematical spacetime (M, g, V )does make Ma Cartan R-manifold, but it is not

enough to make it proper. Thus we give the following

Definition 4.7. A kinematical spacetime (M, g, V )will be called Cartan kinematical space-

time if the R-action induced by the flow of Vmakes Ma Cartan R-manifold and it will be

called proper kinematical spacetime if the R-action induced by the flow of Vis proper on M.

Remark 4.8. Having at hand the definition above, and looking back on Prop. 3.12 and

Prop. 3.16, we can now state the following. Given a spacetime (M, g)and a fixed timelike

vector field V:M→TM, there is a complete timelike vector field V∗:M→TM paral-

lel to Vand a Lorentzian metric g∗conformal to g, such that (M, g∗, V ∗)is a kinematical

spacetime. Hence, if the integral curves of Vare non–partially-imprisoned, (M, g∗, V ∗)is a

Cartan kinematical spacetime. This implies that all results on Cartan kinematical spacetimes

in this, and the following, chapters below, which are conformally invariant—or do only de-

pend on the differentiable and not on the metric structure of (M, g)—hold also for spacetimes

with a fixed global timelike vector field with non–partially-imprisoned integral curves.

Below we will make use of the following lemma by A.W. Wadsley.

Lemma 4.9. Let (M, g)be a Riemannian (Lorentzian) manifold and K:M→TM a

nowhere vanishing (timelike) Killing vector field, i.e., £Kg= 0. There is a conformal factor

f:M→R>0, given by f(x) = [gx(Kx, Kx)]−1(f(x) = [−gx(Kx, Kx)]−1), such that for the

conformally transformed metric ˜g=fg, it holds that £K˜g= 0,˜g(K, K)=1(˜g(K, K) = −1)

and Kis ˜g-geodesic.

Proof. See [Wad75, Lem. 3.1] for the Riemannian case. The Lorentzian case is completely

analogous.

The following proposition is a generalization of known theorems for stationary and confor-

mally stationary spacetimes [Har92] and [JS08], in which the spacetime is required to be

chronological to ensure the Cartan condition. But that is not necessary and we see that the

non–partial-imprisonment of the the flow lines of Vis enough to ensure the Cartan property,

so that spacetimes with possible chronology violations can be included in this notion. The

proof in [JS08] shows that M/Ris Hausdorff if Vis Killing. We will give a slightly different

proof here using item (iv) from Thm. 2.35.

Proposition 4.10. Let (M, g, V )be a Cartan kinematical spacetime. If Vis Killing, i.e.,

£Vg= 0, then (M, g, V )is a proper kinematical spacetime. Particularly, every global triv-

ialization ψof the R-principal bundle associated to (M, g, V ), is an isometry ψ: (M, g)→

(R×S, ψ∗g)with Sdiffeomorphic to M/Rand ψ∗V=∂t. Furthermore, denoting by

t:R×S→Rthe projection on the first factor, which depends on the choice of ψ, and

the canonical projection by pr2:R×S→S, there is a one-form b∈Γ(Λ1S)and a symmet-

ric 2-tensor field γ∈Γ(Σ2S), such that

ψ∗g=−dt⊗dt+ 2pr∗

2(b)∨dt+ pr∗

2(γ).

Proof. By setting g0=g+ 2g(V, ·)⊗g(V, ·)we get a Riemannian metric g0on Mwith V

being Killing for g0, too. This can be seen by computing

£Vg0=£V(g+ 2g(V, ·)⊗g(V, ·)) = 4g([V, V ],·)∨g(V, ·)=0.

Hence, there is a distance dg0associated to g0such that dg0(p, q) = dg0(Φt(p),Φt(q)) for all

p, q ∈Mand all t∈R, where Φis the Killing flow associated to V. We will now show

that for all compact K⊂M, the set ((K, K)) ⊂Ris also compact, thus (M, g, V )is a

proper kinematical spacetime by Thm. 2.35 and Def. 4.7. From Lem. 2.25 we know that

((K, K)) is closed if Kis compact, so we only have to show that ((K, K)) is bounded. Set

cK:= diamdg0(K) := max{dg0(r, s)|r, s ∈K}for any compact set K. Then, as Vis Killing,

we have

ct◦K=diamdg0(t◦K) = cK,

for all t∈R. Hence, it certainly holds that ((t◦K, K)) = ∅for all twith |t|>2cK, such

that we have ((K, K)) ⊂[−2cK,2cK].

As a proper kinematical spacetime, (M, g, V )is in particular an R-principal bundle over

M/R, and by virtue of a diffeomorphism we can set S=M/R, i.e., there is a trivialization

ψ:M→R×S. There is a one-to-one correspondence between (global) trivializations and

(global) sections of the principal bundle (M, S, πM), i.e., we can associate a global section

σ0:S→Mto any trivialization ψ, such that the saturation {t◦σ0(S) =: σt(S)|t∈R}=M.

Then for any p∈Mwith ψ(p)=(t, x), we have that p∈σt(S)and x=πM(p).

Hence, the projection t= pr1depends on the choice of the trivialization ψ, whereas the

canonical projection pr2is the same for all trivializations ψ. We denote by St:= {t}×S=

ψ(σt(S)) ⊂R×Sthe slices associated to the global sections σt(S).

Now we set ˜γ=ψ∗g|Stfor any t∈Rand conclude that £ψ∗V˜γ= 0, as ψ∗Vis transversal to

Stand Killing for ψ∗g, hence ˜γis the pull-back of some symmetric 2-tensor field γon S. In

the same way we conclude that ˜

b= (ψ∗g)(ψ∗V, ·)|Stis the pull-back of some one-form bon

S. Furthermore, this implies that ψ∗maps Vto a multiple of ∂t, say ˜

A−1∂twith ˜

A=A◦pr2

for some function A:S→R>0. As g(V, V )=(ψ∗g)( ˜

A−1∂t,˜

A−1∂t) = −1, we have that

ψ∗g=−˜

A2dt⊗dt+ 2pr∗

2(b)∨dt+ pr∗

2(γ)on R×S. But as ˜

A−1∂tis still Killing for ψ∗ga

short computation of the formula £ψ∗Vψ∗g= 0 yields d˜

A= 0 and the result follows.

Using Lem. 4.9 this result can be generalized to complete timelike Killing vector fields of

arbitrary norm.

Corollary 4.11. Let (M, g)be a stationary spacetime with complete timelike Killing vector

field K:M→TM, such that no integral curve of Kis partially imprisoned. Then (M, g)is

isometric to a spacetime (R×S, g∗)with the Killing vector field given by K∗=∂ton R×S

and

g∗=−(A◦pr2)2dt⊗dt+ 2pr∗

2(b)∨dt+ pr∗

2(γ),

such that b∈Γ(Λ1S)is a one-form on S,γ∈Γ(Σ2S)is a symmetric 2-tensor field and

A:S→R>0.

Proof. Applying Lem. 4.9 to the spacetime (M, g)and the Killing vector field K, yields a

Cartan kinematical spacetime (M, g0, K)with g0= [g(K, K)]−1g. Now we apply Prop. 4.10

to (M, g0, K)and get a Killing vector field K∗=ψ∗K=∂t, as well as

ψ∗g0=−dt⊗dt+ 2pr∗

2(b0)∨dt+ pr∗

2(γ0)

on R×Sfor any trivialization ψ. Next we note that ∇K[g(K, K)] = 0 as Kis Killing. Hence,

there is a function A:S→R>0, such that ψ∗[g(K, K)] = A◦pr2. Thus, ψ∗g= (A◦pr2)2ψ∗g0

and by setting b=A2b0, as well as γ=A2γ0the result follows.

Using Rem. 4.8, the corollary above can be generalized to a complete timelike conformal

vector field K:M→T M as in this case Kis Killing for g∗=−[g(K, K)]−1gand has unit

norm with respect to g∗, which can be inferred similar to Lem. 4.9.

A main generalization to the splitting results in [Har92] and [JS08] is now that the metric γ

on Sis allowed to be semi-Riemannian or even degenerate. With this at hand, the question

of the Riemannian nature of γis intimately tied to the question of causality of (M, g)only,

whereas the question of the topological and metric splitting of the kinematical spacetime

is liberated from all causality assumptions. The stationary splitting results of Prop. 4.10

and Cor. 4.11 are now naturally applicable to non-chronological spacetimes, such as the

totally vicious Gödel spacetime in Example 4.6, or more sophisticated examples like the

torus spacetime in the following example. This example is based on B. Carter’s classical one

in [HE73, Fig. 39, p. 195], and one version also appears in [JS08].

Example 4.12. Let the two-torus T2be given as the coordinate patch (x, y)∈[0,1]×[0,1] ⊂

R2with the identifications (0, y)∼(1, y)for all y∈[0,1] and (x, 0) ∼(x+√2,1). As in

Example 2.28 there is a vector field—∂y= (0,1)Tin this case—the orbits of which are dense

in T2as √2is irrational. We will now consider the spacetime (R×T2, g)with a Lorentzian

metric g, such that the vector field K:= (0, ∂y)Ton R×T2is lightlike for some, or even

all, t∈R. Such a metric is given by

g=−dt2+ 2dtdy+ dx2.

Moreover, (R×T2, g)is stationary with Killing vector field ∂t, and every t=constant slice

{t} × T2contains imprisoned, but non-closed, lightlike curves, the orbits of K, i.e., the

spacetime is causal but causally imprisoning. One could even modify this example to contain

only one slice, say for t= 0, with imprisoned lightlike curves. Let s:R→Rbe a function

with s(0) = 1 and s(t)>1for t6= 0, e.g., s(t) = t2+ 1. Then the metric

g0=−dt2+ 2dtdy+s(t)(dx2+ dy2)−dy2

on R×T2fulfills the required conditions. The spacetime (R×T2, g0)is an example of a

Hubble-isotropic spacetime, which we will study in section 6.1.

We are now ready to give conditions for a Cartan kinematical spacetime (M, g, V )to be

proper, which are much more general and do not depend on the existence of a Killing vector

field parallel to V. Hence, the theorem below can be seen as the most general splitting result

along V, for a given Cartan kinematical spacetime (M, g, V ).

Theorem 4.13. Let (M, g, V )be a Cartan kinematical spacetime. Then the following state-

ments are equivalent:

(i) (M, g, V )is a proper kinematical spacetime.

(ii) There is a one-form ˜uon M, such that Vcd˜u= 0 and ˜u(V)=1.

(iii) Mis an R-principal bundle over S=M/R, the space of flow lines of V.

(iv) There is a Riemannian metric ˜gon M, such that ˜g(V, V )=1,£V˜g= 0 and Vis

˜g-geodesic.

(v) There is a Lorentzian metric ˜gon M, such that ˜g(V, V ) = −1,£V˜g= 0 and Vis

˜g-geodesic.

Proof. “(iv)⇔(v)”: If ˜gis Riemannian, we set ˜u= ˜g(V, ·), and then by ˜g(V, V ) = 1 it

follows that ˜

h= ˜g−2˜u⊗˜uis Lorentzian. If ˜gis Lorentzian the same setting yields that

h= ˜g+ 2˜u⊗˜uis Riemannian. Furthermore, in both cases we have

£V˜g=£V(˜

h±2˜u⊗˜u) = £V˜

h= 0

because of £V˜u=±˙

˜u˜g=±˜g(∇˜g

VV, ·)=0, as Vis geodesic. For the acceleration ˙

˜u, we get

in both cases

˜u˜g=±£V(˜g(V, ·)) = £V(˜

h(V, ·)±2˜u(V)˜u) = ±£V(˜

h(V, ·)) = ±˙

˜u˜

h= 0.

Hence, Vis ˜g-geodesic if and only if it is ˜

h-geodesic.

“(v)⇒(i)”: This is an application or Prop. 4.10.

“(i)⇔(iii)”: This is an application of Prop. 2.53.

“(iii)⇔(ii)”: Using Prop. 2.31, we conclude that Mis a generalized R-principal bundle and

by Prop. 2.33 Mis an R-principal bundle if and only if it admits a connection, say ˜uobeying

˜u(sV ) = sfor all s∈Rand Φ∗

t˜u= ˜ufor all t∈R, where Φis the global flow associated

to the complete vector field V. Hence, these two conditions are equivalent to ˜u(V)=1and

£V˜u= limt→0Φ∗

t˜u−˜u

t= 0, which in turn is equivalent to Vcd˜uand ˜u(V)=1by the formula

£V˜u=Vcd˜u+ d(Vc˜u).

“(iii)⇒(iv)”: Using Lem. 4.9 we only need to find a Riemannian metric g0on Mfor which

Vis Killing. As condition (iii) holds, we know that Mis trivializable as a principal bundle,

i.e., there is ψ:M→R×S. So we set ψ∗g0= dt⊗dt+hon R×Sfor an arbitrary

Riemannian metric hon S. Then we have £∂tψ∗g0= 0, as ∂tis Killing for ψ∗g0. But the

trivialization is an isometry for the chosen metric and the metric g0=ψ−1

∗ψ∗g0on M, such

that we have £Vg0= 0 for V=ψ∗∂t, thus Vis Killing for g0.

Given a kinematical spacetime (M, g, V ), we will now connect the diffeomorphic splitting

properties above to characteristics of the dynamical system (M, Φ) associated to the space-

time by the complete vector field V. Specifically, we will consider the instability of this

dynamical system and the existence of improper saddle points, hence its parallelizability.

Thus, we will arrive at the same splitting result from a different perspective.

Proposition 4.14. Let (M, g, V )be a kinematical spacetime and Φthe global flow associated

to V, such that (M, Φ) is a dynamical system. If the integral curves of Vare non-imprisoned,

which is equivalent to (M, Φ) being unstable, and (M, Φ) has no improper saddle point,

(M, g, V )is a proper kinematical spacetime.

On the other hand, if (M, g, V )is a proper kinematical spacetime, then the associated dy-

namical system (M, Φ) has no improper saddle point.

Proof. By Def. 2.49 a dynamical system is parallelizable if it is unstable and has no improper

saddle point. A dynamical system is unstable if no half-orbit is entirely contained in a

compact set. Comparing this notion to future and past imprisonment from Def. 3.37 shows

that the dynamical system (M, Φ) is unstable if and only if all integral curves of the vector

field Vare non-imprisoned. Hence, using Prop. 2.53, we can conclude that (M, g, V )is a

proper kinematical spacetime.

Again by Prop. 2.53, we can infer that a proper kinematical spacetime (M, g, V )is associated

to a parallelizable dynamical system (M, Φ), which is necessarily without an improper saddle

point.

Remark 4.15. Inspired by Example 3.14, we would like to give some intuition of the meaning

of improper saddle points in the framework of Cartan kinematical spacetimes. Obviously, the

two-dimensional spacetime from Example 3.14 is not proper because of the removed origin

of R2, which leads to a non-Hausdorff orbit space. In some sense, the removed origin is the

improper saddle point in this case. Hence, the absence of improper saddle points signals the

absence of holes in the spacetime. But note that improper saddle points are not always what

we would intuitively consider a hole in spacetime. One could even modifiy Example 3.14

by removing the whole non-negative x-axis. We would get the branched line as orbit space,

instead of the line with two origins, in this case. Hence, the absence of improper saddle

points or the condition of properness assures that such situations as in Example 3.14 do not

occur.

This notion of hole-freeness depends not only on the spacetime (M, g), but also on the com-

plete timelike vector field V, as it is linked with the dynamical system associated to V.

Hence, it is different from other efforts to give a rigorous definition of hole-freeness, e.g., in

[Man09]. Furthermore, a recent clarification of conditions for spacetimes to contain holes in

[Min12], which is based on [Man09], uses Cauchy developments and causality conditions of

the spacetime.

Given a kinematical spacetime (M, g, V ), we can assess the splitting question from two

different sets of conditions. On the one hand, based on Thm. 4.13, we need non–partial-

imprisonment of the integral curves of Vand some one-form on Mthat serves as a princi-

pal connection. On the other hand, based on Prop. 4.14, we need only non-imprisonment

of the integral curves of V, which is a weaker condition than non–partial-imprisonment

(cf. Def. 3.37). But additionally we need the slippery notion of improper saddle points.

Although Rem. 4.15 provides an interpretation of improper saddle points for some cases of

kinematical spacetimes, the question of existence of improper saddle points is usually not

solved straightforwardly for a given kinematical spacetime. So it is comforting that we can

solve the question for improper saddle points in a dynamical system associated to some

Cartan kinematical spacetime by differential geometric means, i.e., a principal connection

one-form. As we will see in Thm. 4.19 below, such a principal connection is provided in

many interesting situations by the Lorentzian metric and the reference frame Vthemselves.

We are now ready to give a generic formula for the Lorentzian metric of a splitting Lorentzian

manifold, i.e., a proper kinematical spacetime.

Theorem 4.16. Let (M, g, V )be a proper kinematical spacetime. Then every trivialization

ψof the R-principal bundle associated to (M, g), is an isometry ψ: (M, g)→(R×S, ψ∗g),

such that ψ∗V=A−1∂tfor some function A:R×S→R>0and S=M/R.

Furthermore, for every trivialization ψthere is a family of one-forms {bt}t∈R⊂Γ(Λ1S)and

a family of Riemannian metrics {ht}t∈R⊂ R(S)on S, both varying smoothly with t, such

that

ψ∗g(t,x)=−A2(t, x)dt⊗dt+ 2pr∗

2(b(t,x))∨dt+ pr∗

2(h(t,x))−pr∗

2(b(t,x))⊗pr∗

2(b(t,x))

A2(t, x),

with the canonical projection pr2:R×S→Sand the projection pr1=t:R×S→R

associated to the trivialization ψ.

Moreover, any other trivialization ˜

ψis given by a gauge transformation in the R-principal

bundle, i.e., by a transformation of the projection t7→ τ=t−t0+f(x)for any function

f:S→Rand any fixed t0∈R. The metric with respect to the trivialization ˜

ψis given by

ψ∗g(τ,x)=−A2(τ, x)dτ⊗dτ+ 2pr∗

2(b(τ,x)+A2(τ, x)dfx)∨dτ+

+pr∗

2(h(τ,x))−pr∗

2(b(τ,x)+A2(τ, x)dfx)⊗pr∗

2(b(τ,x)+A2(τ, x)dfx)

A2(τ, x).

Proof. As a trivialization ψmaps any fiber, i.e., any integral curve γof V, to a line

R×{x} ⊂ R×Swith x=πM(γ)∈S, the push-forward ψ∗Vis parallel to ∂t, i.e., there is

a function A:R×S→R>0, such that ψ∗V=A−1∂t.

Denoting by u=g(V, ·)the one-form metrically associated to V, there is a decomposition

of ψ∗uin R×Sgiven by

(ψ∗u)(t,x)=−A(t, x)dt+β(t,x)

A(t, x),

for all (t, x)∈R×S, as −1 = u(V) = (ψ∗u)(A−1∂t). Here, β(t,x)∈T∗

(t,x)(R×S)is a

one-form obeying β(∂t)=0, hence βis pulled back from some one-form b(t,x)∈T∗

xS, i.e.,

β(t,x)= pr∗

2(b(t,x)). As this holds for all (t, x)∈R×S, there is a family of one-forms

{bt}t∈R⊂Γ(Λ1S), such that

(ψ∗u) = −Adt+pr∗

2(bt)

Based on the analysis in section 3.1, we can decompose gusing uand the projection h, i.e.,

we have g=h−u⊗u, and thus

ψ∗g=ψ∗h−ψ∗u⊗ψ∗u.

Certainly, we have 0 = h(V, ·)=(ψ∗h)(A−1∂t,·), and thus (ψ∗h)(t,x)∈Σ2

(t,x)(R×S)is pulled

back from some symmetric 2-tensor in Σ2

xS, which we will also denote by h. As this holds

for all (t, x)∈R×S, there is a family of symmetric 2-tensor fields {ht}t∈R⊂Γ(Σ2S), such

that

ψ∗h= pr∗

2(ht).

Also from section 3.1, we know that his a Riemannian metric on HM, hence ψ∗his a

Riemannian metric on the bundle horizontal to ∂t, which ensures the htto be Riemannian

metrics on S. Hence, we have

(ψ∗g)(t,x)= pr∗

2(h(t,x))−−A(t, x)dt+pr∗

2(b(t,x))

A(t, x)⊗−A(t, x)dt+pr∗

2(b(t,x))

A(t, x),

and the result follows. Certainly, btand htvary smoothly with t, as gis a smooth metric.

As (M, g, V )constitutes a globally trivial principal bundle, there is a one-to-one correspon-

dence between global trivializations and gauge transformations as stated. We now only have

to compute how ψ∗gchanges under the transformation t7→ τ(t, x) = t−t0+f(x). For

computational simplicity, we will suppress pull-backs, push-forwards and dependencies on

(t, x)in the following. We have

dτ= dt+ df,

hence

dt⊗dt= dτ⊗dτ−2dτ∨df+ df⊗df.

This yields

g=−A2(dτ⊗dτ−2dτ∨df+ df⊗df)+2b∨(dτ−df) + h−b⊗b

A2,

thus

g=−A2dτ⊗dτ+ 2dτ∨(b+A2df) + h−b⊗b

A2−2b∨df−A2df⊗df,

and the result follows.

Using Rem. 4.8, the splitting form of the metric in the theorem above can be generalized to

spacetimes (M, g), together with a complete timelike vector field Vwhich induces a proper

flow. Certainly, in this case, there is a Lorentzian metric g∗in the conformal class of g, such

that (M, g∗, V )is a proper kinematical spacetime. Then the splitting form of the metric as

above can be pulled back to (M, g)by the conformal transformation. We will make use of

this in the sections and chapters below.

Remark 4.17. In the following, if it is no source of confusion we will often omit the pull-

backs when writing the metric in the splitting form, i.e., we will use the same symbol for

objects on the base Sand their pull-back to R×S, but occasionally we will indicate the

dependence on t∈Rby a subscript t. To every trivialization ψ, there corresponds a unique

family of global sections {σt(S)}t∈R⊂M, which form a foliation of Mby submanifolds

diffeomorphic to S. The images of these submanifolds under ψare given by St:= {t}×S:=

ψ(σt(S)) and we will refer to these submanifolds St⊂R×Sas slices, to the foliation

{St}t∈R⊂R×Sas slicing and to a gauge transformation in the principal bundle, which

corresponds to a change of the trivialization, as a change of slicing. In this sense, we can

then talk about geometric objects on a slice as objects that are pulled back from Sand then

restricted to a specific t∈R. Furthermore, making use of Thm. 4.16 above, it is now

possible to refer to an isometry ψas a trivialization of a proper kinematical spacetime if

it is a trivialization of the associated R-principal bundle and maps ψ: (M, g, V )→(R×

S, ψ∗g, A−1∂t).

Moreover, all possible trivializations can be obtained by specific projections onto R.

Lemma 4.18. Let (Mn+1, g, V )be a proper kinematical spacetime. Every surjective function

f:M→R, with df(V)>0(or df(V)<0) and imf(γ) = Rfor all integral curves γof V,

gives rise to a trivialization ψas in the theorem above, such that for the preimages f−1(t)

of f

ψ(f−1(t)) = St,

with the slices St⊂R×Sholds.

Proof. For all t∈Rthe preimage σt:= f−1(t)⊂Mis an n-dimensional submanifold of

M, transversal to V, due to df(V)6= 0. Together with imf(γ) = R, this also implies that all

integral curves of Vare mapped diffeomorphically to Rby f. Thus t6=simplies σt∩σs=∅

and {σt}t∈Ris a foliation of M. Furthermore, because of imf(γ) = R, every flow line of V

intersects every hypersurface σtexactly once, i.e., all σtare diffeomorphic to S=M/R, the

space of flow lines of V. Hence, the map ψ:M→R×Sgiven by

ψ(p)=(f(p), πM(p)),

for all p∈M, is a trivialization and ψ(σt) = (t, S) = {t}×S=St.

Theorem 4.19. Let (M, g, V )be a Cartan kinematical spacetime and ˙u=g(∇VV, ·)the

acceleration one-form associated to V. If there is a function φ:M→Rsuch that ˙u= dφ,

then (M, e−2φg, eφV)is a proper kinematical spacetime. Particularly, if ˙u= 0, i.e., the

vector field Vis geodesic, then (M, g, V )is a proper kinematical spacetime.

Proof. First, we show the assertion for ˙u= 0. Let u=g(V, ·)be the one-form metrically

associated to V. Then ˜u=−uobeys ˜u(V) = 1. Furthermore, the rotation ωof Vis given

by ω= duas ˙u= 0, so that we conclude Vcd˜u=−Vcω= 0 by Prop. 3.21. Hence, (M, g, V )

is a proper kinematical spacetime by item (ii) of Thm. 4.13.

Second, we assume that ˙u= dφand observe that this implies dφ(V)=0as ∇VVis horizontal

(cf. Prop. 3.21). In particular, we have H(dφ)=dφ. Hence, using Prop. 3.23 to compute

the acceleration one-form ˙u∗associated to the conformally transformed metric g∗=e−2φg

and the vector field V∗=eφVyields ˙u∗= 0. Furthermore, we have g∗(V∗, V ∗) = −1and

due to eφ>0and dφ(V) = 0 the vector field V∗is also complete. Thus, using the same

argument as above, (M, e−2φg, eφV)is a proper kinematical spacetime.

Two remarks are in order concerning this theorem.

Remark 4.20. (i) Obviously, the conclusion also holds if d ˙u= 0 and B1(M) = 0, i.e.,

the first Betti number of Mvanishes, as this implies that there is a function φ:M→R,

such that ˙u= dφ, and again we have dφ(V)=0due to Prop. 3.21.

(ii) Using Rem. 4.8, we can further infer that every Cartan kinematical spacetime with an

exact acceleration one-form is of splitting type as in Thm. 4.16.

Of particular concern is now the following equivalent formulation of some results above. It

gives interesting non-existence results for certain classes of manifolds.

Corollary 4.21. Let (M, V )be a Cartan kinematical manifold and assume M/Ris not

Hausdorff. Then there is no Lorentzian or Riemannian metric gon M, such that Vis a

(timelike) g-geodesic vector field or a (timelike) Killing vector field for g.

Proof. Certainly, there is a Lorentzian metric gon Mwith Vbeing timelike geodesic or

Killing if and only if there is a Lorentzian metric g∗in the conformal class of g, such that V

is timelike geodesic or Killing for g∗with unit norm g∗(V, V ) = −1(cf., e.g., Lem. 4.9).

Assume Vwas geodesic and timelike for some Lorentzian metric g∗on Mand g∗(V, V ) = −1.

Then by Thm. 4.19,(M, g∗, V )would by a proper kinematical spacetime, and hence M/R

would be Hausdorff. In the same way, if Vwas Killing for some Lorentzian metric g∗on M

and g∗(V, V ) = −1, then by Prop. 4.10,(M, g∗, V )would be a proper kinematical spacetime

and hence, M/Rwould be Hausdorff.

Hence, by the same arguments as in the proof of Thm. 4.13,Vis geodesic (or Killing) and

timelike for a Lorentzian metric g∗with unit norm if and only if it is geodesic (or Killing)

for the Riemannian metric g∗+ 2g∗(V, ·)⊗g∗(V, ·)and the Riemannian case follows.

We have now generalized known diffeomorphic splitting results of Lorentzian spacetimes, at

the expense of allowing the restriction h−b⊗b

A2of the Lorentzian metric gto the slices of a

trivialization to be Lorentzian or even degenerate. Hence, the question arises in which cases

we can have Riemannian slices, locally or globally. Of course, locally about a point pin the

proper kinematical spacetime this is always possible, but the following proposition shows

that in particular cases we have also a Riemannian slicing locally about an orbit, i.e., in a

tubular neighborhood of an integral curve of Vin a proper kinematical spacetime.

Proposition 4.22. Let (M, g, V )be a proper kinematical spacetime. For every p∈Mthere

is a trivialization ψand a neighborhood Upabout ψ(p)∈R×S, such that the slices Stin R×S

induced by ψare spacelike in Up, i.e., h−b⊗b

A2|St∩Upis Riemannian, whenever St∩Up6=∅.

If the acceleration and the shear of Vvanish, i.e., ˙u= 0 and σ= 0, then for every integral

curve γof Vthere is a trivialization ψand a tubular neighborhood Tγabout ψ(γ)⊂R×S,

i.e., some set R×W⊂R×Swith Wsome open neighborhood about (pr2◦ψ)(γ)∈S, such

that the slices Stin R×Sinduced by ψare spacelike in Tγ, i.e., h−b⊗b

A2|St∩Tγis Riemannian

for all t∈R.

Proof. Let Φbe the global flow associated to V. We show the assertions for global sections

σ0:S→Mand σt:S→Mwith σt(x) = Φt(σ0(x)) for all x∈Sin the R-principal bundle

(M, S =M/R, πM). These sections are in one-to-one correspondence to the slices Stby

virtue of a trivialization.

Take any p∈Mand consider all unit spacelike vectors g-orthogonal to Vp∈TpM, i.e., the

set Σp:= {Ep∈TpM|gp(Vp, Ep)=0, gp(Ep, Ep)=1}. Choose some open neighborhood

U0about 0∈TpM, such that the exponential map exp: U0→Mis a diffeomorphism onto

its image U= exp(U0)in M. Then for all Ep∈Σpwe have spacelike geodesic segments

αE(t) := exp(tEp)⊂Uthrough p, with tin some interval about 0∈R. Denoting the

tangential vector fields as E= ˙αE, these span a spacelike integral manifold Sp⊂Uthrough

p. As Spis spacelike, it is transversal to the vector field V, and hence can be written as

a local section Sp:πM(U)→M. Because Mis trivializable as a principal bundle, we can

extend Spto a global section σ0:S→M, such that σ0|U=Sp. Thus, we get a foliation of M

by the global sections σt= Φt(σ0). It remains to show that there is an ε > 0, such that σt|U

is spacelike for all t∈(−ε, ε). The tangential vector fields E=E(0) at Spare transported

to the tangential vector fields E(t)=Φt∗E(0) on σtby the flow of Vas σt= Φt(σ0). Thus,

all E(t)are vector fields defined in all of the saturation t◦U, all tangential to the sections

σt|t◦U, respectively. Hence, it holds that

£VE(t) = 0,

for all t∈R. Using Prop. 3.21 for u=g(V, ·), we compute £Vg=2

nΘg+2σ−2 ˙u∨u+2

nΘu⊗u,

such that

∇V[g(E(t), E(t))] = (£Vg)(E(t), E(t)) =

nΘg(E(t), E(t)) + 2σ(E(t), E(t)) −2 ˙u(E(t))u(E(t)) + 2

nΘu(E(t))2

holds. This is an ordinary differential equation for all fixed x∈πM(Sp). For every such

fixed xwe denote X(t) = g(E(t), E(t)),ϕ(t) = σ(E(t), E(t)) −˙u(E(t))u(E(t)) + Θ

nu(E(t))2

and s(t) = Rt

Θ(τ)

ndτfor all t∈Rand arrive at the differential equation

X(t) = 2

nΘ(t)X(t)+2ϕ(t).

The solution of this ordinary differential equation is given by

X(t) = e2s(t)X(0) + Zt

2ϕ(τ)e−2s(τ)dτ.

Hence, as the vector fields E=E(0) are spacelike, i.e., X(0) >0, there certainly is some

interval Iabout 0∈Rsuch that X(t)>0for all t∈I.

Furthermore, in the case ˙u= 0 and σ= 0, we consider for every fixed x∈πM(Sp)the

function

α(t) = [g(V, E(t)]2

g(E(t), E(t))

along the orbit γx≃Rover x∈S. As such, αis a function α:R→[0,∞)for all spacelike

vector fields E(t)along the orbit, and it measures the Lorentzian angle between E(t)and

V. It becomes infinite if E(t)becomes lightlike somewhere, but is certainly defined in some

neighborhood of 0∈R, as the vector fields E(0) are considered spacelike on Spin our case.

Hence, if α(t)is defined for all t∈R, i.e., it remains finite for finite times, the vector fields

E(t)remain spacelike in the whole tubular neighborhood t◦Uof the orbit, and so do the

local sections σt|t◦U.

So we compute ∇V[g(V, E(t))] = (£Vg)(V, E(t)) = 0, as ˙u= 0 and ∇V[g(E(t), E(t))] =

(£Vg)(E(t), E(t)) = 2

nΘ[g(E(t), E(t)) + g(V, E(t))2]. This yields the following differential

equation for α(t):

˙α(t) = −α2(t)−2

nΘα(t)

along the orbit γx. As E(0) is assumed spacelike and geodesic, we have α(0) = [g(V, E(0))]2:=

c2. Note that all vectors in the vector field E(0) with g(V, E(0)) = 0 certainly remain space-

like and even perpendicular to V, due to ∇V[g(V, E(t))] = 0. Hence, we only have to

consider initial values c2>0for the differential equation in α. The solution of which is, in

this case, given by

α(t) = e−Rt

nΘ(s)ds

0e−Rs

nΘ(τ)dτds+1

<∞,

for all t∈R.

This proposition will have interesting consequences for the causal properties of shear-free and

acceleration-free proper kinematical spacetimes. These are the Hubble-isotropic spacetimes,

which will be a subject in chapter 6.

4.2 Causality of Splitting Spacetimes

In this section, we assume dim(S)≥1for kinematical spacetimes (R×S, g, V ). The spe-

cial low dimensional case dim(S)=1, together with particularities in the situation when

dim(S)=2or dim(S)=3, for compact Sis the subject of the following section 4.3.

