Singularities in cosmological Bianchi class A models...have also been considered by [Wal83] and...

Diploma Thesis

Singularities in cosmological

Bianchi class A models

Katharina Radermacher

May 2012

Supervisor

Prof. Dr. Frank Loose

Eberhard Karls Universität Tübingen

Mathematisch-Naturwissenschaftliche Fakultät

Fachbereich Mathematik

Hiermit erkläre ich, die vorliegende Arbeit selbstständig verfasst und nur dieangegebenen Quellen verwendet zu haben. Die Arbeit wurde keiner anderen Prü-fungsbehörde vorgelegt.

Tübingen, den 16. Mai 2012Ort, Datum Katharina Radermacher

Contents

Introduction iii

Zusammenfassung in deutscher Sprache vii

1. Introduction to semi-Riemannian geometry 1

1.1. Di�erential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Semi-Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . 51.3. Semi-Riemannian submanifolds . . . . . . . . . . . . . . . . . . . . . 11

2. Lie groups and Lie algebras 17

3. Introduction to the General Theory of Relativity 21

3.1. Lorentz manifolds and Minkowski spacetime . . . . . . . . . . . . . . 213.2. The Einstein equation . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3. Perfect �uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4. Robertson-Walker spacetimes . . . . . . . . . . . . . . . . . . . . . . 273.5. Evolution of Robertson-Walker perfect �uids . . . . . . . . . . . . . . 303.6. Beyond Robertson-Walker models . . . . . . . . . . . . . . . . . . . . 31

4. Cosmological Bianchi models 35

4.1. Milnor bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2. Classi�cation of Bianchi class A models . . . . . . . . . . . . . . . . . 39

5. Ricci curvature in four-dimensional spacetime 43

5.1. Some simpli�ed geometric quantities . . . . . . . . . . . . . . . . . . 445.2. Geometric relations between M and M . . . . . . . . . . . . . . . . . 465.3. Formulae for the Ricci curvature . . . . . . . . . . . . . . . . . . . . . 50

6. Ricci curvature on three-dimensional unimodular Lie groups 57

6.1. The case of a normalised Milnor basis . . . . . . . . . . . . . . . . . . 586.2. The case of a uni�ed Milnor basis . . . . . . . . . . . . . . . . . . . . 60

7. Dynamics on Bianchi class A models 63

7.1. Einstein equations on Bianchi models . . . . . . . . . . . . . . . . . . 637.2. Conserved constraint equations . . . . . . . . . . . . . . . . . . . . . 667.3. div tk and diagonalisation on Bianchi class A . . . . . . . . . . . . . 677.4. Bianchi class I in vacuum . . . . . . . . . . . . . . . . . . . . . . . . 74

i

Contents

7.5. Generalisation to Bianchi class A models in vacuum . . . . . . . . . . 81

A. Appendix � Tensors 89

Bibliography 91

ii

Introduction

At the beginning of the 20th century, Albert Einstein's Theory of Relativity funda-mentally changed our understanding of time, space and matter. The Special Theoryof Relativity, introduced in 1905, describes systems without matter and externalforces. It is based on two seemingly natural postulates, the constancy of the speedof light and the equivalence principle, which states that the laws of physics are thesame in all non-accelerated systems. Nonetheless, these two postulates cause aston-ishing e�ects: Time is not a global absolute value, but depends on the observer. Thenotion of simultaneity depends on the observer as well, i. e. two events can occur syn-chronously in the view of one observer, but consecutively in the view of the other. Amoving clock slows down (time dilation) and the length of a moving rod is shortened(length contraction), if compared to a clock and a rod which are at rest relative tothe observer.

The Special Theory of Relativity, however, fails when it comes to gravity. It becameclear in the 1910s that one needs the mathematical concepts developed in geometryto solve this problem. The General Theory of Relativity postulates that the universecan be described as a four-dimensional semi-Riemannian manifold compliant with theEinstein equation

Ric − 1

2S g + Λ g = 8πT .

This four-dimensional tensor equation relates several curvature quantities to thestress-energy tensor T , which describes the distribution of matter. Time and spaceare no longer two separate variables, but are intertwined in the so-called spacetime.External forces such as gravity are interpreted as pseudo forces, similar to the cen-trifugal or Coriolis force. In the General Theory of Relativity there is only free fallin a curved universe.

Since then, many mathematicians and physicists have dedicated their studies to abetter understanding of this theory. Over the past decades, di�erent approacheshave led to various insights and the establishment of several models. In this thesis weinvestigate models for the whole universe, i. e. cosmological models. We introduce theRobertson-Walker spacetimes and the notion of big bang and big crunch. Our maininterest lies in �nding possible generalisations of these properties to another class ofmodels, the Bianchi models. As a complete treatment of these models requires resultsfrom di�erent areas, we brie�y introduce these foundations in the �rst chapters, beforeturning to a profound analysis of singularities in Bianchi cosmologies.

iii

Introduction

In the �rst chapter we give an overview over semi-Riemannian geometry. After somepreliminary de�nitions on manifolds, we consider semi-Riemannian metrics and geo-metric quantities describing the di�erence between the intrinsic and extrinsic geom-etry of a semi-Riemannian submanifold.

The Bianchi models are constructed out of certain Lie groups, and their classi�cationis based on their associated Lie algebras. In the second chapter, we therefore turnour attention to the theory of Lie groups, Lie algebras and the relation between thetwo.

Chapter three constitutes an introduction to all concepts of the General Theory of Re-lativity which are of importance for this thesis' subject. We de�ne vacuum models andperfect �uids, before thoroughly studying Robertson-Walker spacetimes, the standardmodel in cosmology. The main issue is the qualitative behaviour of these models, inparticular the existence of singularities with special properties. We are able to givean alternative characterisation of big bang and big crunch which di�ers from the onefound in most textbooks. It has a wider application range and will be used as analternative de�nition of big bang in the last chapter.

The main objects of our study are the cosmological Bianchi models. These speci�cspacetimes are constructed out of three-dimensional Lie groups. Using our knowledgeabout Lie groups and Lie algebras, they are introduced and classi�ed in chapter four.

In the �fth chapter, we return to spacetimes which are slightly more general than theBianchi models. We loosen the entanglement of space and time found in arbitraryspacetimes and consider manifolds of the form

M .

.= I ×M

with semi-Riemannian metric

g(t,p)

.

.= −dt2 + tgp .

The three-dimensional Riemannian manifoldM can be thought of as the space, whilethe intervall I is the time-direction. A series of computations then shows that the four-dimensional Ricci curvature can be expressed by time-dependant three-dimensionalgeometric quantities on M .

Our goal is to apply the results of chapter �ve to Bianchi models, i. e. with M a Liegroup. Its Lie algebra possesses a special type of basis, called Milnor basis, if it isunimodular. This basis allows us to obtain easy formulae for the Ricci and scalarcurvature, which we develop in chapter six.

In the �nal chapter we gather all results and apply them to Bianchi models. TheEinstein equation, combined with the formulae from chapter �ve, yield a system

iv

of ordinary di�erential equations with constraints. We state several results aboutexistence and uniqueness of solutions to this �ow, then study the singularities. Inthis, we mainly rely on two di�erential inequalities for the mean curvature, whichhave also been considered by [Wal83] and [Ren95]. The case of Bianchi type I can beintegrated explicitely. Using assumptions motivated by the Hubble constant, we areable to prove existence of a big bang in the sense we developed in chapter three. In thegeneral case of Bianchi class A models, several qualitative statements can be givenconcerning the mean curvature, which prove the existence of an initial singularity.The question whether this singularity is in fact a big bang, remains to be treated infuture works.

v

Zusammenfassung in deutscher

Sprache

Anfang des 20. Jahrhunderts entwickelte Albert Einstein die Relativitästtheorie, dieunser Verständnis von Raum, Zeit und Materie grundlegend verändern sollte. 1905,im "Annus mirabilis", verö�entlichte er seine Überlegungen zur Speziellen Relativi-tätstheorie, welche materie- und kräftefreie Systeme beschreibt. Die Theorie beruhtauf zwei denkbar einfachen Postulaten: Die Lichtgeschwindigkeit wird zur Naturkon-stante erklärt, nachdem Messungen mit immer gröÿerer Genauigkeit keinerlei Schwan-kungen erkennen lieÿen. Darüber hinaus besagt das Relativitätsprinzip, dass jederBeobachter in einem nicht-beschleunigten System, genannt Inertialsystem, dieselbenphysikalischen Gesetze erfährt. Trotz der Einfachheit dieser Postulate sind die re-sultierenden E�ekte erstaunlich: Zeit ist keine absolute Gröÿe, und Gleichzeitigkeithängt vom Betrachter ab. Bewegte Uhren gehen langsamer (Zeitdilatation), bewegteMaÿstäbe sind kürzer (Längenkontraktion) als Uhren und Maÿstäbe, die bezüglichdes Beobachters ruhen.

Die Spezielle Relativitätstheorie bietet jedoch keine Möglichkeit, Gravitationse�ektezu berücksichtigen, und ist deshalb zur Beschreibung des Universums nur einge-schränkt geeignet. Dieser Mangel wurde etwa zehn Jahre später in der AllgemeinenRelativitätstheorie behoben, die sich sehr stark verschiedener Konzepte aus der Geo-metrie bedient. Das Universum wird durch eine vierdimensionale semi-RiemannscheMannigfaltigkeit beschrieben, welche die Einsteingleichung

Ric − 1

2S g + Λ g = 8πT

erfüllt. Diese vierdimensionale Tensorgleichung verknüpft verschiedene Krümmungs-gröÿen mit dem Energie-Impuls-Tensor T , der die Verteilung von Masse im Univer-sum beschreibt. Raum und Zeit sind keine getrennten Gröÿen mehr, sondern werdenin der Raumzeit kombiniert. In der Allgemeinen Relatvitätstheorie werden externeKräfte wie die Gravitation als Scheinkraft interpretiert, ähnlich der Zentrifugal- oderCorioliskraft. Es gibt nur kräftefreien Fall im gekrümmten Raum.

Seit Etablierung der Theorie arbeiten Mathematiker und Physiker an der Entwicklungund Untersuchung verschiedener Modelle. In der vorliegenden Arbeit konzentrierenwir uns auf Mannigfaltigkeiten, die das gesamte Universum beschreiben, also kosmo-logische Modelle. Wir betrachten den Vakuumfall und den Fall, in dem Sterne undGalaxien als Partikel einer Flüssigkeit betrachtet werden.

vii

Zusammenfassung in deutscher Sprache

Ausgehend von den bekannten und gut verstandenen Robertson-Walker-Raumzei-ten untersuchen wir die Bianchi-Modelle und beweisen verschiedene Aussagen überSingularitäten und das Verhalten der mittleren Krümmung in der Nähe dieser Singu-laritäten.

Die ersten drei Kapitel geben einen Überblick über die mathematischen Gebiete,deren Methoden und Ergebnisse für das Verständnis der Bianchi-Modelle nötig sind.Zunächst werden semi-Riemannsche Mannigfaltigkeiten und Untermannigfaltigkeiteneingeführt und mehrere Krümmingsgröÿen de�niert, welche die intrinsische und ex-trinsische Geometrie beschreiben. Danach wird der Zusammenhang zwischen Liegrup-pen und -algebren beleuchtet und schlieÿlich eine Einführung in die mathematischenGrundlagen der Relativitätstheorie gegeben. Von besonderem Interesse sind dieRobertson-Walker Raumzeiten, das Standardmodell in der Kosmologie, und hier-bei speziell die Existenz und Eigenschaften von Singularitäten. Motiviert durchphysikalische Überlegungen werden bestimmte Singularitäten als Big Bang oder BigCrunch bezeichnet. Wir entwickeln eine Charakterisierung, die äquivalent ist zurüblicherweise in der Literatur gefundenen De�nition, aber auch auf allgemeinere Mo-delle angewandt werden kann.

Unter Zuhilfenahme der Kenntnisse aus den ersten Kapiteln de�nieren wir in Kapitelvier die Bianchi-Modelle. Für eine gegebenen dreidimensionale zusammenhängendeLiegruppe G sind dies vierdimensionale Mannigfaltigkeiten

G .

.= I ×G

mit einer semi-Riemannschen Metrik der Form

g(t,p)

.

.= −dt2 + tgp ,

die eingeschränkt auf G linksinvariant ist.

In den folgenden beiden Kapiteln werden verschiedene Krümmungsgröÿen explizitberechnet, einerseits auf speziellen dreidimensionalen Liegruppen, andererseits aufvierdimensionalen Mannigfaltigkeiten, die dieselbe Gestalt aufweisen wie die Bianchi-Modelle, anstelle von G jedoch eine allgemeine Riemannsche Mannigfaltigkeit zu-lassen. Mithilfe dieser Ergebnisse wird im letzten Kapitel die Einsteingleichung alsSystem gewöhnlicher Di�erentialgleichungen mit Zwangsbedingungen interpretiert.Wir untersuchen die Existenz und Eindeutigkeit von Lösungen dieser Flussgleichun-gen und wenden uns dann einer genaueren Betrachtung der Singularitäten zu. Wiein [Wal83] und [Ren95] bedienen wir uns hierbei hauptsächlich zweier Di�erential-Ungleichungen für die mittlere Krümmung. Der Bianchi I-Fall kann explizit inte-griert werden, und unter physikalisch motivierten Annahmen besitzt das Modell eineAnfangssingularität. Wir zeigen, dass dies ein Big Bang im Sinne unser erweitertenDe�nition ist. Für allgemeine Bianchi A-Klassen entwickeln wir verschiedene qua-litative Aussagen und beweisen die Existenz einer Anfangssingularität. Ob es sichhierbei um einen Big Bang handelt, muss in zukünftigen Arbeiten abschlieÿend ge-klärt werden.

viii

1. Introduction to

semi-Riemannian geometry

In this chapter we give a short overview of the theory of semi-Riemannian manifoldsand state all necessary results we use in the following. A more profound introductioncan be found in [O'N83]. Most of the proofs will be omitted due to lack of space. Ifthey are not carried out by [O'N83] in the (same-titled) chapters 3 and 4, we will givean explicite reference.

1.1. Di�erential geometry

De�nition 1.1.1. A topological manifold of dimension n is a connected paracompactHausdor� space M that is locally homeomorphic to Rn, i. e. for every point there isan open neighbourhood U ⊆M , an open subset W ⊆ Rn and a homeomorphism

φ : U −→ W,

called chart. An atlas is a family of charts (Uα, φα) for which⋃α Uα = M .

An atlas (Uα, φα) is called smooth if the chart transitions

φβ ◦ φ−1α : φα (Uα ∩ Uβ)→ φβ (Uα ∩ Uβ)

are C∞. We call two smooth atlases (Uα, φα), (Vβ, ψβ) equivalent if their union isagain a smooth altas. A smooth manifold of dimension n is a pair (M, [(Uα, φα)]),where [(Uα, φα)] denotes the equivalence class.

De�nition 1.1.2. Let M,N two smooth manifolds with altas (Uα, φα), (Vβ, ψβ) re-spectively. A map f : M → N is called smooth if for every two charts φα, ψβ withUα ∩ f−1 (Vβ) 6= ∅ the composition ψβ ◦ f ◦ φ−1α is smooth.

The space of all smooth functions f : M → R, denoted by E(M), is an algebra overR.

Notation 1.1.3. Let M a smooth manifold M of dimension n. We introduce thefollowing notation: For every point p ∈ M we denote TpM the tangent space andT ∗pM its dual, the cotangent space. We further denote TM the tangent bundle andT ∗M the cotangent bundle.

1

1. Introduction to semi-Riemannian geometry

We remark that the tangent bundle TM can be equipped with the structure of asmooth manifold such that the projection

π : TM →M,

which maps any v ∈ TpM onto p, is a smooth map.

De�nition 1.1.4. Let M a smooth manifold M of dimension n. A smooth vector�eld is a smooth map X : M → TM such that Xp

.

.= X(p) ∈ TpM for every p ∈ M .The space of all smooth vector �elds on M is denoted by X(M).A local frame �eld on an open set U ⊂ M is a family X1, . . . , Xn of smooth

vector �elds on U which are linearly independent, i. e. for every p ∈ U the vectorsX1(p), . . . , Xn(p) span TpM .

When talking about manifolds and vector �elds, we will always think of smoothones, but will frequently omit the term.

De�nition 1.1.5. Let M a smooth manifold. The bracket operation or vector �eldcommutator is the map [·, ·] : X(M)×X(M)→ X(M) de�ned by (X, Y ) 7→ X ◦ Y −Y ◦X.Two vector �elds X, Y ∈ X(M) commmute if [X, Y ] = 0.

Lemma 1.1.6. The bracket operation [·, ·] satis�es the following properties:

i) R-bilinearity:

[a1X1 + a2X2, Y ] = a1 [X1, Y ] + a2 [X2, Y ]

[X, b1Y1 + b2Y2] = b1 [X, Y1] + b2 [X, Y2]

ii) skew-symmetry:

[X, Y ] = − [Y,X]

iii) the Jacobi identity:

[[X, Y ] , Z] + [[Y, Z] , X] + [[Z,X] , Y ] = 0

iv) derivative in both arguments

[fX, gY ] = fg [X, Y ] + f ·Xg · Y + g · Y f ·X

where i = 1, 2, X,Xi, Y, Yi ∈ X(M), ai, bi ∈ R, f, g ∈ E(M).

De�nition 1.1.7. Let M a smooth manifold of dimension n. A k-vector bundleE →M overM consists of a smooth manifold E and a smooth projection π : E →Msuch that

2

1.1. Di�erential geometry

i) for every p ∈M the �ber Ep .

.= π−1(p) is a k-dimensional real vector space,

ii) for every p ∈M there is a neighbourhood U and a di�eomorphism

Φ : π−1 (U)→ U × Rk

with pr1 ◦ Φ = π such that for every q ∈ U , the map

Φq.

.= Φ|Eq : Eq → {q} × Rk,

v 7→ Φ (q, v) ,

is a linear isomorphism.

M is called the base manifold, E the total manifold, (Φ, U) a bundle chart.

De�nition 1.1.8. Let π : E → M a k-vector bundle. A section of E is a smoothmap s : M → E with π ◦ s = idM . We denote by Γ(E) the space of sections of E.

The tangent bundle TM is an example of an n-bundle over the n-dimensionalmanifold M . In this case, the space of sections is the space of smooth vector �elds:Γ(TM) = X(M).It is easy to see that for any vector bundle π : E → M the space of sections Γ(E)

becomes a module over E(M) in a natural way.

For every smooth function f ∈ E(M) and every vector �eld X ∈ X(M) the directionalderivative is

(X, f) 7→ Xf,

also denoted by Xf = dXf . It is E(M)-linear in X, R-linear in f and satis�es theproduct rule

X (fg) = Xf · g + f ·Xg.

This concept extends to sections on arbitrary vector bundles by introducing connec-tions.

De�nition 1.1.9. A connection on a vector bundle E →M is a function ∇ : X(M)×Γ(E)→ Γ(E) such that:

(i) (X, σ) 7→ ∇Xσ is E(M)-linear in X:

∇f1X1+f2X2σ = f1∇X1σ + f2∇X2σ for Xi ∈ X (M) , σ ∈ Γ (E) , fi ∈ E (M)

(ii) (X, σ) 7→ ∇Xσ is R-linear in σ:

∇X (a1σ1 + a2σ2) = a1∇X1σ1 + a2∇X2σ2 for X ∈ X (M) , σi ∈ Γ (E) , ai ∈ R

3


(iii) (X, σ) 7→ ∇Xσ satis�es the product rule in σ

∇X (fσ) = f∇Xσ +Xf · σ for X ∈ X (M) , σ ∈ Γ (E) , f ∈ E (M) .

∇Xσ is called the covariant derivative of σ with respect to X.

In particular, we can de�ne connections on the tangent bundle TM of a smoothmanifoldM , as Γ(TM) = X(M). The tangent bundle is the canonical bundle overM ,as it is constructed solely by using the charts, without any need for additional data.Therefore connections on TM are often referred to as connections on the manifoldM itself.Let p ∈M . We want to explore how the connection ∇ depends on the vector �elds

X, Y ∈ X(M). As ∇ is E(M)-linear in the �rst argument, ∇XY (p) only depends onX(p) and is independant of any other value of X. Lemma 1.3 in [Pet98] shows that∇XY (p) only depends on the values of Y in a neighbourhood of p, thus making thecovariant derivative ∇X with respect to X a local quantity.We remark furthermore that on the space of smooth functions E(M) = Γ(M × R)

the covariant derivative coincides with the directional derivative:

Xf = dXf = ∇Xf.

We want to extend a given connection ∇ on E → M to other bundles associatedwith E. These include product bundles, pulled back bundles and the dual bundleE∗, whose �bers are the dual spaces of the �bers of E. We remark that the followingthree constructions yield unique connections on the respective bundles.

De�nition 1.1.10. Let M a smooth manifold, ∇ a connection on a vector bundleE →M . The connection ∇∗ dual to ∇ on the dual bundle E∗ is de�ned by

d (σ, τ ∗) = (∇σ, τ ∗) + (σ,∇∗τ ∗)

for σ ∈ Γ(E), τ ∗ ∈ Γ(E∗), where (·, ·) : E ⊗ E∗ → M denotes the bilinear pairingbetween E and E∗ and d is the directional derivative.

De�nition 1.1.11. LetM a smooth manifold, ∇i a connection on the vector bundlesEi →M , i = 1, 2. Then the induced connection ∇ on E .

.= E1 ⊗ E2 is de�ned by

∇ (σ1 ⊗ σ2) = (∇1σ1)⊗ σ2 + σ1 ⊗ (∇2σ2)

for σi ∈ Γ(Ei), i = 1, 2.

In particular, using these two constructions a given connection ∇ on a smoothmanifold M uniquely induces connections on the tensor bundles TM ⊗ . . . ⊗ TM ⊗T ∗M ⊗ . . .⊗ T ∗M (see the appendix for a de�nition of tensor bundles):

(∇T ) (ω1, . . . , ωr, X1, . . . , Xs) = d (T (ω1, . . . , ωr, X1, . . . , Xs))

−r∑i=1

T (ω1, . . . ,∇ωi, . . . , ωr, X1, . . . , Xs)

−s∑j=1

T (ω1, . . . , ωr, X1, . . . ,∇Xj, . . . , Xs) , (1.1)

4

1.2. Semi-Riemannian geometry

where T ∈ Γ(TM⊗. . .⊗TM⊗T ∗M⊗. . .⊗T ∗M) a tensor of type (r, s), ωi ∈ Γ(TM∗)for i = 1, . . . , r, Xj ∈ X(M) for j = 1, . . . , s.

De�nition 1.1.12. Let M,N smooth manifolds, f : M → N a smooth map and∇E a connection on the vector bundle E → N . We consider the pulled back bundlef ∗E → M , i. e. the bundle over M with bundle charts (f−1(U), φ ◦ f), where (U, φ)are the bundle charts of E. The pullback connection ∇f∗E on f ∗E is de�ned by

∇f∗EX (σ ◦ f) = ∇E

Df(X)σ,

for X ∈ X(M) and σ ∈ Γ(E), where Df denotes the di�erential of f : M → N .

