EXAMENSARBETE INOM TEKNISK FYSIK, AVANCERAD NIVÅ, 30 HP STOCKHOLM, SVERIGE 2020

Hidden symmetries in the vicinity of a cosmological singularity

CARIN JAKOBSSON

KTH SKOLAN FÖR TEKNIKVETENSKAP Master of Science Thesis

Hidden symmetries in the vicinity of a cosmological singularity

Carin Jakobsson

Supervisor: Martin Cederwall

Mathematical Physics, Department of Theoretical Physics School of Engineering Sciences Royal Institute of Technology, SE-106 91 Stockholm, Sweden

Stockholm, Sweden 2020 Typeset in LATEX

Akademisk avhandling f¨oravl¨aggandeav teknologie masterexamen inom ¨amnesomr˚adet teoretisk fysik. Scientific thesis for the degree of Master of Engineering in the subject area of Theoretical physics.

Cover illustration: Weyl chambers of the hyperbolic Kac-Moody `` algebra A1 . Taken from Ref. [28]

TRITA-SCI-GRU 2020:036

c Carin Jakobsson, Mars 2020 Abstract

Recent studies in relativistic cosmology implies that general cosmological systems hide symmetries rooted in the so called Belinskii-Khalatnikov- Lifshitz (BKL) phenomenon. Such symmetries would provide a frame- work for describing how spatial gradients in the general case are encoded in infinite-dimensional algebras, namely hyperbolic Kac-Moody (KM) al- gebras, from which a new formulation of relativistic cosmology and grav- itation would be possible.

This thesis provides a description of the origin of the BKL phenomenon as well as of its connection to hyperbolic KM algebras. Starting from the gravitational field equations for pure gravity in four spacetime-dimensions, the reader is guided through a series of assumptions, including a gener- alized version of the homogeneous particular solution of Kasner (to the field equations), and a hand-waving, heuristic argumentation that the BKL phenomenon, applicable to an inhomogeneous space, can in fact be obtained by studies of homogeneous models of so called Bianchi type (the BKL conjecture). The link between the BKL phenomenon and hy- perbolic KM algebras is obtained by a reparameterization of the metric, such that when inserted into the Lagrangian, a three-dimensional space with Lorentzian signature is constructed. The BKL phenomenon causes motion in the three-dimensional space, which follows the trajectory of a moving ball on a billiard table shaped as the fundamental Weyl chamber of a hyperbolic KM algebra (the billiard picture). A section with focus on Lie algebras, KM algebras, roots and Weyl chambers is provided for the reader who is not yet familiar with the algebraic framework. In the last part of the thesis, the BKL phenomenon and its correspondence with hy- perbolic KM algebras is generalized to higher spacetime-dimensions and to systems including dilatons and Einstein p-form fields. The significance of hyperbolic KM algebras in dimensional reduction of general relativistic systems (where hidden symmetries arise) and extended geometry is dis- cussed, for this reason the reader can also find some theory on Lie groups and symmetries.

i Abstrakt

Senare studier i relativistisk kosmologi visar p˚aatt allm¨annarelativistiska system d¨oljersymmetrier rotade i det s˚akallade Belinskii-Khalatnikov- Lifshitz (BKL) fenomenet. S˚adanasymmetrier skulle tillhandah˚allaett ramverk f¨oratt beskriva hur rumsliga derivator i det generella fallet kan kodas i o¨andlig-dimensionellaalgebror, n¨amligenhyperboliska Kac-Moody (KM) algebror, utifr˚anvilka man i s˚afall skulle kunna hitta en ny formu- lering av relativistisk kosmologi, och av gravitation.