We analyze the possible causality conditions for proper kinematical spacetimes in detail. We

will derive various conditions on the spacetimes for certain steps on the ladder of causality.

We will also determine the causality conditions which cannot (generally) hold for a proper

kinematical spacetime of arbitrary dimension.

Due to the considerations in the previous section, we can now view proper kinematical space-

times in the following way: By virtue of a trivialization, a proper kinematical spacetime is

diffeomorphic to R×S, with Vparallel to ∂tand the push-forward of the metric gto R×S

is given by Thm. 4.16. This push-forward is only fixed up to global gauge transforma-

tions in the R-principal bundle, and the transformation of the metric under such a gauge

transformation—a change of slicing—is again given in Thm. 4.16. These gauge transforma-

tions correspond to the choice of another trivialization of the principal bundle associated

to (M, g, V ). Naturally, the slices St⊂R×Sare pulled back to a foliation {σt}t∈Rof

M, and any such foliation of M, given as the preimages of some function t:M→R, give

rise to a specific trivialization according to Lem. 4.18. Furthermore, as all causality condi-

tions are conformally invariant, it is, here and in the following section, enough to consider

proper kinematical spacetimes for which ψ∗V=∂tholds, with respect to a trivialization

ψ:M→R×S. Here and in the following, we will often omit the pull-back pr∗

2when writing

the metric if this is no source of confusion. Hence, we can fix a trivialization ψ:M→R×S,

such that the proper kinematical spacetime is given as (R×S, g, 1

A∂t)with

g=−A2dt⊗dt+ 2b∨dt+h−b⊗b

and A, b, h as in Thm. 4.16, but then analyze the conformally transformed metric g0=1

A2g,

i.e., the proper kinematical spacetime (R×S, g0, ∂t)with

g0=−dt⊗dt+ 2b0∨dt+h0−b0⊗b0,

for which b0=1

A2band h0=1

A2hholds. All propositions on causality conditions hold for

every member in the conformal class of the Lorentzian metric g. This implies that the

results on causality conditions in this section also hold for spacetimes which are conformal

to a proper kinematical spacetime, i.e., spacetimes (M, g)together with a not necessarily

complete reference frame V, such that Mis diffeomorphic to R×Sand Vpoints along the

factor Rwith respect to the diffeomorphism.

Certainly, we have the following

Lemma 4.23. Let (R×S, g, ∂t)be a spacetime, together with the canonical timelike vector

field ∂t, such that

g(t,x)=−dt⊗dt+ 2pr∗

2(b(t,x))∨dt+ pr∗

2(h(t,x))−pr∗

2(b(t,x))⊗pr∗

2(b(t,x)),

for a family of one-forms {bt}t∈R⊂Γ(Λ1S)and a family of Riemannian metrics {ht}t∈R⊂

R(S), as well as the canonical projection pr2:R×S→Sand t:R×S→R. Then ∂t

is complete, particularly (R×S, g, ∂t)is a proper kinematical spacetime. Furthermore, the

induced metric

g(t,x)|St0=g(t0,x)= pr∗

2(h(t0,x))−pr∗

2(b(t0,x))⊗pr∗

2(b(t0,x))

on the slice St0is Riemannian at some point (t0, x)∈St0if and only if the ht0-norm of the

one-form bt0on Sis strictly smaller than 1at x∈S, i.e.,

kbt0kht0

x:= sup

v∈TxS\{0}

|b(t0,x)(v)|

qh(t0,x)(v, v)

<1.

Proof. Certainly, the transformations ϕt:R×S→R×Sgiven by ϕt(s, x)=(s+t, x)exist

for all t∈R, thus as ∂t= ˙ϕt= (1,0), the vector field ∂tis complete. Furthermore, we have

g(∂t, ∂t) = −1, and therefore, (R×S, g, ∂t)is a proper kinematical spacetime. Moreover, we

have that ht0−bt0⊗bt0is positive definite at x∈Sif and only if

h(t0,x)(v, v)−b(t0,x)(v)2>0,

for all v∈TxS\{0}, i.e., if and only if b(t0,x)(v)2

h(t0,x)(v,v)<1for all v∈TxS\{0}, i.e., if and only if

|b(t0,x)(v)|

qh(t0,x)(v, v)

<1,

for all v∈TxS\{0}. This is equivalent to

|b(t0,x)(v)|<1,

for all v∈TxSobeying h(t0,x)(v, v)=1. As {v|h(t0,x)(v, v)=1} ⊂ TxSis compact, the

supremum exists and is obtained as a maximum smaller than one.

Furthermore, we will use the following

Lemma 4.24. Let (R×S, g, ∂t)be a proper kinematical spacetime with metric

g=−dt⊗dt+ 2bt∨dt+ht−bt⊗bt

and {bt}t∈R,{ht}t∈Ra family of one-forms and Riemannian metrics on S, respectively. For

all (t, x)∈R×Sand all orbits R×{y}, there is a future-directed and a past-directed timelike

curve connecting (t, x)to R× {y}, i.e., for all (t, x)∈R×Sand all y∈Sthere are two

numbers T, ˜

T∈R, as well as a future-directed timelike curve λ: [0,1] →R×Sand a past-

directed timelike curve ˜

λ: [0,1] →R×S, such that λ(0) = ˜

λ(0) = (t, x),λ(1) = (T, y)and

λ(1) = ( ˜

T, y).

Proof. We will prove the future-directed case only. The past-directed case works completely

analogous. Without loss of generality we can assume t= 0. Given x, y ∈S, fix any

curve c: [0,1] →Sobeying c(0) = xand c(1) = y. Consider the lift L={(τ, c(s)) |τ≥

0, s ∈[0,1]} ⊂ R×S, which is a two-dimensional submanifold with boundary in R×S.

Then the velocity ˙cof cin Slifts to a vector field (0,˙c)tangential to L. Set the function

η:L≃R≥0×[0,1] →Rto be

η(τ, s) := qh(τ,c(s))(˙c(s),˙c(s)) + b(τ,c(s))(˙c(s)).

As [0,1] is compact, there exists a function ξ:R≥0→Rdefined by

ξ(τ) = sup

s∈[0,1]

η(τ, s)<∞,

for all τ≥0and we set the vector field Ztangential to Lto be

Z(τ, s)=(ξ(τ)+1,˙c(s)).

Then the integral curve λ: [0,1] →Lof Zobeying λ(0) = (0, x)eventually intersects R×{y}

for s= 1 at the point (T, y)with T=R1

0(ξ(τ) + 1)dτ. It remains to show that the vector

field Zis timelike and future-directed. We compute

g(τ,c(s))(∂t, Z(τ, s)) = −ξ(τ)−1 + b(τ,c(s))(˙c(s)) ≤ −qh(τ,c(s))(˙c(s),˙c(s)) −1<0

as b(τ,c(s))(˙c(s)) ≤ξ(τ)−qh(τ,c(s))(˙c(s),˙c(s)) for all (τ, s)∈R≥0×[0,1], as well as

g(τ,c(s))(Z(τ, s), Z(τ, s)) = −(ξ(τ) + 1) −b(τ,c(s))(˙c(s))2+h(τ,c(s))(˙c(s),˙c(s)).

We observe that, due to the calculation above, g(Z, Z)<0is equivalent to

qh(τ,c(s))(˙c(s),˙c(s)) <(ξ(τ) + 1) −b(τ,c(s))(˙c(s)),

for all (τ, s)∈R≥0×[0,1], which is always true by the construction of ξ. Hence, Zis timelike

and future-directed, and so are all its integral curves.

Remark 4.25. Due to the considerations at the beginning of this section, Lem. 4.24 certainly

also holds for a general proper kinematical spacetime (M, g, V )by pull-back, and because the

property of being timelike is conformally invariant. Hence, any arbitrary point pin (M, g, V )

can be connected to any arbitrary integral curve of Vby a future-directed and a past-directed

timelike curve.

Proposition 4.26. A proper kinematical spacetime (M, g, V )is stably causal if and only if

there is a codimension one foliation (necessarily transversal to V){σt}t∈Rof M, such that

all leaves are spacelike.

Proof. “⇐”: As (M, g, V )is a proper kinematical spacetime and we are interested in the

causality condition of stable causality, we can assume, using the considerations at the be-

ginning of this section, that there is a trivialization, such that the kinematical spacetime is

given by (R×S, g, ∂t)with

g=−dt⊗dt+ 2bt∨dt+ht−bt⊗bt.

The existence of the spacelike foliation transversal to Vin M, then amounts to the slices

St⊂R×Swith the induced metric

g|St=ht−bt⊗bt

being Riemannian, i.e., kbtkht<1on all of R×S, by Lem. 4.23. In the following we suppress

the index tat the objects band h, if this is no source of confusion. Now, let λ:I→R×S

be any future-directed causal curve in R×S, with I⊂Rsome interval. Then we have

λ(s)=(t(s), x(s)) and ˙

λ= (˙

t, ˙x), where s∈Iand the dot means the derivative with respect

to the curve parameter s. As λis assumed future-directed, we have

g(∂t,˙

λ) = −˙

t+b( ˙x)<0⇒b( ˙x)<˙

As λis assumed causal, we have

g(˙

λ, ˙

λ) = −˙

t2+ 2b( ˙x)˙

t+h( ˙x, ˙x) + b( ˙x)2=−(˙

t−b( ˙x))2+h( ˙x, ˙x)≤0,

hence ph( ˙x, ˙x)≤˙

t−b( ˙x). Because of kbtkht<1, this yields

0<ph( ˙x, ˙x) + b( ˙x)≤˙

Thus ˙

tcould only be zero if ˙x= 0. But in these cases, ˙

tcertainly cannot be zero as otherwise

the curve λwould be constant. So, we have ˙

t > 0on all of λ, i.e., the function t:R×S→R

is strictly increasing along every future-directed causal curve in R×Sand thus the spacetime

is stably causal.

“⇒”: Recalling the definition of stable causality, Def. 3.48, we can state the following. If

the proper kinematical spacetime (M, g, V )is stably causal, there is a temporal function

t:M→R, i.e., a smooth function with past-directed timelike gradient ∇t. Particularly,

this means g(V, ∇t)=dt(V)>0and g(∇t, ∇t)>0. As Mis connected, imt(M)is some

connected and open (possibly unbounded) interval I⊂R. Thus, by composition with a

strictly increasing diffeomorphism I→R, we can modify tsuch that imt(M) = Rholds.

Employing Lem. 4.24 and Rem. 4.25, we can infer that for all integral curves γof Vit

must hold that t(γ) = R. Assume there is some p∈M, such that the integral curve γp

through pobeys sup t(γp) = k < ∞for some k∈R. As imt(M) = R, there is some q∈M,

such that t(q) = k+ 1. But as we can connect qto γpby a future- directed timelike curve

and tis certainly non-decreasing along this curve, there is some r∈γpwith t(r)> k, in

contradiction. The same argument applies if we assume t(γp)to be bounded from below

and use a past-directed timelike curve. Due to Lem. 4.18, the function tthen gives rise to a

codimension one foliation {σt}t∈Rof the proper kinematical spacetime (M, g, V ). Certainly,

the gradient ∇tis g-orthogonal to the leaves σteverywhere, and as ∇tis timelike, the leaves

are spacelike.

We recall Def. 3.37 and can state the following

Proposition 4.27. A spacetime (M, g)is non-imprisoning if and only if for all relatively

compact open sets U⊂M, the restricted spacetime (U, g|U)is stably causal.

Proof. This proposition has already been proven by J.K. Beem in [Bee76]. See also [Min09a,

Thm. 1].

For proper kinematical spacetimes, Prop. 4.27 has the following consequence.

Proposition 4.28. A proper kinematical spacetime (M, g, V )is non-imprisoning if and only

if for all compact sets K⊂M(with non-empty, open interior) there is a foliation {σt}t∈Rof

M, such that the partial leaves σt∩˚

Kare spacelike for all t∈R, for which this intersection

is non-empty.

Proof. “⇐”: Let U⊂Mbe a relatively compact set. Then there certainly is a compact

set Kwith U⊂K⊂Mobeying the following condition. Let ψ:M→R×Sbe the

trivialization of (M, g, V )associated to the foliation {σt}t∈Rof M, i.e., ψmaps the leaves

σt⊂Mto the slices St⊂R×S, such that ψ(K) = I×Lwith L⊂Scompact and I⊂Ra

compact interval. Thus the partial slices Lt:= St∩ψ(K)for t∈˚

Iare also spacelike in their

interior. Then we can regard (˚

ψ(K), ψ∗g|˚

ψ(K))=(˚

I×˚

L, ψ∗g|˚

I×˚

L)as conformal to a proper

kinematical spacetime, maybe after applying an isometry, which maps ˚

Idiffeomorphically

to R(see, e.g., the construction below for details). Applying Prop. 4.26 to this restricted

spacetime shows that it is stably causal and so is its pull-back to Mby ψ. Hence, (U, g|U)

is also stably causal as U⊂Kand the result follows from Prop. 4.27.

“⇒”: Assuming (M, g, V )to be non-imprisoning, it follows from Prop. 4.27 that for all

relatively compact open sets U⊂M, the restricted spacetime (U, g|U)is stably causal.

Particularly, we can assume Uto obey the following condition. There is a trivialization

ψ:M→R×Sof (M, g, V ), such that ψ(U) = J×W⊂R×Swith J⊂Ra relatively

compact open interval and W⊂Sa relatively compact open subset. Certainly, there is a

diffeomorphism ρ:J×W→R×W, which maps the slices Wt={t}×W⊂J×Wto slices

Wτ:= ρ(Wt)⊂R×W, such that R3τ= (pr1◦ρ)(t, x)for all (t, x)∈J×W, i.e., ρstems

from a diffeomorphism between Jand R. As R×W, together with the metric pushed forward

by the composition ρ◦ψ|U, can be regarded as conformal to a proper kinematical spacetime,

we can apply Prop. 4.26. Hence, there is a foliation {Rτ}τ∈Rof R×Wby spacelike leaves.

Pulling back these leaves by ρto slices in J×W, i.e., considering Pt:= ρ−1(Rτ)⊂J×W,

yields the {Pt}t∈Jas a family of spacelike sections in R×Sover W⊂S. As Wis relatively

compact, we can consider the family of closed sections {Pt}t∈Jover the closed W. The

extension of Ptto Ptis always possible for all t∈Jas the set U⊂M, with which we

started, can be chosen arbitrarily large, as long as it is relatively compact. Then by [KN63,

Thm. 5.7, p. 58] we extend the closed sections to global ones {St}t∈Jwith Pt⊂St. Hence,

the global sections Stare spacelike for all (t, x)∈Stfor which x∈Wholds. Assume

J= (a, b)⊂R. Now we choose some small ε > 0and set the slices Stfor t>b−εto be the

saturation of the section Sb−εby the principal action in R×Sand slices Stfor t < a +ε

to be the saturation of the section Sa+εby the principal action in R×S. Pulling these

slices back to Myields leaves σt:= ψ−1(St)of a foliation {σt}t∈Rwhich are spacelike in the

interior of the compact set ψ−1([a+ε, b −ε]×W)⊂U, which is only marginally smaller

(depending on how small εis chosen) than U. As we can choose Uarbitrarily large, as long

as it is relatively compact, the result follows.

Having this proposition at hand, we are now able to prove the following theorem, which will

substantially simplify the causal ladder for proper kinematical spacetimes.

Theorem 4.29. Let (M, g, V )be a non-imprisoning, proper kinematical spacetime. Then

(M, g)is reflecting. Particularly, if (M, g)is feebly distinguishing, it is also causally contin-

uous.

Proof. Following the considerations at the beginning of this section, we can assume (M, g, V ) =

(R×S, g, ∂t)with

g=−dt⊗dt+ 2bt∨dt+ht−bt⊗bt

under omission of the pull-back operation pr∗

2. Furthermore, for any compact interval I⊂R

and any compact K⊂Swith open interior, using Prop. 4.28, we can assume that the

induced metric

γ(t,x)=h(t,x)−b(t,x)⊗b(t,x)

on the slice Stis Riemannian, i.e., kbtkht

x<1by Lem. 4.23, if (t, x)∈˚

I×˚

K. This can always

be achieved by the following steps: For a compact L⊂R×Sobeying I×K⊂˚

L, there

is a foliation {σt}t∈Rof R×Sas in Prop. 4.28 with Las the compact set, as we assumed

the spacetime to be non-imprisoning. Then there is a change of slicing ψ: (R×S, g, ∂t)→

(R×S, ψ∗g, ∂t)obeying ψ(σt) = Stand we use the spacetime (R×S, ψ∗g, ∂t).

We will denote the principal action in R×Sby the global flow Φassociated to ∂t, which

operates by Φs(t, x)=(t+s, x)for all s∈Rand all (t, x)∈R×S.

We now have to show that for any two points (t, x)∈R×Sand (s, y)∈R×S

I+((t, x)) ⊃I+((s, y)) ⇔I−((t, x)) ⊂I−((s, y))

holds. We will only prove the implication “⇒” explicitly, as the proof of “⇐” works com-

pletely analogous. We certainly have that (s+ε, y)∈I+((s, y)) for all ε > 0. Hence, assum-

ing I+((s, y)) ⊂I+((t, x)), there is some future-directed timelike curve λε: [0,1] →R×S

with λε(0) = (t, x)and λε(1) = (s+ε, y)for all ε > 0. Now, we choose some compact

I×K⊂R×S, such that (t, x),(s, y)∈I×Kand also λε0(τ),Φ−ε0(λε0(τ)) ∈I×Kfor all

τ∈[0,1] with some fixed ε0>0. This is always possible as the intervals [−ε0, ε0]⊂Rand

[0,1] ⊂Rare compact and so is the image Φ[−ε0,ε0](λε0([0,1])) ⊂R×S. Then we certainly

have that also λε(τ),Φ−ε(λε(τ)) ∈I×Kfor all τ∈[0,1] and all ε > 0obeying ε<ε0.

Now for all ε∈(0, ε0), we define a curve ˜µε: [0,1] →I×Kby ˜µε(τ) = Φ−ε(λε(τ)) for all

τ∈[0,1]. This curve obeys ˜µε(0) = Φ−ε(t, x)=(t−ε, x)and ˜µε(1) = Φ−ε(s+ε, y)=(s, y).

Thus, by setting µε(τ) = ˜µε(1 −τ), we get a curve µε: [0,1] →I×Kobeying µε(0) = (s, y)

and µε(1) = (t−ε, x). We will now show that µεis timelike and past-directed in all its

domain if εis small enough.

We denote λε(τ)=(T(τ), c(τ)) for some function T: [0,1] →Rwith T(0) = tand T(1) =

s+ε, as well as a curve c: [0,1] →Swith c(0) = xand c(1) = y. Denoting the derivative

with respect to the curve parameter τwith a dot, we get ˙

λε= ( ˙

T, ˙c)as a tangential vector

field at the curve λε. As λεis assumed to be future-directed we get

g(∂t,˙

λε) = −˙

T+b(˙c)<0⇔˙

T > b(˙c)

and as it is timelike we get

g(˙

λε,˙

λε) = −˙

T2+ 2 ˙

Tb(˙c) + h(˙c, ˙c)−b(˙c)2<0.

These two conditions together yield ˙

T > ph(˙c, ˙c) + b(˙c). Furthermore, as the slices in the

compact set I×Kare Riemannian, we can infer that

1>sup

(t,x)∈I×Kkbtkht

x>|b(˙c)|

ph(˙c, ˙c),

and hence there is a constant 1> k > 0, such that |b(˙c)|< kph(˙c, ˙c), which yields

T > ph(˙c, ˙c) + b(˙c)≥0.

Therefore, ˙

T > 0and we can reparametrize λεover the interval [t, s +ε], which yields

λε(τ)=(τ, c(τ)) and

1>qh(τ,c(τ))(˙c(τ),˙c(τ)) + b(τ,c(τ))(˙c(τ)) =: η(τ),

for all for τ∈[t, s +ε]. Hence, there is a constant η0with 1> η0= supτ∈[t,s+ε0]η(τ)and

we set

δ:= 1−η0

2>0,

such that surely η(τ) + δ < 1holds for all τ∈[t, s +ε]. Now consider the reparametrized

curves

˜µε(τ)=Φ−ε(λε(τ)) = (τ−ε, c(τ)),

for τ∈[t, s +ε]and ε0>ε>0. The tangential vector field is given by ˙

˜µε= (1,˙c), such

that ˜µεis timelike if

qh(τ−ε,c(τ))(˙c(τ),˙c(τ)) + b(τ−ε,c(τ))(˙c(τ)) <1,(∗)

for all τ∈[t, s +ε]. For all τ∈[t, s +ε]and all ε∈[0, ε0)there is some i(τ, ε)≥0, such

that qh(τ−ε,c(τ))(˙c(τ),˙c(τ)) + b(τ−ε,c(τ))(˙c(τ)) ≤η(τ) + i(τ, ε),

with i(τ, 0) = 0, and thus

qh(τ−ε,c(τ))(˙c(τ),˙c(τ)) + b(τ−ε,c(τ))(˙c(τ)) ≤η(τ) + I(ε),

with 0≤I(ε) := supτ∈[t,s+ε]i(τ, ε). The functions i(τ, ε)and I(ε)can be assumed to vary

continuously with the variables τand ε, as the Riemannian metrics htand the one-forms bt

vary smoothly with t. Hence, as I(0) = 0, there is some specific ε > 0, such that I(ε)< δ,

which implies that (∗) holds and hence ˜µεis timelike. Certainly, ˜µεis also future-directed,

and as µεis just ˜µεwith reversed orientation, it is past-directed and timelike. As this holds

for an arbitrarily small ε > 0and the chronological relation is transitive (cf. Def. 3.35),

we have I−((t, x)) ⊂I−((s, y)).

Furthermore, the remaining assertion follows from Def. 3.49.

The theorem above implies that for proper kinematical spacetimes the steps distinction,

non–partial-imprisonment, strong causality and stable causality are not separable, i.e., either

the spacetime is causally continuous or it is not even feebly distinguishing. Moreover, the

appearance of causal continuity is controlled by the existence of partially imprisoned causal

curves. This leads to the fact that in proper kinematical spacetimes all causal steps, up to

causal continuity, are directly determined by the non-existence of specific types of “badly

behaving” causal curves. As usual, the proper kinematical spacetime is chronological, causal

or non-imprisoning if there is no closed timelike, closed causal or imprisoned causal curve,

respectively, but, particularly, it is causally continuous if there is no partially imprisoned

causal curve.

Recalling Prop. 3.43, we can deduce the following corollary from Thm. 4.29.

Corollary 4.30. Let (M, g, V )be a non-imprisoning, proper kinematical spacetime. Then

there is a homeomorphism ϕ:M→R×S, such that the sets σt=ϕ−1(St),t∈Rare

achronal topological hypersurfaces, i.e., there is an achronal, topological foliation of (M, g).

Proof. As (M, g)is reflecting by Thm. 4.29, we can deduce from Prop. 3.43 that there is

a continuous semi-time function t:M→R, which is strictly increasing along every future-

directed timelike curve. As Mis connected, imt(M)is, in general, some open and connected

(possibly unbounded) interval I⊂R. Modifying tby composition with a strictly increasing

homeomorphism I→R, yields a semi-time function t:M→Robeying imt(M) = R.

Now an argument just as in the proof of Prop. 4.26 and similar to Lem. 4.18 applies. As

any integral curve γpof Vthrough p∈Mis timelike, we certainly have t(r)< t(s)for all

rs∈γ(p), but we also have t(γp) = Rfor all p∈M. Because, assume sup t(γp) = k < ∞,

then there is some q∈Mwith t(q) = k+ 1, as imt(M) = R, but due to Lem. 4.24 and

Rem. 4.25, there is a future-directed timelike curve connecting qto γp, which implies the

existence of some r∈γpwith t(r)> k + 1, as tis strictly increasing along timelike curves,

in contradiction. The same argument applies, mutatis mutandis, if we assume t(γp)to be

bounded from below.

Thus for all c∈R, the preimage σc:= t−1(c)is the disjoint union of one and only one

point from each integral curve of Vin (M, g, V ). As tis continuous and strictly increasing

along timelike curves, the σcare achronal topological hypersurfaces, homeomorphic to the

manifold of flow lines S=M/R. Then setting

ϕ(p)=(t(p), πM(p))

yields a homeomorphism ϕ:M→R×Sobeying ϕ(σc) = {c}×S=Scfor all c∈R, and

{σc}c∈Ris an achronal, topological foliation of (M, g).

It is known that the higher steps on the causal ladder, causal simplicity and particularly

global hyperbolicity, are controlled, in many important cases, by the completeness of Rie-

mannian or Finslerian metrics on specific hypersurfaces (cf. chapters 5and 6below). If we

consider the case of a compact manifold Sfor a proper kinematical spacetime (R×S, g, V ),

the situation is much simpler and global hyperbolicity can be assessed without referring to

the completeness of specific metrics, as the following proposition shows.

Proposition 4.31. Let (M, g, V )be a non-imprisoning, proper kinematical spacetime with

the manifold of flow lines S=M/Rof Vbeing compact, then (M, g)is globally hyperbolic.

Proof. Assume ϕ:M→R×Sis a homeomorphism as in Cor. 4.30, such that the σt=

ϕ−1({t} × S)are achronal topological hypersurfaces for all t∈R. Recalling Def. 3.53

and Thm. 3.54, it is enough to show that every inextendible timelike curve intersects the

hypersurface σT, for some fixed T∈R, to infer that σTis a Cauchy hypersurface and hence

(M, g)is globally hyperbolic.

Assume λ:R→Mis an inextendible timelike curve and assume that λis future-directed

with t(0) < T, but λdoes not intersect σT. The proof for past-directed curves and t(0) > T

works completely analogous and will be omitted. As tis strictly increasing along λwe

can infer that λ(R≥0)⊂ϕ−1([t(0), T]×S). As Sis compact and ϕis a homeomorphism,

ϕ−1([t(0), T]×S)⊂Mis compact, hence λis future imprisoned in a compact set, in

contradiction.

4.3 The Low-dimensional Cases

Based on Prop. 4.31 in the previous section, we will try to answer in this section the following

question: Let (M, g, V )be a proper kinematical spacetime with a compact space of flow

lines S=M/R. When does the condition of (M, g)being causal already, imply the global

hyperbolicity of (M, g)? Certainly, this is equivalent to establishing when such a spacetime

is non-imprisoning if it is causal. We will assess this question for dim(S)∈ {1,2,3}.

In this section, we will often impose the condition of a smooth semi-time function on the

spacetimes under consideration. Recall that a semi-time function t:M→Rfor a spacetime

(M, g)obeys t(p)< t(q)whenever pq. But in this case one assumes tto be, not only

continuous, but also smooth. Certainly, if there exists a semi-time function then the space-

time is chronological. But the circumstances that assure the existence of smooth semi-time

functions are not clear, as reflectivity plus chronology only leads to a continuous function

(cf. Prop. 3.43). However, there are strong indications that in a proper kinematical space-

time (M, g, V )a smooth semi-time function exists in many important cases. Particularly,

if (M, g)is chronological and if, maybe additionally, the space of flow lines S=M/Ris

compact and (M, g)is reflecting (cf. Conj. 7.1, as well as Prop. 7.2 in chapter 7and the

discussion thereafter).

If dim(S) = 1 and Sis compact, we can assume S=S1, and a spacetime (R×S1, g)is a

Lorentzian cylinder. Two-dimensional spacetimes possess the particular feature that with

gbeing a Lorentzian metric, also −gis a Lorentzian metric, while spacelike and timelike

directions are exchanged.

Theorem 4.32. Let (R×S1, g, ∂t)be a proper kinematical Lorentzian cylinder spacetime.

Then one and only one of the following three cases holds.

(i) (R×S1, g)is totally vicious and (R×S1,−g)is globally hyperbolic.

(ii) (R×S1, g)and (R×S1,−g)are non–totally-vicious but non-causal.

(iii) (R×S1, g)is globally hyperbolic and (R×S1,−g)is totally vicious.

Proof. We choose global coordinates (t, x mod 1) ∈R×S1, such that the canonical vector

field ∂t∈Γ(T(R×S1)) is the reference frame of the proper kinematical spacetime and the

canonical vector field ∂x∈Γ(T(R×S1)) induces closed curves, which are not homotopic to

a point. Following Thm. 4.16, we can write the metric gin the following way:

g(t,x)=−dt⊗dt+ 2β(t, x)dt∨dx+ (η(t, x)−β2(t, x))dx⊗dx,

with β:R×S1→Rand η:R×S1→R>0functions, periodic in x, generating one-forms

bt=β(t, x)dxand Riemannian metrics ht=η(t, x)dx⊗dxon S1for all t∈R.

If (R×S1, g)is globally hyperbolic, it is in particular stably causal and by Prop. 4.26 the

slices St={t}×S1are spacelike for all t∈R, i.e, η(t, x)> β2(t, x)for all (t, x)∈R×S1

(maybe after employing a change of slicing). Hence, for the Lorentzian metric

−g= dt⊗dt−2βdt∨dx−(η−β2)dx⊗dx

all integral curves of ∂x, through all points, are CTCs, i.e., (R×S1,−g)is totally vicious

in this case. If we start from global hyperbolicity of (R×S1,−g), we get total viciousness

of (R×S1, g)in the same way. This shows the existence of the cases (i) and (iii) and their

mutual exclusion.

Furthermore, this implies that whenever (R×S1, g)or (R×S1,−g)is globally hyperbolic

or totally vicious, the other one of both must obey the opposite causality condition. Thus,

whenever neither (i) nor (iii) holds, (R×S1, g)and (R×S1,−g)must be both non–totally-

vicious and non–globally-hyperbolic. As S1is compact, employing Prop. 4.31, this implies

that both spacetimes are also causally imprisoning. It remains to show that a Lorentzian

cylinder spacetime is non-causal if it is causally imprisoning.

Let µ:R→R×S1be an inextendible, causal and future-directed curve, which is imprisoned

in a compact set, say K=I×S1⊂R×S1for some compact interval I⊂R. Employing

Prop. 3.39, we can infer that the limit set ω(µ)is compact. Furthermore, we can assume

that (R×S1, g)is chronological, because otherwise there is nothing to show. This implies

that we can employ Prop. 3.4 from [Min08a], which assures that in the chronological case

through all p∈ω(µ), there passes exactly one inextendible lightlike curve that is completely

contained in ω(µ), and is, therefore, also future imprisoned. Let λ:R≥0→R×S1be such a

future imprisoned, lightlike curve, hence, λ(R>0)⊂K. This yields that pr2(λ(R>0)) = S1,

i.e., λintersects every partial orbit I×{x}(x∈S1) in K. This can be seen as follows: We

can write λ(s) = (t(s), c(s)) for s≥0and c(s)is not constant in any finite interval of the

curve parameter, as otherwise λwould be identical, in this interval, to an integral curve of

∂t, which is timelike. As c:R≥0→S1, we can infer that ctakes all values in S1, eventually

even infinitely often, due to the fact that c(s)∈S1, and thus it is periodic.

Now, we analyze the points for which λintersects a specific partial orbit, say I×{x}for some

x∈S1. Let sn(x)∈Rfor all n∈Nbe the curve parameters for which pr2(λ(sn(x))) = x

holds. Now we can assume t(sn+1(x)) > t(sn(x)) for all n∈N, because otherwise we had

already found a closed causal curve by combining λ: [sn(x), sn+1(x)] →Kwith the orbit

piece [t(sn+1(x)), t(sn(x))]×{x}, which would yield a closed causal curve. Thus the sequence

{λ(sn(x))}n∈N⊂I×{x}is strictly increasing and bounded from above on the fiber, such

that λ(sn(x)) converges to some point on I×{x}, which certainly is contained in ω(λ). As

this holds for all x∈S1, we can infer that ω(λ)⊃ {limn→∞(t(sn(x)), x)|x∈S1}. But this

set inclusion is even an equality. This can be seen as follows: Assume (t0, y)∈ω(λ)and

let {σn}n∈NsubsetRbe a strictly increasing sequence, such that λ(σn)→(t0, y)as n→ ∞.

Then, as c(s)is never constant, there is some τn∈R, such that λ(σn−τn)∈I×{y}for

all n∈N. But certainly τn→0as n→ ∞, such that λ(σn−τn)→(t0, y)as n→ ∞ and

{λ(σn−τn)}n∈Nis the sequence in the partial fiber I×{y}, which ensures the equality of

the two sets above.

Therefore, ω(λ)is a section over S1in K⊂R×S1, as for all x∈S1there is exactly one

point on I× {x}in ω(λ). Now, we can again employ Prop. 3.39 to the limit set ω(λ)

and deduce that through all p∈ω(λ)there passes exactly one inextendible lightlike curve

completely contained in ω(λ). This is only possible if ω(λ)is a continuous section generated

by exactly one lightlike curve, which is, therefore, a closed causal curve, i.e, the spacetime

is non-causal. This implies that the only remaining situation, which can happen if neither

(i) nor (iii) holds, is item (ii).

Hence, for dim(S)=1and compact S, causality of (R×S1, g, ∂t)does certainly imply global

hyperbolicity, because the only case that involves a non-causal Lorentzian cylinder, and none

of a higher level of causality, is item (ii) in the theorem above. Hence, a causal Lorentzian

cylinder enforces items (i) or (iii) to hold and, therefore, it is a globally hyperbolic spacetime,

either with metric gor −g.