De�nition 1.1.13. Let M be a manifold. A subset M ⊂ M is a submanifold of Mif the natural inclusion map

ι : M ↪→M

is smooth and the di�erential Dι is injective at each point p ∈M .

Remark 1.1.14. Additionally, we could demand the inclusion ι to be a homeomor-phism, thus equipping M with the subspace topology. We can show that this isequivalent to the following property: For every p ∈ M there is a chart (U, φ) on Msuch that

φ (q) =(φ1 (q) , . . . , φm (q) , 0, . . . , 0

)∀ q ∈M ∩ U.


In semi-Riemannian geometry we want to measure the length of curves and examinethe properties of those curves which minimise the lenght. In order to do so, we equipsmooth manifolds with di�erent types of tensor �elds. See the appendix for a generalintroduction to tensors and tensor �elds. The introduction of these tensor �elds leadsto the notion of Riemannian curvature and several other geometric tensor quantitieswhich we will use extensively in this thesis.

De�nition 1.2.1. A Riemannian metric on a smooth manifoldM is a smooth section

g ∈ Γ (TM∗ ⊗ TM∗)

such that for every p ∈ M , gp .

.= g (p) ∈ T ∗pM ⊗ T ∗pM is a scalar product on thetangent space TpM .A Riemannian manifold is a smooth manifold furnished with a Riemannian metric.

In the context of General Relativity, we will use a more general concept than a Rie-mannian manifold. Instead of a scalar product, a semi-Riemannian manifold possessesa non-degenerate bilinear form on every tangent space. In other words, the positivede�niteness is weakened to non-degeneracy.Let us recall a result about bilinear forms which is discussed in linear algebra.

5


Remark 1.2.2. Let V a �nite-dimensional vector space,

s : V × V → R

a symmetric bilinear form. Then by Sylvester's law of inertia, the dimension of amaximal subspace on which the restriction of s is positive resp. negative de�nitedoes not depend on the choice of subspace.The index ν of a symmetric bilinear form s is the maximal dimension of subspaces

on which s is negative de�nite.A symmetric bilinear form is nondegenerate if

s (v, w) = 0 for every w ∈ V ⇒ v = 0.

De�nition 1.2.3. A semi-Riemannian metric on a smooth manifold M is a smoothsection

g ∈ Γ (TM∗ ⊗ TM∗)

such that for every p ∈ M , gp .

.= g (p) ∈ T ∗pM ⊗ T ∗pM is a non-degenerate bilinearform of constant index ν on the tangent space TpM .A semi-Riemannian manifold is a smooth manifold furnished with a semi-Riemann-

ian metric.

In general, the index ν varies between 0 and n = dimM , with ν = 0 being theRiemannian case.A semi-Riemannian manifold is equipped with a unique special connection which

takes into account the additional structure given by g .

De�nition 1.2.4. A connection ∇ on a semi-Riemannian manifold (M, g ) is called

i) metric if: X g(Y, Z) = g(∇XY, Z) + g(Y,∇XZ) for every X, Y, Z ∈ X(M)

ii) torsion free if: ∇XY −∇YX = [X, Y ] for every X, Y ∈ X(M).

Theorem 1.2.5. On a semi-Riemannian manifold (M, g ) there is a unique metrictorsion free connection, the Levi-Civita connection. It is characterised by the Koszulformula

2 g (∇XY, Z) = X g (Y, Z) + Y g (Z,X)− Z g (X, Y )

+ g ([X, Y ] , Z)− g ([Y, Z] , X) + g ([Z,X] , Y ) (1.2)

Similar to the de�nition of (semi-)Riemannian metrics, we can de�ne metrics onarbitrary tensor bundles by endowing the �bers with scalar products.

6


De�nition 1.2.6. Let π : E → M a vector bundle over a Riemannian manifold(M, g

). A Riemannian bundle metric h is a smooth section

h ∈ Γ (E∗ ⊗ E∗)

such that for every p ∈M , hp .

.= h (p) ∈ E∗p ⊗E∗p is a scalar product on the �ber Ep.It is called metric if

X (h (σ, τ)) = h (∇Xσ, τ) + h (σ,∇Xτ)

for X ∈ X(M), σ, τ ∈ Γ(E).

If we pull back a bundle metric, it inherits the property of being metric, as thefollowing proposition shows.

Proposition 1.2.7. Let M,N smooth manifolds and f : M → N a smooth map. LetE → N a vector bundle over N with connection ∇E and bundle metric h . Considerthe pullback metric f ∗h de�ned on the pullback bundle f ∗E by

f ∗h (f ◦ σ, f ◦ τ) = h (σ, τ) ∀σ, τ ∈ Γ (E) .

Then f ∗h is metric with respect to the pullback connection ∇f∗E if h is metric withrespect to ∇E.

This follows by easy calculation using the de�nition of pullback connection andpullback metric.

In the following, we will not talk about general connections. In fact any connection∇ will always represent the Levi-Civita connection or � if the connection operates onvectors bundles other than the tangent bundle � the unique connection induced bythe Levi-Civita connection on the associated bundle via de�nition 1.1.10, 1.1.11 and1.1.12.

De�nition 1.2.8. Let α : I → M a curve in a semi-Riemannian manifold (M, g ).A vector �eld along the curve α is a section of the pulled back bundle α∗TM → I.

For any vector �eld X ∈ X(M) the restriction X .

.= X ◦α is an example of a vector�eld along the curve α.

De�nition 1.2.9. Let α : I → M a curve in a semi-Riemannian manifold (M, g ).A vector �eld X along the curve is parallel if

∇αX (t) = 0 ∀ t ∈ I.

Remark 1.2.10. For a curve α : I →M and a tangent vector v ∈ Tα(t0)M there is aunique parallel vector �eld X along the curve such that X(α(t0)) = v. The function

P = P tt0

(α) : Tα(t0)M → Tα(t)M,

which maps v ∈ Tα(t0)M to X(α(t)), is called parallel transport along α. This map-ping is a linear isometry.

7


De�nition 1.2.11. A curve α : I → M in a semi-Riemannian manifold (M, g ) iscalled a geodesic provided it satis�es

∇αα (t) = 0 ∀ t ∈ I.

Proposition 1.2.12. Let (M, g ) a semi-Riemannian manifold, p ∈ M , v ∈ TpM .Then there exists ε > 0 and a unique geodesic

αv : (−ε, ε)→M

satisfying αv(0) = p, αv(0) = v. Additionally, αv depends smoothly on p and v.

We can show that geodesics are the curves which locally have minimal length, andevery curve with minimal length is in fact a geodesic. In euclidean space Rn, thegeodesics are the straight lines.When scaling the direction v of the geodesic αv : (−ε, ε)→M with λ > 0, we �nd

that αλv is de�ned on(− ελ, ελ

)and

αv (t) = αλv

(t

λ

)for t ∈ (−ε, ε) .

Using the smooth dependence of the geodesic on v ∈ TpM we can �nd an openneighbourhood V ⊂ TpM of 0 ∈ TpM such that αv is de�ned at time t = 1 forevery v ∈ V . On this neighbourhood we de�ne the exponential map used in the nextproposition.

Proposition 1.2.13. Let (M, g ) a semi-Riemannian manifold, p ∈ M . The map-ping expp : v 7→ αv(1) is called the exponential map of M at p and maps a neighbour-hood of 0 ∈ TpM di�eomorphically onto a neighbourhood of p ∈M .

For a detailed proof, see [Jos05, thm. 1.4.3].

In general, the second derivatives ∇X∇Y and ∇Y∇X for two vector �elds X, Y ∈X(M) do not coincide. We introduce the Riemannian curvature tensor to measuretheir di�erence. In order to obtain a tensor, the correction term ∇[X,Y ] is added.

De�nition 1.2.14. Let (M, g ) a semi-Riemannian manifold and ∇ the Levi-Civitaconnection. The Riemannian curvature tensor R : X(M)× X(M)× X(M)→ X(M)is de�ned as

R (X, Y )Z .

.= ∇X∇YZ −∇Y∇XZ −∇[X,Y ]Z.

We abbreviate

R (X, Y, Z,W ) .

.= g (R (X, Y )Z,W ) .

8


We can show that the Riemannian curvature tensor is indeed a tensor �eld of type(1, 3). Thus, the curvature at a point p ∈ M is a well-de�ned quantity: For tangentvectors x, y, z ∈ TpM we choose arbitrary extensions X, Y, Z ∈ X(M) and set

R p (x, y) z .

.= (R (X, Y )Z) (p) .

As the Riemannian curvature is a tensor, this is independent of the choice of extension.For this reason, whenever we use tangent vectors in the arguments of R , we haveomitted the choice of vector �elds which extend the tangent vectors.

Proposition 1.2.15. The Riemannian curvature tensor R (X, Y, Z,W ) satis�es thefollowing properties:

i) antisymmetry in the �rst two and the last two entries:

R (X, Y, Z,W ) = −R (Y,X,Z,W ) = R (Y,X,W,Z)

ii) symmetry between the �rst two and the last two entries:

R (X, Y, Z,W ) = R (Z,W,X, Y )

iii) Bianchi's �rst identity, a cyclic permutation property for R :

R (X, Y )Z + R (Y, Z)X + R (Z,X)Y = 0 (1.3)

iv) Bianchi's second identity, a cyclic permutation property for ∇R :

(∇ZR ) (X, Y )W + (∇XR ) (Y, Z)W + (∇Y R ) (Z,X)W = 0 (1.4)

As the geometric quantities we consider are independent of our choice of extension,we can choose one that makes calculations easier. We �rst remark that there isan orthonormal basis e1, . . . , en, n = dimM , of the tangent space TpM . It can beconstructed out of the coordinate vectors at the point p of a given chart using theGram-Schmidt process. We can extend this basis to a local frame �eld, i. e. in aneighbourhood U of p we can �nd vector �elds E1, . . . , En such that Ei(p) = ei,i = 1, . . . , n, and for every q ∈ U the tangent vectors E1(q), . . . , En(q) span thetangent space TqM . This is achieved by applying the Gram-Schmidt process tothe coordinate vector �elds on the coordinate neighbourhood. We can demand thefollowing additional properties:

Lemma 1.2.16. Let (M, g ) a semi-Riemannian manifold, p ∈ M and e1, . . . , en abasis of the tangent space TpM . There is a local extension E1, . . . , En of this basis toa frame �eld such that

∇vEi = 0 ∀ v ∈ TpM, i = 1, . . . n.

9


Proof. Choose 1 ≤ i ≤ n. For an arbitrary v ∈ TpM let αv be the geodesic through pwith αv(0) = v. We use parallel transportation to construct the vector �eld Ei alongthe geodesic αv. Obviously Ei satis�es ∇αvEi(t) = 0 at every point of the geodesicαv.We repeat the procedure for every direction w ∈ TpM . Because of proposition 1.2.13,the resulting vector �eld is de�ned on a neighbourhood of p and well-de�ned. Fur-thermore we can show that it is in fact smooth. As is satis�es ∇vEi(p) = 0 byconstruction, this concludes the proof.

De�nition 1.2.17. Let (M, g ) a semi-Riemannian manifold. The Ricci curvatureRic is the contraction of the Riemannian curvature tensor

Ric (X, Y ) (p) .

.= Ricp (Xp, Yp) .

.= tr (z → R p (z,Xp)Yp) (1.5)

for X, Y ∈ X(M), p ∈M .

The Ricci curvature is a tensor �eld of type (0, 2). Due to the properties of theRiemannian curvature, the Ricci curvature tensor is symmetric. Relative to a basis(ei)i=1,...,dimM of TpM , p ∈M , it is given by

Ricp (Xp, Yp) = glk gp (R p (ek, Xp)Yp, el) , (1.6)

where glk .

.= gp(el, ek) and(glk)

.

.= (glk)−1.

Here and in the sequel, we use the Einstein summation convention: An indexoccuring twice in a product, once raised and once lowered, is to be summed from 1up to the dimension of the manifold. More precisely

glk gp (R p (ek, Xp)Yp, el) =dimM∑k,l=1

glk gp (R p (ek, Xp)Yp, el) .

De�nition 1.2.18. Let (M, g ) a semi-Riemannian manifold. The scalar curvatureS is the contraction of the Ricci curvature tensor

S (p) .

.= Sp .

.= tr(Ricp

). (1.7)

Relative to a basis (ei)i=1,...,dimM of TpM , p ∈M , the scalar curvature is given by

Sp = glkRicp (ek, el) ,

where glk .


.

.= (glk)−1.

De�nition 1.2.19. Let (M, g ) a semi-Riemannian manifold. The sectional curvatureof the plane spanned by two linearly independant tangent vectors x, y ∈ TpM , p ∈M ,is

Kp (x, y) .

.=gp (R p (x, y) y, x)

gp (x, x) gp (y, y)− gp (x, y)2.

For every other two tangent vectors x′, y′ ∈ TpM spanning the same plane as x, ywe �nd

Kp (x, y) = Kp (x′, y′) .

10

1.3. Semi-Riemannian submanifolds


Having introduced the metric and the Riemannian curvature tensor, we can closelyexamine the geometric structure of a manifold. In this paragraph we consider a semi-Riemannian manifoldM and a submanifoldM that inherits a metric of its own. Eventhough for vector �elds X, Y ∈ X(M) the metrics coincide, i. e. g(X, Y ) = g(X, Y ),the Levi-Civita connections di�er, thus the curvatures do not coincide. In otherwords, the geometry of the submanifold looks di�erent from the inside than from thesurrounding manifold. In order to compare the two points of view, we introduce thesecond fundamental form II which describes how the submanifold M is curved in M .We distinguish geometric quantities on M from those on M by using overlines.

De�nition 1.3.1. A semi-Riemannian submanifold M ⊂ M is a submanifold M ofa semi-Riemannian manifold

(M, g

)such that the pullback g .

.= ι∗( g ) is a (semi-Riemannian) metric on M .

In that case, the tangent space of M at a point p ∈M decomposes into the directsum

TpM = TpM ⊕ TpM⊥,

with non-degenerate subspaces TpM,TpM⊥.

A vector is called tangent to M or horizontal provided it lies in TpM , whereas itis called normal if it lies in TpM⊥. Because of the direct sum, every vector v ∈ TpM ,p ∈M , has a unique expression

v = tan v + nor v

where we introduced the tangent and normal orthogonal projections onto the sub-spaces

tan : TpM → TpM and nor : TpM → TpM⊥.

Similar arguments can of course be applied to vector �elds: We denote by X(M)the restrictions of smooth vector �elds Z on M to the semi-Riemannian submani-fold M . It is called tangent or horizontal respectively normal to M if Zp ∈ TpMrespectively Zp ∈ TpM⊥, ∀ p ∈M . Applying tan and nor pointwise yields the E(M)-linear projections

tan : X (M)→ X (M) and nor : X (M)→ X (M)⊥

on the direct sum X(M) = X(M) ⊕ X(M)⊥. Note that X(M), the restriction ofsmooth vector �elds to the submanifold, is not to be confused with X(M), the spaceof smooth vector �elds on M .

The Levi-Civita connection ∇ is de�ned for vector �elds on M but gives rise to afunction∇ : X(M)×X(M)→ X(M), the induced connection. A priori, the expression

11


∇XV has no meaning for X ∈ X(M), V ∈ X(M), as X, V are no vector �elds on M .However we can always �nd smooth local extensions X,V ∈ X(M) and de�ne ∇XVas the restriction of ∇XV to M . This construction does not depend on the choice ofextension and leads to a well-de�ned smooth vector �eld.There is an alternative description of the space X(M): Consider the inclusion map

ι : M →M and the tangent bundle TM on M . We see that X(M) is identical to thepulled back bundle ι∗TM . The induced connection is in fact the pullback connection,which is metric due to proposition 1.2.7. This is carried out in the next lemma.

Lemma 1.3.2. The induced connection of the semi-Riemannian submanifoldM ⊂M

∇ : X (M)× X (M)→ X (M)

has the properties of a connection, more precisely for X, Y ∈ X(M), U, V ∈ X(M)

i) ∇XV is E(M)-linear in X

ii) ∇XV is R-linear in V

iii) ∇X(fV ) = Xf · V + f · ∇XV for f ∈ E(M)

iv) [X, Y ] = ∇XY −∇YX

v) X g(U, V ) = g(∇XU, V ) + g(U,∇XV ).

In the special case that both vectors are tangent to the submanifold M , we wantto examine the tangent and normal part of the induced connection. The tangent partturns out to be determined by the geometry of M only.

Lemma 1.3.3. The induced connection on the semi-Riemannian submanifold M ⊂M satis�es

tan∇XY = ∇XY

for X, Y ∈ X(M), where ∇ is the Levi-Civita connection of M .

Proof. Let X, Y, Z ∈ X(M) and choose smooth local extensions X,Y , Z ∈ X(M).The Koszul formula (1.2) yields

2 g(∇XY , Z

)= X g

(Y , Z

)+ Y g

(Z,X

)− Z g

(X,Y

)+ g

([X,Y

], Z)− g

([Y , Z

], X)

+ g([Z,X

], Y).

We use the following relations:

i) g(X,Y )|M = g(X, Y )

ii) X f |M = Xf , for f a smooth local extension of f ∈ E(M)

iii)[X,Y

]|M = [X, Y ]

12


and �nd the Koszul formula for the vector �elds on M :

2 g(∇XY, Z

)= X g (Y, Z) + Y g (Z,X)− Z g (X, Y )

+ g ([X, Y ] , Z)− g ([Y, Z] , X) + g ([Z,X] , Y )

= 2 g (∇XY, Z) .

As Z is tangent to M , we can replace ∇XY by its tangential part tan∇XY and �ndthe requested equation.

The normal part of the induced connection of two tangent vectors is a new quantity.

De�nition 1.3.4. Let M ⊂M a semi-Riemannian submanifold. The second funda-mental form (or shape tensor) II : X(M)× X(M)→ X(M)⊥ is de�ned as

II (X, Y ) = nor∇XY (1.8)

for X, Y horizontal vector �elds.

The second fundamental form is E(M)-bilinear, thus tensorial, and additionallysymmetric. For this reason it makes sense to speak of the second fundamental format a point p ∈M :

IIp (x, y) .

.= II (X, Y ) (p)

for x, y ∈ TpM and smooth local extensions X, Y ∈ X(M).We have seen that the tangential part of ∇XY coincides with the covariant deriva-

tive of M . This relation is expressed by the Gauÿ formula

∇XY = ∇XY + II (X, Y ) . (1.9)

Proposition 1.3.5. Let M a semi-Riemannian submanifold of M , R ,R the respec-tive Riemannian curvature tensors and II the second fundamental form. Then we�nd

R (X, Y, Z,W ) = R (X, Y, Z,W )− g (II (Y, Z) , II (X,W ))

+ g (II (X,Z) , II (Y,W )) (1.10)

for vector �elds X, Y, Z,W ∈ X(M). This result is called the Gauÿ equation.

The Gauÿ equation is a tensor equation and therefore remains valid if the vector�elds X, Y are replaced by individual tangent vectors.Proof. We locally extend all vector �elds to smooth vector �elds onM and calculate

g(∇X∇YZ,W

)= g

(∇X∇YZ,W

)+ g

(∇X II (Y, Z) ,W

)= g (∇X∇YZ,W ) +X g (II (Y, Z) ,W )− g

(II (Y, Z) ,∇XW

)= g (∇X∇YZ,W )− g

(II (Y, Z) ,∇XW

)= g (∇X∇YZ,W )− g (II (Y, Z) , II (X,W )) .

13


We used several times that by de�nition the second fundamental form is normal to ev-ery horizontal vector �eld. The same argument yields g(∇[X,Y ]Z,W ) = g(∇[X,Y ]Z,W ).Altogether, we �nd the desired formula:

g(R (X, Y )Z,W

)= g

(∇X∇YZ,W

)− g

(∇Y∇XZ,W

)− g

(∇[X,Y ]Z,W

)= g (R (X, Y )Z,W )− g (II (Y, Z) , II (X,W ))

+ g (II (X,Z) , II (Y,W )) .

As the Riemannian curvature is a tensor, the result does not depend on the choice ofextension.

De�nition 1.3.6. The Weingarten map of a semi-Riemannian submanifold M ⊂Mis the function

W : X (M)× X (M)⊥ → X (M)

(X, V ) 7→W VX = − tan∇XV

The Weingarten map is closely related to the second fundamental form and yieldsthe same information, as the following lemma shows. In particular, both are tensor�elds. The symmetry of the second fundamental form makes the Weingarten mapself-adjoint.

Lemma 1.3.7. The Weingarten map is the mapping dual to the second fundamentalform relative to the metric g in the following sense

g (W VX, Y ) = g (II (X, Y ) , V ) = g (X,W V Y ) (1.11)

for X, Y ∈ X(M), V ∈ X(M)⊥.

So far we discussed the covariant derivative of a tangent vector �eld with respect toanother tangent vector �eld. In order to understand the geometry of the submanifoldin more detail, we will now consider the change of a normal vector �eld when wemove along the submanifold.

De�nition 1.3.8. The normal connection of a semi-Riemannian submanifold M ⊂M is the function

∇⊥ : X (M)× X (M)⊥ → X (M)⊥

(X, V ) 7→ ∇⊥XV = nor∇XV

Lemma 1.3.9. The normal connection ∇⊥ on the semi-Riemannian submanifoldM ⊂M is a connection on TM⊥, i. e. it satis�es

i) (X, V ) 7→ ∇⊥XV is E(M)-linear in X

ii) (X, V ) 7→ ∇⊥XV is R-linear is V

14


iii) ∇⊥X(fV ) = Xf · V + f · ∇⊥XV for X ∈ X(M), V ∈ Γ(TM⊥), f ∈ E(M).

Furthermore the connection is metric, i. e.

iv) X g(U, V ) = g(∇⊥XU, V ) + g(U,∇⊥XV ) for X ∈ X(M), U, V ∈ X(M)⊥.

The Gauÿ equation provides us with a characterisation of tan R (X, Y )Z for threetangent vectors X, Y, Z. We recall that the second fundamental form is a tensor �eld.Using the normal connection ∇⊥ and formula (1.1), which gives us the covariantdi�erential of a tensor �eld, we �nd

(∇X II ) (Y, Z) = ∇⊥X (II (Y, Z))− II (∇XY, Z)− II (Y,∇XZ) (1.12)

for X, Y, Z ∈ X(M). With this derivative of the second fundamental form we candescribe the normal part nor R (X, Y )Z.

Proposition 1.3.10. Let M a semi-Riemannian submanifold of M , R ,R the respec-tive Riemannian curvature tensors and II the second fundamental form. The normalpart of R (X, Y )Z is given by the Codazzi equation

nor R (X, Y )Z = (∇X II ) (Y, Z)− (∇Y II ) (X,Z) , (1.13)

where X, Y, Z ∈ X(M).