I den h¨aruppsatsen beskrivs uppkomsten av BKL fenomenet och dess kop- pling till hyperboliska KM algebror. Med utg˚angspunkti f¨alt-ekvationerna f¨orren gravitation i fyra rumtids-dimensioner, blir l¨asarenv¨agleddgenom en serie antaganden, bland annat en generaliserad version av Kasners homogena l¨osning till f¨alt-ekvationerna, och en hand-viftande heuristisk argumentation f¨oratt BKL fenomenet, som skall g¨allaf¨oren inhomogen metrik, faktiskt kan h¨arledasutifr˚anstudier av homogena modeller av s˚a kallad Bianchi typ (BKL konjekturen). L¨anken mellan BKL fenomenet och hyperboliska KM algebror erh˚allsgenom en omparametrisering av metriken, som insatt i Lagrangianen definierar ett tre-dimensionellt rum med Lorentzisk signatur. BKL fenomenet ger upphov till en r¨orelsei det tre-dimensionella rummet som f¨oljerbanan hos en biljardboll p˚aett biljardbord format som den fundamentala Weyl kammaren f¨oren hyper- bolisk KM algebra. En del av uppsatsen ¨ardedikerad till det algebraiska ramverket, d¨arbland annat Lie algebror, KM algebror och Weylkammare definieras och diskuteras. I uppsatsens sista del generaliseras l¨anken mel- lan BKL fenomenet och den hyperboliska KM algebran till h¨ogrerumtids- dimensioner, och till system som innefattar dilatoner och Einstein p-form f¨alt. Relevansen av den hyperboliska algebran i dimensionell reduktion av allm¨annarelativistiska system (d¨arg¨omdasymmetrier uppst˚ar)och i utvidgad geometri diskuteras, av denna anledning kan l¨asaren¨aven hitta en del teori om Lie grupper och symmetrier.

ii Contents

Abstract i

Abstrakt (sve) ii

1 Introduction 1 1.1 Historical perspective ...... 1 1.2 Aim and outline ...... 1

2 The BKL phenomenon 3 2.1 The gravitational field equations ...... 3 2.2 Gauge conditions ...... 3 2.3 Gravitational singularities ...... 4 2.3.1 True and fictitious singularities ...... 4 2.3.2 Space-like and time-like singularities ...... 5 2.4 The Kasner solutions ...... 5 2.4.1 Kasner’s particular solution ...... 6 2.4.2 Kasner’s generalized solution ...... 7 2.5 The BKL phenomenon ...... 11 2.5.1 The BKL conjecture ...... 11 2.5.2 Derivation of oscillatory behaviour ...... 12 2.5.3 Discussion on oscillatory behaviour ...... 14

3 Algebraic framework 16 3.1 Groups ...... 16 3.2 Lie Groups ...... 16 3.2.1 Symmetries, Lie groups and Lie algebras ...... 17 3.3 Lie algebras ...... 17 3.3.1 Notation and classification ...... 18 3.3.2 Rigorous definition of a Lie algebra ...... 18 3.4 Kac-Moody algebras ...... 18 3.4.1 Cartan matrix and Dynkin diagram ...... 18 3.4.2 Generators ...... 20 3.4.3 Examples ...... 21 3.4.4 Roots ...... 22 3.4.5 The bilinear form ...... 23 3.4.6 Fundamental weights ...... 24 3.4.7 Examples ...... 24

4 The billiard picture 29 4.1 Iwasawa decomposition of metric ...... 29 4.2 System potential ...... 30 4.3 The limit t Ñ 0...... 31 4.3.1 Potential ...... 31 4.3.2 Interpretation of potential ...... 32

iii 4.3.3 Spatial decoupling ...... 32 `` 4.4 Relation to hyperbolic Kac-Moody algebra A1 ...... 33 5 Generalizations and symmetries 35 5.1 The general case ...... 35 5.2 Pure gravity of higher dimension ...... 37 5.3 11D SUGRA and other systems of interest ...... 39 5.4 Dualities, hidden symmetries, dimensional reduction and extended geometries ...... 40

6 Conclusion 42

A Derivation of gravitational field equations on normal form 43

B Derivation of Kretschmann scalar for Kasner’s particular solu- tion 46

C Homogeneous spaces 48 C.1 Structure constants of homogeneous spaces ...... 48 C.2 Classification of homogeneous spaces ...... 49

D Derivation of metric in β-space for Kasner’s particular solution 51

E Classification of Lie algebras 53

Acknowledgements 56

References 57

iv Carin Jakobsson Introduction

1 Introduction 1.1 Historical perspective The motivation for studies of the BKL phenomenon, discovered in 1970 by Be- linskii, Khalatnikov and Lifshitz (BKL), has shifted, from the urge to find a more general solution to the gravitational field equations, towards establishing symmetries within the gravitational theory in terms of Lie algebras.