For dim(S)=2and compact S, the situation is a bit more involved. We can infer from

Example 4.12 that there is a proper kinematical spacetime which is causal, but causally

imprisoning, if Sis a 2-torus. But we will show in the following that this cannot happen if

Sis not a torus. To this end we need a suitably strong version of the Poincaré–Bendixson

theorem. We recall Def. 2.54, as well as Rem. 2.55 and state the following

Theorem 4.33. Let Sbe a compact, 2-dimensional manifold, which is at least of class C2

and not a torus. Let Φ: R×S→Sbe an R-action on S, which is also at least of class

C2and constitutes a dynamical system on S. Then every Φ-minimal set A⊂Sis either

a fixed point or a closed orbit homeomorphic to S1. Furthermore, let Sbe orientable and

γbe an orbit of Φ, such that ω(γ)is non-empty and contains no fixed points, then ω(γ)is

homeomorphic to S1.

Proof. See the theorem and the corollary in [Sch63].

We would like to apply this version of the Poincaré–Bendixson theorem to a spacetime in

the theorem below. Hence, we will assume that the spacetime is at least of class C2and all

geometric objects are C2-smooth in the following

Theorem 4.34. Assume (M, g, V )is a proper kinematical spacetime of class C2, with the

space of flow lines S=M/Rcompact and orientable, dim(S) = 2 and admitting a smooth

semi-time function. If (M, g)is causal and Sis not a torus, (M, g)is globally hyperbolic.

Proof. Let t:M→Rbe the smooth semi-time function, which we can assume surjec-

tive (maybe by composition with a strictly increasing diffeomorphism). A chain of ar-

gument completely analogous to Lem. 4.18 and Prop. 4.26 yields a smooth trivialization

ψ: (M, g, V )→(R×S, ψ∗g, ψ∗V), such that the slices Stare non-Lorentzian everywhere,

and they are even smooth achronal hypersurfaces, due to the existence of a smooth semi-time

function. Specifically, this yields that any sort of causality violation has to be generated by

some lightlike curve, totally contained in one slice.

As we are only interested in causal properties we can consider the conformal class of the

Lorentzian metric gand assume

ψ∗g=−dt⊗dt+ 2bt∨dt+ht−bt⊗bt,

such that {bt}t∈Ris a family of one-forms on Sand {ht}t∈Ris a family of Riemannian

metrics, with the usual omittance of the pull-back pr∗

2. As the slices are achronal, we infer

completely analogous to Lem. 4.23 that kbtkht≤1for all t∈R, and if a slice Stis degenerate

at some x∈S, we have kb(t,x)kht

x= 1 there.

For all t∈Rthere is a unique vector field Bt∈Γ(T S)given by ht(Bt,·) = bt. The lift of

Btto R×Sis a non-timelike vector field everywhere (as kbtkht≤1) and becomes lightlike

at (t, x)∈R×Sif and only if kb(t,x)kht

x= 1. This can be seen as follows: We denote the

lift of Btto R×Sby ˜

Bt, which is the unique vector field tangential to the slices St(i.e.,

dt(˜

Bt) = 0), that projects to Bton Sunder the action of pr2∗. We get

g(˜

Bt,˜

Bt) = ht(Bt, Bt)−bt(Bt)2=bt(Bt)1−qkbtkht.

As Sis compact, Btis a complete vector field for all t∈R, i.e., there is also a global flow

Φt:R×St→Stassociated to ˜

Btfor all t∈R, which constitutes a dynamical system.

Now, we show that (R×S, ψ∗g)is non-causal if it is causally imprisoning and the same

will hold for (M, g). Employing Prop. 4.31, this shows that (M, g)is globally hyperbolic

if it is causal. Let µ:R→R×Sbe a future imprisoned causal curve, i.e., ω(µ)is non-

empty and compact. We can assume that the spacetime is chronological, as otherwise there

is nothing to show, hence we get from Prop. 3.4 in [Min08a], that through every point

p∈ω(µ)there passes an inextendible lightlike curve completely contained in ω(µ). As we

have a slicing according to the semi-time function t, we can infer that tis non-decreasing

along µ. Therefore, the limit points p∈ω(µ)are all contained in one slice, say St. Thus,

we have an inextendible lightlike curve λ:R→R×Sobeying λ(R)⊂ω(µ)⊂St. Now,

as the only vectors tangential to the slice St, which can become lightlike, are the elements

of the vector field ˜

Bt, we can infer that λis an orbit of Φton St. Hence, λis an orbit of

aC2-dynamical system on a C2-manifold St. Again we can examine ω(λ), which is also

generated by inextendible lightlike curves, due to Prop. 3.4 in [Min08a], and thus contains

no fixed points. Now, we can employ Thm. 4.33 to ω(λ)and deduce that it is homeomorphic

to S1, and hence it constitutes a closed causal curve, i.e., the spacetime is non-causal.

Remark 4.35. Following Rem. 3.6, the orientability of S, in the theorem above, can be

achieved by demanding the orientability of M. The orientability of Myields the orientability

of TM =ξ⊕ν, and as (M, g)is certainly time-oriented as a spacetime by V, the subbundle

ξis oriented, hence, we also have the orientability of ν. But νis isomorphic to the tangent

bundle TS, due to the splitting M=R×S, therefore, Sis orientable.

If we now switch to the case of dim(S) = 3, we are looking for propositions similar to

the Poincaré–Bendixson theorem above, that assure the existence of a closed orbit of a

dynamical system, induced by the one-forms bton Sif btcorresponds to a causal vector field

tangential to the slices St. Unfortunately, the structure of limit sets of dynamical systems

in three dimensions is much harder to assess and there is even the possibility of chaotic

behavior and fractal limit sets (see, e.g., the discussion in Part 2 of [NS60]). A solution in

some special cases is provided by contact geometry, i.e., if the btare contact forms and the

induced dynamical system on Sis generated by the Reeb vector field. Then we can employ

the Weinstein conjecture and, particularly, Thm. 2.14 to infer the existence of a closed orbit.

Proposition 4.36. Let (R×S, g, ∂t)be a proper kinematical spacetime, with dim(S)=3

and Scompact. As usual, there is a family of one-forms {bt}t∈R⊂Γ(Λ1S)and a family of

Riemannian metrics {ht}t∈R⊂ R(S), such that

g(t,x)=−dt⊗dt+ 2pr∗

2(b(t,x))∨dt+ pr∗

2(h(t,x))−pr∗

2(b(t,x))⊗pr∗

2(b(t,x)).

Let Bt∈Γ(TS)be the unique vector field given by bt=ht(Bt,·). Assume there is some

T∈R, such that £BTbT= 0,kbTkhT

x= 1 for all x∈Sand bTis a contact form on S, then

(R×S, g)is not causal. Particularly, this implies that if such a spacetime is causal, then it

is globally hyperbolic.

Proof. As in the proof of Thm. 4.34, the condition kbTkhT

x= 1 implies that the lift of the

vector field ˜

BTto R×S, which is tangential to ST, is a lightlike vector field. In this case we

get g(˜

BT,˜

BT) = 0 for all (T, x)∈R×S, i.e., the integral curves of ˜

BTare lightlike curves

imprisoned in the compact slice ST. Thus, we show that there is at least one closed orbit of

BTon S, i.e., there is a closed causal integral curve of ˜

BTon ST.

As kbTkhT

x= 1, we get bT(BT)=1and

0 = £BTbT=BTcdbT+ d(BTcbT) = BTcdbT,

such that BTis the Reeb vector field associated to the contact form bT. As Sis three-

dimensional and compact, we infer from Thm. 2.14 that there is at least one closed orbit of

BT, i.e., the spacetime is non-causal. The remaining assertion follows from Prop. 4.31.

If we now consider particular proper kinematical spacetimes, we can give a surprising

condition equivalent to the Weinstein conjecture, using Lorentzian causality theory. Let

Y:R→R≥1be a smooth function which obeys Y(T)=1for exactly one T∈Rand

Y(t)>1for all t∈Rwith t6=T.

Proposition 4.37. The Weinstein conjecture holds if and only if all proper kinematical

spacetimes (R×S, g, ∂t)with Scompact, dim(S)=2n+ 1 for n∈Nand metric given by

g=−dt⊗dt+ 2pr∗

2(b)∨dt+Y(t)pr∗

2(h)−pr∗

2(b)⊗pr∗

2(b),

for a contact form bon S, a Riemannian metric hon Sobeying kbkh= 1 and an h-geodesic

vector field B∈Γ(TS)obeying h(B, ·) = b, are non-causal.

Proof. “⇐”: Due to the function Y, we have that kbkY(t)h<1for all t6=T. Hence, the

only violations of causal continuity can occur, for this spacetime, by the behavior of the

integral curves of B, lifted to R×S, on the slice ST, where Bis lightlike. Furthermore, B

is the Reeb vector field for b, as obviously b(B) = 1 and ∇h

BB= 0, where ∇hdenotes the

Levi-Civita connection associated to hon S. Similar to the kinematical quantities defined

in section 3.1, we can decompose the exterior derivative dbinto a part ω⊥, which obeys

Bcω⊥= 0 and a part containing ˙

b:= h(∇h

BB, ·), such that db=ω⊥+b∧˙

bholds. As

h(B, B) = 1, we get Bcdb=˙

b, hence ∇h

BB= 0 implies Bcdb= 0 and Bis indeed the Reeb

vector field associated to b. Now, if (R×S, g)is non-causal, there is a closed causal curve,

which can only exist on the slice STas an integral curve of the lift of B. Thus, there is a

closed orbit of the Reeb vector field Band the Weinstein conjecture holds.

“⇒”: Assume (S, b)is a compact contact manifold and the Weinstein conjecture holds,

i.e., for the Reeb vector field B∈Γ(TS)obeying b(B) = 1 and Bcdb= 0, there exists a

closed orbit. We will now construct a non-causal Lorentzian metric gon R×Sobeying the

conditions in the proposition. We need a Riemannian metric hon Sfor which h(B, ·) = b

holds.

Such a metric can be constructed as follows: Regard Sas the slice for t= 0 in I×S, with I

some (small) open interval containing 0. Then band dbcan be uniquely extended to I×S

and ω= d(etb)is a symplectic form on I×S. It is known (cf., e.g., [McD98]) that there is a

non-empty set of ω-compatible almost complex structures J:T(I×S)→T(I×S),J2=−id

obeying ω(X, Y ) = ω(JX, JY )for all vector fields X, Y on I×Sand ω(X, JX)>0for all

nowhere zero vector fields Xon I×S. This implies that ˜

h(·,·) = ω(·, J·)is a Riemannian

metric on I×S. With respect to this metric we can now choose a vector field normal to S,

i.e., some n∈Γ(NS)obeying ˜

h(X, n) = 0 for all X∈Γ(T S), where we denote the normal

bundle of Sin I×Sby NS. The condition of Xbeing tangential to S, certainly implies

dt(X)=0, so that we can compute

0 = ˜

h(X, n) = ω(X, Jn) = et(dt∧b)(X, Jn) + etdb(X, Jn)=db(X, Jn)−dt(Jn)b(X)(∗)

on the slice S, where t= 0 holds. This yields dt(Jn) = 0 and Jncdb= 0, i.e., Jn is tangential

to Sand proportional to the Reeb vector field B. Hence, by normalizing nproperly and

changing ˜

hby a conformal factor, we can also achieve b(Jn) = 1, i.e., B=Jn. This can be

done as follows: On Swe compute

h(n, n) = ω(n, Jn) = (dt∧b)(n, Jn) = b(Jn)dt(n).

Hence, normalizing nso that dt(n) = 1 holds and switching to a metric h=1

h(n,n)˜

h(certainly

this leaves the condition (∗) unchanged) yields b(Jn)=1. Furthermore, we compute

h(X, Jn) = ω(X, −n) = (dt∧b)(n, X) = b(X)

on Sfor t= 0 and all X∈Γ(T S). Thus, restricting hto the slice Syields a Riemannian

metric obeying h(B, ·) = b.

Now, we take the Riemannian metric hon Sfrom the paragraph above and perform the

following construction. From the condition h(B, ·) = bwe get h(B, B) = kbkh= 1 on Sand,

furthermore, with a chain of argument analogous to the proof of “⇐” above, we can conclude

that Bis h-geodesic. Now consider the product R×Sand any function Y:R×S→R≥1

obeying Y(T)=1for exactly one T∈Rand Y(t)>1for all R3t6=T. Then

g=−dt⊗dt+ 2pr∗

2(b)∨dt+Y(t)pr∗

2(h)−pr∗

2(b)⊗pr∗

2(b)

is a Lorentzian metric on R×S(band dbare nowhere vanishing on S), and (R×S, g, ∂t)

is a proper kinematical spacetime, as ∂tis complete. As the vector field Blifted to ST

is obviously lightlike and has a closed orbit by assumption, the spacetime (R×S, g)is

non-causal.

The metric in Prop. 4.37 defines a Hubble-isotropic spacetime, which we will analyze in more

detail in chapter 6. Although the proposition above constitutes an interesting correspondence

between Lorentzian causality theory and the Weinstein conjecture in arbitrary dimensions,

it seems unlikely that this could provide an alternative route towards a proof of the general

Weinstein conjecture (cf. the detailed discussion in chapter 7).

Chapter 5

SLICED SPACETIMES

In this chapter, we will approach spacetimes with a fixed reference frame from a perspective

which is in some sense dual to to the one adopted in the previous chapters. Instead of setting

out from a Lorentzian manifold Mwith a Cartan flow and, subsequently, determining the

circumstances leading to a splitting M=R×S, as well as analyzing the causality properties

arising in this context, we start now from a spacetime already given in a general splitting

form I×S, with Ian open interval of the real numbers. Furthermore, we will even assume

the metric induced on the slices Stto be Riemannian (with the exception of section 5.3,

where this condition is not necessary). This will lead to the analysis of various special cases

of these sliced spacetimes and, particularly, to the investigation of their global hyperbolicity,

the “nicest” condition on the causal ladder.

Definition 5.1. Let I= (a, b)be an interval of the real numbers with −∞ ≤ a<b≤ ∞. A

spacetime (M, g)is called sliced spacetime if M=I×Sand

g(t,x)=−N2(t, x)dt⊗dt+ 2pr∗

2(b(t,x))∨dt+ pr∗

2(h(t,x)),

with (t, x)∈I×S,{bt}t∈Ia family of one-forms on Svarying smoothly with t, which

are called shift,{ht}t∈Ia family of Riemannian metrics on Svarying smoothly with tand

N:I×S→R>0a function, which is called lapse. We denote by pr2:I×S→Sthe canonical

projection on the second factor and the coordinate tis identified with the projection on the

first factor: t= pr1:I×S→I.

In this chapter, we will again often use the natural identification of objects on the base S

and their pull-back to M, and hence simply write wfor a one-form pr∗

2(w)on M, which

is pulled back from a one-form on Sif this is no source of confusion. For example, the

one-form bton Scan be identified with its pull-back pr∗

2(bt)and can hence also be regarded

as a one-form on the slice St={t}×S. Note that the natural reference frame V=N−1∂t

is not necessarily a complete vector field, but we have the following

Proposition 5.2. Every sliced spacetime (I×S, g)is isometric to a spacetime (R×S, ˜g), and

there is a metric g∗pointwise conformal to ˜g, such that (R×S, g∗, V ), with Vpointing along

the factor R, is a proper kinematical spacetime. Hence, every sliced spacetime is causally

continuous.

Proof. By virtue of a strictly increasing diffeomorphism f:I→R, we get an isometry

ψ: (I×S, g)→(R×S, ˜g),ψ(t, x)=(f(t), x)by setting the Lorentzian metric

˜g(f,x)= (ψ∗g)(f,x)=−(N◦ψ−1)2(f, x)[(f−1)0]2df⊗df+ 2bψ−1(f,x)∨(f−1)0df+hψ−1(f,x)

on R×S. Then setting g∗= [(N◦ψ−1)(f−1)0]2˜g, we have g∗(∂f, ∂f) = −1and ∂fis complete

on R×Sby the same argument as in the proof of Lem. 4.23. Hence, (R×S, g∗, V )with

V=∂fis a proper kinematical spacetime, which is causally continuous by Prop. 4.26 and

Thm. 4.29, as the restriction of g∗to any slice of constant fis a Riemannian metric. Thus,

(I×S, g)is causally continuous, too.

The notion of regularly sliced spacetimes was introduced by Y. Choquet-Bruhat and S. Co-

tsakis in [CBC02] and further investigated in [Cot04] (cf. also [CBC03]). The terminology

regularly sliced spacetime was introduced in [CB09]. The idea is to constrain the geometric

objects lapse, shift and the Riemannian metric constituting a sliced spacetime by some

bounds and regularity conditions in order to have straightforward access to a simple theorem

on global hyperbolicity for these spacetimes. As can be seen from the definition below, the

bounds are quite restrictive, but we will see in the following section that actually many

examples of globally hyperbolic spacetimes occur in this form. We will give a definition of

a regularly sliced spacetime, that is slightly more general than the one used in [Cot04] and

[CB09], as we will allow for different upper and lower metric bounds γand ˜γ(See item (iii)

in the definition below). Indeed, it will turn out in the following section that the condition

of equal upper and lower metric bounds γ= ˜γis very restrictive.

Definition 5.3. A sliced spacetime (M=I×S, g)with metric given by

g(t,x)=−N2(t, x)dt⊗dt+ 2b(t,x)∨dt+h(t,x),

for all (t, x)∈I×Sis called regularly sliced if the following three conditions hold:

(i) The lapse is bounded from below and above by constants n0, n1>0, such that

0< n0≤N(t, x)≤n1,

for all (t, x)∈M.

(ii) The h-norm of the shift is bounded by a constant A > 0, such that

kbkh

(t,x)< A,

for all (t, x)∈M.

(iii) The Riemannian metrics htare bounded from above and below by Riemannian metrics

γ, ˜γon S. That is, there are constants 0< B ≤C < ∞, such that

B˜γx(v, v)≤h(t,x)(v, v)≤Cγx(v, v),

for all (t, x)∈Mand all v∈TxS.

Particularly, we will also allow for the following relaxation of a spacetime being regularly

sliced. We call (I×S, g)regularly sliced from below (above) if (i) and (ii) holds, but (iii) is

modified to possibly hold only with C=∞(B= 0).

5.1 Global Hyperbolicity of Sliced Spacetimes

The following theorem by M. Sánchez and A. Bernal (cf. [BS05]) highlights the particular

relevance of the step of global hyperbolicity on the causal ladder. The theorem shows

that globally hyperbolic spacetimes are a particular class of sliced spacetimes, although not

regularly sliced ones in general.

Theorem 5.4. Let (M, g)be a globally hyperbolic spacetime, then it is isometric to (R×S, g∗)

with

g∗=−N2dt⊗dt+ht,

with N:R×S→R>0and {ht}t∈Ra family of Riemannian metrics on Svarying smoothly

with t. Hence, any globally hyperbolic spacetime is a sliced spacetime with vanishing shift

and I=R.

Proof. See [BS05] and compare to Def. 5.1.

Theorem 5.5. Suppose a regularly sliced spacetime obeys γ= ˜γ, i.e., it is regularly sliced

with the same Riemannian metric γbounding the family {ht}t∈Iof Riemannian metrics

from below and above. Then the spacetime has the slices St={t} × S,t∈Ias Cauchy

surfaces—and is, therefore, globally hyperbolic—if and only if the Riemannian metric γon

Sis complete. Particularly,

(i) a spacetime regularly sliced from below is globally hyperbolic if ˜γis complete, and,

(ii) a globally hyperbolic spacetime which is regularly sliced from above has complete γ.

Proof. See [CBC02], [CBC03] or [CB09] for (i), and [Cot04] for (ii). Then the first assertion

follows from (i) and (ii).

The conditions for a spacetime to be regularly sliced are not conformally invariant, nor is

the completeness condition involved in the theorem on global hyperbolicity. Particularly,

not all globally hyperbolic spacetimes are regularly sliced. This is shown by the following

example.

Example 5.6. Let ˚

S=R3\{0}be the 3-dimensional punctured Euclidean space and M=

R×˚

Sa sliced spacetime with metric

g=−r2dt2+X

i,j

δijdxidxj,

with r2=P3

i=1(xi)2. It is well-known that the standard Euclidean metric δon ˚

Sis incom-

plete. Moreover, the lapse N(xi) = ris unbounded on M, i.e., the spacetime is not regularly

sliced. But if we pass over to the conformal metric

g0=1

r2g=−dt2+1

r2X

i,j

δijdxidxj,

we see that (M, g0)is regularly sliced because now the lapse is bounded and 1

r2δis complete

on ˚

S. Therefore, (M, g0)and (M, g)are globally hyperbolic.

However, subsequently, we will show that necessarily at least one element of the conformal

class of a globally hyperbolic spacetime is regularly sliced.

Lemma 5.7. Let the metric of a regularly sliced spacetime (M=I×S, g), with ˜γ=γ, be

given by

g=−dt⊗dt+ht.

Then (M, g)is globally hyperbolic if and only if all Riemannian metrics hton the slices St

(t∈I)are complete.

Proof. This follows directly from Thm. 5.5. We have Bγ ≤ht≤Cγ in this case. If

(M, g)is globally hyperbolic, γis complete, hence Bγ is complete and all htare complete,

as Bγ ≤ht. If all htare complete, so is Cγ as ht≤Cγ, hence γis complete and (M, g)is

globally hyperbolic.

Below we will need the following lemma established by S. Born and the author in [BD12].

Lemma 5.8. Let T1and T2be two locally compact topological spaces, which are Hausdorff

and countable at infinity. Let F:T1×T2→Rbe a continuous function. Then there are

two continuous functions f1:T1→Rand f2:T2→R, such that F≤f1+f2. Particularly,

if im(F)⊂R>0, there are four functions g1:T1→R>0,g2:T2→R>0and G1:T1→R>0,

G2:T2→R>0, such that g1·g2≤F≤G1·G2, and g1,g2,G1,G2can be chosen to be

smooth if T1and T2are manifolds.

Proof. We can assume that F(x1, x2)≥0for all x1∈T1and x2∈T2. If this is not the

case, we just replace F(x1, x2)by max{F(x1, x2),0} ≥ F(x1, x2).

The statement is obvious if T1or T2is compact: If T1is compact, we have F(x1, x2)≤

maxx1∈T1F(x1, x2) =: f2(x2). The same argument applies if T2is compact. Therefore, we

assume that neither T1nor T2is compact.

By assumption (locally compact and countable at infinity), there are exhaustions

T1=[

i∈N0

Ki,T2=[

i∈N0

Li,

with

∅=K0, Ki⊂˚

Ki+1,

such that the Ki’s are compact subsets of T1for all i∈N0and

∅=L0, Li⊂˚

Li+1,

such that the Li’s are compact subsets of T2for all i∈N0.

Furthermore, we set Ui=˚

Kiwith Ui⊂T1an open set for all i∈N, as well as Vi=˚

with Vi⊂T2an open set for all i∈N. Hence, {Ui}i∈Nis an open cover of T1and {Vi}i∈N

is an open cover of T2. The spaces Mand Nare paracompact, as they are locally compact,

Hausdorff and countable at infinity, hence there are partitions of unity {φi:T1→R}i∈Nresp.

{χi:T2→R}i∈Nthat are subordinate to locally finite refinements of {Ui}i∈Nresp. {Vi}i∈N.

Now we define

ai:= max

(x1,x2)∈Ki×Li

F(x1, x2).

and two functions f1:T1→Rand f2:T2→R:

f1(x1) := X

i∈N

φi(x1)ai

and

f2(x2) := X

i∈N

χi(x2)ai.

Note that in each series we sum only finitely many non-zero terms for every x1∈T1resp. x2∈

T1, and f1and f2are continuous functions.

Now let (x1, x2)be an arbitrary point in T1×T2. Then define mto be the smallest natural

number such that x1∈Umand nto be the smallest natural number such that x2∈Vn.

Hence, x16∈ Uifor i<mand x26∈ Vifor i<n.

This yields

f1(x1) + f2(x2) = X

i∈N

φi(x1)ai+X

i∈N

χi(x2)ai

i≥m

φi(x1)ai+X

i≥n

χi(x1)ai

≥am+an≥F(x1, x2),

as (x1, x2)∈Kk×Lk, where k= max{m, n}.

Obviously, if T1and T2are manifolds, they allow for smooth partitions of unity and the

construction above can be carried out with these smooth partitions of unity, such that the

resulting functions f1and f2are smooth.

Moreover, by taking exponentials we get

F(x1, x2)≤f1(x1) + f2(x2)≤ef1(x1)+f2(x2)=ef1(x1)ef2(x2)=: G1(x1)G2(x2).

Assuming im(F)⊂R>0, we can analyze ˜

F(x1, x2) = [F(x1, x2)]−1and we find two functions

G1and ˜

G2such that ˜

F(x1, x2)≤˜

G1(x1)˜

G2(x2). Setting g1(x1) = [ ˜

G1(x1)]−1and g2(x2) =

[˜

G2(x2)]−1yields

g1(x1)g2(x2)≤F(x1, x2).

Applying this to sliced spacetimes yields the following

Lemma 5.9. For any sliced spacetime ((a, b)×S, g)with −∞ ≤ a<b≤ ∞ and

g=−dt⊗dt+ht,

there is a function Ω2: (a, b)→(0,∞), such that the conformal metric

g∗= Ω2g=−Ω2(t)dt2+ Ω2(t)ht=−Ω2(t)dt2+h∗

t=−dτ2+h∗

τ,

after the transformation τ(t) = Rt

t0Ω(s)dswith arbitrary fixed t0∈(a, b), on (a∗, b∗)×S

with a∗= limt→aτ(t)and b∗= limt→bτ(t), is regularly sliced.

Proof. Fixing any t0∈I= (a, b)we define two norms ||| · |||↑and ||| · |||↓for the family of

Riemannian metrics hton S, depending on h0:= ht0, by setting

|||ht|||↑

x:= sup

kvkh0

x=1

h(t,x)(v, v)

and

|||ht|||↓

x:= inf

kvkh0

x=1

h(t,x)(v, v),

for all (t, x)∈I×S. Then we have for all Riemannian metrics htthat

0<|||ht|||↓

x≤ |||ht|||↑

x<∞,

for all x∈Sand all t∈I.

First, we show that the condition of being regularly sliced

B˜γ≤ht≤Cγ,

for a family of Riemannian metrics ht, Riemannian metrics γ,˜γand constants 0< B ≤C <

∞is equivalent to

D(x)≤ |||ht|||↓

x≤ |||ht|||↑

x≤E(x),

for all (t, x)∈I×Swith D, E :S→R>0two functions, and that this equivalence is

independent of the fixed t0chosen to define the norms |||·|||↑and |||·|||↓.

Certainly, B˜γ≤ht≤Cγ implies D(x)≤ |||ht|||↓

x≤ |||ht|||↑

x≤E(x)by setting D(x) = B|||˜γ|||↓

and E(x) = C|||γ|||↑

x. Obviously, this holds for any chosen fixed t0∈I.

Assume now that there is a function E:S→R, such that |||ht|||↑

x≤E(x)holds for all

x∈S, but there is no Riemannian metric γon S, such that h(t,x)(v, v)≤γx(v, v)for all

(t, x)∈I×Sand all v∈TxS. Hence, there is some x∈S, some v∈TxSand a sequence

{tn}n∈N⊂Iwith tn→bor tn→aas n→ ∞, such that

h(tn,x)(v, v)n→∞

−→ ∞.

It suffices to assume v∈TxSto be an element of the h0-unit sphere in TxS, i.e., h0(v, v)=1.

So we have |||htn|||↑

x≥h(tn,x)(v, v)and thus

|||htn|||↑

n→∞

−→ ∞

is the desired contradiction. If we have defined |||·|||↑by another fixed number, say, t16=t0∈I,

we can rescale v7→ c·vby some c=c(t0, t1), such that vhas unit ht1-length. The same

argument then holds for the norm |||·|||↑based on t1.

For the lower bound assume that there is a function D:S→R>0, such that D(x)≤ |||ht|||↓

holds for all x∈S, but there is no Riemannian metric ˜γon Ssuch that ˜γx(v, v)≤h(t,x)(v, v)

for all (t, x)∈I×Sand all v∈TxS. By a similar argument to the one above, there is now

some x∈S, some v∈TxSwith unit h0-length and a sequence {tn}n∈N⊂Iwith tn→aor

tn→bas n→ ∞, such that

h(tn,x)(v, v)n→∞

−→ 0.

Hence, as |||htn|||↓

x≤h(tn,x)(v, v)we get

|||htn|||↓

n→∞

−→ 0,

the desired contradiction. The argument above for another mapping based on a number

t1∈Iholds, mutatis mutandis, also in this case.

achieved by modifications of normal coordinates about each xn. Hence we find by

Second, starting from the metric g=−dt2+hton (a, b)×S, the norms |||·|||↑and |||·|||↓imply

functions (t, x)7→ |||ht|||↑

xand (t, x)7→ |||ht|||↓

xon (a, b)×Swith values in R>0, such that we

can find four smooth functions f1, F1: (a, b)→R>0and f2, F2:S→R>0obeying

f1(t)·f2(x)≤ |||ht|||↓

x≤ |||ht|||↑

x≤F1(t)·F2(x)

by applying Lem. 5.8.

We now assume that |||ht|||↑

xis not bounded from above as t→band |||ht|||↓

xis not bounded

from below as t→a. If the functions were bounded, there would be nothing to show. All

other cases (e.g., |||ht|||↑

xnot bounded from below as t→b, etc.) are shown by completely

analogous arguments. Hence, we can assume that f1is bounded away from zero from below

in [t0, b)and F1is bounded from above in (a, t0]. Now we set

Ω2(t) := 1

f1(t)·F1(t),

such that

g∗=−(f1(t)·F1(t))−1dt2+h∗

with

h∗

(t,x)=h(t,x)

f1(t)·F1(t).

This yields

|||h∗

t|||↓

x=|||ht|||↓

f1(t)·F1(t)and |||h∗

t|||↑

x=|||ht|||↑

f1(t)·F1(t),

such that we have

B·F1(x)≤ |||h∗

t|||↓

xand |||h∗

t|||↑

x≤C·F2(x),

for constants 0< B ≤C∈Rand t∈[t0, b),t∈(a, t0], respectively. Hence, we can find

functions D(x) = B·F1(x)and E(x) = C·F2(x)such that D(x)≤ |||h∗

t|||↓

x≤ |||h∗

t|||↑

x≤E(x)

holds for all x∈S.

Furthermore, setting

τ(t) = Zt

pf1(s)·F1(s)=Zt

Ω(s)ds,

we find that τis a strictly increasing function in (a, b)with values in (a∗, b∗), with

a∗=Za

Ω(s)dsand b∗=Zb

Ω(s)ds,

in the sense of a limit if necessary, as Ω>0. Thus, τ(t)can be solved for t(τ)and we can

set h∗

τ=h∗

t(τ), such that

g∗=−dτ2+h∗

is a regularly sliced Lorentzian metric on (a∗, b∗)×Sconformal to the original metric gon

(a, b)×S.

Remark 5.10. Note that the metrics γand ˜γthat bound a one-parameter family of Rie-

mannian metrics {ht}t∈(a,b), constituting the spacelike part of a Lorentzian metric g, are

generally not obtained as some kind of limit γ“=” limt→bhtor ˜γ“=” limt→aht. These lim-

its may not be smooth Riemannian metrics, as in example 5.13 below, but this can certainly

be repaired by making the limit metrics a bit “bigger” or “smaller”, respectively.

Theorem 5.11. Every globally hyperbolic spacetime (M, g)is conformal to a regularly sliced

spacetime (I×S, −dt2+ht), such that the Riemannian metrics htare bounded by a complete

Riemannian metric from above, i.e., γ≥htholds for all t∈Iwith γcomplete. Furthermore,

for any globally hyperbolic spacetime (R×S, g)with

g=−N2dt⊗dt+ ¯gt,

we have that all Riemannian metrics h(t,x)=¯g(t,x)

N2(t,x)are complete if the regularly sliced

spacetime, to which (R×S, g)is conformal, obeys ˜γ=γ.

Proof. Applying Thm. 5.4, we conclude that (M, g)is isometric to (R×S, −N2dt2+ht),

with hta family of smoothly varying Riemannian metrics. Thus, gis obviously conformal

to ˜g=−dt2+ht

N2. Hence, we can apply Lem. 5.9 to ˜gand get a regularly sliced metric

g∗=−dt2+h∗

ton I×Swith some interval I⊂R, which is conformal to the original

spacetime (M, g), and hence also globally hyperbolic. By Thm. 5.5, the Riemannian metric

γon Sthat bounds the family htfrom above is complete. The remaining assertion follows

from Lem. 5.7 and from the fact that conformal changes by functions of the same type as

Ω(t), given by Lem. 5.9, do only affect the parameter tof the Riemannian metrics ht, hence

their completeness stays unaffected by such conformal transformations.

Thm. 5.11 can be seen as a regularity result for generic globally hyperbolic spacetimes. On

the one hand, it implies that in the conformal class of a globally hyperbolic metric, there

is at least one element (I×S, −dt2+ht), such that the one parameter family {ht}t∈Iof

Riemannian metrics has a complete metric γas upper bound. This constrains the possible

curves in the space of Riemannian metrics R(S) = {g∈Γ(Σ2S)|gRiemannian}over a

manifold Sthat can serve as such a one-parameter family for globally hyperbolic spacetimes

(cf. [FM78] for an introduction to the notion of the space of Riemannian metrics considered

here, [GMM91] for the construction of a Riemannian manifold of Riemannian metrics in

particular cases, or [Bla00] for a more recent overview). On the other hand, it implies that

globally hyperbolic spacetimes (I×S, −dt2+ht), for which every metric htis complete, are

amongst others (cf. example 5.14), those which are regularly sliced with ˜γ=γ. This has

some interesting consequences.