Proof. We locally extend all vector �elds to smooth vector �elds onM . By de�nitionof the second fundamental form we �nd

R (X, Y )Z = ∇X∇YZ −∇Y∇XZ −∇[X,Y ]Z

= ∇X∇YZ + II (X,∇YZ) +∇X (II (Y, Z))−∇Y∇XZ

− II (Y,∇XZ)−∇Y (II (X,Z))−∇[X,Y ]Z − II ([X, Y ] , Z)

The second fundamental form is bilinear, therefore

II ([X, Y ] , Z) = II (∇XY, Z)− II (∇YX,Z)

as ∇ is torsion free. All terms in the formula for R (X, Y )Z which do not containthe second fundamental form are tangent to M . For the remaining ones we use thederivative of the second fundamental form we de�ned in (1.12) and �nd the desiredformula

nor R (X, Y )Z = nor∇X (II (Y, Z))− II (Y,∇XZ)− II (∇XY, Z)

− nor∇Y (II (X,Z)) + II (X,∇YZ) + II (∇YX,Z)

= ∇⊥X (II (Y, Z))− II (Y,∇XZ)− II (∇XY, Z)

−∇⊥Y (II (X,Z)) + II (X,∇YZ) + II (∇YX,Z)

= (∇X II ) (Y, Z)− (∇Y II ) (X,Z)

15


De�nition 1.3.11. Let M a semi-Riemannian submanifold of M and II the secondfundamental form. The contraction of II is a normal vector �eld and is called themean curvature vector �eld ~H of M ⊂M .

Relative to a basis (ei)i=1,...,dimM of TpM the mean curvature is given by

~H (p) = gkl IIp (ek, el) , (1.14)

where glk .


.

.= (glk)−1.

There exists a second slightly di�erent de�nition of the mean curvature. By someauthors, the contraction of the second fundamental form is de�ned as the meancurvature divided by the dimension of the submanifold. By this means, the meancurvature of the unit n-sphere Sn in the euclidean �at space Rn+1 has constant length1, independent of the dimension n.In the General Theory of Relativity we always consider four-dimensional mani-

folds, therefore we choose the shorter thus easier de�nition and keep in mind thatsome results found in the literature have to be adjusted.

16

2. Lie groups and Lie algebras

The aim of this chapter is to brie�y introduce Lie groups and Lie algebras and stateseveral results we will use in the following. All proofs and a more profound introduc-tion can be found in [War71, chap. 3].

De�nition 2.0.12. A Lie group is a group G which is also a manifold and for whichthe inverse map

G→ G, g 7→ g−1

and the multiplication map

G×G→ G, (g, h) 7→ gh

are smooth.

Let g ∈ G. Consider the left translation by g

lg : h 7→ gh

and the right translation by g

rg : h 7→ hg.

As lg−1 = l−1g and rg−1 = r−1g for every g ∈ G and all translation are smooth byde�nition, they are di�eomorphisms of G. We call a vector �eld X ∈ X(G) leftinvariant if

Dlg ◦X = X ◦ lg ∀ g ∈ G,

and denote by g the set of all left invariant vector �elds on the Lie group G. Analo-gously we call a tensor �eld A of type (0, s) left invariant if

l∗gA = A ∀ g ∈ G.

Right invariance is de�ned the same way.For a given x ∈ TeG, e the identity element of the Lie group G, we can de�ne a

vector �eld by setting

X (g) .

.= Dlg (x)

17


for every g ∈ G. Direct calculation proves left invariance. It can be shown that thisis in fact a smooth vector �eld. This method can be extended to tensor �elds of type(0, s). For a tensor A on the tangent space TeG, one obtains a smooth tensor �eld bysetting

Ag (y1, . . . , ys) .

.= A (Dlg−1y1, . . . , Dlg−1ys)

for every g ∈ G, y1, . . . , ys ∈ TgG. In particular we see that a left invariant tensor�eld is de�ned by giving its values on the tangent space TeG.

De�nition 2.0.13. A Lie algebra is a real vector space a endowed with a bilinearoperation [·, ·] : a× a→ a called Lie bracket which satis�es for every x, y, z ∈ a

• [x, y] = − [y, x] (anti-commutativity)

• [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 (Jacobi identity)

In 1.1.5 we de�ned the bracket operation on smooth vector �elds of a manifold M :

[·, ·] : X (M)× X (M)→ X (M)

(X, Y ) 7→ X ◦ Y − Y ◦X.

As the bracket operation is skew-symmetric and satis�es the Jacobi identity, it makesthe space of smooth vector �elds X(M) a Lie algebra. If dimM ≥ 1, it can be shownthat this Lie algebra is in�nite-dimensional using the fact that E(M) contains allpolynomes de�ned on an open set of M .

Proposition 2.0.14. Let G a Lie group with identity element e, and g the set of leftinvariant vector �elds. Then:

i) g is a real vector space, and the map g→ TeG, X 7→ X(e) is an isomorphism.In particular dimG = dimTeG = dim g.

ii) g endowed with the bracket operation is a Lie algebra.

Because of this proposition, the Lie algebra g of left invariant vector �elds of a Liegroup G is the canonical Lie algebra associated with the Lie group and is called theLie algebra of the Lie group G.

Remark 2.0.15. In the subsequent chapters, we will compute several geometricquantities on Lie groups. On general manifolds equipped with a connection ∇, thecovariant derivative is only de�ned for vector �elds, not for individual tangent vectors.In the setting of Lie groups, we use the identi�cation of any tangent space with TeGand thus with the Lie algebra of left invariant vector �elds and denote by ∇xy thecovariant derivative ∇XY (g), where X, Y ∈ X(M) are left invariant extensions ofx, y ∈ TgG, g ∈ G.

18

Proposition 2.0.16. Let (e1, . . . , en) a basis of the Lie algebra a. The structureconstants Ck

ij de�ned by [ee, ej] = Ckijek satisfy

Ckij + Ck

ji = 0,

CrijC

lrk + Cr

jkClri + Cr

kiClrj = 0.

De�nition 2.0.17. Let G,H Lie groups. A Lie group homomorphism φ : G→ H isa group homomorphism which is smooth. It is an isomorphism if additionally φ is adi�eomorphism.

De�nition 2.0.18. Let a, b Lie algebras. A Lie algebra homomorphism ψ : a→ b isa linear map which preserves the Lie bracket: ψ [x, y] = [ψ(x), ψ(y)] for every x, y ∈ a.It is an isomorphism if additionally ψ is bijective.

It follows immediately that ψ−1 is a Lie algebra homomorphism as well.

Proposition 2.0.19. Let G,H Lie groups, g, h their respective Lie algebras and φ :G→ H a Lie group homomorphism. Then the map dφ : g→ h, such that the diagram

gdφ //

∼=��

h

∼=��

TeGDφe // TeH

commmutes, is a Lie algebra homomorphism.

Thus isomorphic Lie groups have isomorphic Lie algebras. In the case of simplyconnected Lie groups, the following theorem reverses this assertion: If two simplyconnected Lie groups have isomorphic Lie algebras, then they are isomorphic them-selves.

Proposition 2.0.20. Let G,H Lie groups, g, h their respective Lie algebras and ψ :g → h a Lie algebra homomorphism. Let G simply connected. Then there exists aunique Lie group homomorphism ψ : G→ H such that Dφ = ψ.

For a proof of these two propositions, see [War71, 3.14 and 3.27] and the referencesspeci�ed therein.

De�nition 2.0.21. A Lie subgroup of a Lie group G is a subgroup H ⊂ G of G suchthat

i) H is a Lie group,

ii) H is a submanifold of G,

iii) The inclusion ι : H → G is a group homomorphism.

De�nition 2.0.22. A Lie subalgebra of a Lie algebra g is a subspace h ⊂ g which isclosed under the Lie bracket operation.

19


The Lie algebra of left invariant vector �elds of a Lie group G is an example of aLie subalgebra of the space of smooth vector �elds X(M).

Proposition 2.0.23. There is a one-to-one correspondence between connected Liesubgroups of a Lie group and subalgebras of its Lie algebra. It is given by {ι : H →G} 7→ dι(h) ⊂ g.

We de�ne the commutator [·, ·] on n-dimensional matrices by [A,B] .

.= AB−BA forevery A,B ∈ Mat(n,R). Then the set gl(n,R) .

.= (Mat(n,R), [·, ·]) is a Lie algebra,the one associated with the general linear group GL(nR).Every �nite-dimensional Lie algebra is isomorphic to an appropriate subalgebra of

gl(n,R) for some n ∈ N. This result is known as Ado's theorem, see [Jac62, chap.VI], and yields the existence of a Lie subgroup of GL(n,R) such that the given Liealgebra is the Lie algebra associated with this Lie group. The universal covering ofthis Lie group is simply connected and has the same Lie algebra. Combining the lastresults we �nd uniqueness and see:

Theorem 2.0.24. There is a one-to-one correspondence between isomorphism classesof �nite-dimensional Lie algebras and isomorphism classes of simply connected Liegroups. It is given by G 7→ g.

We remark that there are other Lie groups with the same Lie algebra, but whichare not simply connected. They are the quotient groups of the simply connected Liegroup factorised by discrete subgroups of its center.Using this correspondence, we can describe the isomorphism classes of simply con-

nected Lie groups respectively of the universal coverings of general Lie groups by in-vestigating their Lie algebras. The vector space structure and the structure constantsof Lie algebras o�er signi�cant simpli�cations. We will make use of this approach inchapter 4 when classifying special three-dimensional Lie groups by classifying theirLie algebras.

20

3. Introduction to the General

Theory of Relativity

In 1905, the "annus mirabilis", Albert Einstein's Special Theory of Relativity com-pletely changed the understanding of time, space and relative motion. The speed oflight, previously measured with ever-increasing precision without �nding any aber-ration, was postulated to be a natural constant. On the other hand, observers indi�erent inertial systems were postulated to observe the same physical laws. Thetheory, however, as revolutionary as it was, failed when it came to gravitation.One decade later, this defect was corrected with the General Theory of Relativity,

which comprises the Special Theory of Relativity as a special case. Every motion inspacetime is governed by the Einstein equations, a tensor equation for the geometryof a semi-Riemannian manifold. Gravitation then becomes a pseudo force, similar tothe centrifugal or Coriolis force. The Einstein equation determines the curvature ofthe universe, and the only motion is free fall.In this chapter we give an introduction to the mathematical foundations of the

Theory of Relativity based on [O'N83] and [Wal84].

3.1. Lorentz manifolds and Minkowski spacetime

In chapter 1.2 we introduced semi-Riemannian manifolds (M, g ), and in particularde�ned the index of a semi-Riemannian metric g . In the Theory of Relativity weconsider manifolds with index 1, i. e. with one-dimensional maximal subspace onwhich the scalar product gp de�ned by the metric g is negative de�nite.

De�nition 3.1.1. A Lorentz vector space is a four-dimensional vector space V withsymmetric non-degenerate bilinear form 〈·, ·〉 : V × V → R of index 1.

Every vector v ∈ V in a Lorentz manifold is one of the following:

• If 〈v, v〉 > 0, we call v spacelike.

• If 〈v, v〉 = 0, we call v lightlike.

• If 〈v, v〉 < 0, we call v timelike.

The easiest example of a Lorentz vector space is the four-dimensional vector spaceR4 with symmetric bilinear form

〈ξαeα, ζαeα〉 = −ξ0ζ0 +3∑i=1

ξiζ i,

21

3. Introduction to the General Theory of Relativity

where (e0, e1, e2, e3) is the canonical basis of R4. It is often denoted by R1,3. Weremark that the set of timelike vectors consists of two disjoint and open conical setscalled timecones.

De�nition 3.1.2. A Lorentz manifold is a four-dimensional semi-Riemannian mani-fold (M, g ) such that every tangent space TpM , p ∈ M , is a Lorentz vector space,i. e. such that the metric g has index 1.

The notion of space-, light- and timelike generalises to vectors and vector �elds ona Lorentz manifold by applying it pointwise to every tangent space. Furthermore,the metric de�nes timecones in every tangent space.If a Lorentz manifold possesses a timelike vector �eld, we call the manifold time-

orientable. We can show that for two timelike vector �elds X,Y ∈ X(M), the vectorsXp, Y p lie in the same timecone for every p ∈M if and only if

g(X,Y

)< 0.

This de�nes an equivalence relation on the space of timelike vector �elds onM . For atimelike vector �eld X of a time-orientable Lorentz manifold we call the equivalenceclass

[X]a time-orientation on M .

De�nition 3.1.3. A spacetime is a connected Lorentz manifold (M, g ) with a time-orientation.

A time-orientation[X]allows to de�ne the notion of past and future: The vector

�eld X and every timelike vector �eld in the same equivalence class are consideredfuture-pointing, whereas all vector �elds in the equivalence class of −X point into thepast. At a point p ∈ M , the timelike vector Xp sets apart the future timecone fromthe past one. Every tangent vector perpendicular to a timelike vector is spacelike,as otherwise the bilinear form g

pwould be negative de�nite on a two-dimensional

subspace, a contraction to the metric having index 1.Consider the four-dimensional manifold R4. Every tangent space is again R4. We

equip the manifold with the metric η de�ned by

η(ξαEα, ζ

αEα

)= −ξ0ζ0 +

3∑i=1

ξiζ i,

where (E0, E1, E2, E4) is the canonical coordinate frame �eld of R4 and ξα, ζα ∈E(M)for α = 0, 1, 2, 3. Then

(M, η

)is a Lorentz manifold. Equipped with the time-

orientation[E0

], it is the standard example of a spacetime, the Minkowski spacetime.

One can show that the Minkowski spacetime is the unique spacelike complete (i. e.spacelike inextendible geodesics are de�ned on R), simply connected Lorentz manifoldwith vanishing Riemannian curvature tensor R = 0.The Minkowski spacetime is the object investigated in the Special Theory of Rela-

tivity. The two postulates of this theory, the Principle of Relativity and the Principle

22

3.2. The Einstein equation

of Invariant Light Speed can be visualised on this model. The latter corresponds tothe fact that all lightlike vectors v ∈ TpM satisfy g

p(v, v) = 0. The postulates of the

Special Theory of Relativity imply two astounding consequences: time dilation andlength contraction, see for example [O'N83, chap. 6].

3.2. The Einstein equation

The Special Theory of Relativity is a consistent theory, and satisfactory as long as it isapplied to problems in its range. However it does not encompass gravity. In the Gene-ral Theory of Relativity, this problem is overcome by introducing curved spacetimes.Gravity becomes a pseudo force, there is only free fall in a curved universe.

De�nition 3.2.1. Let (M, g ) a Lorentz manifold. We de�ne the Einstein tensor

Ein .

.= Ric − 1

2S g .

Einstein's General Theory of Relativity now postulates that the geometry of four-dimensional spacetime is determined by the Einstein equation

Ein + Λ g = 8πT ,

where T is a tensor of type (0, 2) called the stress-energy tensor. It is symmetric asboth g and Ein have this property. The constant Λ ∈ R is called the cosmologicalconstant. Roughly speaking, the Einstein equation expresses the following: "Mattertells spacetime how to curve; spacetime tells matter how to move" (John A. Wheeler).This quotation makes reference to the entanglement of the stress-energy tensor andthe curvature, they both in�uence each other simultaneaoulsy.

Proposition 3.2.2. The Einstein tensor is divergence free:

div Ein = 0.

Proof. It follows from contracting Bianchi's second identity (1.4) that

div Ric =1

2dS .

On the other hand we �nd for every p ∈M , (e0, e1, e2, e3) a basis of TpM and x ∈ TpMdiv(S g)p

(x) = gαβ∇eα

(S g)

(eβ, x)

= gαβ(

g · ∇eαS + S · ∇eα g)

(eβ, x) ,

where gαβ .

.= gp

(eα, eβ) and(gαβ)

.

.= (gαβ)−1. As the Levi-Civita connection is

metric and ∇eαS = gp

(dS p, eα

), this equals

div(S g)p

(x) = gαβ gp

(dS p, eα

)gp

(eβ, x)

= gp

(dS p, gαβ g

p(eβ, x) eα

)= g

p

(dS p, x

),

23


and therefore

div(S g)

= dS .

In classical mechanics one of the main principles is the conservation of energy, massand momentum. The fact that the Einstein tensor is divergence free is the equivalentof this principle in the General Theory of Relativity.

De�nition 3.2.3. A four-dimensional Lorentz manifold satisfying the Einstein equa-tion with T = 0 is called a vacuum.

Proposition 3.2.4. Let (M, g ) a Lorentz manifold. Then it is a vacuum with cos-mological constant Λ = 0 if and only if the four-dimensional Ricci curvature vanishes

Ric = 0.

Proof. If Ric = 0, then S = trRic = 0 and the assertion follows. Thus supposeEin = 0. Taking the trace yields

0 = trEin

= tr

(Ric − 1

2S g)

= S − 2S= −S .

As the scalar curvature vanishes, we �nd 0 = Ein = Ric , which concludes the proof.

3.3. Perfect �uids

At a large scale, the galaxies in the universe can be seen as small particles in a �uid.In this setting, the behaviour of the universe is dominated by the mass of the galaxiesand the pressure (commonly assumed to be much smaller). We formalise this modeland deduce evolution equations for those quantities.

De�nition 3.3.1. Let(M, g

)a spacetime. A perfect �uid on M is a triple (U, ρ, p)

such that

i) U a timelike future-pointing vector �eld on M with g(U,U) = −1, the �owvector �eld;

ii) ρ ∈ E(M) the energy density function, p ∈ E(M) the pressure function.

24

3.3. Perfect �uids

iii) The manifold satis�es the Einstein equation Ein + Λ g = 8πT with stress-energy tensor

T =(ρ+ p

)U[ ⊗ U [

+ p g ,

where U[denotes the 1-form dual to U , i. e. U

[(Z) = g(U,Z) for every Z ∈

X(M).

This formula is equivalent to

T(U,U

)= ρ, T

(X,U

)= 0 = T

(U,X

), T

(X,Y

)= p g

(X,Y

)for vector �elds X,Y ∈ X(M) satisfying X,Y ⊥ U .

Remark 3.3.2. i) The vacuum models with cosmological constant Λ are the spe-cial cases of perfect �uids (U, ρ, p) with ρ = −p = Λ.

ii) A perfect �uid with cosmological constant can easily be transformed into amodel satisfying

Ein = 8πT = 8π((ρ+ p

)U[ ⊗ U [

+ p g)

by adjusting ρ = ρ+ 18π

Λ and p = p − 18π

Λ. Thus without loss of generality weassume perfect �uids which do not satisfy ρ = −p = const. to have vanishingcosmological constant.

iii) A perfect �uid with p = 0 and ρ > 0 (and Λ = 0, see ii)) is called a dust.

Recall that for a vector �eld Z ∈ X(M) the divergence divZ is given by

divZ (p) = gαβ gp

(∇eαZ, eβ

),

where (e0, e1, e2, e3) is a basis of TpM , gαβ .

.= gp(eα, eβ) and

(gαβ)

.

.= (gαβ)−1. For a

smooth function f ∈ E(M), the gradient is de�ned as the vector �eld grad f satisfying

∇Zf = g(grad f, Z

)for every Z ∈ X(M), or equivalently (see Remark 5.1.1)

grad f (p) = gαβ(∇eαf

)eβ.

Proposition 3.3.3. A perfect �uid (U, ρ, p) on M satis�es

i) Uρ = −(ρ+ p) divU , the energy equation,

ii) (ρ+ p)∇UU = − grad⊥ p, the force equation,

25


where the spatial pressure gradient grad⊥ p denotes the component of grad p orthogonal

to U .

Proof. For p ∈ M we choose a basis (e0 = U(p), e1, e2, e3) of TpM . Using lemma1.2.16 and constant extension in normal direction, we locally extend the spatial vec-tors to vector �elds E1, E2, E3 ∈ X(M) with

Ei ⊥ U

∇vEi = 0

for every v ∈ TpM , i = 1, 2, 3. Additionally, the vector �eld E0 satis�es ∇e0E0, as

we rigorously show in 5.4. For the dual 1-form U[and α, γ = 0, 1, 2, 3 we �nd that

at the point p ∈M

∇eα

(U[)

(eγ) = eα

(U[ (Eγ

))− U [ (∇eαEγ

)= eα g

(U,Eγ

)= gp

(∇eαU, eγ

)+ gp

(e0,∇eαEγ

)= gp

(∇eαU, eγ

).

We can therefore calculate the divergence

div T (eγ) = gαβ∇eαT (eβ, eγ)

= gαβ∇eα

((ρ+ p

)U[ ⊗ U [

+ p g)

(eβ, eγ)

= gαβ(∇eα

(ρ+ p

)U[(eβ)U

[(eγ) +

(ρ+ p

)gp(∇eαU, eβ

)U[(eγ)

+(ρ+ p

)U[(eβ) gp

(∇eαU, eγ

)+(∇eαp

)gp (eβ, eγ) + p

(∇eα g

)(eβ, eγ)

)where we ommited the evaluation of p and ρ at the point p. With U

[(eβ) = −δ0β and

as the connection ∇ is metric, we �nd

div T (eγ) = e0(ρ+ p

)· U [

(eγ) +(ρ+ p

)divU · U [

(eγ) +(ρ+ p

)g(∇e0U, eγ

)+ gp

(grad p, eγ

)= gp

(e0(ρ+ p

)e0 +

(ρ+ p

)divU · e0 +

(ρ+ p

)∇e0U + grad p, eγ

).

This computation is valid for every p ∈ M , thus the equation remains true if wereplace e0 by U . We decompose grad p into its components tangential and orthogonalto U :

grad p = − g(grad p, U

)U + grad⊥ p

= −Up · U + grad⊥ p.

26

3.4. Robertson-Walker spacetimes

Because of linearity, div T = 0 implies

0 = U(ρ+ p

)U +

(ρ+ p

)divU · U +

(ρ+ p

)∇UU + grad p

= Uρ · U +(ρ+ p

)divU · U +

(ρ+ p

)∇UU + grad⊥ p

As U is a unit vector �eld, i. e. g(U,U) = −1, we �nd 0 = g(∇UU,U) which means∇UU ⊥ U . This gives the two equations.


In this paragraph we brie�y introduce the Friedman-Lemaître universes, the standardmodels of cosmology. In these models, the universe is assumed to be isotropic, whichmeans that all spatial directions "look the same".This is motivated by the astro-nomical observation that the distribution of galaxies shows no large asymmetries.We start by giving the Robertson-Walker metrics and then deduce several evolutionequations for perfect �uids on these spacetimes.Consider a connected three-dimensional manifold M and an open interval I ∈ R.

We want to endow the product manifold I ×M with a semi-Riemannian metric gsuch that (I×M, g ) is a spacetime satisfying the Einstein equation of a perfect �uid.Tangent vectors horizontal to M are supposed to be spacelike, while we want thosehorizontal to I to be timelike.Let d

dtthe standard vector �eld on I ⊂ R, and ∂

∂tits lift to I ×M , i. e. the vector

�eld such that D pr1 ( ∂∂t

) = ddt, where pr1 : I ×M → I is the projection. For every

p ∈M we parametrise the product manifold by αp : t 7→ (t, p). As

αp (t) =∂

∂t

∣∣∣∣(p,t)

these are the integral curves of ∂∂t. They are interpreted as the movement in spacetime

of a galaxy that starts at p ∈M at a certain time t ∈ I. The function t is interpretedas the proper time, therefore we require

g(∂

∂t,∂

∂t

)= −1.