Whilst Friedmann’s solutions, proposed in 1922, assumed homogeneity and isotropy of space, BKL asked themselves whether there was a more general solution, where homogeneity and isotropy was not a presumption. Their in- vestigation resulted in the discovery of an intricate oscillatory behaviour in the vicinity of a singularity in four spacetime dimension [4]. Generalization to higher spacetime dimensions showed that the BKL phenomenon persisted for pure gravity up to ten dimensions, but dissolves to a monotonic non-oscillatory behaviour for dimensions ě 11. It was later on discovered, by Damour in 2000, that the oscillatory BKL behaviour could in fact be associated to billiard motion and the fundamental Weyl chamber of a hyperbolic Kac-Moody (KM) algebra [18], [20]. Damour further demonstrated that pure gravity in any spacetime dimension D “ d ` 1 ě 4 is associated with algebras AEd and that pure gravity systems of dimension D ě 11 which don’t exhibit BKL oscillations correspond to an algebra AEd which are not of the hyperbolic type [21]. The relation to hy- perbolic algebras has been established not only for cases of pure gravity, but for superstring models (I, II, HO, HE) of ten dimensions [20] as well as for Einstein- dilaton-p-form systems of arbitrary dimension ě 4 [22] including the case of eleven-dimensional supergravity (the low energy limit of M-theory) [26]. The association of gravitational systems and hyperbolic KM algebras in the vicinity of a space-like singularity have turned out to be somewhat frequent and seems to persist for more general cases, such as eleven-dimensional supergravity (11D SUGRA). In the context of dimensional reduction and extended geometry, the BKL phenomenon and the association with hyperbolic KM algebras is of extra importance because the BKL-phenomenon in some ways resembles dimensional reduction and because hyperbolic algebras are large enough to encode symme- tries generated by reduction of 11D SUGRA to one dimension. Dimensional reduction of gravitational systems to lower dimensions gives rise to so called hidden symmetry groups, and lately it has been speculated that hyperbolic KM algebras could be regarded extensions of the hidden symmetries.

1.2 Aim and outline The aim of this thesis is to introduce the reader to the BKL phenomenon and its association with hyperbolic KM algebras and to provide motivation for the relevance of this theory. The association with hyperbolic KM algebras appear already in the case of pure gravity in four dimensions, which is the focus of the thesis.

1 Carin Jakobsson Introduction

An investigation of how the BKL phenomenon occurs in four-dimensional pure gravity according to the original BKL article from 1970 is provided in section 2. An outline of the correspondence to hyperbolic KM algebras through billiard motion is given in section 4. Generalization to higher-dimensional pure gravity and other general systems of relevance, including supergravity is given in section 5. In the same section, the question of the hyperbolic algebra as an extended hidden symmetry is discussed. Useful framework on Lie groups, Lie algebras and KM algebras is provided in section 3.

2 Carin Jakobsson The BKL Phenomenon

2 The BKL phenomenon

In 1970 it was discovered that the general solution of the gravitational field equations is connected to an oscillatory behaviour close to a singularity. In this section, the analysis leading up to this so called BKL phenomenon is explained. The association with the billiard picture and KM algebras is described in section 4.

2.1 The gravitational field equations BKL studied the gravitational field equations on the form:

a Rb “ 0 (1) where the metric is used to raise and lower indices so that

a ac Rb “ g Rcb, (2) the Ricci tensor is defined by:

c Rab “ Racb, (3) the Riemann tensor by:

a a a a e a e Rbcd “BcΓbd ´ BdΓbc ` ΓceΓbd ´ ΓdeΓbc (4) and the Christoffel symbols by: 1 Γa “ gadpB g ` B g ´ B g q. (5) bc 2 b cd c bd d bc

The metric gab “ gabpxq is a covariant symmetric tensor of order two which depends on the spacetime coordinates xa “ px0, xαq “ px0, x1, x2, x3q.

2.2 Gauge conditions The field equations (1) can be thought of as ten nonlinear differential equations, with ten unknowns gab. Due to diffeomorphism invariance, the gauge symmetry of gravity, the metric is determined by (1) only up to a choice of coordinates. In order to obtain a completely determined metric, we could introduce Gaus- sian/normal coordinates defined by:

00 0α g00 “ g “ ´1, g0α “ g “ 0 (6)

By this condition one assumes a synchronous reference frame, i.e. time is syn- chronized in space. The corresponding line element is given by:

2 a b 2 2 ds “ gabdx dx “ ´dt ` dl (7)

3 Carin Jakobsson The BKL Phenomenon with 2 α β dl “ gαβdx dx (8) and the metric determinant satisfies:

g ” |gab| “ ´|gαβ| (9)