Remark 5.12. Let γand Cγ,C > 0be two Riemannian metrics in the space of Riemannian

metrics R(S). Let γbe complete, then l: [1, C]→ R(S)with s7→ sγ is a line segment of

complete Riemannian metrics connecting γand Cγ. If γand Cγ are the upper and lower

bounds of some family of metrics ht⊂ R(S),t∈[1, C], constituting a regularly sliced and

globally hyperbolic spacetime ((1, C)×S, −dt2+ht), then the metrics htcan not be “too far

away” from the line segment l([1, C]), in some sense. In fact, suppose that C > 1and for the

expansion Θof the reference frame ∂tin the spacetime ((1, C)×S, −dt2+ht)it holds that

Θ>0, then it is straightforward that {ht}t∈[1,C]is a graph over the line segment l([1, C]) in

R(S).

The following example shows that there are globally hyperbolic and regularly sliced space-

times (I×S, −dt2+ht)with htan incomplete Riemannian metric for all t∈I.

Example 5.13. Let gbe the Lorentzian metric

g=−dt2+dx2

(1 + |x|(2−t))2

on (0,1) ×R. Obviously, the spacetime ((0,1) ×R, g)is regularly sliced. From Thm. 5.22 in

the next section, we may already conclude that

ht=dx2

(1 + |x|(2−t))2

is an incomplete Riemannian metric on Rfor all t∈(0,1). If we now take a look at the limit

metric h1=dx2

(1+|x|)2, we see that it is only continuous at x= 0, but complete and ht≤h1for

all t∈(0,1). Of course, we can change h1in a compact set about 0∈Ra little bit, to obtain

a smooth, complete metric, which is “bigger” than any ht, too. As the lower limit metric

h0=dx2

(1+|x|2)2is a smooth, incomplete metric on R, the impossibility to find constants, such

that ((0,1)×R, g)is regularly sliced with the same upper and lower bound metrics, is already

implied by the quadratic growth (1 + |x|2)towards infinity, compared to the linear growth of

(1 + |x|). Nevertheless, ((0,1) ×R, g)is globally hyperbolic. A straightforward calculation

shows that all lightlike vector fields in T((0,1) ×R)are given by

K(t,x)=±1

±(1 + |x|(2−t)).

The lightlike integral curves of Kare easily calculated, and we can conclude that all future-

directed integral curves of Keventually intersect the line t= 1 and all past-directed ones

eventually intersect t= 0. Hence, the sets J+((t, x)) ∩J−((s, y)) are compact or empty for

all (t, x),(s, y)∈(0,1) ×R.

The following example shows that in a globally hyperbolic and regularly sliced spacetime

(I×S, −dt2+ht)with hta family of complete Riemannian metrics and a complete upper

limit metric γ, the lower limit metric ˜γis not necessarily complete.

Example 5.14. Modifying example 5.6, we consider the punctured plane S=R2\ {0} ≃

R>0×S1with the Riemannian metrics

ht=h(c)

t+δ=t

r2dr2+ dφ2+dr2+r2dφ2=t

r2+ 1dr2+ (1 + r2)dφ2,

for t∈(0,1). Hence, h(c)

tis complete on Sfor all t∈(0,1) and the Euclidean metric δis

incomplete, but htis complete for all t∈(0,1). This can be concluded from Lem. 2.7. Thus

we can construct a spacetime ((0,1) ×S, g)with g=−dt2+ht, which is obviously regularly

sliced with complete upper bound γ=1

r2+ 1dr2+ (1 + r2)dφ2and incomplete lower bound

˜γ= dr2+(1+r2)dφ2. Analogous to the previous example, we can solve the lightlike geodesic

equations for fixed φ∈S1and see that none of these geodesics meets the singularity at r= 0

for a finite value of the affine parameter. Hence, the sets J+((t, x)) ∩J−((s, y)) are compact

for all (t, x),(s, y)∈(0,1) ×S.

These two examples correspond well to the fact that both the complete Riemannian metrics

and the incomplete ones are dense in R(S)for some manifold S(cf. [FM78]).

5.2 Cauchy Hypersurfaces in Stationary Spacetimes

In this section, we will investigate particular classes of sliced spacetimes, namely standard

stationary and standard static spacetimes. We are interested in their global hyperbolic-

ity via the existence of Cauchy hypersurfaces. It has been established that this question

is related to the completeness of Riemannian and Randers-type metrics on the spacelike

slices (cf. [CJM11]). There is even a duality result that connects Lorentzian causality of

stationary spacetimes with Randers-type completeness, known as stationary-to-Randers cor-

respondence (cf. [CJS11]). With our results in this section, we refine the existing connection

between Riemannian completeness and causality of stationary spacetimes, mostly by in-

troducing growth conditions and results on pinching Randers-type metrics by Riemannian

metrics. To this end we prove a new result in global Riemannian geometry yielding a rigor-

ous condition for the completeness of a Riemannian metric under conformal transformations

(see Thm. 5.22). Please note that some results in this section have already been published

in [DPS12], but with the results presented in this section some errors contained in [DPS12]

are corrected. Particularly, Thm. 5.22 below is a rectified version of the corresponding of

Thm. 1 in [DPS12] and its subsequent implications are adjusted as well.

Definition 5.15. A sliced spacetime (R×S, g =−N2dt⊗dt+ 2b∨dt+h)is called

standard stationary if the function N, the one-form band the Riemannian metric hare pulled

back from an individual function, one-form and metric on S, respectively by the canonical

projection pr2:R×S→S, i.e., N,band hdo not depend on t. Furthermore, a standard

stationary spacetime is called standard static if b= 0.

Several remarks are in order.

Remark 5.16. (i) Obviously the vector field ∂tis a complete Killing vector field in any

standard stationary or static spacetime and V=∂t

Nis a reference frame, which is also

complete as Ndoes not depend on t(compare to Def. 3.9 and Prop. 5.2). Hence,

(R×S, g, ∂t

N)is a proper kinematical spacetime for a standard stationary metric gas

in Def. 5.15 above, particularly it is causally continuous by section 4.2 (compare to the

results in [JS08]).

(ii) Note that the question if a stationary or static spacetime is standard stationary or

standard static, can be assessed by the result in Cor. 4.11 and the results on causality

in section 4.2.

(iii) A standard static spacetime can obviously be considered to be a Lorentzian warped

product (S, h)×N(R,−dt2).

Note that not even standard static spacetimes are necessarily regularly sliced, as the function

N:S→R>0does not have to be bounded. Nevertheless, standard static spacetimes—in

contrast to standard stationary ones—are conformal to regularly sliced spacetimes by virtue

of a conformal factor 1

N2.

Conditions on the Riemannian slices St:= {t}×Sto be Cauchy hypersurfaces in standard

stationary spacetimes were investigated in [CJM11] and [CJS11]. A crucial point in this

investigation is the construction of a Randers-type metric from the Riemannian metric h

and the one-form bon S.

Definition 5.17. Let (R×S, g =−N2dt⊗dt+ 2b∨dt+h)be a standard stationary

spacetime. The function F:TS →R≥0given by

F(x, v) = shx(v, v)

N2(x)+bx(v)2

N4(x)+bx(v)

N2(x),

for all x∈Sand all v∈TxSis called Fermat metric.

Proposition 5.18. The Fermat metric for a given standard stationary spacetime is a well

defined Finslerian metric of Randers type.

Proof. The function Nis non-zero and with hbeing a Riemannian metric and ba one-form

on S, also h

N2+b⊗b

N4is a Riemannian metric on S. Hence, Fis well defined for all (x, v)∈TS.

The smoothness and continuity conditions for Fare fulfilled on TS, because N,band hare

smooth for all v6= 0, and F(x, v)is continuous at v= 0 as it is a composition of smooth,

respectively continuous, maps. Moreover, Fis positively homogeneous of degree one in the

second argument because we have

F(·, λv) = rλ2h(v, v)

N2+λ2b(v)2

N4+λb(v)

N2=λF(·, v)

if λ > 0. Furthermore, we have

N2bkh+b2

x= sup

v∈TxS\{0}

|b(v)|

pN2h(v, v) + b(v)2<1,

for all x∈S, which assures F(x, v)≥0for all (x, v)∈T S and makes Fa Finslerian metric

of Randers type.

It is known that a slice St(and thus all slices) of a standard stationary spacetime M=R×S

is a Cauchy hypersurface if and only if the associated Fermat metric on Sis forward and

backward complete (see [CJM11] and [CJS11]). Therefore, this is a sufficient condition for M

to be globally hyperbolic. However, it is not a necessary condition in general, since for a slice

in a globally hyperbolic standard stationary spacetime with non-complete Fermat metric—

which is, therefore, not a Cauchy hypersurface—there is always another slicing such that the

slices are Cauchy hypersurfaces, providing a forward and backward complete Fermat metric.

Nevertheless, for any globally hyperbolic standard stationary spacetime, the Riemannian

metric

h=h

N2+b⊗b

is necessarily complete on S. This follows from Lem. 2.20. Moreover, this Riemannian metric

is actually independent of the slicing. This is a straightforward consequence of Thm. 4.16,

as ˜

hcan be identified via pull-back with the projection defined in section 3.1, and thus it is

at-independent Riemannian metric on the horizontal bundle H(R×S).

Now we will establish a new result in global Riemannian geometry.

A ray α: [0, b)→M(or a curve ˜α: (a, b)→Mfor some a≤0< b such that α= ˜α|[0,b)

is a ray) with 0< b ≤ ∞ will be called escaping to infinity on a manifold Mif there is a

sequence {tn}n∈N⊂[0, b)with tn→bas n→ ∞, such that α(tn)→ ∞ as n→ ∞ in the

following sense: there is an exhaustion of Mby compact sets {Kn}n∈N0⊂ P(M)(the power

set of M), M=Sn∈N0Kn,K0={α(0)},Kn⊂˚

Kn+1, such that α(tn)∈Kn\Kn−1for all

n∈N.

Remark 5.19. Note that necessarily all non-imprisoned curves are escaping to infinity.

Compare to Def. 3.37.

Definition 5.20. Let (M, g)be a complete Riemannian manifold.

(i) A function f:M→R>0grows at most linearly towards g-infinity on (M, g)if for

all fixed x0∈Mthere are constants c1, c2>0, such that for all x∈M,f(x)≤

c1dg(x0, x) + c2holds.

(ii) A function f:M→R>0grows superlinearly towards g-infinity on (M, g)if for all

fixed x0∈Mthere are constants ε, c1, c2>0, such that for all x∈M,f(x)≥

c1dg(x0, x)1+ε+c2holds.

(iii) A function f:M→R>0will be called L1on an escaping curve w.r.t. gif there is

x0∈Mand a ray γ: [0,∞)→Mwith γ(0) = x0, such that

Z∞

(f◦γ)(s)qgγ(s)(˙γ(s),˙γ(s))ds < ∞.

This condition is independent of the parametrization of γ, and without loss of gener-

ality, we can assume γto be parametrized by arc length, such that g(˙γ, ˙γ) = 1, and we

have R∞

0(f◦γ)ds < ∞, i.e., f◦γ∈L1(R≥0).

Inspecting item (iii), we see that the condition is equivalent to state that the length of the

escaping ray in the conformally transformed metric f2gon Mis finite. As it is known that a

Riemannian metric is complete if and only if the length of any escaping curve is infinite, the

connection to completeness becomes obvious. But stating the condition in terms of growth

of the conformal factor, instead of in terms of conformally transformed curve lengths, allows

to compare this condition to the linear growth conditions in items (i) and (ii). The precise

relation is clarified in the following

Lemma 5.21. If f:M→R>0grows superlinearly towards g-infinity on a complete Rie-

mannian manifold (M, g), then 1

f:M→R+is L1w.r.t. gon all escaping g-geodesic rays. If

f:M→R>0grows at most linearly towards g-infinity on a complete Riemannian manifold

(M, g), then 1

f:M→R>0is not L1on any escaping curve w.r.t. g.

Proof. Let x0∈Mbe a fixed point and γ: [0,∞)→Many escaping g-geodesic ray with

γ(0) = x0, parametrized by arc length. Assume fto grow superlinearly towards g-infinity,

thus there are constants ε, c1, c2>0such that

f(x)≤1

c1dg(x0, x)1+ε+c2

for all x∈M. Hence, we have for all s∈[0,∞)

(f◦γ)(s)≤1

c1dg(x0, γ(s))1+ε+c2

c1s1+ε+c2∈L1([0,∞))

as ε, c1, c2>0.

Assume now that fgrows at most linearly towards g-infinity on M. Hence, for all x0∈M,

there are constants c1, c2>0such that

f(x)≥1

c1dg(x0, x) + c2

for all x∈M. Let γ: [0,∞)→Mbe any escaping ray emanating at x0∈Mand

parametrized by arc length. Then we have dg(x0, γ(s)) ≤sfor all s∈[0,∞)and thus

(f◦γ)(s)≥1

c1dg(x0, γ(s)) + c2≥1

c1s+c2

This implies R[0,∞)

f◦γ=∞as R∞

c1s+c2=∞for all c1, c2>0.

Theorem 5.22. Let (M, g)be a non-compact and complete Riemannian manifold and

A:M→R>0be a positive function. We denote a conformally transformed metric on Mby

g0=g

A2. Then (M, g0)is complete if and only if 1

A:M→R>0is not L1on any escaping

curve w.r.t. g. Moreover, if Agrows at most linearly towards g-infinity on M, then (M, g0)

is complete and if Agrows superlinearly towards g-infinity on M, then g0is a bounded metric

on M, particularly (M, g0)is incomplete.

Proof. We will show the following statement: 1

Ais L1on an escaping curve if and only if

(M, g0)is incomplete. The first remaining statement then follows easily from Lem. 5.21.

Assume first that there is x0∈Mand a ray γ: [0, b)→Mwith γ(0) = x0escaping to

infinity, such that 1

Ais L1on γw.r.t. g. We can parametrize γby g-arc length, i.e., we have

g(˙γ, ˙γ) = 1 and hence some constant B < ∞such that R∞

(A◦γ)(s)=B. Take any sequence

{sn}n∈N⊂R≥0, with sn→ ∞ and γ(sn)→ ∞ as n→ ∞. Then we compute the distance

between x0and each γ(sn)in the conformally transformed metric g0:

dg0(x0, γ(sn)) ≤Zsn

0pg(˙γ(s),˙γ(s))

A(γ(s)) ds≤Z∞

(A◦γ)(s)ds=B.

Hence, the sequence {γ(sn)}n∈Nis contained in a closed and bounded g0-ball of radius B

about x0. But by the definition of escaping curves the sequence {γ(sn)}n∈Nhas no convergent

subsequence, thus the closed and bounded g0-ball of radius Babout x0is not compact and,

therefore, (M, g0)is incomplete by the Hopf–Rinow theorem.

Assume now that (M, g0)is incomplete, hence there is a point x0∈Mand a g0-geodesic

ray γ: [0, b)→Memanating from x0that is not extendible to the parameter value b. As

ag0-geodesic is parametrized to unit g0-velocity, we conclude the length of γto be b < ∞.

But obviously γescapes to infinity, as there exists no point γ(b)∈M. Now reparametrize

γto unit g-velocity, i.e., g( ˙γ, ˙γ) = 1, then we get γ: [0,∞)→Mas gis assumed complete.

We compute

∞> b =Z∞

0pg(˙γ(s),˙γ(s))

A(γ(s)) ds=Z∞

(A◦γ)(s)ds.

Hence, 1

Ais L1on the escaping curve γw.r.t. g.

As 1

Ais L1w.r.t. gon all escaping g-geodesic rays if Agrows superlinearly towards g-infinity

on Mby Lem. 5.21, we certainly have in this case that g0is incomplete. Furthermore,

computing the distance of a fixed x0∈Mto any x∈Min the g0distance we get for some

finite constant r(ε, c1, c2)<∞

dg0(x0, x)≤Z∞

c1s1+ε+c2

=r(ε, c1, c2).

Hence, g0is bounded as now dg0(x, y)≤dg0(x0, x)+dg0(x0, y) = 2rholds for all x, y ∈M.

Example 5.23. For a simple instructive example we look at the standard metric g0of the

punctured plane R2\{0}∼

=R>0×S1, given in polar coordinates (r, φ)by

g0= dr2+r2dφ2.

This metric is obviously incomplete since radial geodesics would eventually meet the removed

origin. However, the conformally transformed metric

g=g0

r2=1

r2dr2+ dφ2

is isometric to the (complete) flat cylinder R×S1, which may be seen by adopting the new

coordinate ρ:= ln r:

g= dρ2+ dφ2.

Another complete metric g0in the conformal class of g0is then, for example, given by

g0=dρ2+ dφ2

A2,

with A > 0growing at most linearly with respect to ρ:

A(ρ, φ)≤c1|ρ|+c2.

To come full circle, we see that the metric g0=r2gis necessarily incomplete since the

function 1

r= exp(−ρ)clearly grows superlinearly.

As a consequence of Thm. 5.22 we now establish an application to static spacetimes.

Corollary 5.24. Let (M=R×S, g =−N2dt2+h)be a standard static spacetime with

a complete Riemannian manifold (S, h)as base and a lapse function N:S→R>0. Then

(M, g)is globally hyperbolic if and only if the function 1

N:S→R>0is not L1on any escaping

curve on Sw.r.t. h. Particularly, (M, g)is globally hyperbolic if Ngrows at most linearly

towards h-infinity on Sand it is not globally hyperbolic if Ngrows superlinearly towards

h-infinity on S.

Proof. This follows directly from Thm. 5.22 and from Thm. 5.5 applied to the conformally

transformed metric g

N2=−dt2+h

N2.

In the remainder of this section, we will consider standard stationary spacetimes (R×S, g)

with metric given by

g=−dt⊗dt+ 2b∨dt+h.

100

All results on global hyperbolicity below, established for standard stationary spacetimes in

this form also hold for general standard stationary spacetimes if we replace bby b

N2and h

by h

N2. This is due to the fact that the metric g=−dt2+ 2bdt+hmay be seen as derived

from the metric −N2dt2+ 2bdt+hby a conformal transformation with factor 1

N2.

We will repeatedly make use of the following lemma.

Lemma 5.25. Let (S, g)be a Riemannian manifold, ba one-form on Sand kbkg

x<1for all

x∈S, such that h±=g±b⊗bare Riemannian metrics on S. Then, for the corresponding

norms of bthe following holds:

(kbkh±

x)2=(kbkg

x)2

1±(kbkg

x)2

for all x∈S.

Proof. We compute

(kbkg−b⊗b

x)2= sup

v∈TxS\{0}

|bx(v)|2

gx(v, v)−|bx(v)|2= sup

v∈TxS\{0}

|bx(v)|2

gx(v,v)

1−|bx(v)|2

gx(v,v)

=(kbkg

x)2

1−(kbkg

x)2,

since

sup

v∈TxS\{0}

|bx(v)|2

gx(v, v)= (kbkg

x)2<1

holds for all x∈S. Solving for (kbkg

x)2and renaming g=h−+b⊗byields the assertion for

h+.

Theorem 5.26. Let (M=R×S, g)be a standard stationary spacetime with metric g=

−dt2+ 2bdt+h. Then, (M, g)has Cauchy hypersurfaces Stif the Riemannian metric

h:= h+b⊗b

(1 + (kbkh)2)2

is complete on S. Moreover, if (M, g)is globally hyperbolic, then Stis a Cauchy hypersurface

if 1

1+(kbkh

x)2is not L1w.r.t. h+b⊗bon any escaping curve on Sand, particularly, if (kbkh

x)2

grows at most linearly towards (h+b⊗b)-infinity on S.

Proof. For the standard stationary metric g=−dt2+2bdt+hon R×S, the Fermat metric

on Sis given by

F(x, v) = phx(v, v)+(bx(v))2+bx(v),

for all x∈Sand all v∈TxS. Using Lem. 5.25, we compute

F(x, v) = phx(v, v)+(bx(v))2 1 + bx(v)

phx(v, v)+(bx(v))2!≥

101

≥phx(v, v)+(bx(v))21−kbkh+b2

x=phx(v, v)+(bx(v))2 1−s(kbkh

x)2

1+(kbkh

x)2!≥

≥1

2phx(v, v)+(bx(v))2

1+(kbkh)2=1

2qˆ

hx(v, v)

establishing

dF(x, y)≥d1

2ˆ

h(x, y),

for all x, y ∈S, with the Finslerian distance dFassociated to the Fermat metric Fand the

Riemannian distance d1

2ˆ

hassociated to the metric 1

2ˆ

h. As 1

2ˆ

his complete if and only if ˆ

his

complete, the distance d1

2ˆ

his a complete distance if and only if ˆ

his complete. Therefore,

by an argument similar to the one in the proof of Prop. 2.19, the completeness of ˆ

himplies

the forward and backward completeness of the Fermat metric Fand thus the slices Stare

Cauchy surfaces.

If (M, g)is globally hyperbolic, h+b⊗bis complete for any slicing M=R×S. Thus, by

Thm. 5.22,ˆ

his complete on Sif 1

1+(kbkh

x)2is not L1on any escaping curve on Sw.r.t. h+b⊗b

and, particularly, if (kbkh

x)2grows at most linearly towards (h+b⊗b)-infinity and, therefore,

the slices Stare Cauchy hypersurfaces.

By virtue of the stationary-to-Randers correspondence, this theorem also encompasses a

condition for the completeness of a Randers-type metric:

Corollary 5.27. Let (M, h)be a Riemannian manifold and b∈Γ(Λ1M)a one-form on

M, obeying kbkh

x<1for all x∈M. The Randers-type metric R=√h+bis forward and

backward complete if the Riemannian metric 1−(kbkh)22his complete on M.

Proof. Replacing hby h−b⊗bin Thm. 5.26 leads to the completeness of

h:= h

1+(kbkh−b2)22

as a sufficient condition for the forward and backward completeness of R. By Lem. 5.25, the

Riemannian metric ˆ

his equal to 1−(kbkh)22h.

Thm. 5.26 gives a sufficient condition for the existence of Cauchy hypersurfaces and global

hyperbolicity in standard stationary spacetimes based solely on Riemannian completeness

and growth conditions on the base manifold. Moreover, the single fact that

h≤h+b⊗b

yields another simple proof for the necessity of the Riemannian metric h+b⊗bto be complete

in globally hyperbolic standard stationary spacetimes.

Moreover, Thm. 5.26 above gives rise to the following first pinching result.

102

Proposition 5.28. Let F=√h+b⊗b+bbe the Fermat metric associated with a standard

stationary spacetime (R×S, g), with metric g=−dt2+ 2bdt+h. Then we have

2sh+b⊗b

(1 + (kbkh)2)2≤F≤2√h+b⊗b.

Furthermore, a standard stationary spacetime (R×S, −dt2+ 2bdt+h)with complete metric

h+b⊗bon Sis globally hyperbolic if kbkh+b⊗b

kbkh

x2is not L1on any escaping curve on S

w.r.t. h+b⊗b.

And for a globally hyperbolic standard stationary spacetime, the Randers-type metric

kbkh

kbkh+b⊗b2

is complete on S.

Proof. All inequalities are supposed to hold pointwise, but we omit the dependencies

hx(v, v)and bx(v)with x∈Sand v∈TxSfor notational simplicity. We need to es-

tablish the right inequality F≤2√h+b⊗b, as the left inequality follows from the proof of

Thm. 5.26. Starting from

√h+b⊗b+b≤pX2(h+b⊗b),

we would like to find a function X:S→R(i.e., Xdepends on x∈Sonly), such that this

inequality holds for all x∈Sand all v∈TxS. Solving for Xyields

X≥1 +

√h

q1 + b2

As it is true that kbkh≥ | b

√h|, we have that

√h

q1 + b2

≤1.

Thus, the inequality holds if X≥2, as the optimal bound, i.e., in the general case

supx∈Skbkh=∞(equivalent to supx∈Skbkh+b⊗b= 1 by Lem. 5.25), the constant func-

tion X(x)=2is the “smallest” possible function, which makes the right inequality to hold

for all x∈Sand all v∈TxS.

Now, let (R×S, −dt2+ 2bdt+h)be a standard stationary spacetime with complete Rie-

mannian metric h+b⊗b. Consequently, by Lem. 5.25 we have that

kbkh+b⊗b

kbkh

x2

1+(kbkh

x)2,

103

and hence by Thm. 5.26, the slices are Cauchy hypersurfaces and (R×S, g)is globally

hyperbolic if the L1-condition holds.

The left inequality reads

2√h+b⊗b≤(1 + (kbkh)2F=kbkh

kbkh+b⊗b2

by Lem. 5.25. Hence, the completeness of h+b⊗bin the case of global hyperbolicity yields

the completeness of kbkh

kbkh+b⊗b2F.

Now, the question naturally arises if one may find further pinching results for Fermat metrics

on Sin terms of Riemannian metrics on S. By this, we mean inequalities of the form

rg0

Y2≤F=√h+b⊗b+b≤pX2g0,

with g0a (preferably complete) Riemannian metric on Sand X, Y :S→R>0functions

on S. The Fermat metric F(x, v)(like any Randers metric) depends on the direction of

the vector argument v, since the one-form benters its definition. Thus, depending on the

norm of b(with respect to the metric g0) the conformal factors Xand Yhave to alter

the magnitudes of g0(v, v)for a vector vin order to produce Riemannian metrics “larger” or

“smaller” than the Fermat metric. For any Riemannian metric g0there are optimal estimates

of the functions Xand Y, meaning the “smallest” possible functions we may introduce for a

given Riemannian metric g0, such that the inequalities above hold. For example, if we have

g0=g+b⊗b, the optimal estimate for the upper bound is the constant 2as the proof of

Prop. 5.28 above shows.

The obvious choice is g0=h.

Proposition 5.29. Let F=√h+b⊗b+bbe the Fermat metric associated with a standard

stationary spacetime (R×S, g), with metric g=−dt2+ 2bdt+h. Then we have as optimal

estimates √h

p1+(kbkh)2+kbkh≤F≤(q1+(kbkh)2+kbkh)√h.

Thus, furthermore, we have the following two results:

(i) For a standard stationary spacetime (R×S, −dt2+ 2bdt+h)with complete metric

h, the slices Stare Cauchy hypersurfaces if and only if kbkh+b⊗b

kbkh(1+kbkh+b⊗b)is not L1on

any escaping curve on Sw.r.t. h. Particularly, the slices are Cauchy hypersurfaces

if kbkhgrows at most linearly towards h-infinity on Sand they are not if kbkhgrows

superlinearly towards h-infinity on S.

(ii) For a standard stationary spacetime (R×S, −dt2+2bdt+h), for which kbkh+b⊗b

kbkh(1+kbkh+b⊗b)

is not L1on any escaping curve on Sw.r.t. h, the slices Stare Cauchy hypersurfaces

if and only if his complete.

104

Proof. We establish the left inequality starting from

Y2≤√h+b⊗b+b.

The right inequality follows from similar reasoning. Like above, we omit the dependencies

hx(v, v)and bx(v)for simplicity. We get

Y≤r1 + b2

h+b

√h.

Since |b

√h|≤kbkhholds, this inequality is valid if

Y2≥1

(p1+(kbkh)2−kbkh)2= (q1+(kbkh)2+kbkh)2.

For the remaining assertions we first observe that

p1+(kbkh)2+kbkh=1

kbkh

kbkh+b⊗b+kbkh=kbkh+b⊗b

kbkh(1 + kbkh+b⊗b)

by Lem. 5.25.

For item (i) we conclude by Thm. 5.22 that the L1-condition assures the completeness of F

by the left inequality and hence the slices Stare Cauchy hypersurfaces. On the other hand,

if the slices are Cauchy hypersurfaces, the Fermat metric Fis complete and by the right

inequality so is (p1+(kbkh)2+kbkh)2h. As we have that his complete by assumption, this

implies the L1-condition by using Thm. 5.22.

Furthermore, assume that kbkhgrows at most linearly towards h-infinity on S, then so does

p1+(kbkh)2+kbkh. Hence, by using Thm. 5.22, we conclude that Fis complete—and

thus the slices are Cauchy hypersurfaces—by the left inequality under the assumption of

a complete metric h. On the other hand, assume kbkhto grow superlinearly towards h-

infinity on S, then so does p1+(kbkh)2+kbkh. But assuming that the slices are Cauchy

hypersurfaces, and hence Fis complete, yields a complete metric (p1+(kbkh)2+kbkh)2hby

the right inequality and hence an incomplete metric h—again by Thm. 5.22—in contradiction

to the assumption of hbeing complete.

For item (ii) first assume that the slices Stare Cauchy hypersurfaces, i.e., the Fermat metric

Fis complete. This yields a complete metric (p1+(kbkh)2+kbkh)2hby the right inequality

and thus a complete metric hby the L1-condition and Thm. 5.22. Then assuming that his

complete and using the left inequality together with the L1-condition and Thm. 5.22, yields

a complete Fermat metric and hence the slices as Cauchy hypersurfaces.

Now, we would like to establish a pinching result for the Fermat metric using a necessarily

complete Riemannian metric g0. This will result in a purely analytic criterion for global

hyperbolicity of standard stationary spacetimes based on L1- and growth conditions only.

105

The results of K. Nomizu and H. Ozeki [NO61] assure the existence of a complete Riemannian

metric on any manifold. Thus, for the base Sof a standard stationary spacetime (R×

S, g)there necessarily always exists a complete Riemannian metric, generally differing from

the Riemannian metric hinherited from the Lorentzian metric g=−dt2+ 2bdt+hon

R×S. However, a complete Riemannian metric may be analytically constructed from any

Riemannian metric on a manifold Susing proper functions (see [Gor73], [Gor74]). We recall

that a function f:S→Ris said to be proper if f−1(K)is compact whenever K⊂Ris

compact. The following theorem holds:

Theorem 5.30. Let (S, h)be any Riemannian manifold and f:S→Rany proper function

on S. Then (S, ˜

h)is a complete Riemannian manifold, where

h=h+ df⊗df.

Moreover, a Riemannian manifold (S, h)is complete if and only if there is a proper function

f:S→Ron Ssuch that

sup

x∈Skdfkh

x<∞.

Proof. See [Gor73,Gor74] and references therein.

Now, we use Thm. 5.30 to establish the pinching result

rh+ df⊗df

Y2≤F≤pX2(h+ df⊗df),

for a necessarily complete metric h+df⊗dfon the base Sof a standard stationary spacetime

(R×S, g)with f:S→Rbeing any proper function on S.

Theorem 5.31. Let (R×S, g)be a standard stationary spacetime with Lorentzian metric

g=−dt2+2bdt+hand f:S→Ra proper function on S, such that h+df⊗dfis a complete

Riemannian metric on S. Then the pinching of the Fermat metric F=√h+b⊗b+bresults

Y2≥4(1 + (kdfkh)2)(1 + (kbkh)2)

and

X2≥4(1 + (kbkh)2).

Moreover, we have the following two results:

(i) The standard stationary spacetime (R×S, g)is globally hyperbolic if there exists a

proper function f:S→Rsuch that kbkh+b⊗b

kbkh√1+(kdfkh)2is not L1on any escaping curve

on Sw.r.t. h+ df⊗dfand, particularly, if there exists a proper function f:S→R

such that the product kdfkh·kbkhgrows at most linearly towards (h+df⊗df)-infinity.

(ii) For a globally hyperbolic and standard stationary spacetime (R×S, g), with the slices

Stas Cauchy hypersurfaces, the function kbkh+b⊗b

kbkhis not L1on any escaping curve on

Sw.r.t. h+df⊗df, particularly, for all proper functions f:S→R, the function kbkh

does not grow superlinearly towards (h+ df⊗df)-infinity on S.

106

Proof. We first establish the inequality for Y. It must hold that

rh+ df⊗df

Y2≤√h+b⊗b+b.

Omitting the dependencies as in the proof of the previous proposition, this is equivalent to

1 + df2

Y2≤ r1 + b2

h+b

√h!2

which is fulfilled if

1 + df2

Y2≤(q1+(kbkh)2−kbkh)2.

This is equivalent to

Y2≥1 + df2

(p1+(kbkh)2−kbkh)2,

and due to df2

h≤(kdfkh)2and (p1+(kbkh)2+kbkh)2≤4(p1+(kbkh)2)2, this is true if

Y2≥4(1 + (kdfkh)2)(1 + (kbkh)2)

holds. By a similar reasoning the inequality for Xis established. We get

X2≥41+(kbkh)2

1 + df2

which is fulfilled if

X2≥4(1 + (kbkh)2)

holds.

Now we can pinch with the complete Riemannian metric h+ df⊗df:

2sh+ df⊗df

(1 + (kdfkh)2)(1 + (kbkh)2)≤√h+b⊗b+b≤2q(1 + (kbkh)2)(h+ df⊗df).

By Lem. 5.25 we observe that

p1+(kdfkh)2p1+(kbkh)2=kbkh+b⊗b

kbkhp1+(kdfkh)2.