When holding t ∈ I constant we �nd the hypersurfaces

M (t) .

.= {t} ×M.

These are supposed to be normal to ∂∂t. When equipping all hypersurfaces with a Rie-

mannian metric tg , this construction yields the Lorentz manifold (I ×M, g(t,p)

.

.=

−dt2 + tgp .

27


The isotropy condition is formalised in the following way:

De�nition 3.4.1. Let I ×M as above. It is spatially isotropic if for every (t0, p0) ∈I ×M and for every horizontal tangent vectors x, y ∈ Tp0M(t0) with g(t0,p0)(x, x) =g(t0,p0)(y, y) = 1 there is a local isometry Φ = id×ΦM(t0) with DΦ(x) = y.

Spatial isotropy implies in particular that the sectional curvature does not dependon the choice of two-dimensional plane in the three-dimensional tangent space of M .It then follows from Schur's lemma that the sectional curvature is constant on everyslice M(t). Altogether one can show that up to isomorphy the only possible spatiallyisotropic spacetimes are the following:

De�nition 3.4.2. Let (M, g ) be a connected three-dimensional Riemannian mani-fold of constant sectional curvature K ∈ {−1, 0, 1}, I ⊂ R an open interval andf : I → (0,∞) a smooth function. We denote by

M (K , f) .

.= I ×f M

the manifold I ×M with metric

g = −dt2 + f 2 (t) g .

Equipped with time-orientation [ ∂∂t

], this is called a Robertson-Walker spacetime.

Requiring M to be complete and simply connected leads to the three standardchoices for M :

i) the euclidean space R3, with K = 0,

ii) the hyperbolic space H3, with K = −1,

iii) the sphere S3, with K = +1.

The manifold M(K , f) is a special case of a warped product, a semi-Riemannianproduct manifold where the product metric is constructed using a warping functionf that scales one of the two individual metrics with respect to the other. We canexpress the connection and the curvatures on such product manifold in terms of thiswarping function, see [O'N83, chap. 7]. We give the Ricci and scalar curvature in ourcase.

Corollary 3.4.3. Let M(K , f) a Robertson-Walker spacetime, (t, p) ∈M(K , f) andx, y ∈ TpM(t) Then we have the following formulae for the Ricci curvature:

Ric(t,p)(∂

∂t,∂

∂t

)= −3

f

f(t)

Ric(t,p)(∂

∂t, x

)= 0

Ric(t,p) (x, y) =

2

(f

f(t)

)2

+ 2K

f 2 (t)+f

f(t)

g(t,p)

(x, y) ,

28


where f , f denote the �rst and second derivative of f with respect to time. The scalarcurvature thus is

S (t,p) = 6

( ff

(t)

)2

+K

f 2 (t)+f

f(t)

,

and the Einstein tensor satis�es

Ein (t,p)

(∂

∂t,∂

∂t

)= 3

( ff

(t)

)2

+K

f 2 (t)

Ein (t,p)

(∂

∂t, x

)= 0

Ein (t,p) (x, y) =

−2f

f(t)−

(f

f(t)

)2

− K

f 2 (t)

g(t,p)

(x, y) .

Easy calculation now shows that the Robertson-Walker spacetime satis�es the Ein-stein equation for a perfect �uid ( ∂

∂t, ρ, p), where the energy density and pressure are

given by

8π

3ρ =

(f

f

)2

+K

f 2, −8πp = 2

f

f+

(f

f

)2

+K

f 2.

Combining these two equations yields the following relation:

Corollary 3.4.4. A Robertson-Walker manifold M(K , f), which is a perfect �uid( ∂∂t, ρ, p), satis�es

3f

f= −4π

(ρ+ 3p

). (3.1)

Additionally, the energy density ρ and pressure p have to satisfy the energy andforce equation, see proposition 3.3.3. The �rst one is trivially satis�ed, the latteryields another restriction on f . In paragraph 7.1 we will execute the calculations ina more general context.

Corollary 3.4.5. A Robertson-Walker manifold M(K , f), which is a perfect �uid( ∂∂t, ρ, p) satis�es

ρ = −3(ρ+ p

) ff. (3.2)

29


3.5. Evolution of Robertson-Walker perfect �uids

In order to further analyse perfect �uids on Robertson-Walker manifolds, we have alook at astronomical observational data:

i) In our universe distant galaxies move away from us. This causes the redshift oftheir electromagnetic radiation and is measured by the Hubble constant H0. Inthe Robertson-Walker model we �nd that the constant is expressed by

H0 =f (t0)

f (t0),

where t0 is present time. Using di�erent methods, the Hubble constant hasbeen measured to be about 70 km

s ·Mpc, where pc denotes the astronomical unit of

length parsec. In particular H0 > 0.

ii) Energy density dominates pressure in all known forms of matter:

ρ > |p|.

In this setting we can prove the following evolution of Robertson-Walker models:

Proposition 3.5.1. Let M(K , f) a Robertson-Walker spacetime, where we choosethe interval I = (t−, t+) maximal. Then we �nd:

• If H(t0) > 0 for some t− < t0 < t+, then the starting time t− satis�es

t0 −H (t0)−1 < t− < t0.

• Either (1) f > 0 at all times, or (2) f is maximal at some t1 > t0, then t+ <∞and f is monotonically decreasing on (t1, t+).

Thus, if the Robertson-Walker model is accurate, our universe is about 14 billionyears old. If it does not keep on expanding, it will come to an end in �nite time.In the case of a perfect �uid ( ∂

∂t, ρ, p), we further characterise possible initial and

end singularities in terms of the energy density ρ.

De�nition 3.5.2. Let M(K , f) a Robertson-Walker spacetime with maximal inter-val I = (t−, t+), which is a perfect �uid ( ∂

∂t, ρ, p). We call t− resp. t+ a physical

singularity if t− > −∞ and ρ→∞ for t→ t− resp t→ t+.

De�nition 3.5.3. Let M(K , f) a Robertson-Walker spacetime with maximal inter-val I = (t−, t+), which is a perfect �uid ( ∂

∂t, ρ, p). An initial singularity t− is a big

bang if f → 0 and f →∞ for t→ t−. Similarly, a �nal singularity t+ is a big crunchif f → 0 and f → −∞ for t→ t+.

30

3.6. Beyond Robertson-Walker models

Obviously such singularities are physical ones. Under mild assumptions, their ex-istence in Robertson-Walker perfect �uids is ensured by the following theorem. Theproof uses (3.1), (3.2) and basic curve sketching arguments, see [O'N83, thm. 12.17].

Theorem 3.5.4. Let M(K , f) a Robertson-Walker spacetime with maximal intervalI = (t−, t+), which is a perfect �uid ( ∂

∂t, ρ, p). Assume thatM(K , f) has only physical

singularities. If H(t0) > 0 at some time t0, if ρ > 0 and if −13< a ≤ p

ρ≤ A for some

constants a,A ∈ R, we �nd:

i) The initial singularity is a big bang.

ii) If K = −1 or 0, then t+ =∞, and f →∞, ρ→ 0 for t→ t+.

iii) If K = +1, then t+ <∞, and the end singularity is a big crunch.

Let us consider an even more special case: One can show that a Robertson-Walkerspacetime M(K , f) is a dust, i. e. a perfect �uid ( ∂

∂t, ρ, p) with ρ > 0 and p = 0, if

and only if f is not constant and satis�es the Friedmann equation

f(f 2 +K

)= c (3.3)

with positive constant c > 0. Solving this equation leads to explicite cosmologicalmodels for each of the three cases.

Proposition 3.5.5. Let M(K , f) a Robertson-Walker spacetime with maximal in-terval I = (t−, t+) which is a dust. Then the scale function f : I → R is given by thesolutions of the Friedmann equation (3.3). After an appropriate time translation, theinitial singularity occurs at t− = 0 and the three solutions are:

i) If K = 0, then f(t) = 3

√94ct2. Thus the universe expands forever and we �nd

f →∞, f → 0 as t→∞.

ii) If K = 1, then f(τ) = 12c(1 − cos τ) where t = 1

2c(τ − sin τ). The graph is a

cycloid, with maximal expansion at t = πc2and a collapse at t = πc.

iii) If K = −1, then f(τ) = 12c( cosh τ − 1) where t = 1

2c( sinh τ − τ). In particular

f > 1, thus the universe expands forever and we �nd f →∞, f → 1 as t→∞.


In Robertson-Walker models the geometry was de�ned by the metric on the three-dimensional manifold M and the scaling function f : I → R, thus one parameter.The underlying three-dimensional manifold could be one of only three choices if werestricted M to be complete and simply connected. This resulted immediately fromthe universe being isotropic.

31


The 3 K background radiation, discovered in the 1960s, shows anisotropy and thussuggests that realistic cosmological models should not be isotropic. In the followingchapters we want to develop more general models. Instead of spatially isotropicmodels, we consider spatially homogeneous ones which are not necessarily spatiallyisotropic. Spatial homogeneity means that no point in spacetime is privileged.

De�nition 3.6.1. LetM a semi-Riemannian manifold. It is homogeneous if for everytwo points p, q ∈M there is an isometry φ : M →M with φ(p) = q.

De�nition 3.6.2. Consider a Lorentz manifold (I × M) with ∂∂t

as above. It isspatially homogeneous if the manifold M is homogeneous.

We can show that in the case of a simply connected manifold M , spatial isotropyat every point is in fact the stronger condition, as it implies spatial homogeneity (see[Wal84, chap. 5]).

Remark 3.6.3. Let M a Riemannian manifold. According to the Myers-Steenrodtheorem, the groupG of isometries onM is a Lie group. Then the spatial homogeneitycondition in de�nition 3.6.1 is equivalent to G acting transitively, i. e. for every p, q ∈M there is an element g : M →M of G such that g(p) = q.

Restricting the action of G to be simply transitive, i. e. requiring the element g ∈ Gto be unique, leads to dimG = 3. As a result, the action G ×M → M correspondsto left multiplication. It can be shown that in fact the manifold M itself can be seenas the Lie group.

Proposition 3.6.4. Up to isomorphy all spatially homogeneous spacetimes with sim-ply transitive action are four-dimensional semi-Riemannian manifolds G .

.= I × G,where I ⊂ R is an open interval, G is a three-dimensional Lie group with a family( tg )t∈I of left invariant metrics, and G is equipped with the metric

g(t,p)

= −dt2 + tgp .

We want to analyse the evolution of these models and investigate possible singu-larities. In the Robertson-Walker perfect �uid those are de�ned either via the scalingfunction f or via the energy density ρ. Both concepts will be of little use to us, aswe will consider vacuum models, i. e. ρ ≡ 0, and will not have one single functiondetermining the four-dimensional geometry. The concept of big bang and big crunchwill have to be adapted to the more general situation.Recall that in the Robertson-Walker spacetime the Hubble constant satis�es

H0 =f (t0)

f (t0).

Easy calculation (see 7.4.2) shows that this quotient has a deeper geometrical mean-ing, as

−3f (t)

f (t)= tH . (3.4)

32


Here − tH ∂∂t

.

.= t~H , where t~H denotes the mean curvature (de�ned in 1.3.11) of thesubmanifold ({t} ×M, f 2(t) g ) in (M(K , f), g ).Furthermore, consider the volume form vol( g ) of a given Riemannian manifold

(M, g ). Relative to a basis (e1, . . . , en) of TpM , it is given by

vol(

gp)

=√

det gijdx1 ∧ . . . ∧ dxn,

with dxi the 1-form dual to ei, gij .

.= gp(ei, ej). Easy calculation shows that on theRobertson-Walker spacetime the volume form of the submanifold ({t} ×M, f 2(t) g )satis�es vol( tg ) = f 3(t) vol( t0g ), therefore

f (t) =

(vol(tg)

vol(t0g)) 1

3

. (3.5)

The knowledge of equations (3.4) and (3.5) yields an equivalent characterisation ofbig bang and big crunch that does not require the scaling function:

Lemma 3.6.5. Let M(K , f) a Robertson-Walker spacetime with maximal intervalI = (t−, t+), which is a perfect �uid ( ∂

∂t, ρ, p), t0 ∈ I an arbitrary time. Set tg .

.=f 2(t) g ) and denote by vol( tg ) the volume form of the submanifold (M, f 2(t) g ). Aninitial singularity t− is a big bang if and only if

vol(tg)→ 0, tH

(vol(tg)

vol(t0g)) 1

3

→ −∞ for t→ t−.

A �nal singularity t+ is a big crunch if and only if

vol(tg)→ 0, tH

(vol(tg)

vol(t0g)) 1

3

→∞ for t→ t+.

Remark 3.6.6. The property vol( tg ) → 0 can be interpreted as a collapse, as itmeans

volt (K) =

∫K

vol(tg)→ 0

for every compact set K ⊂M .

We will use this as a generalised de�nition of big bang and big crunch, whichenables us to extend the notion of these singularities to arbitrary spacetimes of theform M = I ×M .

33

4. Cosmological Bianchi models

In this chapter we give an introduction to cosmological Bianchi models, which gen-eralise the Robertson-Walker spacetimes to spatially homogeneous models. They areconstructed out a three-dimensional Lie groups with left invariant metrics. We there-fore start our study by classifying these three-dimensional Lie groups and their Liealgebras, more speci�cally the unimodular ones. In doing so, we follow [Mil76] and[Ung09].

4.1. Milnor bases

We recall the cross product operation on three-dimensional oriented euclidean vectorspaces:

De�nition 4.1.1. Let V a three-dimensional oriented vector space with scalar prod-uct 〈·, ·〉. The cross product

× : V × V → V

(u, v) 7→ u× v

is the bilinear and skew-symmetric mapping such that u × v is orthogonal to both

u and v and has length√〈u, u〉〈v, v〉 − 〈u, v〉2. It is uniquely de�ned by demanding

the triple (u, v, u× v) to be positively oriented for u, v linearly independent.

It follows immediately that u × v = 0 for linearly dependant vectors. We nowadditionally endow a euclidean vector space with a Lie bracket to obtain a euclideanLie algebra (a, [·, ·] , 〈·, ·〉). They appear in the form of the Lie algebra of a Lie groupwith left invariant metric, which is the main example we apply our results to.

Proposition 4.1.2. Let G a three-dimensional euclidean Lie algebra (a, [·, ·] , 〈·, ·〉)with given orientation. Then there is a uniquely de�ned linear mapping L : a −→ asatisfying

[u, v] = L (u× v) .

Proof. We choose a positively oriented orthonormal basis (e1, e2, e3) on a whichby de�nition satis�es e1 × e2 = e3, e2 × e3 = e1, e3 × e1 = e2. On this basis wede�ne L by L(e1) = [e2, e3], L(e2) = [e3, e1], L(e3) = [e1, e2]. This obviously means

35


L(ei × ej) = [ei, ej] for all basis vectors ei, ej and therefore L(u × v) = [u, v], usinglinearity. Uniqueness follows from surjectivity of the cross product.

The map L will be of use in the following. We investigate its properties in the specialcase where the Lie algebra is unimodular. In order to do so, let G a Lie group andrecall that it possesses � up to a multiplicative constant � a unique left invariant Haarmeasure, i. e. a measure µ on the Borel σ-algebra Bσ of G

µ : Bσ → R≥0

such that

i) µ(gB) = µ(B) for every g ∈ G, B ∈ Bσ;

ii) µ(K) <∞ for every compact set K;

iii) µ(B) = inf{µ(U) : E ⊂ U, U open} for every B ∈ Bσ (outer regularity of Borelsets);

iv) µ(U) = sup{µ(K) : K ⊂ U, K compact} for every open set K (inner regularityof open sets).

De�nition 4.1.3. A Lie group G is called unimodular if its left invariant Haarmeasure is also right invariant.

There is an equivalent characterisation of unimodularity which we need in the follow-ing.

Lemma 4.1.4. Let G a Lie group with Lie algebra (g, [·, ·] ). Then the followingstatements are equivalent:

i) G is unimodular.

ii) For every g ∈ G the automorphism Ad(g) : G→ G, h 7→ ghg−1 has determinant±1.

If additionally G is connected, we have another equivalent statement.

iii) For every x ∈ g the linear mapping ad(x) : g→ g, y 7→ [x, y] has trace zero.

For a proof of this lemma, see [Mil76, chap. 6]. The last statement allows to extendthe notion of unimodularity to Lie algebras.

De�nition 4.1.5. A euclidean Lie algebra (a, [·, ·] , 〈·, ·〉) is called unimodular if themapping ad(x) : a→ a, y 7→ [x, y] has trace zero.

Lemma 4.1.6. The mapping L in proposition 4.1.2 is self-adjoint if and only if theLie algebra a is unimodular.

36

4.1. Milnor bases

Proof. Let (e1, e2, e3) a positively oriented orthonormal basis and suppose

L (ei) = αijej.

Direct calculation then shows that

tr ad (e1) = −α23 + α32

tr ad (e2) = −α31 + α13

tr ad (e3) = −α12 + α21,

which means that a is unimodular if and only if the matrix (αij)i,j=1,2,3 is symmetric.This is equivalent to L being self-adjoint.

De�nition 4.1.7. Let (a, [·, ·] ) a 3-dimensional Lie algebra. A basis (e1, e2, e3) of ais said to have Milnor property if

[e1, e2] = λ3e3, [e2, e3] = λ1e1, [e3, e1] = λ2e2,

with λi ∈ R, i = 1, 2, 3.

In this thesis we consider Lie algebras that arise from Lie groups equipped with aleft invariant metric. In addition to the Lie bracket, these Lie algebras are naturallyendowed with a scalar product, the one induced by the metric. A Milnor basis onthese Lie algebras should take into account this additional euclidean structure. Inaccordance with Milnor [Mil76] we therefore de�ne:

De�nition 4.1.8. Let (a, [·, ·] , 〈·, ·〉) a three-dimensional euclidean Lie algebra withgiven orientation. A basis (e1, e2, e3) of a is called a Milnor basis if the basis ispositively oriented, orthogonal and has Milnor property

[e1, e2] = λ3e3, [e2, e3] = λ1e1, [e3, e1] = λ2e2,

with λi ∈ R, i = 1, 2, 3.We designate two special cases: If the basis is even orthonormal, it is called nor-

malised Milnor basis. If the structure constants satisfy λi ∈ {−1, 0, 1}, i = 1, 2, 3,then (e1, e2, e3) is called uni�ed Milnor basis.

We now study the behaviour of the structure constants of a basis with Milnorproperty under scaling. The structure of the underlying Lie algebra, i. e. the Liebracket, is �xed.

Lemma 4.1.9. Let (a, [·, ·] , 〈·, ·〉) a three-dimensional euclidean Lie algebra with givenorientation, and (e1, e2, e3) a basis which has Milnor property with λ1, λ2, λ3 ∈ R. Letξ, η, ζ be positive real numbers such that the basis

(ηζe1, ξζe2, ξηe3)

is orthonormal. Then the new basis has Milnor property with structure constants

ξ2λ1, η2λ2, ζ

2λ3.

37


Remark 4.1.10. Obviously, if the original basis (e1, e2, e3) is orthogonal, then byscaling one can achieve orthonormality. On the other hand, scaling the basis vectorsonly scales the structure constants but does not create additional non-vanishing ones.Thus this transformation can orthonormalise a basis (e1, e2, e3) whilst preserving itsMilnor property if and only if the basis is orthogonal.

Proof of lemma 4.1.9. Obviously the new basis possesses the Milnor property. Wecan determine the structure constants by easy calculation, but only carry out the �rstone:

〈[ξζe2, ξηe3] , ζηe1〉 =⟨ξ2ζη [e2, e3] , ζηe1

⟩= ξ2λ1〈ζηe1, ζηe1〉= ξ2λ1.

Lemma 4.1.9 allows us to rescale a given Milnor basis in order to obtain a basis withspeci�c properties: We see that multiplying the structure constants λi with positivereal numbers only scales the Lie bracket and thus preserves the structure of the Liealgebra. In particular, starting with a normalised Milnor basis we can reverse theprocess. By setting

ξ2 =1

|λ1|if λ1 6= 0, ξ2 = 1 otherwise

and similarly for η, ζ, we can generate structure constants in {−1, 0, 1}.

Proposition 4.1.11. (Existence of Milnor bases) Every unimodular three-dimensionaloriented euclidean Lie algebra possesses a normalised Milnor basis and (after scaling)a uni�ed Milnor basis.

Proof. Let L : a −→ a be the linear mapping de�ned in proposition 4.1.2. Accordingto lemma 4.1.6, L is self-adjoint. Therefore the Lie algebra a possesses an orthonor-mal basis of eigenvectors (e1, e2, e3). We may assume without loss of generality thatthe basis is positively oriented (replace e1 by −e1 if necessary). In other words, thereare real numbers λ1, λ2, λ3 such that

[e1, e2] = L (e1 × e2) = L (e3) = λ3e3,

[e2, e3] = λ1e1,

[e3, e1] = λ2e2,

thus (e1, e2, e3) is a normalised Milnor basis. By scaling this basis as described inlemma 4.1.9, we may gain λi ∈ {−1, 0, 1} for i = 1, 2, 3 while the basis remainsorthogonal. Thus we have found a uni�ed Milnor basis.

38

4.2. Classi�cation of Bianchi class A models


Using the results above, we will now classify all three-dimensional oriented euclideanunimodular Lie algebras. Theorem 2.0.24 then ensures the existence of a simplyconnected Lie group which is unique up to isomorphy. It is equipped with a leftinvariant Riemannian metric obtained by left translating the scalar product on itsLie algebra. All other Lie groups with the same Lie algebra are quotient groups ofthe simply connected Lie group factorised by discrete subgroups of its center, seechapter 2.According to proposition 4.1.11, we can choose a uni�ed Milnor basis (e1, e2, e3)

with structure constants λi ∈ {−1, 0, 1}, i = 1, 2, 3. Without loss of generality we mayassume that at most one of the structure constants is negative, otherwise consider(−e1,−e2,−e3).

λ1 λ2 λ3 Lie algebra exemplary Lie group0 0 0 a(3) R⊕ R⊕ R+ 0 0 hei(3,R) Hei(3,R)+ - 0 e(1, 1) E(1, 1)+ + 0 e(2) E(2)+ + - sl(2,R) SL(2,R)+ + + su(2) SU(2)

Table 4.1.: Classi�cation of three-dimensional oriented euclidean unimodular Lie al-gebras and three-dimensional unimodular Lie groups with left invariantmetric.