Thus, if the L1-condition is fulfilled, the left inequality of the pinching above, yields the

completeness of Fby Thm. 5.22 as h+ df⊗dfis complete. Particularly, if kdfkh· kbkh

grows at most linearly towards (h+ df⊗df)-infinity, so does p1+(kdfkh)2p1+(kbkh)2

and again by Thm. 5.22 and the completeness of h+ df⊗df, we get the completeness of

F. This proves item (i).

107

For item (ii) assume the slices Stare Cauchy hypersurfaces, hence the Fermat metric is

complete. By the right inequality of the pinching result we now get that

(1 + (kbkh)2)(h+ df⊗df) = kbkh

kbkh+b⊗b2

(h+ df⊗df)

is complete, using Lem. 5.25. As fis a proper function, h+ df⊗dfis complete and thus

kbkh+b⊗b

kbkhis not L1on any escaping curve on Sw.r.t. h+df⊗dfby Thm. 5.22. Similarly, the

superlinear growth of kbkhtowards (h+df⊗df)-infinity on Sis impossible by Thm. 5.22.

Thus, we conclude that the linear growth of kbkhtowards h-infinity on the base Sof a

standard stationary spacetime (R×S, g)alone is neither a necessary nor a sufficient condition

for (R×S, g)to be globally hyperbolic—as one might guess from Prop. 5.29—but only the

linear growth of kbkhtowards (h+ df⊗df)-infinity for any proper function fon Sis

necessary. This is a weaker condition. Of course, in the case of a complete Riemannian

metric hon S, the linear growth of the product kdfkh·kbkhtowards (h+ df⊗df)-infinity

leads to the condition of linear growth of kbkhtowards (h+ df⊗df)-infinity, which is

sufficient for global hyperbolicity in this case.

Further remarkable results are the completeness of the Randers-type metric

G:= kbkh

kbkh+b⊗b2

in the case of global hyperbolicity from Prop. 5.28 and the L1-condition of kbkh+b⊗b

kbkhin the

case of Cauchy hypersurfaces Stfrom Thm. 5.31. It is worth comparing the completeness

of Gto the results on the stationary-to-Randers correspondence in [CJS11]. Therein, it has

been established that a standard stationary spacetime (R×S, g)with g=−dt2+ 2bdt+h

and Fermat metric F=√h+b⊗b+bis globally hyperbolic if and only if there is a function

f:S→Rsuch that ˜

F=F−dfis forward and backward complete on S. See Thms. 4.3

and 5.10 in [CJS11]. Hence, with the results above we have the following

Corollary 5.32. Let (M, g)be a Riemannian manifold and ba one-form on M, such that

kbkg<1and R=√g+bis a Randers metric. If there is a function f:M→R, such that

R=R−dfis a complete Randers metric on M, the Randers metric

R0=kbkg−b⊗b

kbkg2

is complete.

The L1-condition of kbkh+b⊗b

kbkhgives a criterion to check for the slices Stto be Cauchy hyper-

surfaces, based purely on the growth of this function along curves.

108

5.3 Lorentzian Bochner Technique

The Bochner technique in Riemannian geometry yields important non-existence results for

Killing vector fields on compact Riemannian manifolds. For example, a compact (without

boundary), oriented Riemannian manifold (M, g)with non-positive Ricci curvature, Ric ≤0,

admits only geodesic Killing vector fields, and if the Ricci curvature is negative, Ric < 0,

then there are no non-trivial Killing vector fields. See, e.g., [Wu80] for a survey. The

Riemannian Bochner technique relies on the ellipticity of the Laplace–Beltrami operator;

so generalizations to the semi-Riemannian setting are ambiguous. Nevertheless, A. Romero

and M. Sánchez developed a Bochner technique for compact Lorentzian manifolds in [RS96]

and [RS98], using an integral formula. Similar ideas were also developed by S.E. Stepanov

in [Ste93], [Ste99] and [Ste00]. Already A. Lichnerowicz has obtained some of the results in

this section about Ricci flat stationary Lorentzian manifolds with compact or asymptotically

flat spacelike slices in [Lic55] using similar considerations.

We will need the following particular class of semi-Riemannian submersions. The standard

references for the theory of submersions are [O’N66] and [Gra67], where we have to apply a

straightforward generalization from the Riemannian to the Lorentzian case at some places.

Some theory on Lorentzian submersions is contained in [FIP04], but we will not base this

section on that reference, as our focus here is a different one.

Definition 5.33. A mapping π: (Mn+1, g)→(Nn, h)will be called a Lorentz-to-Riemann

submersion if it is a surjective submersion, (M, g)is a Lorentzian, (N, h)is a Riemannian

manifold and π∗preserves the norms of horizontal vectors. A vector v∈TpMfor some

p∈Mis called vertical if v∈ker(dπp), i.e., dπp(v) = 0 and it is called horizontal if it is

in the g-orthocomplement ker(dπp)⊥of ker(dπp). The manifold Nis called the base of the

Lorentz-to-Riemann submersion, the preimages π−1(x)⊂Mare called fibers over x∈N.

If π: (M, g)→(N, h)is a Lorentz-to-Riemann submersion, we necessarily have that all

vertical vector fields are timelike and all horizontal vector fields are spacelike. Particularly,

for any horizontal vector field X∈Γ(TM)we have

gp(Xp, Xp) = hπ(p)(πp∗Xp, πp∗Xp),

for all p∈M. It is not difficult to see that any Lorentz-to-Riemann submersion π: (M, g)→

(N, h)has the splitting structure of a proper kinematical spacetime if the fibers are diffeo-

morphic to R. Indeed, in this case, as Nis a manifold and ker(dπp)is timelike for all p∈M,

there is a free R-action on every fiber π−1(x)for x∈N, as well as a global trivialization.

Therefore, there naturally exists a unique vertical unit vector field V, with Vp∈ker(dπp)

for all p∈M, which serves as a reference frame and there is a complete timelike vector

field V∗parallel to V(cf. Thm. 6.2 for details). We adopt the designations V M and HM

from section 3.1 for the horizontal and vertical distributions, as well as Vand Hfor the

projections on them.

A vector field X∈Γ(HM)on any surjective submersion π:M→Nis called basic if it

projects to a unique vector field X−∈Γ(TN)on N, i.e., we can write X−=π∗X. Hence, for

109

any vector field X∈Γ(TN), there is a unique vector field ˜

X∈Γ(HM)called its horizontal

lift, given by the unique basic vector field on Mthat projects to X. Generally, we will

denote lifted objects with a tilde in this section. Particularly, we also have for functions:

if j:N→Ris any function on the base Nof the Lorentz-to-Riemann submersion, then

j:= j◦π:M→Ris the lifted function on M. We now have the following

Lemma 5.34. Let π: (M, g)→(S, h)be a Lorentz-to-Riemann submersion. For a function

F:M→R, there is a function f:S→R, such that

(˜

f=)F=f◦π:M→R

if and only if dF(V) = 0 for some (and hence for all) nowhere vanishing vertical vector

fields Von M.

Proof. Obviously, as ker(dπp)is one-dimensional for all p∈Mby definition, we see that

dF(V)=0⇔dF(ϕV ) = ϕdF(V)=0for any nowhere vanishing function ϕon M, shows

that dF(V)=0for one nowhere vanishing vector field V∈Γ(V M)implies the same for all

nowhere vanishing vertical vector fields.

“⇒”: If F=f◦π, then dF= d(f◦π) = π∗(df), and hence for any vector field X:M→TM

we have for all p∈M

(dF)(X)|p=π∗(df)(X)|p= dfπ(p)(πp∗Xp),

which is obviously zero if Xis vertical.

“⇐”: Let x∈Sand π−1(x)⊂Mbe the fiber over x. As dF(V) = 0,Fis constant

along any fiber, i.e., F(π−1(x)) = F(p)for any p∈π−1(x). Hence, f:S→Rgiven

by f(x) := F(π−1(x)) is a well defined function on S. But this obviously implies that

F(p) = f(π(p)) = (f◦π)(p)for all p∈M.

The following theorem shows that Lorentz-to-Riemann submersions occur naturally as spa-

tially stationary spacetimes.

Theorem 5.35. Let (M, g, V )be a proper kinematical spacetime with the usual projection

πM:M→S=M/R.πMis a Lorentz-to-Riemann submersion if and only if (M, g, V )is

spatially stationary.

Proof. Following section 4.1 there is a trivialization, such that a proper kinematical space-

time can be written as (M=R×S, g, V ), with πM= pr2:R×S→Sbeing a surjec-

tive submersion and dim(S) = dim(R×S)−1. Denote by Xany horizontal vector field

X∈Γ(HM), i.e., g(X, V )=0. Following section 3.1, the projection ˜

h=g+u⊗uwith

u=g(V, ·)is a Riemannian metric on HM and g(X, X) = ˜

h(X, X).

“⇒”: If pr2acts as a Lorentz-to-Riemann submersion, we have ˜

h= pr∗

2(h)for some Rieman-

nian metric hon S. In this case £V˜

h= 0 holds and (R×S, g, V )is spatially stationary by

definition (cf. Def. 3.28).

110

“⇐”: Suppose the vector fields X, Y ∈Γ(HM)are basic, i.e., lifted from some X−, Y −∈

Γ(TS). In this case we have

V(˜

h(X, Y )) = (£V˜

h)(X, Y )=0

as £VX=£VY= 0. By the same argument as in Lem. 5.34, there is a function η:S→R

such that ˜

h(X, Y )◦pr2=ηand thus there is a Riemannian metric hon Ssuch that

η=˜

h(X, Y )◦pr2=h(X−, Y −). As this holds for all basic vector fields, the submersion

pr2: (R×S, g)→(S, h)preserves the norms of horizontal vectors.

In particular, stationary spacetimes, which play a key role in this section, can be regarded

as Lorentz-to-Riemann submersions, as the following corollary shows.

Corollary 5.36. Any stationary spacetime (R×S, g)with a Killing vector field ∂tand metric

g=−(A◦pr2)2dt⊗dt+ 2pr∗

2(b)∨dt+ pr∗

2(h)−pr∗

2(b)⊗pr∗

2(b)

(A◦pr2)2,

such that his a Riemannian metric on S,b∈Γ(Λ1S)and A:S→R>0is a Lorentz-to-

Riemann submersion π= pr2: (R×S, g)→(S, h).

Proof. Obviously, π= pr2:R×S→Sis a surjective submersion. Moreover, (R×S, g)is

Lorentzian and (S, h)is Riemannian with dim(S) = dim(R×S)−1, by assumption. Now, for

any p= (t, x)∈R×Sand a vector v∈Tp(R×S), we denote by v0:= Vp(v)∈Vp(R×S)≃R

its vertical part and by ~v := πp∗(v)∈TxSits projection to S. The vector vis horizontal if

and only if gp(∂t|p, v)=0, i.e., if and only if

gp(∂t|p, v) = −(A◦π)2(p)v0+π∗

p(b)(v) = −A2(x)v0+bx(~v)=0.

Hence, we have

gp(v, v) = −(A◦π)2(p)v2

0+ 2π∗

p(b)(v)v0+π∗

p(h)(v, v)−π∗

p(b)(v)π∗

p(b)(v)

(A◦π)2(p)=

=−A2(x)v0−bx(~v)

A2(x)+hx(~v,~v) = hx(~v,~v).

These two results allow proper kinematical spacetimes (R×S, g, V )to be called (spatially)

stationary Lorentz-to-Riemann submersions in the following if Vis parallel to a Killing

vector field (Vis a rigid reference frame). Compare to section 3.2.

Now we would like to establish a relation between the Raychaudhuri equations introduced

in section 3.3 and the Lorentzian Bochner technique, which was investigated in [RS96] and

[RS98]. To this end, we assume for the moment that (M, g)is any Lorentzian manifold and

111

X:M→TM any complete timelike vector field. Defining the operator AX: Γ(TM)→

Γ(TM)by

AX(Y) = −∇YX,

for all Y∈Γ(TM)yields the equation

Xdiv(X) = −Ric(X, X) + div(∇XX)−Tr(AX⊗AX).

As can be checked by inspection, this is nothing more than the Raychaudhuri equation

for the expansion of X(cf. item (i) in Prop. 3.33), generalized to non-unit vector fields

X. Now, we assume (M, g)to be a compact Lorentzian manifold and use the identities

Tr(AX) = −div(X), as well as div(div(X)X) = Xdiv(X)+(div(X))2to obtain the integral

[Ric(X, X) + Tr(AX⊗AX)−(Tr(AX))2]µg= 0,

with µgthe canonical measure induced by gon M. Unfortunately, as the metric gis

Lorentzian, the difference Tr(AX⊗AX)−(Tr(AX))2has no fixed sign. But we can con-

sider a reference frame V=|g(X, X)|−1

2Xon (M, g), which gives rise to the orthogonal

decomposition of the tangent bundle as in section 3.1. Hence, we can define an operator

V: Γ(TM)→Γ(HM)on HM =V⊥by

V(Y) = H(AV(Y)),

for all vector fields Yon M. As it holds that Tr(A0

V) = Tr(AV), we have

[Ric(V, V ) + Tr(A0

V⊗A0

V)−(Tr(A0

V))2]µg= 0.

Thus, a decomposition of A0

Vinto a symmetric part S0

Vand an anti-symmetric part H0

such that A0

V=S0

V+H0

Vyields

[Ric(V, V )−(Tr(S0

V))2+ Tr(S0

V⊗S0

V) + Tr(H0

V⊗H0

V)]µg= 0.(∗)

This integral was used in [RS96] and [RS98] as a Lorentzian Bochner technique to deduce

non-existence results of particular vector fields, mostly Killing vector fields, on compact

Lorentzian manifolds. Our goal in this section is to broaden this argument to non-compact

Lorentzian manifolds—particularly, to stationary spacetimes—and certain related classes

of Lorentz-to-Riemann submersions. Therefore, let (Mn+1, g, V )be a proper kinematical

spacetime and consider the Raychaudhuri equation for the expansion of V(cf. item (i) in

Prop. 3.33) in the form

Ric(V, V ) + |σ|2−|ω|2−n−1

nΘ2=−div(ΘV−∇VV),

which follows from div(ΘV) = dΘ(V) + Θ2. Some implications of this form of the Ray-

chaudhuri equation for physically motivated questions were recently also investigated in

[AV11].

112

Remark 5.37. Of course, the Raychudhuri equation above also holds if we assume (M, g, V )

to be a compact kinematical spacetime. In this case, we get the integral

[Ric(V, V ) + |σ|2−|ω|2−n−1

nΘ2]µg= 0,

which is equivalent to (∗). Hence, we can translate the Bochner quantities depending on S0

and H0

Vto the corresponding kinematical quantities appearing in the Raychaudhuri equation.

We get the following Raychaudhuri-Bochner dictionary:

Rauchudhuri Bochner

nh+σ=−g(·, S0

V(·))

ω=g(·, H0

V(·))

Θ=Tr(S0

|ω|2=−Tr(H0

V⊗H0

|σ|2+Θ2

n=Tr(S0

V⊗S0

Moreover, in the case of n= 1, i.e., if (M2, g, V )is a compact spacetime, which is necessarily

a torus in the oriented case, we have that ω= 0 and σ= 0, as the shear only consists of its

trace Θ, and we recover the Lorentzian Gauss–Bonnet formula

ZM2

Ric(V, V )µg=ZM2Kµg= 0,

where K=−Ric(V, V )is the Gaussian curvature of M2.

Proposition 5.38. Let (R×Sn, g, V )be a stationary Lorentz-to-Riemann submersion with

π: (R×S, g)→(S, h). Then

(i) u=g(V, ·) = −(A◦π)dt+π∗(b

A)and ˙u=g(∇VV, ·) = π∗(dA

A),

(ii) ω=π∗(d( b

A2)A), hence ω= 0 if and only if d( b

A2)=0,

(iii) (R×S, g)with reference frame Vis standard static if ω= 0 and the first Betti number

of Svanishes, i.e., B1(S) = 0,

(iv) there are functions r:S→R>0and Ω2:S→R>0, such that Ric(V, V ) = r◦πand

|ω|2= Ω2◦π,

(v) ∇VVis a basic vector field,

(vi) div(∇VV) = [k∇VVk2

h+divh(π∗(∇VV))]◦π, where k∇VVk2

h=h(π∗(∇VV), π∗(∇VV))

and divhis the divergence on Swith respect to h.

Proof. The spacetime (R×S, g)is stationary as in Cor. 5.36, we have V=∂t

A◦πand ∂t

is Killing, with dt(∂t)=1, based on the metric splitting results in section 4.1, particularly

Cor. 4.11.

113

(i)

u=1

A◦πg(∂t,·) = 1

A◦π−(A◦π)2dt+π∗(b) + π∗(b)(∂t)dt+π∗(γ)(∂t)=

=−(A◦π)dt+π∗(b

A).

As ω= du+u∧˙u, we have ˙u=Vcdu. Thus,

˙u=∂t

A◦πc−d(A◦π)∧dt−d(A◦π)

(A◦π)2∧π∗(b) + dπ∗(b)

(A◦π)=d(A◦π)

(A◦π)=π∗(dA

as d(A◦π)(∂t)=0and ∂tcdπ∗(b) = 0.

(ii) We compute

ω= du+u∧˙u=−d(A◦π)−d(A◦π)

(A◦π)2∧π∗(b)+ dπ∗(b)

A◦π+−(A◦π)dt+π∗(b)

A◦π∧d(A◦π)

A◦π=

=dπ∗(b)

A◦π+ 2π∗(b)∧d(A◦π)

(A◦π)2=π∗db

A+ 2b∧dA

A2=π∗(d( b

A2)A),

hence ω= 0 if and only if d( b

A2)=0follows from A > 0.

(iii) By Def. 5.15,(R×S, g)is standard static if b= 0. We show that there is an isometry (a

change of slicing according to section 4.1)φ: (R×S, g)→(R×S, g0), such that (R×S, g0)

is standard static if ω= 0 and B1(S)=0. We showed above in (ii) that ω= 0 implies

d( b

A2) = 0 on S. Thus, as B1(S) = 0, there is a function f:S→R, such that b=A2df

if the rotation vanishes. As (R×S, g), together with the reference frame V, constitutes a

proper kinematical spacetime, we can apply Thm. 4.16 to (R×S, g)using a trivialization φ

that transforms b7→ b−A2df. This is certainly an isometry and φ∗(b) = 0 in (R×S, g0),

hence (R×S, g0)is standard static.

(iv) By Lem. 5.34, it suffices to show that £∂t[Ric(V, V )] = 0 and ∇V|ω|2= 0. As ∂tis

Killing we certainly have £∂tg= 0, which also implies £∂tRic = 0. This together with

[V, ∂t] = 0 implies the first equation. For the second equation, we will use the Raychaudhuri

equation for the rotation from Prop. 3.33. As (R×S, g)is stationary the expansion Θand

the shear σfor the reference frame Vboth vanish. Moreover, we have d ˙u= 0 because of

(i), hence the Raychaudhuri equation for the rotation reads in this case

∇Vω=u∧(∇VVcω).

Thus, we have

∇V|ω|2=∇VTr(ω⊗ω) = Tr((∇Vω)∨ω) = Tr ([u∧(∇VVcω)] ∨ω) =

i,j

[u∧(∇VVcω)](Ei, Ej)ω(Ei, Ej) = 0

as u(Ei)=0(i∈ {1, . . . , n}) for any horizontal vector field in some pseudo-orthonormal

frame {V, E1, . . . , En}and ω(V, ·) = 0.

114

(v) Let {Ei}i=1,...,n be any h-orthonormal frame on S. Then, the Ei’s lift to basic vector fields

Eion R×S, such that {V, ˜

Ei}i=1,...,n is a g-pseudo-orthonormal frame on R×S. Following

Lem. 1.2 in [Ej75], for a horizontal vector field Xon R×Sto be basic, it suffices to show that

gp(X, ˜

Ei) = gq(X, ˜

Ei)for all p, q ∈π−1(x)in the fiber over any x∈S. (Actually, a proof

for this assertion is given in [Ej75] for Riemannian submersions only, but a generalization

to the Lorentzian case, at hand here, is straightforward.) Clearly, ∇VVis horizontal. As in

our case the fibers are one-dimensional, it suffices to show that V[g(∇VV, ˜

Ei)] = 0 for all

the ˜

Ei’s. As the ˜

Ei’s are basic we have [V, ˜

Ei] = 0 and thus

V[g(∇VV, ˜

Ei)] = £V( ˙u(˜

Ei)) = (£V˙u)( ˜

Ei) = (Vcd ˙u)( ˜

Ei)=0

due to d ˙u= 0. Hence, there is a unique and well-defined vector field π∗(∇VV)on Sthat

lifts to the acceleration of Von R×S.

(vi) As above, let {V, ˜

Ei}i=1,...,n be a g-pseudo-orthonormal frame on R×S, that results

from the lift of an h-orthonormal frame {Ei}i=1,...,n on S. We denote by ∇hthe Levi-Civita

connection associated to hon S. Following Thm. 3.2 in [Gra67], we have for all basic vector

fields X, Y ∈Γ(H(R×S)), related to X−, Y −∈Γ(TS), that ∇XYis basic and related to

∇h

X−Y−on S. Then we compute

div(∇VV) = −g(∇V∇VV, V ) + X

g(∇˜

Ei(∇VV),˜

Ei) = g(∇VV, ∇VV)+

g(^

∇h

Ei[π∗(∇VV)],˜

Ei) = h(π∗(∇VV), π∗(∇VV)) ◦π+X

h(∇h

Ei[π∗(∇VV)], Ei)◦π,

and the result follows.

Now we are ready to state a main result of this section.

Theorem 5.39. Let (R×S, g, V )be a stationary Lorentz-to-Riemann submersion with

π:R×S→Sand compact S. Then there are two functions r:S→R≥0and Ω2:S→R≥0

with Ric(V, V ) = r◦πand |ω|2= Ω2◦π, such that

[r−Ω2−k∇VVk2

h]µh= 0,

as well as ZS

rµh≥0

holds. Thus, the timelike Ricci curvature cannot be non-positive, and negative somewhere,

i.e., for any stationary Lorentz-to-Riemann submersion with compact S,Ric(X, X)≤0, and

Ric(X, X)<0somewhere, for all timelike X∈Γ(T(R×S)) cannot occur. Furthermore,

(i) if (R×S, g)is Ricci flat, then it is static and Vis geodesic, if additionally B1(S) = 0,

then (R×S, g)is standard static and

(ii) if Ric(X, X)≥0, and Ric(X, X)>0somewhere, for all timelike X∈Γ(T(R×S))

then (R×S, g)is not static or Vis not geodesic.

115

Proof. We start from the Raychaudhuri equation for the expansion of V, which reads

Ric(V, V )−|ω|2= div(∇VV)

in the stationary case. Based on (iv) and (vi) in Prop. 5.38, there are two functions rand

Ω2on Sas stated and we have

r◦π−Ω2◦π=k∇VVk2

h◦π+ divh(π∗(∇VV)) ◦π,

hence

r−Ω2−k∇VVk2

h= divh(π∗(∇VV))

holds on S. As Sis compact, integration with respect to the measure µhinduced by hyields

[r−Ω2−k∇VVk2

h]µh= 0.

Clearly, we have Ω2≥0and k∇VVk2

h≥0on all of S, such that RSrµh≥0follows. Now

suppose that for all timelike X∈Γ(T(R×S)), we have Ric(X, X)≤0, and Ric(X, X)<0

somewhere. This would imply RSrµh<0, a contradiction. In the Ricci flat case we have

[Ω2+k∇VVk2

h]µh= 0,

and due to Ω2≥0and k∇VVk2

h≥0this implies Ω2= 0 and k∇VVk2

h= 0. Hence, because

of (v) in Prop. 5.38 we have that k∇VVk2

h= 0 if and only if |˙u|2= 0 and because of

Ω2=|ω|2◦πwe have that Ω2= 0 if and only if |ω|2= 0. This yields ˙u= 0 and ω= 0 in

the Ricci flat case. The standard static case follows from (iii) in Prop. 5.38. For the last

assertion suppose that (R×S, g)is static and Vis geodesic, i.e., Ω2=k∇VVk2

h= 0. This

implies ZS

rµh= 0,

hence Ric(V, V )≥0, and Ric(V, V )>0somewhere, yields a contradiction, as this would

imply RSrµh>0.

Remark 5.40. Based on Thm. 5.39, we can conclude that negative timelike Ricci curvature

on spacetimes (R×S, g)with gbeing timelike along the factor Rand Scompact is an

obstruction for the existence of a timelike Killing vector field along the factor R.

In a next step, we will generalize the result in Thm. 5.39 in some directions. There are two

directions in which the assumptions in the theorem could be relaxed. The first one is to

generalize the existence of a Killing vector field parallel to V, to the vector field parallel

to Vbeing a homothetic vector field. The problem that arises in this case is that the

mapping π: (R×S, g)→(S, h)will still be a submersion, but generally not a Lorentz-

to-Riemann submersion any more. The second direction of relaxation is to consider non-

compact manifolds S, while imposing some asymptotic conditions on the geometry of (S, h)

116

towards the ends of S. This leads to consider Riemannian manifolds (S, h)as a base which

possess a specific asymptotic behavior of the metric towards infinity. This generalization,

we will explore first.

A manifold Snis said to have m(m∈N)ends, denoted by N(k)⊂S,k= 1, . . . , m, if there

is a compact set K⊂S, such that S\K=Fm

k=1 N(k)and each end N(k)is diffeomorphic

to R>1×Sn−1(or R>1if n= 1).

Definition 5.41. Let (Sn, h)be a Riemannian manifold. We say that (Sn, h)is of order p

(at infinity) if Shas mends diffeomorphic to R>1×Sn−1(or R>1if n= 1), and in each end

there exists a coordinate chart (r, φ)∈R>1×Sn−1(with φbeing the canonical coordinates

on the spheres Sn−1) or r∈R>1if n= 1, such that for the components of the metric hin

these charts

hij =δij +O(1

rp)

holds (i, j = 1, . . . , n), for some p≥0and with δ= dr2+r2dφ2being the flat metric on

R>1×Sn−1or δ= dr2being the flat metric on R>1if n= 1.

Remark 5.42. Being of order p > 0is a weaker condition for a Riemannian manifold (S, h)

than being p-asymptotically flat, in which case one additionally requires the first derivatives

of the metric to be of order p+ 1, i.e., ∂khij =O(1

rp+1 ), and the second derivatives to be

of order p+ 2, i.e, ∂l∂khij =O(1

rp+2 ), on all ends (cf. [Wal84, Ch. 11]). But hence, a

p-asymptotically flat Riemannian manifold is certainly of order p > 0at infinity.

Definition 5.43. Let (R×S, g, V )be a stationary Lorentz-to-Riemann submersion. We say

that the reference frame Vis q-spatially asymptotically geodesic if there is some q > 0, such

that

kπ∗(∇VV)k2

h=O(1

rq)

holds on all ends of S.

In the case when (S, h)is additionally of order pwe have the following

Proposition 5.44. Let (R×S, g, V )be a stationary Lorentz-to-Riemann submersion with

(S, h)being of order pat infinity. Then the reference frame Vis q-spatially asymptotically

geodesic if ∂iA

A=O(1

rp+q

)

holds for the lapse function Aand all indices i= 1, . . . , n, on all ends of S.

Proof. If (S, h)is of order p, the components of the inverse metric h]∈Γ(TS ∨TS)

associated to hon any end of Sbehave as

h]=δ+O(rp),

where δis the flat metric. Hence, clearly

kπ∗(∇VV)k2

h=h](dA

A,dA

A) = δ(dA

A,dA

A) + O(rp)O(dA

A)2

117

is of order 1

rqif ∂iA

Ais of order 1

rp+q

on all ends of S.

Theorem 5.45. Let (R×Sn, g, V )be a stationary Lorentz-to-Riemann submersion, such

that Shas m > 0ends diffeomorphic to R>1×Sn−1or R>1if n= 1. Then there are two

functions r:S→R≥0and Ω2:S→R≥0with Ric(V, V ) = r◦πand |ω|2= Ω2◦π, and

[r−Ω2−k∇VVk2

h]µh= 0,

as well as ZS

rµh≥0

hold if

(i) (S, h)is of order pand

∂iA

A=O(1

rp+n−1+ε

)

holds for the lapse function A, all indices i= 1, . . . , n and some ε > 0on all ends of

S, or

(ii) the function kπ∗(∇VV)k2

hdecreases faster towards infinity on all ends of S, than the

(n−1)-volume of the embedded (n−1)-spheres of radius Rgrows towards infinity, i.e.,

max

Sn−1(R)kπ∗(∇VV)k2

h·Volh

n−1(Sn−1(R)) R→∞

−→ 0,

(iii) n= 1 and kπ∗(∇VV)k2

R→∞

−→ 0towards infinity on all ends of S.

In all these cases, the same conclusions on the Ricci curvature, and on the spacetime being

(standard) static and/or Vbeing geodesic, hold as in Thm. 5.39.

Proof. As in the proof of Thm. 5.39 above, the existence of the functions rand Ω2follow

from (iv) and (vi) in Prop. 5.38 and we get

r−Ω2−k∇VVk2

h= divh(π∗(∇VV)),

on Sfrom the Raychaudhuri equation. Now we integrate both sides of this equation over

a compact set KR⊂S, the boundary of which is a disjoint union of membedded (n−1)-

dimensional spheres in the ends of S, all of which have radius R > 1in the coordinates (r, φ)

on the ends of S. We denote these spheres by Sn−1

k(R)for k= 1, . . . , m and get

ZKR

[r−Ω2−k∇VVk2

h]µh=ZKR

divh(π∗(∇VV))µh=

k=1 ZSn−1

k(R)∗h(h(π∗(∇VV),·)),

where we denote by ∗hthe Hodge operator on Swith respect to h. This yields

ZKR

[r−Ω2−k∇VVk2

h]µh≤

k=1

max

Sn−1

k(R)kπ∗(∇VV)k2

h·Volh

n−1(Sn−1

k(R)).

118

As clearly RKRµg→RSµgwhen R→ ∞, this proves the condition in item (ii). Moreover,

if n= 1 we have Volh

0(S0(R)) = 1 and the condition in item (iii) follows.

Now assume that (S, h)is of order pat infinity and the condition for the order of ∂iA

Ato

decrease towards infinity on all ends of Sholds as in item (i). Using Prop. 5.44 this yields

that the Lorentz-to-Riemann submersion is (n−1 + ε)-spatially asymptotically geodesic,

i.e.,

kπ∗(∇VV)k2

h=O(1

rn−1+ε)

on all ends of S. As (S, h)is of order p, the volume of an embedded (n−1)-sphere in an

end of Swith radius R > 1increases as (1 + 1

Rp)Rn−1towards infinity. Hence, we have

max

Sn−1(R)kπ∗(∇VV)k2

h·Volh

n−1(Sn−1(R)) = O(1

Rε),

for some ε > 0, which approaches zero as R→ ∞. This proves the condition in item (i). All

remaining assertions follow from reasoning identical to the proof of Thm. 5.39. Note that

we allow RSrµh=∞if necessary.

It is worth comparing the results in Thms. 5.39 and 5.45 to Thm. 0.1 by M.T. Anderson

in [And00], which says that any stationary, chronological, geodesically complete and Ricci

flat spacetime, of dimension four, is (a finite spacelike quotient of) flat Minkowski space.

The crucial step in proving this theorem is to conclude from the Raychaudhuri equation

divh(π∗(∇VV)) = Ω2+k∇VVk2

hon the base (S, h)of a stationary spacetime (R×S, g),

and other constraint equations for the kinematical quantities that follow from Ric = 0, that

ω= 0 and ˙u= 0. If Sis compact or of order pat infinity—the cases we investigated

in this work—the assertion follows from the developed Lorentzian Bochner technique. In

fact in the compact case we can conclude that S(if it is assumed orientable) can only

be isometric to the flat 3-torus. Based on further propositions shown in this thesis, we

can conclude the following about the scope of Anderson’s theorem: Once the vanishing

rotation and acceleration are established, the flatness of the spacetime, i.e., the vanishing

of the Weyl curvature, follows from Remark 3.34. Certainly, this shows that the theorem

cannot be generalized to spacetime dimensions ≥5, as only in dimension four the Weyl

tensor is completely determined by the kinematical quantities and the trace-free part of

Ricci curvature. The assumption of geodesic completeness certainly cannot be relaxed (just

consider the spacetime arising from cutting pieces from Minkowski space). But we can see

that the chronology condition is not necessary. In fact Anderson states in [And00] that it is

unknown if the theorem holds for non-chronological stationary spacetimes. We can fill this

gap based on the considerations in chapter 4. Indeed, the chronology condition is used in

the proof of the theorem only for assuring the diffeomorphic splitting of the spacetime as

R×Salong the Killing vector field, based on the considerations in [Har92]. But looking

at Prop. 4.10 and its corollary, reveals that the only necessary assumptions that cannot be

relaxed are the completeness of the Killing vector field and the non–partial-imprisonment of

its integral curves. As a result the spacetimes can be arbitrary finite quotients of Minkowski

space.

119

We will now investigate the scope of the particular version of the Lorentzian Bochner tech-

nique, which was developed in this section, for spacetimes admitting a timelike homothetic

vector field. To this end we will consider spacetimes (R×S, g)already given in a splitting

form, together with the complete canonical vector field ∂twhich we assume timelike and ho-

mothetic for g. Certainly, there is a timelike vector field Vparallel to ∂twhich is a reference

frame with respect to g, but is not necessarily complete. Hence, we call a triple (R×S, g, V )

with the properties as stated a homothetic splitting spacetime. Subsequently, we will only

consider the special case of a compact base manifold S.