In table 4.1 we list the six possibilities of structure constants with the associatedLie algebra:

• a(3) =

λ1 0 0

0 λ2 00 0 λ3

∈ gl(3,R) : λ1, λ2, λ3 ∈ R

• hei(3,R) =

0 λ1 λ2

0 0 λ30 0 0

∈ gl(3,R) : λ1, λ2, λ3 ∈ R

• e(1, 1) =

0 0 0

λ2 0 λ1λ3 λ1 0

∈ gl(3,R) : λ1, λ2, λ3 ∈ R

• e(2) =

0 0 0

λ2 0 λ1λ3 −λ1 0

∈ gl(3,R) : λ1, λ2, λ3 ∈ R

39


• sl(2,R) = {a ∈ gl(2,R) : tr (a) = 0}

• su(2) = {a ∈ gl(2,C) : ta+ a = 0, tr (a) = 0}

The exemplary Lie groups listed are the following:

• Hei(3,R): The Heisenberg group, the group of all 3 × 3 upper triangular ma-trices with 1 on the diagonal.

• E(1, 1): The group of all rigid motions of the Minkowski space R1,1.

• E(2): The group of all rigid motions of the Euclidean space R2.

• SL(2,R): The special linear group, i. e. the group of all unitary 2× 2 matricesof determinant 1.

• O(1, 2): The Lorentz group, i. e. the group of all linear transformations in R3

preserving the quadratic form t2 − x2 − y2.

• SU(2): The special unitary group, i. e. the group of unitary 2 × 2 matrices ofdeterminant 1.

• SO(3): The special orthogonal group, i. e. the group of all rotations in three-dimensional space.

As we argued in chapter 2, these Lie groups are unique up to isomorphy if they aresimply connected. This is the case for all given Lie groups apart from E(2) andSL(2,R), which are chosen here because they are well known examples.As soon as two of the Lie algebras admit bases with identical structure constants,

they are isomorphic. It remains to show that these six cases are in fact pairwisedistinct up to isomorphy. This can be done by considering the Killing form

β (x, y) .

.= tr (ad (x) ◦ ad (y)) .

For a proof of this statement, see for example [Ung09, chap. 3].At the end of the last chapter we mentioned that three-dimensional Lie groups

with a family of left invariant metrics are used to construct spatially homogeneousspacetimes. They are named after Luigi Bianchi, who classi�ed the associated Liealgebras at the end of the 19th century.

De�nition 4.2.1. Let I ⊂ R an open interval, G a three-dimensional connected Liegroup with a family ( tg )t∈I of left invariant metrics. The semi-Riemannian manifoldG .

.= I ×G with metric

g(t,p)

= −dt2 + tgp

and time-orientation[∂∂t

]is called Bianchi model.

If the Lie group G is unimodular, then the spacetime (G, g ,[∂∂t

]) is a Bianchi

model of class A, otherwise of class B.

40


Bianchi type λ1 λ2 λ3I 0 0 0II + 0 0V I0 + - 0V II0 + + 0V III + + -IX + + +

Table 4.2.: Bianchi class A models.

The unimodular Lie groups are classi�ed using the structure constants of a Milnorbasis of their respective Lie algebra. In accordance with this result, the Bianchi classA models are divided into six types, see table 4.2. The missing numbers, types III,IV , V Ih and V IIh, are the Bianchi models of class B.

41

5. Ricci curvature in

four-dimensional spacetime

Throughout this chapter, M is a three-dimensional manifold with a family of Rie-mannian metrics tg , for t ∈ I, I an open intervall in R. We add a normal time-like direction and thus construct a four-dimensional semi-Riemannian manifold byendowing

M .

.= I ×M,

with the metric

g(t,p)

.

.= −dt2 + tgp .

Then (M, g ) is a semi-Riemannian manifold, even a Lorentz manifold. On (M, tg )we denote by ∇ the Levi-Civita connection, and by tR , tRic and tS the Riemannian,Ricci and scalar curvature. The Levi-Civita connection depends on t as well, but thetime is suppressed for the bene�t of easier readability. The corresponding quantitieson M are denoted by ∇, R , Ric and S .The manifold (M(t) .

.= {t} ×M, g(t, ·)) is isometrically di�eomorphic to (M, tg ).The tangent space at (t, p) ∈M is

T(t,p)M ∼= TtI ⊕ TpM ∼= R⊕ TpM,

thus we can identify X(M) ∼= E(M) ⊕ X(M). We identify a vector �eld X ∈ X(M)with its equivalent on M(t) and de�ne the constant extension in time X .

.= (0, X) ∈X(M) which is horizontal to every slice {t} ×M .On M we de�ne the normal direction E0

.

.= ∂∂t

to be the vector �eld which, at apoint (t0, p0) ∈M , is tangential to the curve

(−ε, ε)→M

t 7→ (t0 + t, p0) .

Is is easy to see that this is indeed a smooth vector �eld which satis�es

g(∂

∂t,∂

∂t

)= −1.

We want to compute the Ricci curvature at an arbitrary point (t0, p0) ∈ M . Let(e1, e2, e3) a basis of the tangent space Tp0M and consider the scalar product t0gp0 onTp0M

∼= Tp0M(t0) as well as the bilinear form g(t0,p0)

on T(t0,p0)M .

43

5. Ricci curvature in four-dimensional spacetime

We set e0 .

.= ∂∂t

(t0, p0). Then g(t0,p0)

(e0, x) = 0 for every vector x horizontal to{t0} ×M , notably for the basis (e1, e2, e3). By construction, (e0, e1, e2, e3) forms abasis of the four-dimensional tangent space T(t0,p0)M . Using lemma 1.2.16 we extende1, e2, e3 in a neighbourhood U ⊂ M(t0) of p0 to vector �elds E1, E2, E3 ∈ X(U)satisfying

∇vEi = 0 (5.1)

for every v ∈ Tp0M(t0) and i = 1, 2, 3. We identify these vector �elds with theirconstant extensions in normal direction

Ei.

.= (0, Ei) ∈ X (I × U) i = 1, 2, 3,

For the sake of simplicity, we will often write ∂∂t

instead of ∂∂t

(t, p), when the contextclari�es at which point (t, p) ∈M we evaluate the vector �eld. By construction

Eα (t0, p0) = eα α = 0, 1, 2, 3

and (E0 =

∂

∂t

)⊥ Ei i = 1, 2, 3.

We will point out several other properties of such a basis in lemma 5.2.2. The followingnotation will be used throughout this chapter:

tg kl .

.= tg (Ek, El)

gαβ .

.= g (Eα, Eβ)

and

gkl .

.= t0gp0 (ek, el)

gαβ .

.= g(t0,p0)

(eα, eβ) .

The inverse matrices are denoted by(gkl)

.

.= (gkl)−1(

gαβ)

.

.= (gαβ)−1

and t0g kl denotes the function p 7→ gkl.

5.1. Some simpli�ed geometric quantities

The manifold M(t0) is a hypersurface in the semi-Riemannian manifold M , thereforethe normal space has dimension 1. Having chosen the normalised normal �eld ∂

∂t,

44

5.1. Some simpli�ed geometric quantities

several of the geometric quantities we de�ned in the very �rst chapter have an easierform. Furthermore, we want to deduce their representation in our basis (e0, e1, e2, e3).In the following, we will use di�erent alphabets in our Einstein summation conven-

tion in order to distinguish between the four-dimensional and the three-dimensionalcase: latin indices k, l, . . . denote a sum over 1, 2, 3, whereas greek indices α, β, . . .are summarised over 0, 1, 2, 3. In addition, we remark that the trace tr contracts inthree dimensions with respect to tg , while tr denotes the four-dimensional trace withrespct to g .

We start by simplifying the second fundamental form tII on the submanifold (M(t), tg ) ⊂(M = I ×M, g = −dt2 + tg ). Being the normal part of ∇XY , tII (X, Y ) has to be ascalar multiple of ∂

∂t. Thus, the scalar second fundamental form tk : X(M)×X(M)→

E(M)

tk (X, Y ) .

.= g(tII (X, Y ) ,

∂

∂t

)∀X, Y ∈ X (M)

is a tensor carrying the same information as tII . We will often make use of thefollowing equivalent description for the second fundamental form:

tII (X, Y ) = − tk (X, Y )∂

∂t.

Remark 5.1.1. For a basis (e1 . . . , en) of an n-dimensional vector space V with scalarproduct 〈·, ·〉 we de�ne gij .

.= 〈ei, ej〉 and (gij) .

.= (gij)−1. We deduce the coordinate

expression v = µjej for a vector v ∈ V . First we see

〈v, ei〉 = µk〈ek, ei〉= µkgki,

and after multiplication with the inverse matrix

〈v, ei〉gij = µkgkigij

= µkδjk

= µj.

Altogether, this yields

v = 〈v, ei〉gijej.

We will frequently use this result in this chapter's computations.

We recall that the mean curvature vector �eld t~H is the vector �eld to M(t) givenby the contraction of tII . We de�ne the scalar mean curvature

tH .

.= g(t~H ,

∂

∂t

).

45


Relative to a basis (e1, e2, e3) of the tangent space Tp0M(t) we recall

t~Hp0 = gij tIIp0 (ei, ej) ,

where gij .

.= tgp0(ei, ej) and (gij) .

.= (gij)−1, thus the scalar mean curvature is given

by

tHp0 = g(t,p0)

(t~Hp0 ,

∂

∂t

)= gij g

(t,p0)

(tIIp0 (ei, ej) , e0

)= gij tk p0 (ei, ej) . (5.2)

The natural choice of normal direction allows a similar simpli�cation for the Wein-garten map. We use the identi�cation M ∼= M(t), thus X(M) ∼= X(M(t)), and viewthe Weingarten map as the tensor

tW : X (M)→ X (M)

X 7→W (X) = − tanM(t)∇X∂

∂t.

The relation (1.11) between the second fundamental form and the Weingarten mapallows us to express tW in terms of our basis:

tk p0 (ei, ek) = g(t,p0)

(tIIp0 (ei, ek) ,

∂

∂t

)= tgp0

(tWp0 (ei) , ek

).

As tWp0 (ei) is orthogonal to ∂∂t, i. e. g

(t,p0)( tWp0 (ei), e0) = 0, we �nd

tWp0 (ei) = g km tk p0 (ei, ek) em. (5.3)

5.2. Geometric relations between M and M

Our goal in this section will be to express the Einstein equations at the point (t0, p0) interms of the scalar products ( tgp0 )t∈I , the scalar second fundamental forms ( tk p0 )t∈I ,and several geometric quantities. Following Zeghib's approach in [Zeg11] we relatethe geometry or more speci�cally the connection and the curvatures of M and M bycomputing them in the basis we constructed at the beginning of the chapter. It thenbecomes obvious that the computed terms correspond to several geometric tensorsand are therefore independent of the choice of basis.We start by giving several equations relating the Levi-Civita connection and the

Riemannian curvature of M to the scalar products ( tgp0 ).

46


Proposition 5.2.1. Let M = I ×M with metric g(t,p)

= −dt2 + tgp . We choose a

basis (e1, e2, e3) of Tp0M∼= Tp0M(t0), local horizontal extensions E1, E2, E3 ∈ X(M)

such that ∇ejEi = 0, 1 ≤ i, j ≤ 3, and denote by E1, E2, E3 their constant extensions

in normal direction. Setting E0.

.= ∂∂t, e0 .

.= ∂∂t

(t0, p0) we �nd:

∇E0E0 = 0 (5.4)

∇e0Ei =1

2gklgikel i = 1, 2, 3 (5.5)

tk = −1

2tg ∀ t ∈ I (5.6)

R p0 (e0, eα, eβ, e0) = 0 α = 0 or β = 0 (5.7)

R p0 (e0, ei, ej, e0) = −1

2gij +

1

4gklgikgjl i, j = 1, 2, 3, (5.8)

where glk .

.= gp0(el, ek),(glk)

.

.= (glk)−1 and gij, gij denote the �rst and second

derivative at t = t0 of tgp0(ei, ej) with respect to time.

In order to proof these equations, we need the following lemma.

Lemma 5.2.2. Let X ∈ X(M) and X = (0, X) ∈ X(M) its constant extension innormal direction. Then X and ∂

∂t= E0 commute:[X,E0

]= 0,

and therefore

∇XE0 = ∇E0X.

We �nd a similar equation for E0 = ∂∂t:

∇E0E0 = 0,

which means that the trajectories of ∂∂t

are geodesics.

Proof. Choose E1, E2, E3 and E1, E2, E3 as in proposition 5.2.1. Then the horizontalvector �eld X can be locally expressed as X =

∑3i=1 ξiEi with ξi : M → R, i = 1, 2, 3.

Its constant extension in normal direction has the local expression

X = ξiEi = ξi (0, Ei)

with ξi : M → R, i = 1, 2, 3. On the other hand X = (0, X) = (0,∑3

i=1 ξiEi), whichmeans that the functions ξi are constant in normal direction:

E0ξi = 0 i = 1, 2, 3.

As the vector �elds Ei commute with ∂∂tby construction, this yields the �rst equation.

47


The second equation follows immediately, as the Levi-Civita connection∇ is torsionfree, i. e. [

X,E0

]= ∇XE0 −∇E0

X.

In order to prove the last equation, we recall that E0 was de�ned orthogonal to themanifold M(t0) and satis�es g(E0, E0) = −1. More speci�cally

g(E0, Eα

)∈ {0,−1}

is constant for 0 ≤ α ≤ 3. Therefore we �nd

0 = eα g(E0, E0

)= 2 g

(t0,p0)

(e0,∇eαE0

)which leads to

0 = e0 g(E0, Eα

)= g

(t0,p0)

(∇e0E0, eα

)+ g

(t0,p0)

(e0,∇e0Eα

)= g

(t0,p0)

(∇e0E0, eα

)+ g

(t0,p0)

(e0,∇eαE0

)= g

(t0,p0)

(∇e0E0, eα

),

since E0 and Eα commute. This concludes the proof.

Proof of proposition 5.2.1. The �rst equation has already been proven in lemma5.2.2.In order to prove the second equation, we express the connection on M = I ×M

using the Koszul formula (1.2). Let i, k = 1, 2, 3:

2 g(t0,p0)

(∇e0Ei, ek

)= e0 g

(Ei, Ek

)+ ei g

(Ek, E0

)− ek g

(E0, Ei

)+ g

(t0,p0)

([E0, Ei

]p0, ek

)− g

(t0,p0)

([Ei, Ek

]p0, e0

)+ g

(t0,p0)

([Ek, E0

]p0, ei

)The forth and the last summand vanish due to commutation, see lemma 5.2.2. Thescalar product g(Ek, E0) equals zero, as E0 is orthogonal to any horizontal vector�eld. Thus, the second and third summand in the Koszul formula vanish as well.Finally, the commutator of two horizontal vectors is again horizontal. Therefore itsscalar product with the normal vector e0 equals zero. In total we have shown that

2 g(t0,p0)

(∇e0Ei, ek

)= e0 g

(Ei, Ek

)=

∂

∂t

∣∣∣∣t=t0

tgp0 (ei, ek)

= gik.

48


Combined with the equation

2 g(t0,p0)

(∇e0Ei, e0

)= 0

we deduced in the proof of lemma 5.2.2, this is equivalent to the desired formula

∇e0Ei =1

2gklgikel.

The third equation can be deduced similarly: Let x, y ∈ Tp0M ∼= Tp0M(t0) horizon-tal tangent vectors, X, Y ∈ X(M(t0)) smooth local extensions and X .

.= (0, X), Y .

.=(0, Y ) ∈ X(M) their constant extensions in normal direction. Then

t0gp0 (x, y) =∂

∂t

∣∣∣∣t=t0

tgp0 (x, y)

= e0 g(X,Y

)= g

(t0,p0)

(∇e0X, y

)+ g

(t0,p0)

(x,∇e0Y

)= g

(t0,p0)

(∇xE0, y

)+ g

(t0,p0)

(x,∇yE0

)= − g

(t0,p0)

(e0,∇xY

)− g

(t0,p0)

(e0,∇yX

),

since E0 commutes with horizontal vectors (see lemma 5.2.2) and is orthogonal toany horizontal vector �eld. By de�nition of the second fundamental form this equals

t0gp0 (x, y) = − g(t0,p0)

(e0,

t0IIp0 (x, y))− g

(t0,p0)

(e0,

t0IIp0 (y, x))

= −2 g(t0,p0)

(e0,

t0IIp0 (x, y))

= −2 t0k p0 (x, y) ,

where we used the symmetry of the second fundamental form.The fourth equation follows immediately from

R (X, Y )Z = −R (Y,X)Z

and

R (X, Y, Z, V ) = R (V, Z, Y,X)

for arbitrary vector �elds X, Y, Z, V ∈ X(M).In order to prove the �fth and last equation, we use the de�nition of the Riemannian

curvature tensor. Let i = 1, 2, 3:

R p0 (e0, ei) e0 = ∇e0∇EiE0 −∇ei∇E0

E0 −∇[E0,Ei]p0

E0

= ∇e0∇EiE0

= ∇e0∇E0Ei

49


because of equation (5.4) and commutation (lemma 5.2.2). Thus, we can use equation(5.5) and �nd for i, j = 1, 2, 3

R p0 (e0, ei, ej, e0) = −R p0 (e0, ei, e0, ej)

= − g(t0,p0)

(R p0 (e0, ei) e0, ej

)= − g

(t0,p0)

(∇e0∇E0

Ei, ej)

= −e0 g(∇E0

Ei, Ej

)+ g

(t0,p0)

(∇e0Ei,∇e0Ej

)= − ∂

∂t

∣∣∣∣t=t0

tgp0(

1

2gklgikel, ej

)+ g

(t0,p0)

(1

2gklgikel,

1

2gmngjmen

)= − ∂

∂t

∣∣∣∣t=t0

(1

2gklgikglj

)+

1

4gklgikg

mngjmgln

= − ∂

∂t

∣∣∣∣t=t0

(1

2gikδ

kj

)+

1

4gklgikgjmδ

ml

= −1

2gij +

1

4gklgikgjl.

This concludes the proof.

5.3. Formulae for the Ricci curvature

We now compute the Ricci curvature of the four-dimensional manifold using formula(1.6)

Ric(t0,p0) (x, y) = gαβ g(t0,p0)

(R p0 (eα, x) y, eβ

)for the basis (e0, e1, e2, e3) of T(t0,p0)M = R×Tp0M(t0) we constructed at the beginningof this chapter. We express the four-dimensional quantity Ric in terms of the three-dimensional geometry of M , more precisely the metric and the second fundamentalform on M(t), their derivatives with respect to time and several deduced quantities.The expressions we �nd are tensorial, and therefore independant of our choice ofbasis.Part 1: We start with the Ricci curvature of two horizontal basis vectors. For

i, j, k, l = 1, 2, 3 the Gauÿ equation (1.10) yields

R p0 (ek, ei, ej, el) = t0R p0 (ek, ei, ej, el)− g(t0,p0)

(t0IIp0 (ei, ej) ,

t0IIp0 (ek, el))

+ g(t0,p0)

(t0IIp0 (ek, ej) ,

t0IIp0 (ei, el))

= t0R p0 (ek, ei, ej, el) + t0k p0 (ei, ej)t0k p0 (ek, el)

− t0k p0 (ei, el)t0k p0 (ej, ek) ,

50


since g( ∂∂t, ∂∂t

) = −1, thus with kij .

.= t0k p0(ei, ej) we can write

Ric(t0,p0) (ei, ej) = −R (t0,p0) (e0, ei, ej, e0) + gkl t0R p0 (ek, ei, ej, el)

= −R (t0,p0) (e0, ei, ej, e0) + t0Ricp0 (ei, ej) + gklkijkkl − gklkilkjk.

Equation (5.8) provides us with an explicit representation of R p0(e0, ei, ej, e0). Weuse the relation tk = −1

2tg and the symmetry of gkl to calculate

Ric(t0,p0) (ei, ej) = −kij + t0Ricp0 (ei, ej) + gklkklkij − 2gklkikkjl.

The third term contains the scalar mean curvature we de�ned in (5.2). The termgklkikkjl is the basis representation of the tensor �eld

tW 2 : X (M)× X (M)→ E (M)

(X, Y ) 7→ tW 2 (X, Y ) .

.= tg(tW (X) , tW (Y )

)as the following computation makes clear (see equation (5.3))

t0Wp02 (ei, ej) = t0gp0

(gkm t0k p0 (ei, ek) em, g

ln t0k p0 (ej, el) en)

= gkm t0k p0 (ei, ek) gln t0k p0 (ej, el) gmn

= gkmglngmnkikkjl

= gklkikkjl.

We have thus found the following equation for horizontal vectors x, y ∈ Tp0M(t0)

Ric(t0,p0) (x, y) = − t0k p0 (x, y) + t0Ricp0 (x, y) + t0Hp0t0k p0 (x, y)− 2 t0Wp0

2 (x, y) .

(5.9)

Remark 5.3.1. The right-hand side of this equation is only valid for tangent vectorsand vector �elds on M ∼= M(t0), whereas the four-dimsional Ricci curvature Ric isde�ned on the tangent bundle of M and allows non-horizontal arguments. Thus,in order to formulate a correct tensor equation, we have to replace Ric by t0ι

∗ Ric ,where

t0ι : M →M

is the embedding that maps M onto M(t0) ∼= M . We will frequently make use of thisnotation in the following.

Applying this notation, we achieve the following equation on Tp0M :

t0ι∗ Ric(t0,p0) = − t0k p0 + t0Ricp0 + t0Hp0

t0k p0 − 2 t0Wp02. (5.10)

51


Part 2: Since R (t0,p0)(e0, e0, e0, e0) vanishes, we �nd

Ric(t0,p0) (e0, e0) = gkl g(t0,p0)

(R (t0,p0) (ek, e0) e0, el

)= gklR (t0,p0) (e0, el, ek, e0)

= gkl(−1

2gkl +

1

4gmngknglm

)= gklkkl + gklgmnkknklm.

The term gklgmnkknklm is the tensor |k |2g expressed in our basis (see the appendix).As the following computation makes clear, it is the trace of t0Wp0

2:

tr t0Wp02 = gmn t0Wp0

2 (en, em)

= gmngklknkkml

= tr tr23(t0k p0 ⊗ t0k p0

)=∣∣ t0k p0

∣∣2g .

The �rst term is the trace of the �rst derivative of the scalar second fundamentalform:

gklkkl = tr t0k p0 .

The following lemma yields a formula for the contraction of tensor �elds.

Lemma 5.3.2. Let ( tA)t∈I , a family of symmetric tensors on M . Then(tr tA

)˙ = tr

(tA)

+ 2 tr tr23(k ⊗ tA

)where the dot denotes di�erentiation with respect to t.

Proof. Let t0 ∈ I and (e1, . . . , en) a basis of Tp0M , p0 ∈ M . We remark thatgijgjk = δik for every t ∈ I, thus gijgjk + gij gjk = 0 and equivalently

gij = −gikgklglj.

Recall tk = −12tg . Then(

tr tAp0

)˙ =

∂

∂t

∣∣∣∣t=t0

(gij tAp0 (ei, ej)

)= gij t0Ap0 (ei, ej) + gij t0Ap0 (ei, ej)

= tr(t0Ap0

)− gikgklglj t0Ap0 (ei, ej)

= tr(t0Ap0

)+ 2gikkklg

lj t0Ap0 (ei, ej)

= tr(t0Ap0

)+ 2 tr tr23

(t0k p0 ⊗ t0Ap0

).