Proposition 5.46. Let (R×Sn, g, V )be a homothetic splitting spacetime, such that £∂tg=

2c2

ngfor some constant c > 0. Then there is a metric g∗in the conformal class of g, such

that (R×S, g∗, ∂t)is a proper kinematical spacetime, and the metric gcan be written as

g=e2c2

nt[−(A◦π)2dt⊗dt+ 2π∗(b)∨dt+π∗(h)−π∗(b⊗b)

(A◦π)2],

with π= pr2:R×S→Sand t= pr1:R×S→Rbeing the natural projections onto the

factors, b∈Γ(Λ1S),A:S→R>0and ha Riemannian metric on S. The reference frame

Vis given by

V=e−c2

nt∂t

A◦π,

and the expansion associated to Vreads

Θ = c2

A◦πe−c2

nt(p).

Furthermore, the projection π: (R×S, g)→(S, h)is a surjective submersion and for hori-

zontal vectors Xp∈ker(dπp)⊥we have

gp(Xp, Xp) = e2c2

nthπ(p)(πp∗Xp, πp∗Xp)

for all p∈R×S.

Proof. We set

g∗=g

|g(∂t, ∂t)|,

which yields g∗(∂t, ∂t) = −1and g∗is a smooth Lorentzian metric as ∂tis timelike by

assumption. Thus ∂tis a complete reference frame for g∗and (R×S, g∗, ∂t)is a proper

kinematical spacetime as pr2projects on the manifold S. Hence, the splitting of the metric g∗

given in Thm. 4.16 for a proper kinematical spacetime can be pulled back to the homothetic

splitting spacetime (R×S, g, V ), which yields

g=−˜

A2dt⊗dt+ 2π∗(bt)∨dt+π∗(ht)−π∗(bt)⊗π∗(bt)

A2.

Here, ˜

A:R×S→R>0and the families {bt}t∈R,{ht}t∈Rare one-forms and Riemannian

metrics respectively on S, varying smoothly with t, hence the splitting is understood to hold

pointwise for all (t, x)∈R×S. Now we use £∂tg=2c2

ngtogether with £∂t(dt)=0and get

−2˜

A(∂t˜

A)dt⊗dt+ 2π∗(∂tbt)∨dt+π∗(∂tht)−

120

−(2π∗(∂tbt)∨π∗(bt)) ˜

A−2˜

A(∂t˜

A)(π∗(bt)⊗π∗(bt))

A4=

=2c2

n[−˜

A2dt⊗dt+ 2π∗(bt)∨dt+π∗(ht)−π∗(bt)⊗π∗(bt)

A2].

This yields

∂t˜

A=c2

n˜

A, ∂tbt=2c2

nbt, ∂tht=2c2

nht.

Hence, there is a function A:S→R>0, such that ˜

A=ec2

nt(A◦π)and a one-form b∈Γ(Λ1S)

such that bt=e2c2

ntb, as well as a Riemannian metric hon Ssuch that ht=e2c2

nth, and the

desired result follows.

Obviously, g(V, V ) = −1implies V=e−c2

nt∂t

A◦πand the formula for the expansion follows

from Thm. 3.25.

The fact that a vector field Xon R×Sis horizontal if and only if

g(V, X) = ec2

nt[−(A◦π)dt(X) + π∗(b)(X)

A◦π]=0,

yields for horizontal vector fields the formula

dt(X) = π∗(b)(X)

(A◦π)2.

A straightforward computation for any p∈R×Syields

gp(Xp, Xp) = e2c2

ntπ∗

p(hπ(p))(Xp, Xp) = e2c2

nthπ(p)(πp∗Xp, πp∗Xp).

The notions of basic vector fields and lifts carry over to homothetic splitting spacetimes.

Particularly, this implies for an h-orthonormal frame {Ei}i=1,...,n, i.e., h(Ei, Ej) = δij, on

the base Sof such a spacetime that its lift {˜

Ei}i=1,...,n is horizontal and obeys g(˜

Ei,˜

Ej) =

e2c2

ntδij. Hence, there is a g-orthonormal frame on R×Sgiven by {V, ˆ

Ei=e−c2

nt˜

Ei}i=1,...,n.

Remark 5.47. The homothetic splitting spacetimes are special cases of conformal Lorentzian

submersions, which we will analyze in greater detail in the following chapter.

Lemma 5.48. For a homothetic splitting spacetime (R×Sn, g, V )it holds that the expansion

Θof Vobeys

∇VΘ + Θ2

n= 0.

Proof. Based on the formulas for Vand Θfrom Prop. 5.46, we compute

∇VΘ + Θ2

n=e−c2

A◦π∂tc2

A◦πe−c2

nt+1

(A◦π)2e−2c2

nt=

121

=c2e−c2

(A◦π)2−c2

ne−c2

nt+1

(A◦π)2e−2c2

nt= 0.

Now, we establish a few facts about homothetic splitting spacetimes similar to the statements

about stationary Lorentz-to-Riemann submersions shown in Prop. 5.38. All proofs below

work similar to the ones in Prop. 5.38 and we refer to the proof of that proposition for

missing details.

Proposition 5.49. Let (R×Sn, g, V )be a homothetic splitting spacetime. Then

(i) u=g(V, ·) = e−c2

nt−(A◦π)dt+π∗(b)

A◦πand ˙u=g(∇VV, ·) = π∗c2

A2+dA

A,

(ii) ω=ec2

ntπ∗db

A2Aand d ˙u=Θω

(iii) there are functions Ω2:S→R≥0and r:S→R, such that |ω|2=e−2c2

nt(Ω2◦π)and

Ric(V, V ) = e−2c2

nt(r◦π),

(iv) there is a vector field W:S→TS, such that ∇VV=e−2c2

nt˜

Wwith the lift ˜

W∈

Γ(H(R×S)), i.e., ∇VVis proportional to a basic vector field,

(v) with the divergence divhand the norm k·khassociated to hon Swe have div(∇VV) =

e−2c2

nt[kWk2

h+ divh(W)] ◦π.

Proof. (i) Using V=e−c2

nt∂t

A◦π, we compute

u=g(V, ·) = −(A◦π)ec2

ntdt+e−c2

A◦ππ∗(b),

and hence

˙u=Vcdu=Vcc2

nec2

ntdt∧u+ec2

ntd[−(A◦π)dt+π∗(b)

A◦π]=

=c2

∂t

A◦πcdt∧π∗(b)

A◦π+π∗(dA

A) = c2

nπ∗(b

A2) + π∗(dA

A).

(ii) In (i) we have already partially used

du=c2

nec2

ntdt∧π∗(b)

A◦π+ec2

nt−π∗(dA)∧dt−π∗(dA∧b)

(A◦π)2+π∗(db)

A◦π.

Combining this with

u∧˙u=ec2

ntc2

n−dt∧π∗(b)

A◦π−dt∧π∗(dA) + π∗(b∧dA)

(A◦π)2

122

yields

ω= du+u∧˙u=ec2

ntπ∗db

A+ 2b∧dA

A2=ec2

ntπ∗(d( b

A2)A).

Based on (i) we compute

d ˙u=π∗(c2

nd( b

A2)) = ω

ne−c2

A◦π

and using Prop. 5.46 the result d ˙u=Θω

nfollows.

(iii) Let {V, ˆ

Ei}i=1,...,n be a g-orthonormal frame, such that ˆ

Ei=e−c2

nt˜

Eifor all i= 1, . . . , n

and the ˜

Ei’s are the lifts of some h-orthonormal frame {Ei}i=1,...,n on S. Then we have

|ω|2=X

i,j

ω(ˆ

Ei,ˆ

Ej)ω(ˆ

Ei,ˆ

Ej) = X

i,j

e−2c2

ntπ∗(d( b

A2)A)( ˜

Ei,˜

Ej)2

i,j

e−2c2

nt[d( b

A2)(Ei, Ej)A]2◦π.

Now setting Ω2:= Pi,j[d( b

A2)(Ei, Ej)A]2:S→R≥0, yields the desired |ω|2=e−2c2

nt(Ω2◦π).

Furthermore, any homothetic vector field is indeed an affine vector field, i.e., the curvature

tensor and hence the Ricci tensor is constant along it (cf. [KY61]). Thus, we have

∂tRic(V, V ) = £∂t(Ric(V, V )) = 2Ric(£∂tV, V ).

Now using £∂tV=£∂t(e−c2

A◦π∂t) = −c2

nV, yields

∂tRic(V, V ) = −2c2

nRic(V, V ),

so there is a function r:S→R, such that Ric(V, V ) = e−2c2

nt(r◦π)solves this ordinary

differential equation.

(iv) Due to the formula for ˙uin (i),u(∇VV)=0and the structure of the metric in Prop. 5.46,

we get

e2c2

nt(π∗h)(∇VV, ·) = π∗c2

A2+dA

A.

Due to the right hand side of this equation being a one-form on R×Spulled back from

S, we infer that there is a vector field W:S→TS, such that its lift ˜

Wto R×Sobeys

W=e2c2

nt∇VV, which gives the desired result.

(v) Denote by ∇hthe Levi-Civita connection associated to hon S. Then A. Gray’s result

(cf. Thm. 3.2 in [Gra67]) holds in the same way as it does for Lorentz-to-Riemann submer-

sions and we have that for any two basic vector fields X, Y ∈Γ(H(R×S)) related to the

123

vector fields X−, Y −∈Γ(TS), the vector field ∇XYis basic and related to ∇h

X−Y−. Let

{V, ˆ

Ei}i=1,...,n be a g-orthonormal frame as in the proof of (iii), then we compute

div(∇VV) + g(V, ∇V∇VV) = X

g(ˆ

Ei,∇ˆ

Ei∇VV) = e−2c2

ntX

g(˜

Ei,∇˜

Ei∇VV) =

=e−4c2

ntX

g(˜

Ei,∇˜

W) = e−2c2

ntX

h(Ei,∇h

EiW)◦π=e−2c2

ntdivh(W)◦π.

Furthermore,

g(V, ∇V∇VV) = −g(∇VV, ∇VV) = −e−4c2

ntg(˜

W, ˜

W) = −e−2c2

nth(W, W )◦π

and the result follows.

Now, we establish results about the Ricci curvature of homothetic splitting spacetimes, which

are essentially similar to the ones in Thm. 5.39. Hence, there is a Bochner technique which

works in the same way for stationary and homothetic splitting spacetimes with a compact

base.

Theorem 5.50. Let (R×Sn, g, V )be a homothetic splitting spacetime and Scompact. Then

(i) the timelike Ricci curvature of (R×S, g)cannot be non-positive, and negative some-

where, as RSrµh≥0,

(ii) if (R×S, g)is Ricci flat, then ω= 0 and ˙u= 0, if additionally B1(S)=0and the

homothetic vector field is proper, i.e., c6= 0, then (R×S, g)is isometric to a warped

product (R>0,−dτ2)×τ2(S, c4

0n2h)for some a0∈R,

(iii) if Ric(X, X)≥0, and Ric(X, X)>0somewhere, for all timelike vector fields Xon

R×S, then the rotation of Vdoes not vanish or Vis not geodesic.

Proof. Using Lem. 5.48 together with items (iii) and (v) in Prop. 5.49 yields the Raychaud-

huri equation for the expansion Θof Vto read

e−2c2

ntdivh(W)◦π=e−2c2

nt(r◦π)−e−2c2

nt(Ω2◦π)−e−2c2

ntkWk2

h◦π.

Hence, as Sis compact and e−2c2

nt>0for all t∈R, we get

[r−Ω2−kWk2

h]µh= 0.(∗)

Thus, RSrµh≥0follows, as well as the first assertion, as r◦πand Ric(V, V )share the same

sign everywhere.

Ricci flatness enforces RS[Ω2+kWk2

h]µh= 0, thus Ω = 0 and W= 0 and hence ω= 0

and ∇VV= 0. With the same argument as in the proof of Thm. 5.39, vanishing first Betti

124

number now ensures b= 0, by isometry. But the spacetime is not static in this case, we

rather have a metric of the form

g=e−2c2

nt[−a2

0dt⊗dt+π∗(h)]

on R×S, where we have used that vanishing acceleration now ensures the function A◦π

to be equal to a constant, which we denote by a0. Now, consider an isometry (which

is admissible if c6= 0) fostered by the transformation t7→ a0n

c2e−c2

nt=τ, which yields

dτ⊗dτ=a2

0e−2c2

ntdt⊗dtand the transformed metric reads

g=−dτ⊗dτ+τ2π∗(c4

n2a2

h).

This is the desired warped product metric on R>0×Sas τ > 0.

The third assertion follows from the integral (∗), as positive timelike Ricci curvature certainly

enforces r≥0and r > 0somewhere.

To conclude this section, we state the following. Based on Thm. 5.50 we see that, for

spacetimes (R×S, g)with compact Sand gtimelike along the factor R, negative timelike

Ricci curvature is not only an obstruction to the existence of a timelike Killing vector field

along the factor R, but also to the existence of a timelike homothetic vector field along the

factor R.

125

Chapter 6

CONFORMAL LORENTZIAN SUBMERSIONS

In this chapter, we will consider spacetimes (M, g)which carry the structure of a surjective

submersion π:Mn+1 →Nn(cf. Def. 5.33), such that there is a function s:M→R>0and

we have

gp(Xp, Xp) = s2(p)hπ(p)(π∗Xp, π∗Xp),

for all p∈M, some Riemannian metric hon Nand all horizontal vector fields Xon

M. Obviously, this is a generalization of the Lorentz-to-Riemann submersions, and the

homothetic splitting spacetimes, considered above. We will prove that this submersion

structure enforces the spacetime to be conformal to a proper kinematical one in the most

important cases (hence, they split as in Thm. 4.16), and that these submersions can be

unequivocally characterized by kinematical quantities (see Thm. 6.2 below). Some theory

on semi-Riemannian conformal submersions is already contained in [Gra67]. Analogues

to O’Neill’s fundamental tensors for submersions were developed in [OR93] and [Gud92]

for the Riemannian case. In section 6.1 we will introduce the notion of Hubble-isotropic

spacetimes as a particular case of these submersions, which is motivated by considerations

from cosmology. Subsequently, in sections 6.2 and 6.3 we will analyze topological and causal

properties of these spacetimes.

Just as in the case of Lorentz-to-Riemann submersions above, we will use the designations

HM and V M for the horizontal and vertical vector bundle and H∗Mand V∗Mfor the

horizontal and vertical covector bundle, respectively, as well as Hand Vfor the projections

onto these bundles.

Definition 6.1. A submersion π: (Mn+1, g)→(Nn, h), with (M, g)a spacetime and (N, h)

a Riemannian manifold will be called a conformal Lorentz-to-Riemann submersion, or con-

formal Lorentzian submersion, if there is a function s:M→R>0, such that for all horizontal

vector fields X∈Γ(HM)and all p∈M

gp(Xp, Xp) = s2(p)hπ(p)(π∗Xp, π∗Xp)

holds. Furthermore, the conformal Lorentzian submersion will be called horizontally homo-

thetic if H(ds)=0.

Theorem 6.2. Let π: (M, g)→(N, h)be a conformal Lorentzian submersion with fibers

diffeomorphic to R. Then there is a uniquely determined future-directed reference frame

V:M→TM tangential to the fibers, as well as a future-directed, timelike and complete

vector field V∗parallel to V, such that (M, g∗, V ∗)is a proper kinematical spacetime with g∗

being a Lorentzian metric conformal to g.

126

Furthermore, a spacetime (M=R×N, g)together with a reference frame Vparallel to ∂t

gives rise to pr2:R×N→Nbeing a conformal Lorentzian submersion if and only if it is

spatially conformally stationary, i.e., σ= 0 for V.

Proof. Assume π: (M, g)→(N, h)is a conformal Lorentzian submersion. For all p∈

M, let VpM= ker(πp∗)⊂TpM. Then VpMis a timelike subspace, by the definition of

the conformal Lorentzian submersion (the orthocomplement of VpMwith respect to gis

spacelike). Moreover, dim(VpM)=1, hence there is a uniquely defined reference frame

V:M→TM, given by Vp∈VpM,gp(Vp, Vp) = −1and by the requirement of future-

directedness for all p∈M. Certainly, (M, N, π)is a fiber bundle, with fibers diffeomorphic

to R, hence, there is a global section (see [KN63, Ch. 1, Thm. 5.7]). As Nis a manifold,

it remains to show that (M, N, π)is principal in order to get a global trivialization, and

hence a proper kinematical spacetime, i.e., we need to construct a free R-action on M. The

vector field Vconstructed above is not necessarily complete, i.e., its associated flow is not

necessarily global. But just as in Prop. 3.12, with the help of a complete Riemannian metric

gRon M, we can construct a complete vector field V∗parallel to V. The flow associated

to V∗is global and its flow lines coincide with the fibers. Thus, (M, N, π)is a globally

trivializable principal fiber bundle, i.e., (M, g∗, V ∗)is a proper kinematical spacetime if we

choose g∗from the conformal class of g, such that g∗(V∗, V ∗) = −1holds.

Now we consider the spacetime (M=R×N, g)with a reference frame V:M→TM

pointing along the factor R, i.e., Vis parallel ∂t. Certainly, pr2:M→Nis a surjective

submersion. Denote by Xa horizontal vector field X∈Γ(HM), i.e., g(X, V )=0. Following

section 3.1, the projection ˜

h=g+u⊗uwith u=g(V, ·)is a Riemannian metric on HM

and g(X, X) = ˜

h(X, X).

“⇒”: If pr2acts as a conformal Lorentzian submersion, we have

h(X, X) = s2[h(pr2∗X, pr2∗X)◦pr2],

for some Riemannian metric hon Nand thus

h=s2pr∗

as ˜

his purely horizontal. In this case we get

£V˜

h=V(s2pr∗

2h) = (2s˙s)pr∗

2h=2 ˙s

s˜

denoting ˙s=∇Vs. Hence, (R×N, g, V )is spatially conformally stationary by definition,

setting ϕ:= 2 ˙s

s, and the shear vanishes (cf. Def. 3.28 and Lem. 3.30).

“⇐”: Suppose the vector fields X, Y ∈Γ(HM)are basic, i.e., lifted from some X−, Y −∈

Γ(TN). In this case, we have £VX=£VY= 0. Hence

V(˜

h(X, Y )) = (£V˜

h)(X, Y ) = ϕ˜

h(X, Y ),

for some function ϕ:M→R>0. Now we have that for any two basic vector fields X, Y , the

function ˜

h(X, Y ): R×N→Rchanges according to the differential equation

(∂t˜

h)(X, Y )(t, x) = A(t, x)ϕ(t, x)˜

h(X, Y )(t, x)

127

as Vis parallel to ∂t, i.e., there is a function A:R×M→R>0, such that A·V=∂t. This

is an ordinary differential equation in every fiber π−1(x), which has the solution

h(X, Y )(t, x) = ˜

h(X, Y )(t0, x) exp(Zt

A(t0, x)ϕ(t0, x)dt0),

for some fixed t0∈R. Obviously, the initial value ˜

h(X, Y )(t0, x)does not depend on the

position in the fiber. Using an argument analogous to Lem. 5.34, this implies that there is

a function h(X−, Y −): N→R>0, such that ˜

h(X, Y )(t0, x) = h(X−, Y −)◦pr2. But as this

holds for all vector fields X−, Y −on N, this defines a Riemannian metric on N, and by

setting s2(t, x) = exp(Rt

t0A(t0, x)ϕ(t0, x)dt0), we get the desired result.

Furthermore, this implies the following

Proposition 6.3. Let π: (Mn+1 =R×Nn, g)→(Nn, h)be a conformal Lorentzian sub-

mersion and V:M→TM the future-directed reference frame parallel to ∂t. Then there is a

function ¯s:N→R>0, such that ¯π: (M, g)→(N, ¯sh)is a horizontally homothetic conformal

Lorentzian submersion if and only if the spacetime (M, g)together with the reference frame

Vis spatially homothetic.

Proof. Let Θbe the expansion associated to V, and as we have a conformal Lorentzian

submersion the shear associated to Vvanishes due to Thm. 6.2 above. Following Lem. 3.31,

a spacetime (M, g)with reference frame Vis spatially homothetic if and only if H(dΘ) = 0,

i.e., the reference frame Vis spatially homothetic. Furthermore, from the computations in

the proof of Thm. 6.2, we get

£V˜

h= 2 ˙s

s˜

for ˜

h=g+u⊗uin the spatially conformally stationary case, which implies Θ = n˙s

s, with

the dot denoting the derivative ∇V. Thus, we have

H[d( ˙s

s)] = 0

if and only if Vis spatially homothetic.

Investigating this expression, we can infer that it is fulfilled if and only if shas product form

s=s1·s2with H(ds1) = 0 and V(ds2) = 0. But this is the case if and only if there is

a function ¯s:N→R>0, such that s2◦π= ¯s. Thus, a horizontally homothetic conformal

Lorentzian submersion π: (M, g)→(N, h = ¯sh)is determined by

gp(Xp, Xp) = s2

1(p)hπ(p)(π∗Xp, π∗Xp),

for all horizontal vector fields X∈Γ(HM)if and only if shas a product structure as above,

i.e., if and only if Vis spatially homothetic.

Particularly, the conclusions in the theorem and the proposition above hold for kinematical

spacetimes (M, g, V ), but in general we do not have to require Vto be complete. Moreover,

128

we see that a spatially conformal (and even a spatially homothetic) kinematical spacetime

does not uniquely determine the conformal Lorentzian submersion π: (M, g)→(N, h)as-

sociated to it. Rather, there is the freedom of choosing the Riemannian metric hup to

a conformal factor λ:N→R>0, which in turn then determines s:M→R>0up to the

multiplication with a function Λ: M→R>0given by Λ◦π=1

λ.

6.1 Hubble-isotropic Spacetimes

In the remaining sections of this chapter, we will consider a particular class of conformal

Lorentzian submersions only. To this end, we perform the following

Definition 6.4. An ordered triple (M, g, V )is called Hubble-isotropic spacetime if (M, g)is

a spacetime together with a future-directed reference frame V:M→TM, and the shear and

the acceleration of Vvanish, i.e., σ= 0 and ˙u= 0.

In general, we do not require a Hubble-isotropic spacetime (M, g, V )to be a (Cartan) kine-

matical spacetime, as Vis not necessarily complete. But below we will primarily consider

Hubble-isotropic spacetimes given in a splitting form (R×N, g, V )with Vparallel to ∂t.

Obviously, the notion of Hubble-isotropic spacetimes do naturally include conformally sta-

tionary and stationary ones with vanishing acceleration (cf. Prop. 6.5 below). We will

sometimes refer to a Hubble-isotropic spacetime which is not conformally stationary, and

hence also not stationary, as being properly Hubble-isotropic.

Proposition 6.5. A Hubble-isotropic spacetime (M, g, V )is

(i) a proper kinematical spacetime if (M, g, V )is a Cartan kinematical spacetime,

(ii) a conformal Lorentzian submersion with totally geodesic fibers if M=R×Nfor some

manifold Nand Vparallel to ∂t,

(iii) conformally stationary if d(Θu) = 0 holds for u=g(V, ·), and

(iv) stationary if Θ = 0.

Proof. (i) This is an application of Thm. 4.19.

(ii) This follows from Thm. 6.2. As the fibers are the integral curves of V, vanishing

acceleration implies totally geodesic fibers.

(iii) and (iv) For vanishing shear and acceleration, the conditions d(Θu) = 0 resp. Θ=0lead

to conformally stationary resp. stationary spacetimes as can be deduced from Thm. 3.25.

The following definition and proposition clarify the term Hubble-isotropic for the spacetimes

defined above. These spacetimes are of particular interest in physical applications, especially

in cosmology. Nevertheless, their global properties have scarcely been analyzed up to now.

The standard references for Hubble-isotropic spacetimes are [Has91] and [HP99]. But already

J. Ehlers et al. (see [Ehl93]1) have analyzed spacetimes of this kind.

1Note that this a reprint of a 1961 article.

129

Definition 6.6. For a spacetime (M, g)together with a future-directed reference frame

V:M→TM, we denote by K∈Γ(T M)a lightlike vector field, i.e., g(K, K)=0, such that

g(V, K)6= 0 and by (z, k)∈TM, with z∈Mand k:= Kz∈TzM, its coordinates in the

tangent bundle. Let

CM := {(z, k)∈T M |gz(k, k)=0, gz(Vz, k)6= 0}

be the lightlike subset of the tangent bundle TM. Then, we define a functional on CM,

called the Hubble functional, by

H:CM →R, H(z, k) := gz((∇kV)z, k)

gz(Vz, k)2.

Let P:TM →Mbe the natural projection. Then the Hubble functional His called isotropic

if it is the composition of a Hubble function ˜

H:M→Rand the natural projection restricted

to CM, such that

H(z, k) = ˜

H(z)◦P|CM .

The Hubble functional appears naturally in cosmology as the first coefficient H1in a series

expansion of the relation of the redshift ζto the distance Dof galaxies, firstly given by

J. Kristian and R.K. Sachs in [KS66]:

ζ=H1D+···+HnDn+O(Dn+1).

Hence, there is a strong motivation from physics to analyze these spacetimes. If H1is

isotropic, it is usually called Hubble constant, but may depend on the position. Thus, the

Hubble-isotropic spacetimes are exactly those which have an isotropic linear coefficient in

the series expansion of the ζ−Drelation (see [HP99] for a detailed analysis).

The following proposition is contained in [HP99], but has already been implicitly shown in

[Ehl93]. We give a full explicit proof here for the sake of completeness.

Proposition 6.7. (Mn+1, g, V )is a Hubble-isotropic spacetime if and only if its Hubble

functional is isotropic: H=˜

H◦P. Furthermore, we have in this case ˜

H=Θ

Proof. From Prop. 3.21 we get

gz((∇kV)z, k) = σz(k, k)−˙uz(k)gz(Vz, k) + Θ

nhz(k, k),

for all (z, k)∈CM. As h=g+u⊗u, it follows that hz(k, k) = (gz(Vz, k))2. Hence, we have

H(z, k) = σz(k, k)

(gz(Vz, k))2−˙uz(k)

gz(Vz, k)

| {z }

=:Y(z,k)

+Θ(z)

Thus, we observe that His independent of kif the term Y(z, k)is zero for all (z, k)∈CM.

In this case, we also get ˜

H(z) = Θ(z)

n. We observe that Y(z, k)=0if and only if σ= 0 and

˙u= 0. This can be seen as follows: suppose σand ˙uis such that

σz(k, k) = ˙uz(k)gz(Vz, k) = ˙uz(k)uz(k),

130

for all (z, k)∈CM. Now let e∈TzMbe some spacelike vector obeying gz(Vz, e)=0and

gz(e, e)=1and consider two lightlike vectors k1=Vz+e, as well as k2=−Vz+ein TzM.

Then we have

σz(e, e) = σz(k1, k1) = ˙uz(k1)uz(k1) = −˙uz(e) = −˙uz(k2)uz(k2) = −σz(k2, k2) = −σz(e, e).

As this holds for any spacelike vector e∈TzM, we have σ= 0 and hence ˙u= 0. Furthermore,

this also shows that Y(z, k)cannot assume some non-zero value, which is constant with

respect to k, hence we have that His independent of kif and only if Y(z, k)is zero for all

(z, k)∈CM.

6.2 Topological and Causal Properties

Having established that a Hubble-isotropic spacetime is a conformal Lorentzian submersion if

the underlying manifold splits appropriately as a product, and that it is a proper kinematical

spacetime if it is a Cartan kinematical spacetime, in the previous section, it is natural to

ask for the splitting structure of the metric implied by Thm. 4.16 in these cases.

Theorem 6.8. Let (Mn+1 =R×Nn, g, V )be a Hubble-isotropic spacetime of splitting type,

with the reference frame Vparallel to ∂t. Then there are two functions A, s:R×N→R>0

and a Riemannian metric hon N, such that V=1

A∂tand the metric is given by

g(t,x)=−A2(t, x)dt⊗dt+ 2pr∗

2(b(t,x))∨dt+s2(t, x)pr∗

2(hx)−pr∗

2(b(t,x))⊗pr∗

2(b(t,x))

A2(t, x)

with x∈N,t= pr1:R×N→R,pr2:R×N→Nand {bt}t∈Ra family of one-forms on

Nobeying

b(t,x)=A(t, x)βx+Zt

t0H(dA)(t0,x)dt0,(∗)

for some t0∈Rand a one-form β∈Γ(Λ1N). The expansion Θof Vis given by

Θ(t, x) = n∇Vs

s(t, x) = n(∂ts)(t, x)

A(t, x)s(t, x).

Furthermore, the Hubble-isotropic spacetime is locally isometric to (I×N, ˜g, ∂t), with

˜g=−dτ⊗dτ+ 2pr∗

2(β)∨dτ+ ˜s2pr∗

2(h)−pr∗

2(β)⊗pr∗

2(β),

where s= ˜s◦ϕ, with ϕbeing the local isometry and I⊂Ra, possibly unbounded, open

interval.

Proof. Certainly, ∂tis complete and there is a Lorentzian metric g∗in the conformal class

of g, such that (R×N, g∗, ∂t)is a proper kinematical spacetime. Hence, the splitting of the

metric as in Thm. 4.16 can be applied to gby pullback.

131

As usual we have u=g(V, ·) = −Adt+pr∗

2(b)

Aand the splitting yields

g(t,x)=−A2(t, x)dt⊗dt+ 2pr∗

2(b(t,x))∨dt+ pr∗

2(˜

h(t,x))−pr∗

2(b(t,x))⊗pr∗

2(b(t,x))

A2(t, x),

for a family of Riemannian metrics {˜

ht}t∈Ron N. We will show that ˙u= 0 implies (∗) to

hold for the one-forms btand σ= 0 implies the existence of a function s:R×N→R>0

such that ˜

h(t,x)=s2(t, x)hx. We have

˙u=∇Vu=Vcdu=∂t

Ac−dA∧dt−dA

A2∧pr∗

2(b) + d[pr∗

2(b)]

A=

=−∂tA

Adt+dA

A−∂tA

A3pr∗

2(b) + pr∗

2(∂tb)

where we denoted by ∂tbthe derivative of the family of one-forms {bt}t∈Ron Nwith respect

to their parameter t. As

dA= (∂tA)dt+H(dA),

it follows that

˙u=−∂tA

A3pr∗

2(b) + H(dA)

A+pr∗

2(∂tb)

A2.

Hence, as ˙u= 0, we have

pr∗

2(∂tb) = ∂tA

Apr∗

2(b) + AH(dA).

But this is an ordinary differential equation in each fiber R×{x} ⊂ R×N, and we have for

all fixed x∈Ndbx

dt(t) = d log(Ax)

dt(t)bx(t) + Ax(t)H(dA)x(t)

with Ax(t) := A(t, x). The general solution of this differential equation is easily computed

to be

bx(t) = Ax(t)βx+Zt

t0H(dA)x(t0)dt0

where bx(t0) = βxis the initial value.

The proof of ˜

h(t,x)=s2(t, x)hxworks similar to the proof of Thm. 6.2 above. Following

section 3.1, we have g= pr∗

2(˜

h)−u⊗uand, as the acceleration vanishes, £Vu= 0 holds.

Hence, by Prop. 3.21 we get

£Vpr∗

2(˜

h) = £Vg= 2σ+2

nΘpr∗

2(˜

h).

As pr∗

2(˜

h)is horizontal, we get £Vpr∗

2(˜

h) = £∂t

pr∗

2(˜

h) = 1

A£∂tpr∗

2(˜

h) = 1

Apr∗

2(∂t˜

h). Thus,

σ= 0 implies the ordinary differential equation

d˜

dt(t) = 2

nΘx(t)Ax(t)˜

hx(t)(∗∗)

132

in each fiber R×{x} ⊂ R×N. The general solution of this differential equation is easily

found to be

hx(t) = exp( 2

nZt

Θx(t0)Ax(t0)dt0)hx,

for some initial value hx=˜

hx(t0). As the choice of t0∈Ris arbitrary, we can define the

function sas

s(t, x) = exp( 1

nZt

Θ(t0, x)A(t0, x)dt0),

which is the desired result. Furthermore, computing

∂ts

s=1

nΘA

yields the formula for the expansion as stated.

Locally the Hubble-isotropic spacetime is certainly diffeomorphic to Rn+1, hence also to

I×Rn, with a (possibly unbounded) open interval ˜

I⊂Rand a splitting metric given as

above. Then, locally, we can always find a diffeomorphism ϕ:˜

I×Rn→I×Rn, with

possibly another bounded or unbounded open interval I⊂R, which acts by transforming

the t-coordinate according to

t7→ τ(t, x) = Zt

A(t0, x)dt0,

for some fixed t0∈˜

Iand all t∈˜

I. Hence, we have

Adt= dτ−Zt

t0H(dA)(t0,x)dt0

as H(d Rt

t0A(t0, x)dt0) = Rt

t0H(dA)(t0,x)dt0can be assumed to hold due to the local nature of

this analysis by application of Lebesgue’s dominated convergence theorem. This yields

˜g(τ,x)=−dτ−Zt

t0H(dA)(t0,x)dt02

A(t, x)pr∗

2(b(t,x))∨dτ−Zt

t0H(dA)(t0,x)dt0+

+s2(t, x)pr∗

2(hx)−pr∗

2(b(t,x))⊗pr∗

2(b(t,x))

A2(t, x)=

=−dτ⊗dτ+ 2 pr∗

2(b(t,x)

A(t, x)+Zt

t0H(dA)(t0,x)dt0∨dτ+s2(t, x)pr∗

2(hx)−

−pr∗

2(b(t,x))⊗pr∗

2(b(t,x))

A2(t, x)−(Zt

t0H(dA)(t0,x)dt0)⊗(Zt

t0H(dA)(t0,x)dt0).

Using (∗) and s(t, x) = (˜s◦ϕ)(t, x)yields

˜g(τ,x)=−dτ⊗dτ+ 2pr∗

2(βx)∨dτ+ ˜s2(τ, x)pr∗

2(hx)−pr∗

2(βx)⊗pr∗

2(βx).

133

Remark 6.9. The investigation of Hubble-isotropic spacetimes and particularly the ana-

lytic approach taken in this section, as well as the theorem above, are inspired by [GPS+10].

Therein, a toolbox was given for constructing spacetimes of splitting type with given kinemat-

ical quantities (e.g. σ= 0,∇VV= 0, but Θ6= 0 as in the Hubble-isotropic case), by starting

from a stationary spacetime or a Riemannian metric on the spacelike factor of the spacetime.

This can be seen as the complementary synthetic approach to our analysis of Hubble-isotropic

spacetimes as conformal Lorentzian submersions. At last, the common ground of both ap-

proaches is the differential equation (∗∗)in the proof of Thm. 6.8 above. The general solution

of this differential equation can be regarded to hold necessarily in any Hubble-isotropic space-

time, due to the conformal submersion condition, as in our approach here. Or it can be

seen as evolving from a given initial condition metric and in this process constructing a

Hubble-isotropic spacetime from the initial Riemannian metric as in [GPS+10].

It is now natural to ask under which circumstances the local isometry in the theorem above

holds even globally. Certainly, this is not true in general, as the following example shows.

Example 6.10. Let (t, x, y)be coordinates on R3,s:R→R>0a smooth function and

(R3, g, etx∂t)the Hubble-isotropic spacetime given by

g(t,x)=−e−2txdt2−2e−tx(1 + tx)−1

x2dxdt+s2(t)(dx2+ dy2)−[e−tx(1 + tx)−1]2

x4dx2,

which is also defined and smooth at x= 0 in the sense of taking the limit x→0. Computing

H(de−tx) = −te−txdxand

−Zt

t0e−t0xdt0=(e−tx(1+tx)−1

x2if x6= 0

−t2

2if x= 0,

which is actually smooth at x= 0, we see that this metric is indeed Hubble-isotropic with

β= 0. But now we observe that

Z∞

e−t0xdt0=(e−t0x

xif x > 0

∞if x≤0.

Hence, a time-coordinate τas in Thm. 6.8, can never be globally defined on all of R.

Corollary 6.11. Let (M=R×N, g, V )be a Hubble-isotropic spacetime as in Thm. 6.8.

Then there is an isometry ϕ: (M, g)→(I×N, ˜g), such that (M, g, V )is isometric to

(I×N, ˜g, ∂τ), with ˜gas in Thm. 6.8 and ∂τ=ϕ∗V, if there are two constants c1, c2obeying

−∞ ≤ c1< c2≤ ∞, such that R∞

0A(t0, x)dt0=c2for all x∈Nand R0

−∞ A(t0, x)dt0=c1

for all x∈N.

Proof. From Thm. 6.8 we have V=∂t

Aon M=R×N. We need an isometry

ϕ: (R×N, g)→(I×N, ϕ∗g= ˜g),

134

which acts by

t7→ τ(t, x) = Zt

A(t0, x)dt0,

for a fixed t0∈Rand also maps ∂t

Ato the canonical vector field ∂τon I×N(τ∈I) by

push-forward. Without loss of generality we can choose t0= 0. Such an isometry exists if

the range of τ(t, x)in tis equal for all x∈N, i.e., if the conditions in the corollary are met.

As in this case we have

c1= lim

t→−∞τ(t, x)<lim

t→∞τ(t, x) = c2,

for all x∈N, thus every fiber R×{x}is mapped to I×{x}:= (c1, c2)×{x}by ϕ. Obviously,

the choice of the metric ˜g=ϕ∗gand the vector field ∂τ=ϕ∗(∂t

A), on I×Nassures that

ϕis indeed an isometry, according to the computations in the proof of Thm. 6.8. At last,

we have to address the problem of interchanging the integral and the partial derivatives in

the computation of Adt= dτ−Rt

t0H(dA)(t0,x)dt0in the global case. We need to ensure

that the integral Rt

t0H(dA)(t0,x)dt0exists for all (t, x)∈R×N, so that H(d Rt

t0A(t0, x)dt0) =

t0H(dA)(t0,x)dt0holds for all (t, x)∈R×N. But that this is indeed the case, can be seen

from the condition (∗) in Thm. 6.8 above that holds for every Hubble-isotropic spacetime.

Now we will examine the causal properties of Hubble-isotropic spacetimes. First we give

several examples, which show that all steps on the causal ladder, that are allowed by the

analysis in section 4.2 for proper kinematical spacetimes, can indeed appear for properly

Hubble-isotropic spacetimes.

Example 6.12. Consider global coordinates (t, x, y)on R3, the reference frame V=∂tand

the metric

g∗

¨o=−dt2+ 2exdydt+s2(t)dx2+1

2(s2(t)−2)e2xdy2

with s:R3→R>0a function that only depends on the t-coordinate. This metric can be

considered to be a modification of the Gödel spacetime from example 4.6 with possibly non-

vanishing expansion. A metric of this form was first given in [GPS+10], where it was also

shown that (R3, g∗

¨o, ∂t)is indeed Hubble-isotropic. Now obviously, the signature of the metric

induced on the slices Stof constant tin R3depends on the magnitude of s. We have that

a slice R2≃St0⊂R3is Riemannian if s(t0)>√2, it is Lorentzian if s(t0)<√2and

it is degenerate if s(t0) = √2. Then following section 4.2, and for example considering

the function s2(t) = sin2(t) + 1

2, leads to a properly Hubble-isotropic spacetime (R3, g∗

¨o, ∂t)

which is totally vicious, but for example considering s2(t) = t2+ 3 leads to (R3, g∗

¨o, ∂t)being

causally continuous. Total viciousness can be shown analogously to the Gödel spacetime by

considering large enough closed curves on the slices. Moreover, if s(t)is somewhere smaller

and somewhere bigger than 2, for example s2(t) = t2+ 1, we have a region of (R3, g∗

¨o, ∂t)

which is non-chronological (−1<t<1) and two regions (t < −1and t > 1) which are

causal. The regions are bounded by degenerate hypersurfaces. Furthermore, in the case of

Riemannian slices as above, i.e., with s(t)uniformly bounded away from 2, we can assert by

Thm. 6.18 below that (R3, g∗

¨o, ∂t)is even globally hyperbolic, as the metric h= dx2+e2xdy2

is the complete hyperbolic Riemannian metric on R2.

135

Example 6.13. Consider the manifold R×Z2, with Z2being a two-dimensional cylinder

obtained by using coordinates (x, y)∈R2and the identification x∼x+1. We use coordinates

(t, x mod 1, y)on R×Z2. Consider the metric

g=−dt2+ 2 sin(y)dxdt+ (t2+ 1)2(dx2+ dy2)−sin2(y)dx2

on R×Z2, which is clearly properly Hubble-isotropic as it is of the splitting form given in

Thm. 6.8 and Θ = 2∂t(t2+1)

t2+1 =4t

t2+1 is non-zero almost everywhere. As any slice of constant

tin this spacetime is a copy of Z2, the integral curves of the canonical vector field ∂xare

closed. Furthermore, we have

g(∂x, ∂x) = (t2+ 1)2−sin2(y),

which is zero if and only if t= 0 and yis an odd integer multiple of π

2. Hence, only on the

t= 0 slice we have CCCs, namely the integral curves of ∂xat the values {yj= (2j+1)π

2}j∈Z.

As everywhere else the slices of constant tare Riemannian, this spacetime is an example of

a properly Hubble-isotropic spacetime which is chronological, but non-causal.

Example 6.14. We recall the torus spacetime from example 4.12 given by the metric

g0=−dt2+ 2dtdy+ (t2+ 1)(dx2+ dy2)−dy2

on (R×T2), with respect to the identifications (0, y)∼(1, y)for all y∈[0,1] and (x, 0) ∼

(x+√2,1) for all x∈[0,1], which is clearly Hubble-isotropic. In example 4.12 it has already

been stated that this spacetime is causal but causally imprisoning.

Example 6.15. Consider coordinates (t, x, y)∈R3with x < 1, i.e., the manifold R×R<1×R

and transform xand yto polar coordinates rand ϕon R2by setting x=rcos ϕand

y=rsin ϕ. Then we have rcos ϕ < 1. Consider the Euclidean metric h= dr2+r2dϕ2on

R<1×Rand the vector field

Y=B(r)p1−c2(r)

r∂ϕ+c(r)∂r.

Here B(r)is a bump function which obeys B(0) = 0 and B(r) = 1 for all r > 1

2, and c(r)is

the smooth bump function

c(r) = (exp(−1

1−r2),0≤r < 1

0, r ≥1,

which obeys c(r)<1for all r≤0. Hence, the vector field Yis globally well-defined and we

even have h(Y, Y ) = 1 if r > 1

2. We observe that the integral curves of Yare spirals in the

region 1

2< r < 1converging to the limit curve r= 1. But as rcos ϕ < 1, this limit curve is

not closed, i.e., there is the point (x, y) = (1,0) missing in R<1×R. This implies that the

integral curves of Yin the region 1

2< r < 1are partially imprisoned but non-imprisoned.

We will now construct a Hubble-istropic metric on R×R<1×Rfor which the integral curves

of Y, lifted to the spacetime, are lightlike on one slice, i.e., we get a partially imprisoning

but non-imprisoning spacetime.

136

We set the one-form b∈Γ(Λ1(R>0×R)) to be

b=h(Y, ·) = c(r)dr+rB(r)p1−c2(r)dϕ

and define a Hubble-isotropic metric gon R×R<1×Rby

g=−dt⊗dt+ 2b∨dt+ (t2+ 1)h−b⊗b=

=−dt2+ 2 hc(r)dr+rB(r)p1−c2(r)dϕidt+

+t2+ 1 −c2(r)dr2−2rB(r)c(r)p1−c2(r)drdϕ+t2+ 1 −B2(r)r2(1 −c2(r))dϕ2.

As h(Y, Y ) = 1, we have that g(˜

Y , ˜

Y) = 0 for the lift ˜

Yof Yto R×R<1×Rif r > 1

2and

t= 0, as the induced metric (t2+1)h−b⊗bon the t=const slices degenerates only for t= 0.

Hence, the slice t= 0 in R×R<1×Rcontains partially imprisoned lightlike curves, which are

not imprisoned in some compact set, namely the integral curves of ˜

Yand (R×R<1×R, g, ∂t),

with gas above, is a partially imprisoning but non-imprisoning Hubble-isotropic spacetime.

By the analysis in section 4.2 it is not feebly distinguishing, either.

For the further analysis of the causality conditions of Hubble-isotropic spacetimes in this

section, we will limit ourselves to metrics that can be cast into the form

g=−dt⊗dt+ 2pr∗

2(b)∨dt+s2pr∗

2(h)−pr∗

2(b)⊗pr∗

2(b)

on I×N, with I⊂Ra possibly unbounded interval, pr2:I×N→Nthe natural projection,

s:I×N→R>0,b∈Γ(Λ1N)and h∈ R(N), by use of Cor. 6.11. Then the reference frame

Vis given by the canonical vector field ∂talong the factor I. We will call such a Hubble-

isotropic spacetime specially Hubble-isotropic and we will usually omit the pull-back by pr2

if this is no source of confusion. Note that V=∂tis not necessarily a complete reference

frame as Imay be bounded, but it certainly is complete if I=R. Furthermore, we set

|||b|||h:= supx∈Nkbkh

x, and we will usually assume |||b|||h>0, as otherwise we had b= 0 and

the Hubble-isotropic metric would simply be a twisted product metric.

Proposition 6.16. Let (I×N, g, ∂t)be a specially Hubble-isotropic spacetime with

g=−dt⊗dt+ 2b∨dt+s2h−b⊗b.

s2(t, x)>(kbkh

x)2

holds for all (t, x)∈I×N, then (I×N, g)is causally continuous and the induced metric

γ=s2h−b⊗b

on the slice Nt={t} × Nis Riemannian for all t∈I. Furthermore, if a given specially

Hubble-isotropic spacetime (R×N, g, ∂t)is causally continuous, then there is a slicing (ac-

cording to Thm. 4.16), such that the metric gis given as above and s2(t, x)>(kbkh

x)2holds

for all (t, x)∈R×N.

137

Proof. Let τ:I→Rbe a monotonically increasing diffeomorphism, which implies an

isometry ϕ: (I×N, g)→(R×N, ˜g)by setting

˜g=ϕ∗g=−dτ⊗dτ

(τ0)2+ 2pr∗

2(b)∨dτ

τ0+ (s◦τ−1)2pr∗

2(h)−pr∗

2(b)⊗pr∗

2(b).

Setting V:= ϕ∗∂t=τ0∂τyields a Hubble-isotropic spacetime (R×N, ˜g, V ), and there

certainly is a vector field V∗parallel to Vwhich is complete and a reference with respect to

some Lorentzian metric g∗in the conformal class of ˜g, such that (R×N, g∗, V ∗)is a proper

kinematical spacetime. By the analysis in section 4.2, the spacetime (R×N, g∗)is causally

continuous if the slices Nτ=N×{τ} ⊂ R×Nare spacelike for all τ∈Rwith respect to

g∗. As g∗is conformal to g, this is the case if and only if the induced metric

γ(t,x)=s2(t, x)hx−bx⊗bx

on all slices Nt(for all t∈I) is Riemannian, i.e., if and only if

γ(t,x)(v, v)>0,

for all (t, x)∈I×Nand all 06=v∈TxN(This is a situation similar to Lem. 4.23). This is

equivalent to the condition

s2(t, x)>|bx(v)|2

hx(v, v),

for all (t, x)∈I×Nand all 06=v∈TxN, as his certainly Riemannian. Now, the supremum

with respect to all 06=v∈TxNof the right hand side of this inequality exists and is finite

for all x∈N, but the left hand side does not depend on v. Therefore, the condition is

equivalent to

s2(t, x)>sup

v∈TxN\{0}

|bx(v)|2

hx(v, v)= (kbkh

x)2.

And with the spacetime (R×N, g∗)being causally continuous, also the spacetime (I×N, g)

is causally continuous.

Furthermore, if (R×N, g, ∂t)is specially Hubble-isotropic, the reference frame ∂tis complete

and (R×N, g, ∂t)is a proper kinematical spacetime with the metric given by

g=−dt⊗dt+ 2b0∨dt+s2h−b0⊗b0.

By Thm. 4.16, a change of slicing acts on (R×N, g, ∂t)by t7→ τ=t+f(x)for a function

f:N→Rand the metric reads

g=−dτ⊗dτ+ 2(b0+ df)∨dτ+s2h−(b0+ df)⊗(b0+ df)

after the transformation. By the analysis in section 4.2, there is at least one slicing, such

that γ=s2h−(b0+ df)⊗(b0+ df)is Riemannian on Nfor all τ∈Rand the result follows

by considering b=b0+ dfand inferring s2(τ, x)>(kbkh

x)2from γbeing Riemannian as

above.

138

We can now proceed to give a first assessment of global hyperbolicity of specially Hubble-

isotropic spacetimes by introducing uniform bounds on s(t, x)and kbkh

xand using the com-

pleteness of the Riemannian metric h. To this end, we will need the following

Lemma 6.17. Let (N, h)be a complete Riemannian manifold and α: [a, b)→Na curve

of finite length l(α) = C > 0. Then αis extendible to the value b, i.e., there exists a point

x∈N, such that α(t)→xas t→b.

Proof. By the Hopf–Rinow theorem, the metric ball K={y∈N|dh(α(a), y)≤C}is

compact. Hence, for any strictly increasing sequence {tn}n∈N⊂[a, b)with tn→bas

n→ ∞, the sequence {α(tn)}n∈N⊂Kconverges to some x∈K. As αhas finite length,

this x∈Kis uniquely determined for all such sequences {tn}n∈N⊂[a, b), thus α(t)→xas

t→band we can uniquely extend αto the value b.

The metric structure of specially Hubble-isotropic spacetimes, together with the uniform

bound on sand kbkh, does now imply necessary and sufficient conditions for global hyper-

bolicity, which can be proved similarly to the conditions for global hyperbolicity in warped

product spacetimes (cf. [BEE96, Sec. 3.6]).

Theorem 6.18. Let (I×N, g, ∂t)be a specially Hubble-isotropic spacetime, with

g=−dt⊗dt+ 2b∨dt+s2h−b⊗b.

Assume

S0(t) := inf

x∈Ns2(t, x)>(|||b|||h)2= sup

x∈N

(kbkh

x)2>0

holds for all t∈I.

(i) If the Riemannian metric hon Nis complete, then (I×N, g)is globally hyperbolic.

(ii) If (I×N, g)is globally hyperbolic, then the Riemannian metrics s2hon Nare complete

for all t∈Iand if additionally the function s(t, ·): N→R>0is bounded from above

on Nfor all fixed t∈I, then even his complete. Particularly, this is the case if

pr2: (I×N, g)→(N, h)is a horizontally homothetic Lorentzian submersion.

Proof. (i) We assume (I×N, g)is not globally hyperbolic and have to show that his not

complete in this case. The condition S0(t)>(|||b|||h)2certainly implies that (I×N, g)is

causally continuous and s2h−b⊗bis Riemannian for all t∈Iby Prop. 6.16. Hence, if

(I×N, g)is not globally hyperbolic, there are two values r < ˜r∈Iand two points xr, x˜r∈N,

such that J+((r, xr)) ∩J−((˜r, x˜r)) is not compact. Thus, there is a future-directed causal

curve γ: [t1, t2)→J+((r, xr)) ∩J−((˜r, x˜r)) with r≤t1< t2≤˜r, which is not extendible

to the value t2. This follows from Thm. 3.52. As (I×N, g)is causally continuous, we

can assume without loss of generality that γis parametrized by t, i.e., γ(t)=(t, c(t)) and

˙γ= (1,˙c)with c: [t1, t2)→Na curve in Nand the dot denoting the derivative with respect

to t. As we have u=g(∂t,·), the condition of γbeing future-directed causal amounts to

u(˙γ) = −1 + b(˙c)<0

139

and

g(˙γ, ˙γ) = s2h(˙c, ˙c)−(1 −b(˙c))2≤0,

which yields

sph(˙c, ˙c)≤1−b(˙c).

By the definition of the norm kbkh

x, it certainly holds that |b(˙c)| ≤ kbkhph(˙c, ˙c), so that we

get

sph(˙c, ˙c)≤1 + |b(˙c)| ≤ 1 + kbkhph(˙c, ˙c).

Hence, we infer

phx(˙c, ˙c)≤1

s(t, x)−kbkh

x≤1

infx∈N(s(t, x)−kbkh

x)≤1

infx∈Ns(t, x)−supx∈Nkbkh

x≤

≤1

S0(t)−|||b|||h.

As [t1, t2]⊂Iis compact, there certainly is some k∈Rwith k= inft∈[t1,t2]S0(t)>|||b|||h,

such that phx(˙c, ˙c)≤1

k−|||b|||h.

This implies that the h-length of cin Nis finite. Furthermore, as γis inextendible to the

value t2, so is c. By Lem. 6.17, this implies that his not complete.

(ii) We will show that (I×N, g)is not globally hyperbolic if s2(t, x)hxis an incomplete

Riemannian metric on Nfor some t∈I. To this end, we first prove that s2hincomplete

for some t∈I, implies that the Riemannian metric ˜

h:= (|||b|||h)2his also incomplete on N.

Assume (|||b|||h)2hwas complete, then we would have s2(t, x)hx(v, v)≥(|||b|||h)2hx(v, v)for

all v∈TxNand even all t∈I. Hence, the distances ds2hand d˜

hfor points x, y ∈Nare

also related by ds2h(x, y)≥d˜

h(x, y). But this implies (similar to Prop. 2.19) that any ds2h-

Cauchy sequence is also a d˜

h-Cauchy sequence, which converges as ˜

his assumed complete,

contradicting the incompleteness of s2h.

Now, assume (|||b|||h)2his an incomplete Riemannian metric on N. The Hopf–Rinow theorem

assures the existence of a (|||b|||h)2h-geodesic ray c: [0, r)→N, parametrized by arc length

(|||b|||h)2h(˙c, ˙c)=1, which is not extendible to the value r. We call x=c(0) and choose some

p= (t1, x)∈I×N. Next, we will construct a future-directed causal curve α: [0, r)→I×N,

with α(σ)=(t(σ), c(σ)) and α(0) = p. Let ˙α= (˙

t, ˙c), with the dot denoting the derivative

with respect to σin this case, and similar to the proof of (i) the future-directedness amounts

to ˙

t−b(˙c)>0. We compute

g( ˙α, ˙α) = −˙

t2+ 2˙

tb(˙c) + h(˙c, ˙c)−b(˙c)2=−(˙

t−b(˙c))2+1

(|||b|||h)2,

thus we need a function t(s)obeying

t≥1

|||b|||h+b(˙c),

140

for αbeing causal. Certainly, ˙

t > 0, as by assumption the spacetime is causally continuous.

Hence, as |b(˙c)|≤kbkh

xph(˙c, ˙c)≤ |||b|||hph(˙c, ˙c)=1, this can be accomplished by setting

t=Kwith

K=1

|||b|||h+ 1.

Thus, we have α(σ)=(t1+Kσ, c(σ)). Setting q= (t2, x)∈I×Nwith t2> t1, we get

by an analogous reasoning a past-directed causal curve β: [0, r)→I×N, with β(0) = q

and β(σ) = (t2−Kσ, c(σ)). Then assuming t2−t1≥2Kr, implies α(σ)≤β(σ)for all

σ∈[0, r)and particularly α⊂J+(p)∩J−(q). As cis not extendible to the value r,αis not

extendible, either. By the use of Thm. 3.52, this implies that J+(p)∩J−(q)is not compact

and thus (I×N, g)is not globally hyperbolic. The only remaining obstacle is, if t1and t2

can always be chosen such that t2−t1≥2Kr holds. The problem is that t1, t2∈I, which

is possibly bounded. As |||b|||his a fixed constant depending on g, so is K. Let Lbe the

(possibly infinite) length of I. Then by cutting the geodesic cshorter, such that r < L

2Kand

after reparametrization choosing a new initial point x, the choice t2−t1≥2Kr is always

possible.

To prove the remaining assertions we will use Lem. 2.7. If for all t∈I, the function

s(t, ·): N→R>0is bounded from above,

inf

x∈N

s2(t, x)>0,

for all t∈I. Hence, with s2hbeing complete for all t∈I, also h=1

s2s2his complete.

Particularly, if we have a horizontally homothetic Lorentzian submersion at hand, then

H(ds)=0, which implies that sdepends on tonly and is, therefore, constant over N.

Obviously, for all t∈Ithere is a Randers-type metric

R=s√h+b,

given on the factor Nin a specially Hubble-isotropic spacetime (I×N, g, ∂t), with the usual

metric gif s2(t, x)>(kbkh

x)2holds for all (t, x)∈I×N, because in this case

1>(kbkh

x)2

s2(t, x)= sup

v∈TxN\{0}

|bx(v)|2

s2(t, x)hx(v, v)= (kbks2h

x)2,

i.e., Ris indeed a well-defined Finslerian metric of Randers-type. Due to Prop. 2.19, the

conditions “hcomplete” and “S0(t)>|||b|||h” in the theorem above imply the forward and

backward completeness of R. It is now natural to ask if, in analogy to the case of stationary

spacetimes, already the weaker condition of forward or backward completeness of Rfor all t∈

I, does imply global hyperbolicity of the specially Hubble-isotropic spacetime (I×N, g, ∂t).

To prove the theorem below, we need a lemma similar to Lem. 6.17 for Finslerian metrics.

Lemma 6.19. Let (S, F)be a forward or backward complete Finslerian manifold and α: [a, b)→

Sa curve of finite length

lF(α) = Zb

F(α(t),˙α(t))dt=C > 0.

141

Then αis extendible to the value b, i.e., there exists a point x∈Ssuch that α(t)→xas

t→b.

Proof. Using the Hopf–Rinow type theorem for Finslerian metrics (cf. Thm. 2.17), the

proof is completely analogous to Lem. 6.17. We have to make use of the closed and forward

bounded metric balls Kf={y∈S|dF(α(a), y)≤C}or the closed and backward bounded

metric balls Kb={y∈S|dF(y, α(a)) ≤C}, which are compact in the forward or backward

complete cases, respectively. And the extendibility follows.

Theorem 6.20. Let (I×N, g, ∂t)be a specially Hubble-isotropic spacetime with

g=−dt⊗dt+ 2b∨dt+s2h−b⊗b

and s2(t, x)>(kbkh

x)2for all (t, x)∈I×N. If the Randers-type metric

R(t, x) = s(t, x)phx+bx

on Nis forward complete for all t∈Ior backward complete for all t∈I, then (I×N, g)is

globally hyperbolic.

Proof. We assume, that (I×N, g)is not globally hyperbolic and have to show that there

is some τ∈I, such that R(τ)is forward and backward incomplete. Just as in the proof of

Thm. 6.18 we have that (I×N, g)is causally continuous and s2h−b⊗bis Riemannian for all

t∈Iby assumption. Hence, there are two values r < ˜r∈Iand two points xr, x˜r∈N, such

that J+((r, xr)) ∩J−((˜r, x˜r)) is not compact. Thus, by Thm. 3.52, there is a future-directed

causal curve γ: [t1, t2)→J+((r, xr)) ∩J−((˜r, x˜r)) with r≤t1< t2≤˜r, which is not

extendible to the value t2. As the spacetime is causally continuous, we have γparametrized

by t, i.e., γ(t) = (t, c(t)) and ˙γ= (1,˙c), with c: [t1, t2)→Na curve in N, and

u=g(∂t,˙γ) = u( ˙γ) = −1 + b(˙c)<0.

As γis causal,

g(˙γ, ˙γ) = s2h(˙c, ˙c)−(1 −b(˙c))2≤0

yields

s(t, c(t))qhc(t)(˙c(t),˙c(t)) + bc(t)(˙c(t)) ≤1,(∗)

for all t∈[t1, t2). Now as [t1, t2]⊂Iis compact, there is some τ∈[t1, t2]such that

s(τ, x) = mint∈[t1,t2]s(t, x). Hence, for the Randers-type metric R(τ, x) = s(τ, x)√hx+bx,

the inequality (∗) still holds, and we get

R(τ, c(t),˙c(t)) = s(τ, c(t))qhc(t)(˙c(t),˙c(t)) + bc(t)(˙c(t)) ≤1,

for the curve c: [t1, t2)→N. But this implies that the Finslerian length lR(τ)(c)is finite,

and cis not extendible to t2as γis not extendible to t2. Thus by Lem. 6.19,R(τ)is forward

and backward incomplete.

142

We conclude this section with the following

Remark 6.21. As all causality conditions of a spacetime are conformally invariant, all

propositions in this section also hold for kinematical spacetimes arising from specially Hubble-

isotropic ones by conformal transformations. Following Prop. 3.23, the shear of a confor-

mally transformed Hubble-isotropic spacetime will still be zero, but the vanishing acceleration

transforms to H(dφ)for a pointwise conformal transformation g7→ e2φg. Hence, the space-

times (R×N, g), conformal to specially Hubble-isotropic ones—and thus obeying the same

conditions for global hyperbolicity given by the propositions in this section—can be charac-

terized by the existence of a function φ:R×N→R, such that V=e−φ∂tis a reference

frame for gwith σ= 0 and ˙u= dφ, as well as a lapse function in the splitting according to

Thm. 4.16 given by A=eφ.

6.3 Completeness and Singularities

We recall that in a Lorentzian manifold (M, g), geodesic completeness comes in three different

types: timelike, lightlike and spacelike completeness, referring to the completeness of the

geodesics of the respective causal type. The three types of geodesic completeness are logically

independent, in general (cf. [BEE96, Sec. 6.2]), and do not depend on causality conditions

in a direct or obvious way. Therefore, propositions that connect completeness of different

type with one another or with steps on the causal ladder for specific classes of spacetimes

are particularly interesting. In the case of Hubble-isotropic spacetimes we have the following

Proposition 6.22. Let (R×N, g, V )be a proper kinematical spacetime (with Vparallel to

∂t), which is Hubble-isotropic. If (R×N, g)is spacelike geodesically complete, it is causally

continuous.

Proof. As (R×N, g, V )is Hubble-isotropic, the shear and the acceleration of Vvanish.

Thus, we can employ Prop. 4.22 to (R×N, g, V ). Hence, for any (t, x)∈R×Nwe set

U(t,x)⊂T(t,x)(R×N)to be

U(t,x)={E(t,x)∈T(t,x)(R×N)|g(t,x)(V(t,x), E(t,x))=0, g(t,x)(E(t,x), E(t,x))=1}.

Now, as (R×N, g)is spacelike geodesically complete, the spacelike geodesic rays γE(τ) =

exp(τE(t,x)),τ∈Rare all complete, hence exp(U(t,x))⊂R×Nis a spacelike submanifold

diffeomorphic to N. As σ= 0 and ∇VV= 0, we can use the flow Φ: R×R×N→R×N

associated to Vto transport exp(U(t,x)), such that the saturation {Φ(s, exp(U(t,x))) |s∈R}

is diffeomorphic to R×Nand in fact constitutes a trivialization of (R×N, g, V )with slices

that are spacelike everywhere, as it is a foliation of (R×N, g)by spacelike hypersurfaces.

By Prop. 4.26 and Thm. 4.29, this implies that (R×N, g)is causally continuous.

In the light of this proposition and item (i) from Thm. 6.5, we can conclude that all Hubble-

isotropic spacetimes, which are not causally continuous must be spacelike geodesically in-

complete if the reference frame induces a Cartan flow. Particularly, this applies to the

various examples given in section 6.2, which illustrate the lower steps on the causal ladder.

143

In a Hubble-isotropic spacetime the integral curves of the reference frame Vare geodesics.

Thus, we can give criteria for the timelike geodesic incompleteness of a Hubble-isotropic

spacetime, i.e., it being singular, by deriving conditions for the geodesic incompleteness of

the integral curves of V. As we have the Raychaudhuri equations at hand in kinematical

spacetimes, this leads to a specific singularity theorem for Hubble-isotropic spacetimes.

As it is obvious that for specially Hubble-isotropic spacetimes (I×N, g, ∂t), the integral

curves of ∂tare complete if I=Rand incomplete if I( R, we will consider Hubble-

isotropic spacetimes (M, g, V ), which are not special for the remainder of this section only.

Furthermore, we will assume M=R×Nwith the reference frame Vparallel to ∂t, i.e., we

consider Hubble-isotropic spacetimes which split as a product manifold.

Theorem 6.23. Let (R×Nn, g, V )be a Hubble-isotropic spacetime, such that the expansion

of Vis given by

Θ = n˙s

where s:R×N→R>0is the function obtained from the splitting of gaccording to Thm. 6.8

and the dot denotes the derivative ∇Vs. Then

W0:= |ω|2s4≥0

is constant along every fiber, i.e., ∇VW0= 0 and the function sobeys

¨s=∇V∇Vs=W0

s3−Ric(V, V )s,

which is an ordinary differential equation of Ermakov type in every fiber. Then we have: if

n·Ric(V, V )>|ω|2

everywhere along a fiber or n·Ric(V, V )≥ |ω|2everywhere, and ˙s6= 0 somewhere, along

a fiber, this fiber is an incomplete geodesic of the reference frame Vand the spacetime is

timelike geodesically incomplete. Furthermore, assuming |ω|2>0and the Ricci curvature

Ric(V, V )constant along a fiber, this fiber is a complete geodesic of the reference frame V.

Proof. The acceleration ˙u=g(∇VV, ·)and the shear σare zero by definition in (R×

Nn, g, V ). Then following Prop. 3.33, the Raychudhuri equations for the expansion Θand

the rotation ωare given by

Θ + Θ2

n=|ω|2−Ric(V, V )

and

˙ω=−2Θ

nω.

As we have |ω|2=Pi,j ω(Ei, Ej)ω(Ei, Ej), for any pseudo-orthonormal frame {Ei}on R×N

by Def. 3.22, it follows that

∇V|ω|2= 2 X

i,j

˙ω(Ei, Ej)ω(Ei, Ej) = −4Θ

i,j

ω(Ei, Ej)ω(Ei, Ej) = −4Θ

n|ω|2.

144

Using Thm. 6.8 yields Θ = n˙s

s, and hence for W0=|ω|2s4, we can compute

∇VW0=V(|ω|2s4) = s4∇V(|ω|2)+4|ω|2s3˙s=−s44Θ

n|ω|2+ 4|ω|2s3·Θ

ns= 0.

Furthermore, we have

Θ = ∇V(n˙s

s) = n¨ss −˙s2

s2and 1

nΘ2=n˙s2

s2,

thus the Raychaudhuri equation for the expansion reads

n¨s

s=|ω|2−Ric(V, V ),

and hence

¨s=W0

s3−Ric(V, V )s.