52


As tH = tr tk , this yields

tH = tr(tk)

+ 2∣∣ tk ∣∣2g , (5.11)

and we have found the following two expressions for Ric(t0,p0)(e0, e0) in terms of thescalar second fundamental form and the metric

Ric(t0,p0) (e0, e0) = tr t0k p0 +∣∣ t0k p0

∣∣2g (5.12)

= t0Hp0 −∣∣ t0k p0

∣∣2g . (5.13)

We want to gain an alternative expression which does not contain the derivative oftk p0 or tHp0 . It can be eliminated using equation (5.10) for Ric we found in the �rstpart:

Ric(t0,p0) (e0, e0) = tr t0k p0 +∣∣ t0k p0

∣∣2g

= tr(−Ric(t0,p0) + t0Ricp0 + t0Hp0

t0k p0 − 2 t0Wp02)

+∣∣ t0k p0

∣∣2g

= −Ric(t0,p0) (e0, e0)− S (t0,p0) + t0Sp0 +(t0Hp0

)2 − 2∣∣ t0k p0

∣∣2g +

∣∣ t0k p0

∣∣2g

= −Ric(t0,p0) (e0, e0)− S (t0,p0) + t0Sp0 +(t0Hp0

)2 − ∣∣ t0k p0

∣∣2g .

In total we get the third relation

Ric(t0,p0) (e0, e0) =1

2

(t0Sp0 − S (t0,p0) +

(t0Hp0

)2 − ∣∣ t0k p0

∣∣2g

). (5.14)

Part 3: For the remaining Ricci curvatures, let i, k, l = 1, 2, 3. The derivative of thesecond fundamental form satis�es

∇eit0II = ∇ei

(− t0k · E0

)= −

(∇ei

t0k)· e0 − t0k p0 · ∇eiE0

= −(∇ei

t0k)· e0 − t0k p0 · ∇e0Ei,

because E0, Ei commute due to lemma 5.2.2. We have seen in (5.5) that ∇e0Ei ishorizontal to M(t0), thus

g(t0,p0)

((∇ei

t0II)

(el, ek) , e0)

=(∇ei

t0k)

(el, ek) .

In order to compute the missing Ricci curvatures we use the Codazzi equation (1.13)

R (t0,p0) (ei, el, ek, e0) = g(t0,p0)

(R (t0,p0) (ei, el) ek, e0

)= g

(t0,p0)

(nor R (t0,p0) (ei, el) ek, e0

)= g

(t0,p0)

((∇ei

t0II)

(el, ek) , e0)− g

(t0,p0)

((∇el

t0II)

(ei, ek) , e0)

=(∇ei

t0k)

(el, ek)−(∇el

t0k)

(ei, ek) .

53


This shows

Ric(t0,p0) (e0, ei) = −R (t0,p0) (e0, e0, ei, e0) + gklR (t0,p0) (ek, e0, ei, el)

= gklR (t0,p0) (ei, el, ek, e0)

= gkl(∇ei

t0k)

(el, ek)− gkl(∇el

t0k)

(ei, ek) ,

where the very �rst summand R (t0,p0)(e0, e0, ei, e0) vanishes due to equation (5.7).The divergence divω of a symmetric tensor of type (0, 2) ω : TM(t0)× TM(t0)→E(M(t0)) is given by

divω (x) = tr (∇ω) (x) = glk (∇elω) (x, ek)

for x ∈ Tp0M(t0), thus using the symmetry of t0gp0 we see

gkl(∇el

t0k)

(ei, ek) = div t0k (ei) .

Now we want to �nd a tensor relation for the �rst term of Ric(t0,p0)(e0, ei). To beginwith we remark that(

∇eit0k)

(el, ek) = eit0k (El, Ek)− t0k p0 (∇eiEl, ek)− t0k p0 (el,∇eiEk)

= eit0k (El, Ek)

= eit0k lk

as by our choice of extension, the covariant derivative of Ek in horizontal directionvanishes in p0. Furthermore, the Levi-Civita connection preserves the metric, i. e.∇ g = 0, and therefore

0 =(∇ei g

)(ek, el)

= eit0g (Ek, El)− t0gp0 (∇eiEk, el)− t0gp0 (ek,∇eiEl)

= eit0g kl

for the same reason. This yields that the inverse function has vanishing derivative inp0 as well:

0 = eiδkm

= ei(t0g kl t0g lm

)= glm · ei t0g kl + gkl · ei t0g lm= glm · ei t0g kl

and therefore

0 = eit0g kl .

54


This leads to

gkl(∇ei

t0k)

(el, ek) = gkl · ei(t0k lk

)= ei

(t0g kl t0k lk

)−(ei

t0g kl)· klk

= ei(t0g kl t0k lk

)= ei

t0H= d t0H (ei)

where we used the symmetry of (gkl) and equation (5.2).Altogether, we see the following equation on Tp0M

t0ι∗Ric(t0,p0) (e0, ·) = d t0H − div t0k , (5.15)

where we used the notation we introduced in remark 5.3.1.In total, we have found equations for the four-dimensional Ricci curvature Ric at

an arbitrary point (t0, p0) ∈ M in terms of the geometry of M at p0 ∈ M and timet0 ∈ I. As all terms are tensor �elds, our computations are independant of the point(t0, p0). They are tensor equations and remain true anywhere on the four-dimensionalLorentz manifold. In the following theorem we summarise these tensor equations wefound for the Ricci curvature.

Theorem 5.3.3. Let M = I ×M with metric g(t,p)

= −dt2 + tgp and set E0.

.=∂∂t∈ X(M) the normalised normal direction. For t ∈ I denote by tι : M → M the

embedding that maps M onto M(t) ∼= M . Then we have the tensor equations

tι∗Ric

(E0, E0

)= tr tk +

∣∣ tk ∣∣2g= ˙tH −

∣∣ tk ∣∣2gtι∗ (Ric

(E0, E0

)+ S

)=

1

2

(tS +

(tH)2 − ∣∣ tk ∣∣2g )

tι∗Ric

(E0, ·

)= d tH − div tk

tι∗Ric (·, ·) = − tk + tRic + tH tk − 2 tW 2.

55

6. Ricci curvature on

three-dimensional unimodular

Lie groups

In the last chapter we were able to express the four-dimensional Ricci curvature RiconM .

.= I×M in terms of time-dependant quantities on the three-dimensional mani-fold M . All terms are composed of the metric tg and the scalar second fundamentalform tk obtained when immersing M into M via

ιt : M →M

p 7→ (p, t) ,

which maps M onto M(t) = {t} ×M ∼= M .

In this thesis we want to analyse the dynamics of Bianchi class A models. Lettherefore G a three-dimensional Lie group with left invariant metric g . Let e ∈ Gthe identity element, g ∼= TeG the associated Lie algebra and ge the scalar producton g. Recall that the Lie group G in a Bianchi class A model G .

.= I×G is connectedand unimodular, thus its Lie algebra possesses a uni�ed Milnor basis and a nor-malised Milnor basis, see proposition 4.1.11. We now develop expressions for thethree-dimensional Ricci curvature using these two types of basis. It is notable thatin the case of a normalised Milnor basis, these expression take into account only thestructure constants λi of the normalised Milnor basis, whereas in the case of a uni�edMilnor basis, they only take into account the signs of the λi and the scalar productge .

We have seen in Chapter 2 that the Lie algebra of a Lie group G can be expressedin two di�erent ways: It can be seen as the set of left invariant vector �elds g on theLie group or as the tangent space TeG in the identity element. The Lie bracket is thebracket operation de�ned on vector �elds: [·, ·] : X(M) × X(M) → X(M)(X, Y ) 7→X ◦ Y − Y ◦X. We explained in Remark 2.0.15 that in Lie groups it makes sense toconsider the covariant derivative ∇xy for tangent vectors x, y ∈ TeG, as we implicitelyconsider their left invariant extensions.

57

6. Ricci curvature on three-dimensional unimodular Lie groups

6.1. The case of a normalised Milnor basis

Recall that a normalised Milnor basis of a three-dimensional oriented euclidean Liealgebra is a positively oriented orthonormal basis (e1, e2, e3) such that

[e1, e2] = λ3e3, [e2, e3] = λ1e1, [e3, e1] = λ2e2,

with λi ∈ R, i = 1, 2, 3.We introduce the structure constants Cijk .

.= ge( [ei, ej] , ek), which generalise theλi:

Cijk = ge ([ei, ej] , ek)

= εijkλk,

with the Levi-Civita symbol

εijk =

+1, (i, j, k) an even permutation of (1, 2, 3)−1, (i, j, k) an odd permutation of (1, 2, 3)0, otherwise

.

Using the Levi-Civita symbols, the positive orientation is expressed by

ei × ej =∑i=1

3εijkek.

Theorem 6.1.1. Let G a three-dimensional Lie group with left invariant metric, andg its Lie algebra. Let e ∈ G the identity element, ge the scalar product on TeG ∼= gand (e1, e2, e3) a normalised Milnor basiswith

[e1, e2] = λ3e3, [e2, e3] = λ1e1, [e3, e1] = λ2e2, λi ∈ R.

Then this basis diagonalises the Ricci tensor which is then given by

Rice (e1, e1) = 2µ2µ3,

Rice (e2, e2) = 2µ1µ3,

Rice (e3, e3) = 2µ1µ2,

with µi = 12(λ1 + λ2 + λ3)− λi, i = 1, 2, 3.

The scalar curvature is given by

Se = 2 (µ2µ3 + µ1µ3 + µ1µ2) .

Proof. Let X, Y, Z left invariant vector �elds on G. Then g(X, Y ) is constant an Gand thus the Koszul formula (1.2) simpli�es to

2 g (∇XY, Z) = g ([X, Y ] , Z)− g ([Y, Z] , X) + g ([Z,X] , Y ) .

58

6.1. The case of a normalised Milnor basis

For the left invariant extensions (e1, e2, e3) of the Milnor basis (e1, e2, e3) this yields

ge (∇eiej, ek) =1

2(Cijk − Cjki + Ckij) .

As the Milnor basis is orthonormal, i. e. gkl = δkl, this is equivalent to

∇eiej =1

2δkl (Cijk − Cjki + Ckij) el

=3∑

k=1

1

2εijk (λk − λi + λj) δ

klel

=3∑

k=1

εijkµiek

= µiei × ej.

Note that we sum over k and l but cannot use the Einstein summation convention inboth cases, as the lowered index k occurs twice. Using the linearity of the connectionwe have thus shown that

∇eiv = µiei × v

for every tangent vector v ∈ g ∼= TeG, V ∈ X(G) its left invariant extension.If we endow the vector space TeG with the cross product instead of the bracket

operation, we obtain another Lie algebra. The Jacobi identity on (g,×) then givese1 × (e2 × v) + e2 × (v × e1) + v × (e1 × e2) = 0 which leads to

e1 × (e2 × v)− e2 × (e1 × v) = (e1 × e2)× v.

Therefore, we �nd for the Riemannian curvature tensor

R e (e1, e2) v = ∇e1∇e2v −∇e1∇e2v −∇[e1,e2]v

= µ1e1 × (µ2e2 × v)− µ2e2 × (µ1e1 × v)− λ3µ3e3 × v= µ1µ2 (e1 × (e2 × v)− e2 × (e1 × v))− λ3µ3e3 × v= (µ1µ2 − λ3µ3) e3 × v

and analogously

R e (e2, e3) v = (µ2µ3 − λ1µ1) e1 × vR e (e3, e1) v = (µ1µ3 − λ2µ2) e2 × v.

For i, j, k pairwise distinct this means R e(ek, ei)ej = 0 and proves the diagonalform of the Ricci tensor, as with equation (1.5) and the asymmetry of the Riemann-ian curvature tensor (proposition 1.2.15), clearly

Rice (ei, ej) = δkl ge (R e (ek, ei) ej, el)

= ge (R e (ei, ei) ej, ei) + ge (R e (ej, ei) ej, ej)

= 0.

59


The diagonal elements can be computed the same way:

Rice (e1, e1) = δkl ge (R e (ek, e1) e1, el)

= ge (R e (e2, e1) e1, e2) + ge (R e (e3, e1) e1, e3)

= ge ((λ3µ3 − µ1µ2) e3 × e1, e2) + ge ((µ1µ3 − λ2µ2) e2 × e1, e3)= λ3µ3 − µ1µ2 − µ1µ3 + λ2µ2

= λ3µ3 +1

2(λ1 − λ2 − λ3)µ2 +

1

2(λ1 − λ2 − λ3)µ3 + λ2µ2

= 2µ2µ3

and similarly

Rice (e2, e2) = 2µ1µ3

Rice (e3, e3) = 2µ1µ2.

The formula for the scalar curvature follows immediately, as

Se = δijRic (ei, ej)

for an orthonormal basis.

6.2. The case of a uni�ed Milnor basis

A basis (e1, e2, e3) of a three-dimensional oriented euclidean Lie algebra is a uni�edMilnor basis if it is positively oriented, orthogonal and satis�es

[e1, e2] = λ3e3, [e2, e3] = λ1e1, [e3, e1] = λ2e2,

with λi ∈ {−1, 0, 1}, i = 1, 2, 3.Using the results for normalised Milnor bases, we prove formulae for the Ricci and

scalar curvature in terms of a uni�ed Milnor basis.

Theorem 6.2.1. Let G a three-dimensional Lie group with left invariant metric, andg its Lie algebra. Let e ∈ G the identitey element, ge the scalar product on g ∼= TeGand (e1, e2, e3) a uni�ed Milnor basis with

[e1, e2] = λ3e3, [e2, e3] = λ1e1, [e3, e1] = λ2e2, λi ∈ {−1, 0, 1},

and

(g (ei, ej))1≤i,j≤3 =

g11 0 00 g22 00 0 g33

.

60

6.2. The case of a uni�ed Milnor basis

Then this basis diagonalises the Ricci tensor which is then given by

Rice (e1, e1) =g11

2g11g22g33

((λ1g11

)2−(λ2g22 − λ3g33

)2)Rice (e2, e2) =

g222g11g22g33

((λ2g22

)2−(λ3g33 − λ1g11

)2)Rice (e3, e3) =

g332g11g22g33

((λ3g33

)2−(λ1g11 − λ2g22

)2).

The scalar curvature is given by

Se =λ1λ2g33

+λ2λ3g11

+λ3λ1g22− 1

2

λ21g11g22g33

− 1

2

λ22g22g11g33

− 1

2

λ23g33g11g22

.

Proof. We want to use the formulae we proved in proposition 6.1.1. The basis(e1, e2, e3) is orthogonal, thus after normalisation we obtain a normalised Milnor basis

(e1, e2, e3) .

.=

(e1√g11

,e2√g22

,e3√g33

).

This already proves the diagonal form of the Ricci tensor with respect to the basis(e1, e2, e3). The new structure constants are

λ1 = ge ([e2, e3] , e1)

=1

√g11g22g33

ge ([e2, e3] , e1)

=1

√g11g22g33

g11λ1

and likewise

λ2 =1

√g11g22g33

g22λ2

λ3 =1

√g11g22g33

g33λ3.

For i = 1, 2, 3 this means

µi =1

2(λ1 + λ2 + λ3)− λi

=1

2

1√g11g22g33

(λ1g11 + λ2g22 + λ3g33 − 2λigii

).

61


Using proposition 6.1.1 we �nd

Rice (e1, e1) = Rice (√g11e1,

√g11e1)

= g11Rice (e1, e1)

= 2g11µ2µ3

=g11

2√g11g22g33

2

(λ1g11 − λ2g22 + λ3g33

)·(λ1g11 + λ2g22 − λ3g33

)=

g112g11g22g33

((λ1g11

)2−(λ2g22 − λ3g33

)2).

The expressions for Rice(e2, e2) and Rice(e3, e3) follow immediately by permutationof the indices.For the scalar curvature proposition 6.1.1 yields

Se = 2 (µ2µ3 + µ1µ3 + µ1µ2)

=1

2

1

g11g22g33

((λ1g11

)2−(λ2g22 − λ3g33

)2+(λ2g22

)2−(λ3g33 − λ1g11

)2+(λ3g33

)2−(λ1g11 − λ2g22

)2)=

1

2

1

g11g22g33

(2λ1λ2g11g22 + 2λ2λ3g22g33

+2λ3λ1g33g11 − λ21g112 − λ22g222 − λ23g332)

=λ1λ2g33

+λ2λ3g11

+λ3λ1g22− 1

2

λ21g11g22g33

− 1

2

λ22g22g11g33

− 1

2

λ23g33g11g22

.

62

7. Dynamics on Bianchi class A

models

In this chapter we further investigate the dynamics on Bianchi models using theresults of the last two chapters. We consider the four-dimensional manifold

G .

.= I ×G,

where G is a three-dimensional connected unimodular Lie group and I ⊂ R an openinterval. Let e ∈ G the identity element, ( tge )t∈I a family of scalar products onthe Lie algebra g ∼= TeG of G and ( tg )t∈I the family of left invariant Riemannianmetrics on G de�ned by left translation of these scalar products. Following the sameconstruction as in chapter 5 we endow G with the metric

g(t,p)

.

.= −dt2 + tgp

to obtain a four-dimensional Lorentz manifold. This choice of metric simpli�es theform of the scalar product on T(t,e)G ∼= R × g. With respect to this isomorphy, weobtain the representation

g(t,e)

=

(−1

tge

).

Just like in the earlier chapters, we distinguish between the geometry of G and G byoverlining geometric quantities of the four-dimensional manifold G.Our goal is to transform the Einstein �eld equations

Ein + Λ g = 8πT

into a system of (ordinary) di�erential equations for the family of scalar products( tge )t and to solve these in several special cases. We analyse perfect �uid models,including the vacuum case with cosmological constant Λ.

7.1. Einstein equations on Bianchi models

In earlier chapters we investigated the more general case with a three-dimensionalRiemannian manifold M instead of the Lie group G. The formulae we developed for

63

7. Dynamics on Bianchi class A models

the Ricci curvature apply to the Lorentz manifold (G, g ):

tι∗Ric

(E0, E0

)= tr tk +

∣∣ tk ∣∣2g= ˙tH −

∣∣ tk ∣∣2gtι∗ (Ric

(E0, E0

)+ S

)=

1

2

(tS +

(tH)2 − ∣∣ tk ∣∣2g )

tι∗Ric

(E0, ·

)= d tH − div tk

tι∗Ric (·, ·) = − tk + tRic + tH tk − 2 tW 2,

with tι : G→ G the embedding that maps G onto G(t) .

.= {t}×G ∼= G, tk the scalarsecond fundamental form of (G(t), tg ) in (G, g ), tH its mean curvature and tW 2

the square of the Weingarten map (see Chapter 5). The dot denotes di�erentiationwith respect to time. The normal direction E0

.

.= ∂∂t

is the vector �eld which, at thepoint (t0, p0) ∈M , is tangential to the curve

(−ε, ε)→ G

t 7→ (t0 + t, p0) .

Additionally, we found

tg = −2 tk .

Using the identity tH = tr ( tk ) + 2| tk |2g , see (5.11), the relation for the timelike

Ricci curvature Ric(E0, E0) immediately yields the four-dimensional scalar curvature

tι∗S = tS + tH 2 − 2 tr tk − 3

∣∣ tk ∣∣2g= tS + tH 2 − 2 tH +

∣∣ tk ∣∣2g .As we consider a left invariant metric, the second fundamental form and its trace,

the mean curvature, are left invariant as well. Thus, the mean curvature tH isconstant on the Lie group G for every t ∈ I and we �nd d tH = 0. The Einsteintensor Ein = Ric − 1

2S g thus simpli�es to

tι∗Ein

(E0, E0

)=

1

2

(tS +

(tH)2 − ∣∣ tk ∣∣2g )

tι∗Ein

(E0, ·

)= − div tk

tι∗Ein (·, ·) = − tk + tRic + tH tk − 2 tW 2

− 1

2

(tS + tH 2 − 2 tr tk − 3

∣∣ tk ∣∣2g ) tg

= − tk + tRic + tH tk − 2 tW 2

− 1

2

(tS + tH 2 − 2 tH +

∣∣ tk ∣∣2g ) tg .

64

7.1. Einstein equations on Bianchi models

According to de�nition 3.3.1 and remark 3.3.2, the Lorentz manifold (G, g , ∂∂t

) isa perfect �uid as soon as it satis�es

Ein = 8π((ρ+ p

)dt⊗ dt+ p g

),

where ρ, p ∈ E(M). In the case of a cosmological constant Λ and vacuum we haveρ = −p = Λ .We rearrange the formula given by the Einstein equation for two horizontal vector

�elds and �nd that the dynamics of Bianchi models which form a perfect �uid isgoverned by the system of di�erential equations

tg = −2 tk (7.1)

tk = tRic + tH tk − 2 tW 2 − 1

2

(tS + tH 2 − 2 tH +

∣∣ tk ∣∣2g − p)

tg . (7.2)

Remark 7.1.1. We notice that equation (7.2) contains the derivative of the scalarsecond fundamental form on both sides, as tH = tr ( tk ) + 2| tk |2g . We want to applythe Picard-Lindelöf theorem, therefore need to �nd a system of di�erential equationsof the form

∂

∂t

(tgtl

)= F

(t, tg , tl

).

This can be achieved by setting

tl .

.= tk − tr tk · tg = tk − tH tg ,

as one easily computes tl = tk − tH tg + 2 tH tg .

Rearranging the Einstein tensor for time-like and mixed vector �elds results in thefollowing two constraint equations

0 = div tk (7.3)

0 =1

2

(tS + tH 2 −

∣∣ tk ∣∣2g )− ρ. (7.4)

Additionally, any perfect �uid ( ∂∂t, ρ, p) satis�es the energy equation

∂

∂tρ = −

(ρ+ p

)div

∂

∂t

and the force equation (ρ+ p

)∇ ∂

∂t

∂

∂t= − grad⊥ p,

which arise from div Ein = 0, see proposition 3.3.3. The latter is trivially satis�ed asthe pressure p is supposed to be left invariant thus constant on every horizontal slice

65


{t} ×G, and ∇ ∂∂t

∂∂t

vanishes due to equation (5.4). Choose an arbitrary basis ( ∂∂t

=

e0, e1, e2, e3) of T(t0,p0)G with e1, e2, e3 ⊥ e0 and smooth local extensions satisfyingEi ⊥ E0 and

∇vEi = 0

for every v ∈ T(t0,p0)G horizontal to G(t0), i = 1, 2, 3, as we did in (5.1).Computingthe divergence div ∂

∂treveals

div∂

∂t(t0, p0) = gαβ g

(t0,p0)

(∇eαE0, eβ

)= gij t0gp0

(∇eiE0, ej

)= gij

(eit0g(E0, Ej

)− t0g(t0,p0)

(e0,∇eiEj

))= −gij t0gp0

(e0,∇eiEj + t0II(t0,p0) (ei, ej)

)= −gijkij= − t0Hp0

where we used (5.4): ∇E0E0 = 0. We have thus found another constraint equation

ρ =(ρ+ p

)tH . (7.5)

We now want to further analyse the �ow given by these equations on Bianchi class Amodels, i. e. with G being a connected unimodular Lie group. To that end, we �rstexplore under which conditions the three constraint equations are conserved by this�ow. This is done without the restriction to unimodular Lie groups.