Obviously, for all fixed x∈N, this is an ordinary differential equation for s(·, x): R×{x} →

R>0in the fiber R×{x} ⊂ R×Nover x. Furthermore, this equation is of the type of an

Ermakov2equation (see, e.g., [PZ03] or particularly [LA08] and the references therein for

various applications of this type of equation). The general solution of this type of equation

can be given (see [PZ03]), and will be used for the analysis in the case of constant Ricci

curvature below. In the general case, we analyze this equation qualitatively. Certainly, a

fiber, i.e., a geodesic integral curve of V, is incomplete if s(t, x)assumes the value zero for

a finite value of the parameter t(note that the parameter tdoes not coincide, in this case,

with the first value of points (τ, x)∈R×N, but is an affine parameter of the timelike

geodesic integral curve of V, i.e., a proper time). Consider any initial value problem of the

Ermakov-type equation above with s(t0, x)>0. Then n·Ric(V, V )>|ω|2implies

¨s=W0

s3−Ric(V, V )s < W0

s3−|ω|2

ns= 0

as W0=|ω|2s4. Therefore, we have ¨s < 0along all the fiber. Together with the initial

value s(t0, x)>0, this implies that s(r, x)=0for some finite r∈R, hence this fiber is an

incomplete geodesic. If only n·Ric(V, V )≥ |ω|2holds along the fiber, we have ¨s≤0and

need the additional condition, that ˙s6= 0 somewhere, in order for s(t, x)to become zero in

finite proper time, because otherwise we could have s(t, x) = s(t0, x) = const along the fiber

for all t∈R.

If Ric(V, V )is constant along a fiber, the solution of the Ermakov-type equation can be

computed explicitly. The general solution is given in the following way (see [PZ03]). Let

w:R→Rbe any non-trivial solution of the differential equation

¨w+Ric(V, V )w= 0

2sometimes also: Yermakov

145

in some fiber R× {x} ⊂ R×N. Then the general solution of the Ermakov equation

¨s+Ric(V, V )s=W0

s3in this fiber is given by

c1s2(t, x) = w2(t, x)"W0

n+c2+c1Zt

dt0

w2(t0, x)2#,

where the constants c1, c2∈Rare determined by the initial values s(t0, x)and ˙s(t0, x). If

Ric(V, V )is constant along R×{x}, a function wis easily computed explicitly. In the case

Ric(V, V ) = 0 along the fiber, we get

s(t, x) = sna1(x)2W0+ (t+a2(x))2

na1(x),

with a1(x), a2(x)determined by the initial values. Due to the initial value s(t0, x)>0, we

have na2

1W0>0, hence the solution exists for all t∈Rand obeys s(t, x)>0.

If Ric(V, V ) =: −ρ2<0for some ρ > 0, we have w(t) = eρt as a solution for ¨w−ρ2w= 0.

Without loss of generality, we can choose t0= 0 as the initial point and the general solution

can be written as

c1s2(t) = e2ρt "W0

n+c2+c1

2ρ−c1

2ρe−2ρt2#.

Certainly, this solution exists for all t∈Rand a positive initial value s(0) >0ensures

s(t)>0for all t∈Ras W0, ρ > 0.

If Ric(V, V ) =: ρ2>0for some ρ > 0, we can use w(t) = cos(ρt)as a solution for

¨w+ρ2w= 0. Assuming again t0= 0, the general solution is

c1s2(t) = cos2(ρt)"W0

n+c2+c1

ρtan(ρt)2#.

Now we observe that

lim

t→(2k+1) π

2ρ

cos(ρt) = 0 and lim

t→(2k+1) π

2ρ

tan(ρt) = ∞,

for all k∈Z. But computing

lim

t→(2k+1) π

2ρ

(c1s2(t)) = c1

ρ2

cos2(ρt) tan2(ρt) = c1

ρ2

shows that the general solution exists for all t∈ {(2k+ 1) π

2ρ}k∈Zand is positive everywhere

as W0, c1>0. We also see that this only holds if W0>0, i.e., |ω|2>0along the fiber.

Thm. 6.23 constitutes a singularity theorem for Hubble-isotropic spacetimes (cf. [HE73,

Ch. 8], [Pen72]). We observe that in this case, no assumption on the causality of the space-

time is needed to ensure the appearance of a singularity. Instead, we need the timelike Ricci

146

curvature to be (strictly) larger than the rotation scalar, which is a different (and in many

cases stronger) condition than the energy and timelike convergence conditions (cf. [HE73,

Sec. 4.3]) usually used in the standard singularity theorems. Moreover, this result clearly

shows that a large enough rotation can avoid the formation of singularities (cf. [Obu00] and

the references therein). This result becomes most evident in the case of constant Ricci cur-

vature Ric(V, V )along a fiber. As soon as we add a, possibly very small but non-vanishing,

amount of rotation, the fiber is necessarily always complete.

147

Chapter 7

OUTLOOK

We will conclude this work with pointing out some interesting directions of further research

and some open problems, based on the propositions established in this thesis. Some of the

annotations below are based on remarks already given in the previous chapters.

With kinematical spacetimes, we considered free, timelike R-actions on non-compact Lo-

rentzian manifolds. An interesting extension of this concept could be to consider compact

spacetimes, which are necessarily non-chronological, with an S1-action. Similar splitting

questions can be asked in this case, i.e., when is a compact kinematical spacetime (M, g, V ),

with the integral curves of Vbeing embedded circles, a product manifold S1×Nwith N

the space of flow lines of V. Certainly, such a spacetime is always an S1-principal bundle,

which is trivial if Nis simply connected, but can we find conditions for this to happen

based on geometric quantities derived from gand V? And what can be derived about the

causal nature of (global) sections in this bundle? As a compact manifold must have Euler

characteristic zero if it admits Lorentzian metric, a natural class of manifolds to analyze,

which are non-trivial S1-principal bundles, would be odd-dimensional Lorentzian Berger

spheres. Then one can, for example, classify Hubble-isotropic metrics on these spheres or

embedded Hopf tori, which are timelike submanifolds.

In the beginning of section 4.1, we introduced a possible alternative definition for the non-

triviality of closed timelike curves. With example 4.5, we have already given an impression

of the difference of this definition to B. Carter’s classical one in [Car68]. It could be an

interesting task to further determine the differences and similarities of these two notions of

triviality of CTCs, for example by the construction of more, maybe physically interesting,

examples.

In Rem. 4.15, we gave a possible definition of “hole freeness” of kinematical spacetimes, based

on the notion of improper saddle points from the theory of dynamical systems. As mentioned

in this remark, there are other notions of “hole freeness” established in [Man09] and [Min12].

It would be an interesting task to work out the detailed similarities and discrepancies of

these different ideas of holes in a spacetime.

In section 4.3, the question of the existence of a smooth semi-time function in a proper

kinematical spacetime was introduced. Recall from section 3.4 that a continuous semi-time

function exists in every chronological and reflecting spacetime. To the best knowledge of the

author there exist no examples of proper kinematical spacetimes, which are chronological,

but do not admit a smooth semi-time function. Therefore, we formulate the following

Conjecture 7.1. A proper kinematical spacetime (M, g, V )(with a compact manifold S=

M/Rof flow lines of V) is reflecting and admits a smooth semi-time function if it is chrono-

148

logical.

The conjecture seems in any case simpler to assess if a compact Sis assumed, but we believe

it to be true also in the general case. Evidence for this hypothesis is provided by the following

Proposition 7.2. Let γ:R→Mbe a partially future (or past) imprisoned timelike curve

in a chronological, proper kinematical spacetime (M, g, V ). Then ψ◦γis not contained in a

single slice St⊂R×S, with respect to any trivialization ψ: (M, g, V )→(R×S, ψ∗g, ψ∗V).

Proof. We will show the assertion for the partially future imprisoned case. The partially

past imprisoned case works completely analogous and will be omitted. From the considera-

tions in section 4.2, we can assume

ψ∗g=−dt⊗dt+ 2bt∨dt+ht−bt⊗bt

and ψ∗V=∂twith respect to a trivialization ψ. Consider λ:R≥0→R×Sgiven by

λ(s) = (ψ◦γ)(s), to be partially future imprisoned, i.e., there is a monotonically increasing

sequence {sn}n∈N⊂R≥0, such that λ(sn)is contained in some compact set K⊂R×S.

Hence, switching to a subsequence we can assume λ(sn)→(t0, x0)∈Kas n→ ∞. Assume

now that all of λis contained in some slice, i.e., λ(s) = (t0, c(s)) ∈St0for all s≥0, with

c:R≥0→Sthe projection of λto S. Particularly, this implies that the induced metric

ht0−bt0⊗bt0on St0is Lorentzian on λ⊂St0, i.e., by Lem. 4.23

kbt0kht0

λ(s)>1.

As the set {(t, x)∈R×M|kbtkht

x>1}is certainly open, there is a neighborhood Uof λin

R×Ssuch that kbtkht

x>0for all (t, x)∈U. Thus, for all n∈N, we can define a future-

directed timelike curve µn: [sn, sn+1]→R×Sin the following way. Let τn∈(sn, sn+1)be

the value of the curve parameter for which λre-enters Kto approach λ(sn+1). If λis not

only partially- but totally-imprisoned in K, we can just set τn=sn. Then we set

µn(s) = (λ(s)=(t0, c(s)), s ∈[sn, τn)

(t0−ε(s−τn), c(s)), s ∈[τn, sn+1],

with some ε > 0. As Uis open, we can choose εso small that µnis future-directed and

timelike in all its domain for all n∈N. This can be achieved as follows: Denote by a dot

the derivative with respect to the curve parameter, then we have g(∂t,˙

λ) = b(˙c)<0as λis

assumed future-directed. Now we set

−δ:= sup

{s|λ(s)∈K}

gλ(s)(˙

λ(s),˙

λ(s)) = sup

{s|λ(s)∈K}h(t0,c(s))(˙c(s),˙c(s)) −b(t0,c(s))(˙c(s))2<0

and

α:= sup

{s|λ(s)∈K}|b(t0,c(s))(˙c(s))|>0.

Then we compute

g( ˙µn,˙µn)≤ −ε2+ 2αε −δ,

149

and this expression is certainly negative if ε= 0 as −δ < 0, and hence it is also negative in

a neighborhood of ε= 0, from where we can pick a positive εto assure µnis timelike. Then,

certainly, also g(∂t,˙µn) = −ε+b(˙c)<0and µnis future-directed.

Furthermore, by making the compact set Kmaybe a bit larger we can achieve that sn+1 −τn

is bounded from below for all n∈N, say by some k > 0, such that sn+1 −τn> k > 0. This

implies that

pr1(µn(sn+1)) = t0−ε(sn+1 −τn)< t0−kε,

for all n∈N. But this implies that there must be some N∈N, for which λ(sN)∈

I+(µN(sN+1)), thus there is a closed timelike curve in contradiction to the assumption of

chronology. That such an Nmust exist can be inferred as follows: As εis small pr1(µn(sn+1))

is also bounded from below. Since c(sn)→x0as n→ ∞, we can infer that, maybe

after switching to a subsequence, also µn(sn+1)converges to a point (t1, x0)in the fiber

over x0with t1≤t0−kε. Now we set T=t1+t0

2and analyze the point (T, x0)in the

same fiber. Certainly, (t1, x0)∈I−((T, x0)) and (t0, x0)∈I+((T, x0)), but because the

chronological future and past sets are open, there certainly is some N∈N, such that

µN(sN+1)∈I−((T, x0)) and λ(sN)∈I+((T, x0)), as the sequences converge to (t1, x0)and

(t0, x0), respectively. Then choose a future-directed timelike curve connecting µN(sN+1)to

(T, x0)and a future-directed timelike curve connecting (T, x0)to λ(sN). This yields a future-

directed timelike curve connecting µN(sN+1)to λ(sN), and hence λ(sN)∈I+(µN(sN+1)).

The proposition above could make it possible to find a trivialization ψ: (M, g, V )→(R×

S, ψ∗g, ψ∗V), such that the slices St={t} × S⊂R×Sare achronal hypersurfaces, and

hence, are the associated preimages of a smooth semi-time function, which is strictly in-

creasing along all future-directed timelike curves. But there are various technical difficulties

to overcome in order to construct such a trivialization. Another possible idea to resolve the

conjecture would be to independently prove the reflectivity of chronological proper kinemat-

ical spacetimes, which would then yield a continuous semi-time function. Then one can try

to apply a smoothing procedure to this continuous function, which could work similar to the

smoothing of time functions established in [BS05].

Prop. 4.37, establishing a relation between the Weinstein conjecture and Lorentzian causality,

is particularly intriguing. In order to use this relation to prove a general version of the

Weinstein conjecture one would have to view causality from a perspective which does not

directly involve CTCs. Such a perspective exists, using various topologies on the space

of Lorentzian metrics on a manifold (see, e.g., [Min09c] for a recent account). Usually

in this approach, metrics with a causality conditions from the lower part of the causal

ladder are identified by some topological boundary construction on the space of Lorentzian

metrics. In order to assess the Weinstein conjecture, one would need a topology fine enough

to distinguish between non-chronological and non-causal metrics. It is unclear if such a

reasonable topology exists.

In Rem. 5.12, the possibility to regard spacetimes as graphs over lines in the space of

Riemannian metrics R(S)was established. It would be an interesting task to investigate

which spacetimes can appear as such graphs and to classify them. Then one could even add

150

a non-trivial twist to the timelike reference frame in the resulting spacetime and analyze

spacetimes, which are graphs over lines in R(S)×Λ1(S), where Λ1(S)is the space of one-

forms on S.

In section 5.3, we established the new version of the Lorentzian Bochner technique for

stationary spacetimes in the case of a compact and, in a reasonable sense, asymptotically

flat base manifold S. The homothetic case was only established for a compact base S.

After performing the appropriate definitions of asymptotic flatness for the homothetic case,

conditions similar to Thm. 5.45 should hold.

In theorems 6.18 and 6.20, we established conditions for the global hyperbolicity of specially

Hubble-isotropic spacetimes. There are at least two ways how these results could possibly be

expanded. The obvious first way is to include general Hubble-isotropic spacetimes in these

theorems. In this case one has to deal with a non-trivial lapse function. The second way is to

try to derive an appropriate converse proposition from Thm. 6.20, i.e., to deduce from global

hyperbolicity of a Hubble-isotropic spacetime the completeness of particular Randers-type

metrics.

Furthermore, as we can view Hubble-isotropic spacetimes as conformal Lorentzian submer-

sions (R×N, g)→(N, h), it could be an interesting task to analyze generalizations of

these submersions with a Finslerian manifold of Randers type as base, i.e., maps of the

type π: (R×N, g)→(N, √h+b), where his a Riemannian metric and ba one-form on

N. Stationary spacetimes, therefore, occur as such Lorentz-to-Randers submersions by de-

manding, among other things, that for any p= (t, x)∈R×Nthe bounded causal future

(past) J+

T(p) := J+(p)∩([t, t +T]×N)(J−

T(p) := J−(p)∩([t−T, t]×N)) is mapped by π

to a forward (backward) geodesic ball of √h+bon N. Moreover, it could be interesting to

analyze such submersion structures, which could also be regarded as Lorentzian submetries,

and their generalizations, e.g., for a generic Finslerian base manifold, under global view-

points and also to consider the question of a possible relation between the curvatures of the

Finslerian base manifold and the Lorentzian total space.

Another fascinating result obtained from regarding shear-free kinematical spacetimes as con-

formal Lorentzian submersions is the possibility to formulate a long-standing conjecture from

theoretical physics in a completely geometrical way. This is the shear-free fluid conjecture

(see, e.g., [VdB99] and the references therein for a statement of the conjecture in a physical

context and the known special cases), which can be re-formulated as follows.

Conjecture 7.3. Let π: (M4=R×N3, g)→(N3, h)be a conformal Lorentzian submersion.

Denote by Hthe horizontal projection and the horizontal distribution. Assume there are two

functions µ, p:M→R, such that µ+p6= 0 and p=p(µ), the exterior curvature of the

fibers is given by −H(∇p)

µ+pand the Ricci curvature of (M, g)obeys

Ric(V, V ) = 3

2(µ+ 2p), Ric(V, E)=0, Ric(E, E) = −1

2(µ+ 4p),

for all vertical unit vector fields Vand all horizontal unit vector fields E. Then His integrable

or there is a constant c > 0, such that π: (M4, g)→(N3, c ·h)is a Lorentz-to-Riemann

submersion.

151

The problems for a general proof of this conjecture arise from the fact that the exterior

curvature can vary relatively arbitrarily along the fibers. Although the exterior curvature

is essentially a horizontal gradient, one has in general no control over its evolution along

the fiber. Every known special case of the conjecture provides, in one way or another, such

a control over the evolution of the exterior derivative and allows for a proof in this way.

The geometric formulation of the shear-free fluid conjecture above, also allows for possible

generalizations, for example by considering arbitrary dimensions. But it particularly points

to a possible alternative direction towards a proof of the conjecture, firstly, by wielding

B. O’Neill’s curvature formulas for submersions ([O’N66]), as well as their generalizations

to the conformal case (see, e.g., [Gud92]) and, secondly, by employing, adapting and maybe

developing known results about semi-Riemannian submersions with constraints on the Ricci

curvature (see [KY97]).

152

BIBLIOGRAPHY

[And00] M.T. Anderson, On stationary vacuum solutions to the Einstein equations, Ann.

Henri Poincare 1(2000), 977–994.

[AV11] G. Abreu and M. Visser, Some generalizations of the Raychaudhuri equation,

Phys. Rev. D 83 (2011), 104016.

[BCS00] D. Bao, S.S. Chern, and Z. Shen, An introduction to Riemann–Finsler geometry,

Springer, 2000.

[BD12] S. Born and A. Dirmeier, Boundedness of functions on product spaces by sums of

functions on the factors, Preprint, 2012.

[Bee76] J.K. Beem, Conformal changes and geodesic completeness, Comm. Math. Phys.

49 (1976), 179.

[BEE96] J.K. Beem, P.E. Ehrlich, and K.L. Easley, Global Lorentzian geometry, Marcel

Dekker, Inc., 1996.

[BF72] N.P. Bhatia and L.M. Franklin, Dynamical systems without separatrices, Funk-

cialaj Ekvacioj 15 (1972), 1–12.

[Bla00] D.E. Blair, Spaces of metrics and curvature functionals, Handbook of Differential

Geometry, Vol. 1 (F.J.E. Dillen and L.C.A. Verstraelen, eds.), Elsevier, 2000,

pp. 153–185.

[BS05] A.N. Bernal and M. Sánchez, Smoothness of time functions and the metric split-

ting of globally hyperbolic spacetimes, Commun. Math. Phys. 257 (2005), 43–50.

[BS07] A.N. Bernal and M. Sánchez, Globally hyperbolic spacetimes can be defined as

“causal” instead of “strongly” causal, Class. Quant. Grav. 24 (2007), 745–750.

[Car49] H. Cartan, Seminaire Henri Cartan, Vol. 1, 1948-1949.

[Car68] B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev.

174 (1968), 1559–1571.

[Car71] , Causal structure in space-time, Gen. Rel. Grav. 1(1971), 349–391.

153

[CB09] Y. Choquet-Bruhat, General relativity and the Einstein equations, Oxford Science

Publications, 2009.

[CBC02] Y. Choquet-Bruhat and S. Cotsakis, Global hyperbolicity and completeness, J.

Geom. Phys. 43 (2002), 345–350.

[CBC03] , Completeness theorems in general relativity, Proceedings of the 10th

Hellenic relativity conference on recent developments in gravity (K.D. Kokkotas

and N. Stergioulas, eds.), 2003.

[CJM11] E. Caponio, M.A. Javaloyes, and A. Masiello, On the energy functional on Finsler

manifolds and applications to stationary spacetimes, Math. Ann. 351 (2011), 365–

392.

[CJS11] E. Caponio, M.A. Javaloyes, and M. Sánchez, On the interplay between Lorentzian

causality and Finsler metrics of Randers type, Rev. Mat. Iberoamericana 27

(2011), 919–952.

[Con01] L. Conlon, Differentiable manifolds, 2nd ed., Birkhäuser, 2001.

[Cot04] S. Cotsakis, Global hyperbolicity of sliced spaces, Gen. Rel. Grav. 36 (2004), 1183–

1188.

[CS08] A.M. Candela and M. Sánchez, Geodesics in semi-Riemannian manifolds:

Geometric properties and variational tools, Recent Developments in pseudo-

Riemannian Geometry (D.V. Alekseevsky and H. Baum, eds.), ESI Lect. Math.

Phys., Eur. Math. Soc., Zürich, 2008, pp. 359–418.

[Die69] J. Dieudonné, Foundations of modern analysis III, Academic Press, 1969.

[DPS08] A. Dirmeier, M. Plaue, and M. Scherfner, On conformal vector fields parallel to

the observer field, Adv. Lorentz. Geom. (2008), 21–36.

[DPS12] , Growth conditions, Riemannian completeness and Lorentzian causality,

J. Geom. Phys. 62 (2012), 604–612.

[DS99] K.L. Duggal and R. Sharma, Symmetries of spacetimes and Riemannian mani-

folds, Kluwer Academic Publishers, 1999.

[DS10] K.L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds,

Birkhäuser, 2010.

[EGS68] J. Ehlers, P. Geren, and R. Sachs, Isotropic solutions of Einstein–Liouville equa-

tions, J. Math. Phys. 9(1968), 1344–1349.

154

[EH84] J.-H. Eschenburg and E. Heintze, An elementary proof of the Cheeger–Gromoll

splitting theorem, Ann 2(1984), 141–151.

[Ehl93] J. Ehlers, Contributions to the relativistic mechanics of continuous media, Gen.

Rel. Grav. 25 (1993), 1225–1266, reprint of the original German article from 1961

in Abh. Math.-Nat.wiss. (Kl. Akad. Wiss. Lit., Mainz).

[Ej75] R.H. Escobales jr., Riemannian submersions with totally geodesic fibers, J. Diff.

Geom. 10 (1975), 253–276.

[FIP04] M. Falcitelli, S. Ianus, and A.M. Pastore, Riemannian submersions and related

topics, World Scientific, 2004.

[FM78] H.D. Fegan and R.S. Millman, Quadrants of Riemannian metrics, Michigan Math.

J. 25 (1978), 3–7.

[Gal89] G. J. Galloway, The Lorentzian splitting theorem without completeness assump-

tion, J. Diff. Geom. 29 (1989), 373–387.

[Gei08] H. Geiges, An introduction to contact topology, Cambridge Univ. Press, 2008.

[Ger70] R.P. Geroch, Domain of dependence, J. Math. Phys. 11 (1970), 437–449.

[GMM91] O. Gil-Medrano and P.W. Michor, The Riemannian manifold of all Riemannian

metrics, Q. J. Math. 42 (1991), 183–202.

[GO03] M. Gutiérrez and B. Olea, Splitting theorems in presence of an irrotational vector

field, arXiv:math/0306343v1, 2003.

[GO09] , Global decomposition of a Lorentzian manifold as a generalized

Robertson–Walker space, Diff. Geom. Appl. 27 (2009), 146–156.

[Gor73] W.B. Gordon, An analytical criterion for the completeness of Riemannian man-

ifolds, Proc. Amer. Math. Soc. 37 (1973), 221–225.

[Gor74] , Corrections to “An analytical criterion for the completeness of Rieman-

nian manifolds”, Proc. Amer. Math. Soc. 45 (1974), 130–131.

[GPS+10] M. Gürses, M. Plaue, M. Scherfner, T. Schönfeld, and L.A.M. de Sousa jr., On

spacetimes with given kinematical invariants: Construction and examples, Gödel-

type Spacetimes: History and New Developments (M. Plaue and M. Scherfner,

eds.), Publ. Series of the K. Gödel Society X, 2010.

155

[Gra67] A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. of

Mathematics and Mechanics 16 (1967), 715–737.

[GRK96] E. García-Río and D.N. Kupeli, Singularity versus splitting theorems for stably

causal spacetimes, Ann. Glob. Anal. Geom. 14 (1996), 301–312.

[Gud92] S. Gudmundsson, The geometry of harmonic morphisms, Ph.D. thesis, University

of Leeds, 1992.

[Har92] S. Harris, Conformally stationary spacetimes, Class. Quant. Grav. 9(1992), 1823–

1827.

[Has91] W. Hasse, Geometrische Auswertungsmethoden für kosmologische Beobachtungen,

Ph.D. thesis, TU Berlin (in German), 1991.

[HE73] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cam-

bridge University Press, 1973.

[Her60] R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be

a fiber bundle, Proc. Amer. Math. Soc. 11 (1960), 236–242.

[Hic65] N.J. Hicks, Note on differential geometry, Van Nostrand Reynhold, 1965.

[HOW12] S. Hervik, M. Ortaggio, and L. Wylleman, Minimal tensors and purely electric or

magnetic spacetimes of arbitrary dimension, arXiv: 1203.3563v1 [gr-qc], 2012.

[HP88] W. Hasse and V. Perlick, Geometrical and kinematical characterization of

parallax-free world models, J. Math. Phys. 29 (1988), 2064–2068.

[HP99] , On spacetimes with an isotropic Hubble law, Class. Quant. Grav. 16

(1999), 2559–2576.

[Hus94] D. Husemöller, Fibre bundles, Springer, 1994.

[Hut10] M. Hutchings, Taubes’ proof of the Weinstein conjecture in dimension three, Bull.

Amer. Math. Soc. 47 (2010), 73–125.

[JS08] M.A. Javaloyes and M. Sánchez, A note on the existence of standard splittings for

conformally stationary spacetimes, Class. Quant. Grav. 25 (2008), 168001–7.

[KN63] S. Kobayashi and K. Nomizu, Foundations of differential geometry I, Interscience

Publishers, 1963.

156

[Kro68] L. Kronecker, Näherungsweise ganzzahlige Auflösung linearer Gleichungen, Werke

3(1884, (reprint 1968)), 47–109.

[KS66] J. Kristian and R.K. Sachs, Observations in cosmology, Astrophys. J. 143 (1966),

379–399.

[KS07] S. Kar and S. Sengupta, The Raychaudhuri equations: A brief review, Pramana

Journal of Physics 69 (2007), 49–76.

[KY61] M.S. Knebelmann and K. Yano, On homothetic mappings of Riemann spaces,

Proc. Amer. Math. Soc. 12 (1961), 300–303.

[KY97] J.-H. Kwon and J.S. Young, On sectional and Ricci curvatures of semi-

Riemannian submersions, Kodai Math. J. 20 (1997), 53–66.

[LA08] P.G.L. Leach and K. Andriopoulos, The Ermakov equation: A commentary, Appl.

Anal. Discrete Math. 2(2008), 146–157.

[Lic55] A. Lichnerowicz, Théories relativiste de la graviation et de l’électromagnetisme,

Masson, 1955.

[Man09] J.B. Manchak, Is spacetime hole-free?, Gen. Rel. Grav. 41 (2009), 1639–1643.

[Mar69] L. Markus, Parallel dynamical systems, Topology 8(1969), 47–57.

[McD98] D. McDuff, Symplectic structures – a new approach to geometry, Notices of the

AMS 45 (1998), 952–960.

[Min08a] E. Minguzzi, The causal ladder and the strength of K-causality I, Class. Quant.

Grav. 25 (2008), 015009.

[Min08b] , The causal ladder and the strength of K-causality II, Class. Quant. Grav.

25 (2008), 015010.

[Min08c] , Non-imprisonment conditions on spacetime, J. Math. Phys. 49 (2008),

062503.

[Min08d] , Weak distinction and the optimal definition of causal continuity, Class.

Quant. Grav. 25 (2008), 075015.

[Min09a] , Chronological spacetimes without lightlike lines are stably causal, Com-

mun. Math. Phys. 288 (2009), 801–819.

157

[Min09b] , K-causality coincides with stable causality, Commun. Math. Phys. 290

(2009), 239–248.

[Min09c] , Widening the lightcones on subsets of spacetime: Some variations to

stable causality, Class. Quant. Grav. 26 (2009), 175004.

[Min10] , Time functions as utilities, Comm. Math. Phys. 298 (2010), 855–868.

[Min12] , Causally simple inextendible spacetimes are hole-free, arXiv:1204.3012v1

[gr-qc], 2012.

[MS08] E. Minguzzi and M. Sánchez, The causal hierarchy of spacetimes, Recent Develop-

ments in pseudo-Riemannian Geometry (D. V. Alekseevsky and H. Baum, eds.),

ESI Lect. Math. Phys., Eur. Math. Soc., Zürich, 2008, pp. 299–358.

[NO61] K. Nomizu and H. Ozeki, The existence of complete Riemannian metrics, Proc.

Amer. Math. Soc. 12 (1961), 889–891.

[NS60] V.V. Nemytskii and V.V. Stepanov, Qualitative theory of differential equations,

Princeton University Press, 1960.

[Obu00] Y.N. Obukhov, On physical foundations and observational effects of cosmic ro-

tation, Colloquium on Cosmic Rotation, Wissenschaft & Technik Verlag, 2000,

pp. 23–96.

[OD77] D.R. Oliver and W.R. Davis, On certain timelike symmetry properties and the

evolution of matter field space-times that admit them, Gen. Rel. Grav. 8(1977),

905–914.

[O’N66] B. O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13

(1966), 459–469.

[O’N83] , Semi-Riemannian geometry with applications to relativity, Academic

Press, 1983.

[OR93] L. Ornea and G. Romani, The fundamental equations of conformal submersions,

Beiträge zur Algebra Geom. 34 (1993), 233–243.

[Pal61] R.S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann.

Math. 73 (2) (1961), 295–323.

[Pen72] R. Penrose, Techniques of differential topology in relativity, Cbms-Nsf Regional

Conference Series in Applied Mathematics, Philadelphia: SIAM, 1972.

158

[Per90] V. Perlick, On redshift and parallaxes in general relativistic kinematical world

models, J. Math. Phys. 31 (1990), 1992–1971.

[Pet98] P. Petersen, Riemannian geometry, Springer, 1998.

[Pla12] M. Plaue, Causality-violating Lorentzian manifolds admitting a shear-free timelike

flow, Ph.D. thesis, TU Berlin, 2012.

[PR93] R. Ponge and H. Reckziegel, Twisted products in pseudo-Riemannian geometry,

Geom. Ded. 48 (1993), 15–25.

[PZ03] A.D. Polyanin and V.F. Zaitsev, Handbook of exact solutions for ordinary differ-

ential equations, 2nd ed., Chapman & Hall/CRC, 2003.

[Ray55] A.K. Raychaudhuri, Relativistic cosmology I, Phys. Rev. 98 (1955), 1123–1126.

[Rov98] V. Rovenskii, Foliations on Riemannian manifolds and submanifolds, Birkhäuser,

1998.

[RS96] A. Romero and M. Sánchez, An integral inequality on compact Lorentz manifolds

and its applications, Bull. London. Math. Soc. 28 (1996), 509–513.

[RS98] , Bochner’s technique on Lorentz manifolds and infinitesimal conformal

symmetries, Pacific J. Math. 186 (1998), 141–148.

[S´

11] M. Sánchez, Recent progress on the notion of global hyperbolicity, Advances in

Lorentzian Geometry – Proceedings of the Lorentzian Geometry Conference in

Berlin (M. Plaue, A. Rendall, and M. Scherfner, eds.), AMS and International

Press, 2011.

[Sch63] A.J. Schwartz, A generalization of a Poincaré–Bendixson theorem to closed two-

dimensional manifolds, Am. J. Math. 85 (1963), 453–458.

[SS78] L.A. Steen and J.A. Seebach, Counterexamples in topology, Springer, 1978.

[ST51] H. Seifert and W. Threlfall, Variationsrechnung im Großen, Chelsea, 1951.

[Ste93] S.E. Stepanov, Bochner techniques and cosmological models, Russ. Phys. J. 36

(1993), 578–581.

[Ste99] , The Bochner technique for an m-dimensional compact manifold with

SL(m, R)-structure, St. Petersburg Math. J. 10 (1999), 703–714.

159

[Ste00] , An analytic method in general relativity, Theoret. Math. Phys. 122

(2000), 402–414.

[Str86] R.S. Strichartz, Sub-Riemannian geometry, J. Diff. Geom. 24 (1986), 221–263.

[Tau07] C.H. Taubes, The Seiberg–Witten equations and the Weinstein conjecture, Geom.

Topol. 11 (2007), 2117–2202.

[tD87] T. tom Dieck, Transformation groups, de Gruyter, 1987.

[Tho03] K. Thorne, Warping spacetime, The Future of Theoretical Physics and Cosmology

– Celebrating Stephen Hawking’s 60th Birthday (G.W. Gibbons, E.P.S. Shellard,

and S.J. Rankin, eds.), Cambridge Univ. Press, 2003, pp. 74–104.

[VdB99] N. Van den Bergh, The shear-free perfect fluid conjecture, Class. Quant. Grav. 16

(1999), 1–13.

[vQ76] B. von Querenburg, Mengentheoretische Topologie, Springer, 1976.

[Wad75] A.W. Wadsley, Geodesic foliations by circles, J. Diff. Geom. 10 (1975), 541–549.

[Wal84] R. Wald, General Relativity, Chicago University Press, 1984.

[Wei79] A. Weinstein, On the hypotheses of Rabinowitz’ periodic orbit theorems, J. Differ-

ential Equations 33 (1979), 353–358.

[Wu80] H. Wu, The Bochner technique, Proceedings of the 1980 Beijing Symposium on

Differential Geometry and Differential Equations, Vol. 2 (Beijing, 1980), 929–

1071.