7.2. Conserved constraint equations

We consider the vacuum case with cosmological constant Λ = 0, i. e. both the pressurep and the energy ρ vanish. Equations (7.3) and (7.4) then simplify to

0 = div tk

0 =1

2

(tS + tH 2 −

∣∣ tk ∣∣2g ) .The third constraint, equation (7.5), is trivially satis�ed.Wald shows in [Wal84, chap.s 7.2, 10.2] that in this the �ow given by equations (7.1)

and (7.2) preserves these two constraint equations. A convenient choice of "gauge",i. e. coordinate system, transforms the Einstein equation into a quasilinear, diagonal,second order hyperbolic system. General theory of partial di�erential equations thenstates that the General Theory of Relativity possesses an initial value formulation.This means that for given initial values satisfying the constraint equations, thereexists a solution of the �ow, and this solution satis�es the constraint equations for alltimes. We state this result for Bianchi models.

66

7.3. div tk and diagonalisation on Bianchi class A

Theorem 7.2.1. Let G a three-dimensional Lie group. For a given family ( tge )t∈Iof scalar products on its Lie algebra g ∼= TeG set tg the induced left invariant metric,

(G .

.= I × G, g(t,p)

.

.= −dt2 + tgp ) and tk the scalar second fundamental form of

{t} ×G in G. Consider the �ow

tg = −2 tktk = tRic + tH tk − 2 tW 2

with constraints

0 = div tk

0 =1

2

(tS + tH 2 −

∣∣ tk ∣∣2g ) .Then for given initial conditions t0ge , t0k e compatible with these constraints, there isa unique solution ( tg ) to the �ow, and it conserves the constraints.

In the non-vacuum case, existence of an initial value formulation critically dependson the stress-energy tensor T . The case of perfect �uids is further investigated in[HE73]. For appropriate choices of p = p(ρ) an initial value formulation is known toexist.We do not give the proof of these statements here, as this would require a de-

tailed introduction into the theory of di�erential equations. It is, however, ratherstraightforward to show that the constraint equation

0 = div tk

is preserved on Bianchi class A models, even in the non-vacuum case. We will carryout this proof in the next paragraph.


In the following, we restrict ourselves to Bianchi class A models, i. e. where the Liegroup G is unimodular. We not only show that the constraint

0 = div tk

is preserved by the �ow de�ned by (7.1) and (7.2), but also that it allows substantialsimpli�cation, as it reduces the dimension of the solution space.We start by studying the conditions that ensure the existence of a basis which

orthogonalises several mappings at the same time.

Proposition 7.3.1. Let V a �nite-dimensional euclidean K-vector space with innerproduct 〈·, ·〉, K = C or R, and A,B : V → V self-adjoint linear operators. Thenthere exists an orthonormal basis of eigenvectors for both A and B if and only if Aand B commute, i. e. A ◦B = B ◦ A.

67


Proof. Suppose there is a basis (ei)i=1,...,n of V satisfying for every i = 1, . . . , n

A (ei) = λiei

B (ei) = µiei

with λi, µi ∈ K the eigenvalues of A resp. B. We see

A ◦B (ei) = λiµiei = µiλiei = B ◦ A (ei)

and therefore A ◦B = B ◦ A follows from linearity.Conversely, suppose A ◦ B = B ◦ A. Using the spectral theorem for self-adjoint

operators we decompose V into the direct sum of eigenspaces of A

V =r⊕i=1

Eig (A;λi) ,

where λi ∈ R are the eigenvalues of A. These eigenspaces are invariant under B, asfor an eigenvector v ∈ Eig(A;λi) we have

A (B (v)) = A ◦B (v)

= B ◦ A (v)

= λi ·B (v) ,

thus B(v) ∈ Eig(A;λi). Using the spectral theorem for self-adjoint operators againfor every B|Eig(A;λi), i = 1, . . . n, we �nd an orthonormal basis (v

(i)1 , . . . , v

(i)ni ) of eigen-

vectors for B|Eig(A;λi) : Eig(A;λi)→ Eig(A;λi). Then (v(1)1 , . . . , v

(1)n1 , v

(2)2 , . . . , v

(r)nr ) is

the desired basis of V .

We now apply this proposition to the �ow de�ned by the Einstein equations on Bianchimodels of class A. We recall that the Lie algebra of a unimodular Lie group alwayspossesses a Milnor basis, see proposition 4.1.11. We have seen in chapter 6 that thecovariant derivate relative to these basis vectors is essentially the cross product:

∇eiv = µiei × v

for every v ∈ g, where × denotes the cross product relative to the orentation of theMilnor basis. Once again we used the identi�cation between the tangent space TeGand the space of left invariant vector �elds g of G, see Remark 2.0.15. This leads toeasy expressions of the divergence of the scalar second fundamental form div tk . Theconstraint equation div tk = 0 then ensures the existence of a basis with simultaneousdiagonalisation properties.

Proposition 7.3.2. Let G a three-dimensional Lie group, ( tge )t∈I a family of scalarproducts on its Lie algebra g ∼= TeG and ( tg )t∈I the family of induced left invariant

68


metrics. Consider (G .

.= I ×G, g(t,p)

.

.= −dt2 + tgp ) and let (e1, e2, e3) a normalised

Milnor basis of TeG, with

[e1, e2] = λ3e3, [e2, e3] = λ1e1, [e3, e1] = λ2e2, λi ∈ R.

Then the divergence of the scalar second fundamental form tk of (G(t), tg ) in (G, g )satis�es

div tk (e1) = (λ3 − λ2) tk (e2, e3)

div tk (e2) = (λ1 − λ3) tk (e3, e1)

div tk (e3) = (λ2 − λ1) tk (e1, e2) .

Proof. The basis (e1, e2, e3) is orthonormal, therefore gkl = δkl and we compute fori = 1, 2, 3 and Ei ∈ X(G) the left invariant extension of ei:

div tk (ei) = gkl(∇ek

tk)

(ei, el)

= gkl(ek

tk (ei, el)− tk e (∇ekei, el)− tk e (ei,∇ekel))

= δkl(ek

tk (Ei, El)− tk e (∇ekei, el)− tk e (ei,∇ekel)).

The �rst term vanishes due to left invariance of the metric and thus of the second fun-damental form. In the proof of theorem 6.1.1, we have shown that ∇eiej = µiei × ejwith µi = 1

2(λ1 + λ2 + λ3)− λi. This means δkl∇ekel = 0 which yields

δkl tk (ei,∇ekel) = 0

due to linearity of tk , and therefore

div tk (ei) = −3∑

k=1

tk (µkek × ei, ek) .

The three individual expressions are

div tk (e1) = − tk (µ1e1 × e1, e1)− tk (µ2e2 × e1, e2)− tk (µ3e3 × e1, e3)

= 0 + µ2tk (e3, e2)− µ3

tk (e2, e3)

= (µ2 − µ3)tk (e2, e3)

= (λ3 − λ2) tk (e2, e3)

and similarly

div tk (e2) = (λ1 − λ3) tk (e3, e1)

div tk (e3) = (λ2 − λ1) tk (e1, e2) .

69


Recall that the Weingarten operator of (G(t), tg ) in (G, g ) de�ned by tW : X(G)→X(G), X 7→ W (X) = − tanG(t)∇X

∂∂t

is the mapping dual to the scalar second fun-damental form relative to the metric g , see (1.11):

g(

W ∂∂tX, Y

)= g

(tII (X, Y ) ,

∂

∂t

)= tk (X, Y )

for every X, Y ∈ X(M). We restrict the Weingarten map to the Lie algebra g ∼= TeG.

Theorem 7.3.3. Let G a three-dimensional unimodular Lie group, ( tge )t∈I a familyof scalar products on its Lie algebra g ∼= TeG and ( tg )t∈I the family of induced left

invariant metrics. Consider (G .

.= I ×G, g .

.= −dt2 + tg ) and tk the scalar second

fundamental form of {t} ×G in G. Then the following statements are equivalent:

i) There exists a uni�ed Milnor basis of g which diagonalises the scalar secondfundamental form t0k at t0 ∈ I.

ii) There exists an orthonormal basis of g which simultaneously diagonalises theWeingarten operator t0W : g → g at t0 ∈ I, and the unique map L : g → gsatisfying [u, v] = L(u× v).

iii) div t0k = 0 at t0 ∈ I.

In order to prove this theorem, we need the following properties of the Weingartenmap and the Lie bracket.

Lemma 7.3.4. Let G a three-dimensional unimodular Lie group, ( tge )t∈I a family ofscalar products on its Lie algebra g ∼= TeG and tg the induced left invariant metric.

Consider (G .

.= I×G, g .

.= −dt2+ tg ). Then both the Weingarten operator tW : g→g, t ∈ I, and the unique map L : g→ g satisfying [u, v] = L(u× v) are self-adjoint.

Proof. Both tW and L are linear mappings. Let x, y ∈ g. We know that theWeingarten operator satis�es

g(t,e)

(tWe (x) , y

)= g

(t,e)

(tIIe (x, y) ,

∂

∂t

)= g

(t,e)

(x, tWe (y)

).

By lemma 4.1.6 the mapping L is self-adjoint if and only if the Lie group is unimo-dular.

Proof of proposition 7.3.3 It is obvious by de�nition that diagonalisation of theWeingarten operator and diagonalisation of the second fundamental form are equiv-alent. A basis is a normalised Milnor basis if and only if it is orthonormal anddiagonalises L. Therefore, equivalence between the �rst two assertions results fromscaling the basis, see lemma 4.1.9 and the subsequent remark.

70


As both t0W and L are self-adjoint, we can show the remaining equivalence usingproposition 7.3.1. We choose a normalised Milnor basis (e1, e2, e3) in accordance withproposition 4.1.11. The matrix representation of L in this basis is

L←→

λ1 0 00 λ2 00 0 λ3

with λi ∈ R, i = 1, 2, 3. The Weingarten operator t0We is represented by the matrix(kij)i,j=1,2,3, as the following computation shows:

g(t0,e)

(ei,

t0We (ej))

= g(t0,e)

(t0IIe (ei, ej) ,

∂

∂t

)= t0k e (ei, ej) .

Our goal is to show that these two matrices commute if and only if div t0k = 0.Recall (proposition 7.3.2) that for our normalised Milnor basis vectors, the divergenceis given by

div t0k (e1) = (λ3 − λ2) t0k (e2, e3)

div t0k (e2) = (λ1 − λ3) t0k (e3, e1)

div t0k (e3) = (λ2 − λ1) t0k (e1, e2) .

First case: λ1 = λ2 = λ3. The matrix representation of L is a scalar multiple of theidentity matrix and thus commutes with every matrix. On the other hand, div t0kvanishes on all three basis vectors, and by linearity on the whole Lie algebra g.Second case: λ1 = λ2 =.

. λ 6= λ3. The divergence div t0k (e3) vanishes invariably.The divergence div t0k vanishes on the whole Lie algebra if and only if t0k e(e1, e3) =0 = t0k e(e2, e3). Let's turn to the matrix representations. Easy computation shows

L ◦ t0We ←→

λk11 λk12 λk13λk12 λk22 λk23λ3k31 λ3k32 λ3k33

t0We ◦ L←→

λk11 λk12 λ3k13λk12 λk22 λ3k23λk13 λk23 λ3k33

where we abbreviated kij = t0k e(ei, ej). We see that t0We and L commute if andonly if t0k e(e1, e3) = 0 = t0k e(e2, e3), i. e. if and only if div t0k = 0.Third case: λi pairwise distinct. Computing the matrix representations and ar-

guments similar to the ones in the second case show that t0We and L commute ifand only if the matrix ( t0k e(ei, ej))i,j=1,2,3 is diagonal, which is satis�ed if and onlyif div t0k = 0.

71


We are now able to prove that the constraint equation div tk = 0 is preserved underthe �ow

tg = −2 tktk = tRic + tH tk − 2 tW 2 − 1

2

(tS + tH 2 − 2 tH +

∣∣ tk ∣∣2g − p)

tg .

In other words we show that as soon as div t0k = 0 for a given t0ge , t0k e at time t0,div tk = 0 remains true for any t ∈ I, provided ( tge ), ( tk e ) comply with the �owequation.Let div t0k = 0 for a given time t0 ∈ I. According to theorem 7.3.3 we �nd a basis

(e1, e2, e3) of T(e,t0)G ∼= R×g with the uni�ed Milnor property which diagonalises thescalar second fundamental form. More explicitely,

i) (e1, e2, e3) diagonalises the scalar product t0ge ;

ii) (e1, e2, e3) diagonalises the scalar second fundamental form t0k e ;

iii) (e1, e2, e3) simpli�es the Lie bracket [·, ·] to

[e1, e2] = λ3e3, [e2, e3] = λ1e1, [e3, e1] = λ2e2, λi ∈ {−1, 0,+1}.

In this basis, the given initial conditions t0ge , t0k e are represented by diagonal ma-trices

t0ge ←→

G1,0

G2,0

G3,0

, t0k e ←→

K1,0

K2,0

K3,0

,

with Gi,0 ∈ R>0, Ki,0 ∈ R.We now restrict the �ow to the scalar products tge relative to which the basis

remains orthogonal. That is, we consider all geometric quantities in matrix represen-tations in the basis (e1, e2, e3) and restrain the �ow to scalar products

tge ←→

G1 (t)G2 (t)

G3 (t)

.

As tge = −2 tk e , the scalar second fundamental form inherits this diagonal structure

tk e ←→

K1 (t)K2 (t)

K3 (t)

.

The same is true for tk e , as the following arguments show: Any uni�ed Milnor basisof g diagonalises the Ricci curvature tRic , see theorem 6.2.1. The scalar product tge

72


and the scalar second fundamental form tk e are diagonalised by the basis, thereforethe Weingarten operator tW satis�es

tWe2 (ei, ej) = tge

(tWe (ei) ,

tWe (ej))

= tge(gkm tk e (ei, ek) em, g

ln tk e (ej, el) en)

= gkm tk e (ei, ek) gln tk e (ej, el) gmn

= gkl tk e (ei, ek)tk e (ej, el) .

If i 6= j, we �nd tWe2(ei, ej) = 0, thus tWe

2 is diagonal as well. As tk e is composedof diagonalised terms, this proves the diagonal shape of tk e . In total, we �nd thatboth tge and tk e are diagonalised by the basis (e1, e2, e3) as soon as tge is diagonal.Now suppose we start with initial conditions t0ge , t0k e which are diagonal with

respect to (e1, e2, e3), i. e. which satisfy the divergence constraint. We consider thesystem of di�erential equations restricted to diagonalised scalar products. As both�ow equations inherit the diagonal shape, we can apply the Picard-Lindelöf theoremto each of the three diagonal entries (recall also remark 7.1.1). It then ensures the(unique) existence of a solution tge with diagonal shape. Let us view this solution inthe general context, where we drop the restriction. The family ( tge )t∈I still satis�esthe �ow equations as well as the initial condition. It is thus the unique solution tothe initial value problem.Starting with initial condition div t0k = 0, we have seen that the basis (e1, e2, e3)

remains orthogonal with respect to every tge , t ∈ I, and the scalar second fundamen-tal form tk e is diagonalised by this basis as well. The Lie bracket [·, ·] is invariantunder any change of t, thus the basis has the uni�ed Milnor property at all times.Using proposition 7.3.3, this means div tk = 0 at all times. In total, we have provedthe following proposition for Bianchi class A models.

Proposition 7.3.5. Let G a three-dimensional unimodular Lie group. For a givenfamily ( tge )t∈I of scalar products on its Lie algebra g ∼= TeG set tg the induced left

invariant metric, (G .

.= I ×G, g(t,p)

.

.= −dt2 + tgp ) and tk the scalar second funda-

mental form of {t} ×G in G. Then the �ow de�ned by

tg = −2 tktk = tRic + tH tk − 2 tW 2 − 1

2

(tS + tH 2 − 2 tH +

∣∣ tk ∣∣2g ) tg

preserves the constraint equation

0 = div tk .

Furthermore, this constraint restricts the �ow to scalar products tge which are diag-onal with respect to a uni�ed Milnor basis of g.

73


7.4. Bianchi class I in vacuum

We know that the General Theory of Relativity has an initial value formulationin vacuum. In our scenario of Bianchi models this means that out of a given Liegroup G and suitable initial conditions t0ge , t0k e we can construct a four-dimensionalspacetime

(G, g

)which satis�es the vacuum Einstein equation with Λ = 0.

We want to carry out this construction in the simplest case of Bianchi type I andvacuum. Therefore set G .

.= R3, which is a Lie group when equipped with additionas the group operation. We construct all left invariant metrics

g(t,p)

= −dt2 + tgp

on G = R × G, such that the resulting four-dimensional semi-Riemannian mani-fold satis�es the Einstein vacuum equation with vanishing cosmological constant:Ein = Ric − 1

2S g = 0.

By de�nition, all structure constants vanish, see 4.2.1. The Ricci curvature andscalar curvature of G are determined by theorem 6.1.1 and thus vanish as well:

tRic = 0tS = 0.

We want to solve the vacuum case which is equivalent to considering Ric = 0, ac-cording to Proposition 3.2.4. Therefore, the �ow equations simplify to

tg = −2 tk (7.6)tk = tH tk − 2 tW 2 (7.7)

with constraints

0 = div tk (7.8)

0 =1

2

(tH 2 −

∣∣ tk ∣∣2g ) . (7.9)

We start by remarking that the relations for the four-dimensional Ricci curvature wefound in theorem 5.3.3 yield an evolution equation for the mean curvature:

tH = Ric(E0, E0

)+∣∣ tk ∣∣2g . (7.10)

As Ric = 0 in the vacuum case, equation 7.9 yields

tH =∣∣ tk ∣∣2g = tH 2.

Together with the initial condition t0H = H0 this leaves two possibilities for the meancurvature, namely

tH = 0 or tH = − 1

t− a

74


for a = 1H0

+ t0, which we will treat seperately. Obviously, the second function showsa singularity at time 1

H0+t0. Therefore, only one of the two branches of the hyperbola

gives a useful solution, depending on the sign of H0. Applying a translation in time,we can assume a = 0 without loss of generality, when specifying whether we considert > 0 or t < 0.First case: tH = −1

t. We have shown that the constraint equations are conserved

by the �ow, thus in particular div tk = 0. In accordance with theorem 7.3.5, we�nd a basis (e1, e2, e3) of TeG which is orthogonal and diagonalises the scalar secondfundamental form at all times t. Thus, the matrix representation of both the scalarproduct tge and the scalar second fundamental form tk e is diagonal:

tge ←→

G1 (t)G2 (t)

G3 (t)

and tk e ←→

K1 (t)K2 (t)

K3 (t)

.

In particular, in this basis the inverse of tge is given by the matrix with inverse entries:

tge −1 ←→

G−11 (t)G−12 (t)

G−13 (t)

.

The �ow equations (7.6) and (7.7) give two di�erential equations for the matrixentries:

Gi = −2Ki,

Ki = tHKi − 2G−1i Ki2,

where i = 1, 2, 3.The derivative of the inverse function is(

Gi−1). = −Gi

−2Gi.

Consider the quantity Gi−1Ki. Using the two �ow equations, we �nd:(

Gi−1Ki

).

=(Gi−1).Ki +Gi

−1 d

dtKi

= −Gi−2GiKi +Gi

−1 ( tHKi − 2Gi−1Ki

2)

= 2Gi−2Ki

2 + tHGi−1Ki − 2Gi

−2Ki2

= tHGi−1Ki.

With tH = −1t, we �nd ∂

∂t(Gi−1Ki) = −1

tGi−1Ki, thus there are constants pi ∈ R,

i = 1, 2, 3, such that

Gi−1Ki =

pit.

75


Because of the �rst �ow equation, this is equivalent to Gi = −2pitGi and �nally yields

Gi = ci+t−2pi t > 0,

Gi = ci− (−t)−2pi t < 0,

with constants ci+, ci− ∈ R, and t > 0 if and only if H0 < 0. The scalar secondfundamental form is thus given by

Ki = −1

2Gi = ci+pit

−2pi−1 t > 0,

Ki = −1

2Gi = −ci−pi (−t)−2pi−1 t < 0

The constants pi, ci cannot be chosen at will but subject to restrictions. First we havea relation for the mean curvature:

−1

t= tH =

3∑i=1

Gi−1Ki =

3∑i=1

pit,

which is valid in both cases, and thus

3∑i=1

pi = −1.

Additionally, the constraint equation (7.9), expressed in coordinates, yields(3∑i=1

Gi−1Ki

)2

=3∑i=1

Gi−2Ki

2,

which is equivalent to (3∑i=1

pi

)2

=3∑i=1

p2i .

We can scale the basis (e1, e2, e3) without losing the simultaneous diagonalisation ofthe scalar product and the scalar second fundamental form. We can thus choose ourbasis to be orthonormal at t = 1 resp. t = −1. By doing so, we �x

1 = Gi (±1) = ci±,

which is the last of the restrictions on pi, ci±.The resulting metrics are called the Kasner metrics. They are de�ned by the scalar

products on T(t,e)G

g(t,e)

= −dt2 +3∑i=1

t−2pidx2i t > 0

g(t,e)

= −dt2 +3∑i=1

(−t)−2pi dx2i t < 0

76


with∑3

i=1 pi = −1 and∑3

i=1 p2i = (

∑3i=1 pi)

2 = 1, where dxi denotes the 1-form dualto ei.The Kasner metrics were constructed such that they satisfy equations (7.6), (7.8)

and (7.9). Direct calculation shows that equation (7.7) is satis�ed as well. Thus, theKasner metrics are not only the candidates for solutions, but each of them gives anexample of a Bianchi type I vacuum spacetime.Second case: tH = 0. Just as we did in the �rst case, we choose a basis (e1, e2, e3)

of TeG which is orthogonal and diagonalises the scalar second fundamental form atall times t. The quantity Gi

−1Ki satis�es(Gi−1Ki

).

= tHGi−1Ki = 0,

therefore

Gi−1Ki = qi

with constants qi ∈ R for i = 1, 2, 3. Using the �rst �ow equation, this is equivalentto Gi = −2qiGi and yields

Gi = di exp (−2qit)

Ki = diqi exp (−2qit)

with constants di ∈ R.The mean curvature vanishes, therefore we �nd

0 = tH =3∑i=1

Gi−1Ki =

3∑i=1

qi.

With this result, the constraint equation (∑3

i=1Gi−1Ki)

2 =∑3

i=1Gi−2Ki

2 simpli�esto

0 =

(3∑i=1

qi

)2

=3∑i=1

q2i ,

which means that all qi vanish:

qi = 0, i = 1, 2, 3.

We choose the basis to be orthonormal at time t = 0 (and thus at all times), which�xes

1 = Gi (0) = di.

The resulting metric is the Minkowski metric de�ned by

g(t,p)

= −dt2 +3∑i=1

dx2i .

77


As the scalar second fundamental form vanishes, equations (7.6) to (7.9) are satis�ed,thus the Minkowski metric is indead a solution to the initial value problem.On the Lie algebra R3, the group operation is addition. Therefore, all tangent

spaces can be trivially identi�ed, and the left invariant metrics are given by identicalformulae at every point of the Lie group. Altogether, we have shown the followingtheorem.

Theorem 7.4.1. Up to isometry, the maximal solutions of Bianchi type I in a vacuumuniverse are

i) the Minkowski metric, de�ned by the scalar products

g(t,p)

= −dt2 +3∑i=1

dx2i ;

ii) the Kasner metrics, de�ned by the scalar products

g(t,p)

= −dt2 +3∑i=1

t−2pidx2i t > 0

g(t,p)

= −dt2 +3∑i=1

(−t)−2pi dx2i t < 0

with constants p1, p2, p3 ∈ R that satisfy∑3

i=1 pi = −1 and∑3

i=1 p2i = 1.

The 1-forms dxi are the 1-forms dual to an orthonormal basis of the three-dimensionaltangent space TpG.

We want to compare this result with the Robertson-Walker perfect �uids we dis-cussed in chapter 3. Recall that the metric of a Robertson-Walker spacetimeM(K , f) =I ×f M is given by

g(t,p)

= −dt2 + f 2 (t) gp .

We de�ned the following physical singularities:

• Big bang: t− > −∞, and f(t)→ 0, f(t)→∞ for t→ t−;

• Big crunch: t+ <∞, and f(t)→ 0, f(t)→ −∞ for t→ t+.

The Minkowski and Kasner metrics are given by the scalar products

g(t,p)

= −dt2 +3∑i=1

fi (t) dx2i ,

with several restrictions for the functions fi. Therefore, the geometry of the four-dimensional spacetime is de�ned by three parameters instead of just one. In the

78


Minkowski case we �nd fi ≡ 1 for i = 1, 2, 3. This metric is de�ned for all timest ∈ R. In particular, there are no singularities.In the case of the Kasner metrics we have fi(t) = t−2pi with

∑3i=1 pi = −1 and∑3

i=1 p2i = 1. We see that if we suppose t > 0 and

−1

2< pi < 0,

then we �nd a singularity at t = 0 with

fi (t)→ 0, fi (t)→∞, t→ 0.

If all three constants p1, p2, p3 satisfy this relation, this can be seen as the equivalent ofa big bang. In the other cases there are no apparent parallels between the propertiesof fi and the de�nition 3.5.3 of big bang and big crunch. Note furthermore that notall three constants pi have to comply to the restriction. If this happens, in some sensethe universe exhibits a big bang in only some of the three spatial directions.

Remark 7.4.2. In the Robertson-Walker models M(K , f) we �nd that the scalar

mean curvature of {t} ×M satis�es tH = −32ff(t). This becomes clear by the fol-

lowing: Choose a basis (e1, e2, e3) of TpM , p ∈ M , that complies with the metric,i. e. tgp(ei, ej) = δijf(t). The Robertson-Walker models are special cases of the onesdiscussed in chapter 5, thus we can use the formula tg = −2 tk and see

tH = gijkij

= −1

2gij gij

= −1

2

3∑i=1

2ff

f 2

= −3f

f.

The behaviour of the scaling function f leads to the following properties of tH :

• As f > 0 by de�nition, we have f(t) > 0 if and only if tH < 0.

• If the Robertson-Walker models has a big bang at t−, then tH → −∞ fort→ t−.

• If the Robertson-Walker models has a big crunch at t+, then tH → +∞ fort→ t+.

At the end of chapter 3 we argued that on more general spacetimes , the de�nitionof singularities should involve the mean curvature tH and the volume form vol( tg )of the submanifolds {t} ×M . We state this de�nition in the case of unimodular Liegroups.

79


De�nition 7.4.3. Let G a three-dimensional unimodular Lie group. Consider afamily ( tge )t∈I , I = (t−, t+) maximal, of scalar products on its Lie algebra g ∼= TeG,tg the induced left invariant metric, such that the spacetime (G .

.= I × G, g(t,p)

.

.=

−dt2 + tgp , ∂∂t) is a perfect �uid ( ∂∂t, ρ, p). An initial singularity t− is a big bang if

vol(tg)→ 0, tH

(vol(tg)

vol(t0g)) 1

3

→ −∞ for t→ t−.

A �nal singularity t+ is a big crunch if

vol(tg)→ 0, tH

(vol(tg)

vol(t0g)) 1

3

→∞ for t→ t+.

In Chapter 3 we proved that on Robertson-Walker spacetimes the two de�nitionsof big bang and big crunch coincide.In the Minkowski case the mean curvature vanishes for all times: tH ≡ 0. This

model possesses no singularities. In the Kasner cases, the mean curvature satis�es

tH = −1

t

after a convenient time translation, thus a singularity at t = 0. In an expandinguniverse the Hubble constant is positive, thus in accordance with the Robertson-Wal-ker model we suppose that the mean curvature t0H at present time t0 is negative.Therefore, only the branch of t 7→ −1

twith negative values, i. e. t > 0, is of importance

to us, and the singularity occuring at t− = 0 is an initial singularity. The volumeform can be computed using the results of the previous construction. We know√

det gij =√G11 (t)G22 (t)G33 (t)

=√t−2(p1+p2+p3)

=√t2

= t,

therefore, when considering time t = 1 we �nd

vol(tg)

= t vol(1g).

In total, this yields

vol(tg)→ 0 t→ t− = 0

and

tH

(vol(tg)

vol(t0g)) 1

3

= −1

tt13 = −t−

23 → −∞ t→ t− = 0,

thus t− = 0 is a big bang.

80

7.5. Generalisation to Bianchi class A models in vacuum

Remark 7.4.4. Consider the vacuum Bianchi type I models with vanishing cosmo-logical constant and suppose that the initial value of the mean curvature is negative:t0H = H0 < 0. Then the following properties ensue.

i) The mean curvature is negative at all times.

ii) The mean curvature increases monotonically.

iii) The mean curvature vanishes asymptotically: tH ↗ 0 as t→∞.

iv) The universe admits an initial singularity at ∞ < t− ≤ t0 with tH → −∞ ast→ t−.

v) The initial singularity t− is a big bang.

In the following chapter we show that the �rst four properties generalise to Bianchiclass A models with non-positive scalar curvature.

7.5. Generalisation to Bianchi class A models in

vacuum

In the case of Bianchi type I in vacuum, we are able to state an explicit solution tothe initial value problem. This is due to the simpli�cation we gain as the Ricci andscalar curvature vanish on G = R3. We want to investigate the remaining Bianchicosmologies and turn our attention to Bianchi class A models, where we can applythe results from chapter 6.

Bianchi type λ1 λ2 λ3 scalar curvature SI 0 0 0 = 0II + 0 0 < 0V I0 + - 0 < 0V II0 + + 0 ≤ 0V III + + - ≤ 0IX + + + ∈ R

Table 7.1.: Bianchi class A. The signs of the structure constants are given with respectto a Milnor basis. Theorem 6.2.1 yields the scalar curvature.

In table 7.1 we list the di�erent types of Bianchi class I with the sign of theirrespective scalar curvature we computed using theorem 6.2.1. Apart from type IX,all Bianchi class A models have non-positive scalar curvature. For type V II0 and

81


V III, i. e. λ1 = λ2 = +1, we used the following identity:

S =λ1λ2g33

+λ2λ3g11

+λ3λ1g22− 1

2

λ21g11g22g33

− 1

2

λ22g22g11g33

− 1

2

λ23g33g11g22

=2

g11g22g33

(2g11g22 + 2λ3g22g33 + 2λ3g11g33 − g112 − g222 − λ23g332

)=

2

g11g22g33

(− (g11 − g22)2 + 2λ3g22g33 + 2λ3g11g33 − λ23g332

).

As we consider the vacuum case, we have the di�erential equation for the meancurvature (7.10)

tH =∣∣ tk ∣∣2g

and the constraint equation (7.4)

0 =1

2

(tS + tH 2 −

∣∣ tk ∣∣2g ) .Combining these two, we �nd that all Bianchi class A models except some of typeIX satisfy:

tH = tS + tH 2 ≤ tH 2. (7.11)

This inequality gives us information on the development of the sign of the meancurvature tH , as the following proposition shows.

Proposition 7.5.1. Consider Bianchi class A models with non-positive scalar cur-vature, i. e. all Bianchi class A models except those type IX models where the scalarcurvature admits positive values. Let t0,H0 ∈ R and t 7→ ( tg , tk ) the maximal solu-tion of the Einstein vacuum equation with cosmological constant Λ = 0


with constraints

0 = div tk

0 =1

2

(tS + tH 2 −

∣∣ tk ∣∣2g )such that the initial value is t0H = H0. Then we �nd the following development ofthe mean curvature tH :

• If H0 < 0, then tH < 0 for t ≥ t0.

• If H0 > 0, then tH > 0 for t ≤ t0.

82


• If H0 = 0, then tH ≥ 0 for t ≤ t0, andtH ≤ 0 for t ≥ t0.

For the proof, we use the Lemma of Gronwall, see [Gro19].

Theorem 7.5.2. (Lemma of Gronwall) Let Φ : [a, b] → R di�erentiable and h :[a, b]→ R continuous such that

Φ′ ≤ hΦ.

Then for every t ∈ [a, b]

Φ (t) ≤ Φ (a) exp

(∫ t

a

h (s) ds

).

Corollary 7.5.3. In particular, Φ′ ≤ hΦ with Φ(a) = 0 implies Φ(t) ≤ 0 for allt ∈ [a, b].Using the re�ection t 7→ −t, the Lemma can be applied to the case Φ′ ≤ hΦ with

Φ(b) = 0. It then implies Φ(t) ≥ 0 for all t ∈ [a, b].

Proof of proposition 7.5.1. We discussed the solutions of tH = tH 2 in the previousparagraph:

tH = 0 or tH = − 1

t− a.

Suppose that H0 < 0 and choose a such that the function tH : (a,∞)→ (−∞, 0), t 7→− 1t−a admits the same initial value: t0H = − 1

t0−a = H0. The geometric quantities aresupposed to be smooth functions. As(

tH − tH)

.

= tH − t ˙H

≤ tH 2 − tH 2

=(tH + tH

)(tH − tH

),

and t0H − t0H = 0, we can apply the Lemma of Gromwall to Φ(t) .

.= tH − tH ,

h(t) .

.= tH + tH . This yields tH − tH ≤ 0 for t ≥ t0, thus we �nd

tH ≤ tH < 0 ∀ t ≥ t0.

Similar arguments prove the other two cases.

In the last proposition we used an inequality bounding tH from above. We can alsobound it from below, as we will show now.Let (e1, e2, e2) a basis of the tangent space TpM . The coordinate expressions of the

mean curvature tHp and the tensor | tk p |2g are given by∣∣ tk p

∣∣2g = tr tr23

(tk p ⊗ tk p

)= gmngklknkkml

83


and

tHp = gklkkl,

see chapter 5. Choose an orthonormal basis which diagonalises the scalar secondfundamental form tk p . Note that this is possible in the general case we discussed inchapter 5, thus in particular applies to all Bianchi models, not only class A. In thisbasis we calculate ∣∣ tk p

∣∣2g = δmnδklknkkml = k11

2 + k222 + k33

2,

as both tgp and tk p are diagonal. From the binomial theorem we deduce the inequal-ity a2 + b2 ≥ 2|ab| and compute

tH 2p =

(δklkkl

)2= (k11 + k22 + k33)

2

= k112 + k22

2 + k332 + 2k11k22

+ 2k11k33 + 2k22k33

≤ 3(k11

2 + k222 + k33

2)

= 3∣∣ tk p

∣∣2g .

This leads to the required lower bound

tH =∣∣ tk ∣∣2g ≥ 1

3tH 2

and immediately proves monotony:

Corollary 7.5.4. Consider M = I × M with metric g(t,p)

= −dt2 + tgp and set

E0 = ∂∂t∈ X(M) the normalised normal direction. Then the mean curvature tH

increases monotonically: tH ≥ 0.

The solutions of tH = 13tH 2 are

tH = 0 or tH = − 3

t− a.

We remark that the latter function shows the same qualitative behaviour as thefunction t 7→ −1

twe analysed above. It has two separate branches with vertical

asymptotes where both the function and its �rst derivative diverge. Using the samereasoning as in the proof of the preceding proposition, we �nd that the mean curvaturecannot exist at all times but has a �nite past or future, depending on the initialconditions.

84


Proposition 7.5.5. Consider M = I ×M with metric g(t,p)

= −dt2 + tgp and set

E0 = ∂∂t∈ X(M) the normalised normal direction. Let t0,H0 ∈ R and t 7→ tH on

I = (t−, t+) such that

tH ≥ 1

3tH 2

with initial value t0H = H0 and I chosen maximal. Then we �nd the followingdevelopment of the mean curvature tH :

• If H0 < 0, then t0 + 3H0≤ t− ≤ t0. The mean curvature satis�es tH < 0 for

t ≤ t0 and tH → −∞ for t→ t−. Additionally, for every b < 0 there is a timetb ∈ R such that b < tbH < 0.

• If H0 > 0, then t0 ≤ t+ ≤ 3H0. The mean curvature satis�es tH > 0 for t ≥ t0

and tH → +∞ for t→ t+. Additionally, for every b > 0 there is a time tb ∈ Rsuch that 0 < tbH < b.

• If H0 = 0, then tH ≤ 0 for t ≤ t0 and tH ≥ 0 for t ≥ t0.

Proof. We prove the case H0 < 0: Set with a .

.= 3H0

+ t0. Applying the Lemma ofGronwall and Corollary 7.5.3 to

Φ (t) .

.= − 3

t− a− tH

h (t) .

.= −1

3

(3

t− a− tH

)yields − 3

t−a ≤tH for t ≤ t0 and − 3

t−a ≥tH for t ≥ t0.

The function t 7→ − 3t−a has a vertical asymtote at 3

H0+ t0 where its values tend

to −∞. As the mean curvature lies beneath this function, it exhibits the sameasymptotic behaviour, and 3

H0+ t0 is the earliest time for its vertical asymptote. As

t → ∞, the function t 7→ − 3t−a tends to 0. The mean curvature lies above this

function, thus exceeds every negative value.The case H0 > 0 is proved similarly, and the third case H0 = 0 is an exact analogue

of the one in proposition 7.5.1.

In [Ren95] Rendall examined the relation between the mean curvature and existenceand properties of singularities of the �ow de�ned by the Einstein equations. In thecase of vacuum Bianchi models with vanishing cosmological constant the followingassertion ensues:

Proposition 7.5.6. Consider the �ow given by the vacuum Einstein equations withcosmological constant Λ = 0 on Bianchi models. If a solution is de�ned on the interval(t−, t+) and cannot be extended, then t− or t+ �nite implies that the mean curvaturetH is unbounded in a neighbourhood of t−, t+ respectively.

85


In particular, the solution to the �ow equations exists as soon as the mean curvatureis bounded.We combine all the results we found on Bianchi class A models with non-positive

scalar curvature in the following theorem.

Theorem 7.5.7. Consider Bianchi class A models with non-positive scalar curvature,i. e. all Bianchi class A models except those type IX models where the scalar curvatureadmits positive values. Let t0,H0 ∈ R and t 7→ ( tg , tk ) the maximal solution of theEinstein vacuum equation with cosmological constant Λ = 0


with constraints

0 = div tk

0 =1

2

(tS + tH 2 −

∣∣ tk ∣∣2g )such that the initial value is t0H = H0. Then we �nd the following development ofthe mean curvature tH :

• If H0 < 0, then −∞ < t− ≤ t0. The mean curvature increases monotonicallyand satis�es tH < 0 at all times, tH → −∞ for t → t− and tH ↗ 0 fort→ +∞.

• If H0 > 0, then t0 ≤ t+ < +∞. The mean curvature increases monotonicallyand satis�es tH > 0 at all times, tH → +∞ for t → t+ and tH ↘ 0 fort→ −∞.

• If H0 = 0, then tH = 0 at all times t.

As the Hubble constant satis�es H0 > 0, suppose H0 < 0. Then we have proventhat the mean curvature in Bianchi class A models with non-positive scalar curvaturein vacuum show the same qualitative behaviour as the Kasner metrics, which solvethe Einstein equations in the Bianchi type I case:

i) The mean curvature is negative at all times.

ii) The mean curvature increases monotonically.

iii) The mean curvature vanishes asymptotically: tH ↗ 0 as t→∞.

iv) The universe admits an initial singularity at −∞ < t− ≤ t0 with tH → −∞ ast→ t−.

86


In order to determine whether t− is a big bang, we have to consider the volume form.As (

vol(tgp0))

.

=(√

det gijdx1 ∧ dx2 ∧ dx3

).

=1

2

1√det (gij)

det (gij). dx1 ∧ dx2 ∧ dx3

=1

2

1√det (gij)

det (gij) tr((gij)

(gij))dx1 ∧ dx2 ∧ dx3

=1

2

√det (gij)g

ij gijdx1 ∧ dx2 ∧ dx3

= −gijkij vol(tgp0)

= − tHp0 vol(tgp0)

for every (t, p0) ∈ G, (e1, e2, e3) a basis of Tp0G(t), we see that the volume formsatis�es the di�erential equation(

vol(tg))

.

= − tH vol(tg).

At the present state, this does not enable us to decide whether or not

vol(tg)→ 0 and tH

(vol(tg)

vol(t0g)) 1

3

→ −∞, t→ t− = 0.

This question remains to be treated in future works.

87

A. Appendix � Tensors

De�nition A.1. Let V an R-vector space, V ∗ its dual space, and r, s ≥ 0 two integersnot both zero. A tensor of type (r, s) is an R-multilinear mapping

A : (V ∗)r × V s → R.

In the context of Riemannian geometry, the word tensor is often used to describetensor �elds, which additionally respect the smooth structure of the manifold. For amanifoldM consider the tangent bundle TM and the dual bundle T ∗M . Additionally,denote by X(M) the set of smooth vector �elds, and by X∗(M) its pointwise dual,the set of smooth one-forms. Similarly to the tangent bundle TM →M , the sets

TM ⊗ . . .⊗ TM ⊗ T ∗M ⊗ . . .⊗ T ∗M .

.=∐p∈M

TpM ⊗ . . .⊗ TpM ⊗ TpM∗ ⊗ . . .⊗ TpM∗

with π : TM⊗ . . .⊗TM⊗T ∗M⊗ . . .⊗T ∗M →M can be endowed with the structureof a vector bundle overM . For reasons which the following de�nition motivates, theyare called tensor bundles.

De�nition A.2. Let M a smooth manifold of dimension n and r, s ≥ 0 two integersnot both zero. A tensor �eld of type (r, s) is a E(M)-multilinear mapping

A : X∗ (M)r × X (M)s → E (M) .

Equivalently, one could de�ne a tensor �eld of type (r, s) as a smooth section ofthe vector bundle

TM ⊗ . . .⊗ TM︸︷︷︸r

⊗T ∗M ⊗ . . . T ∗M.︸︷︷︸s

A tensor �eld A is E(M)-multilinear by de�nition. The set T rs of all tensor �elds

of type (r, s) is then a module over E(M). In order to prove that a function B :X∗(M)r × X(M)s → E(M) is a tensor �eld, one has to show E(M)-linearity in eachargument.

Lemma A.3. There is a unique E(M)-linear function tr : T 11 → E(M) such that

tr (X ⊗ σ) = σ (X) ∀X ∈ X (M) , σ ∈ X∗ (M) .

It is called (1, 1) contraction.

89

A. Appendix � Tensors

Let A a tensor �eld of type (r, s) and 1 ≤ i ≤ r, 1 ≤ j ≤ s. Choose σk, k =1, . . . , i− 1, i+ 1, . . . r and Xl, l = 1, . . . , j − 1, j + 1, . . . s. The function

(σ,X)→ A (σ1, . . . , σi−1, σ, σi+1, . . . , σr, X1, . . . , Xj−1, X,Xj+1, . . . , Xs)

is a tensor �eld of type (1, 1), thus can be contracted. This construction yields atensor �eld of type (r − 1, s− 1), the contraction of A over i, j, which we denote bytrij A .Consider a tensor �eld A of type (r, s) on a semi-Riemannian manifold (M, g ).

There is a canonical isomorphism between the tangent bundle and its dual, the cotan-gent bundle.Using this isomorphism, a tensor �eld of type (r, s) with s ≥ 1 can be interpreted

as a tensor �eld of type (r+1, s−1). These two tensor �elds are metrically equivalent.Contracting this tensor �eld of type (r+1, s−1) yields a tensor �eld of type (r, s−2),the metric contraction of A .For 1 ≤ a, b ≤ s, p0 ∈ M and a basis (e1, . . . , en) of the tangent space Tp0M the

metric contraction is given by

trabA (σ1, . . . , σr, X1, . . . , Xa−1, Xa+1, . . . , Xb−1, Xb+1, . . . Xs) (p0) =

gklAp0 (σ1 (p0) , . . . , σr (p0) , X1 (p0) , . . . , ek, . . . , el, . . . , Xs (p0)) .

De�nition A.4. Let (M, g ) a semi-Riemannian manifold and ∇ the Levi-Civitaconnection. Let A a symmetric tensor �eld of type (0, s). The divergence of A is themetric contraction of ∇A

div A .

.= tr12∇A = tr (X 7→ ∇XA) .

We used the relation (∇A)(X1, X2, . . . , Xs+1) = (∇X1 A)(X2, . . . , Xs+1) to de�nethat the tensor �eld has to be contracted over the additional covariant variable andthe �rst entry of the tensor �eld. As A is symmetric, the latter could be replaced byany other entry of the tensor �eld.Relative to a basis (e1, . . . , en) of Tp0M , p0 ∈M , we �nd

div (Ap0) (x) = gij (∇eiAp0) (ej, x)

for every x ∈ Tp0M .

De�nition A.5. Let (V, 〈·, ·〉) a vector space with inner product and A a tensor oftype (0, 2). The contraction of A with itself is

|A|〈·,·〉 .

.= tr tr23 (A⊗ A) .

De�nition A.6. Let (M, g ) a semi-Riemannian manifold and A a tensor �eld oftype (0, 2). The contraction of A with itself is the function |A |2g : M → R with

|A |2g .

.= tr tr23 (A ⊗ A) .

Let (e1, . . . , en) a basis of Tp0M , p0 ∈M . Then we see

|Ap0|2g = tr tr23 (Ap0 (p0)⊗ Ap0)

= gijgklAp0 (ei, ek) Ap0 (ej, el) .

90

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http://www.mathematik.uni-tuebingen.de/~loose/studium/Diplomarbeiten/Dipl.Ungaenz.pdf

http://www.mathematik.uni-tuebingen.de/~loose/studium/Diplomarbeiten/Dipl.Ungaenz.pdf

http://arxiv.org/pdf/1106.4720v2

http://arxiv.org/pdf/1106.4720v2

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