Inserting these metric requirements into (5) and defining the entity:

α αγ καβ “B0gαβ, κβ “ g κγβ (10) one can express the Christoffel symbols as:

0 α 0 Γ00 “ Γ00 “ Γα0 “ 0 (11)

α 1 α Γβ0 “ 2 κβ (12) 0 1 Γαβ “ 2 καβ (13) α 1 αδ Γβγ “ 2 g pBβgγδ ` Bγ gβδ ´ Bδgβγ q (14) and obtain the vacuum field equations on normal form as:

1 B 1 R0 “ κα ` κακβ “ 0 (15) 0 2 Bt α 4 β α 1 R0 “ pκβ ´ κβ q “ 0 (16) α 2 β;α α;β 1 B ? Rα “ P α ` ? p ´gκαq “ 0 (17) β β 2 ´g Bt β α where Pβ is the spatial Ricci tensor, i.e. the Ricci tensor of three-dimensional space at fixed t. A detailed derivation of equations (15) - (17) is provided in appendix A.

2.3 Gravitational singularities The BKL phenomenon describes the behaviour of the solution close to a grav- itational singularity, that is a true physical singularity. This section gives a description on different types of singularities, how to distinguish between them, and clarifies which are of relevance w.r.t the BKL phenomenon.

2.3.1 True and fictitious singularities A gravitational singularity is a point in spacetime where the gravitational field strength is predicted infinite independent of choice of coordinates. The gravita- tional field strength is measured by the scalar invariant curvatures of spacetime, ab abcd such as such as R, R Rab and the so called Kretschman invariant R Rabcd. Gravitational singularities are so called true physical singularities as opposed to fictitious or coordinate singularities, which can be removed by a change of

4 Carin Jakobsson The BKL Phenomenon coordinates. For example, the Schwarzschild solution in spherically symmetric coordinates pt, r, θ, φq, has line element:

´1 rs 2 2 rs 2 2 2 2 2 g “ ´ 1 ´ r c dt ` 1 ´ r dr ` r pdθ ` sin θdφ q (18) ´ ¯ ´ ¯ where 2GM r “ (19) s c2 is the event horizon or Schwarzschild radius (G is the gravitational constant and M the mass of the body exerting the gravitational field). Note that (18) becomes infinite for r “ 0 and r “ rs. One may thus suspect that r “ 0 and r “ rs are both singularities of the Schwarzschild solution but this is not the case. The Kretschman invariant becomes: 12r2 RabcdR “ s (20) abcd r6 which is infinite at r “ 0 but finite at r “ rs. Thus, r “ 0 is considered a grav- itational singularity whereas r “ rs a coordinate singularity. Indeed, a change of coordinates to Eddington–Finkelstein coordinates removes the apparent sin- gularity of (18) at the event horizon [14].

2.3.2 Space-like and time-like singularities A singularity of the gravitational field equations on a hyperplane is said to be of space-like character if the hyperplane is space-like (contains space-like vectors) and of time-like character if the hyperplane is time-like. If the singularity is space-like, one can always choose a reference frame in which the Gaussian gauge (6) holds, i.e. one can choose a coordinate system such that the singularity occurs simultaneously in all space. In cosmology, (true) space-like singularities are of most interest and it is also this type of singularity which gives rise to the BKL phenomenon [4]. The vacuum field equations on normal form (15)-(17) will constitute as the basis from which the BKL phenomenon is derived.

2.4 The Kasner solutions The BKL phenomenon is based upon analysis of the vacuum field equations (15) - (17) w.r.t. Kasner’s generalized solution (see section 2.4.2), a generalization of Kasners particular solution (see section 2.4.1). Note that, the analysis is applicable also for the case including matter ([3] section 3.2), in which the field equations on normal form are given by:

1 B 1 1 R0 “ κα ` κακβ “ T 0 ´ T (21) 0 2 Bt α 4 β α 0 2 1 R0 “ pκβ ´ κβ q “ T 0 (22) α 2 β;α α;β α

5 Carin Jakobsson The BKL Phenomenon

1 B ? 1 Rα “ P α ` ? p ´gκαq “ T α ´ δαT (23) β β 2 ´g Bt β β 2 β where T ab is the energy-momentum tensor. This section however, for simplicity, provides the definitions and derivations of the Kasner solutions to the vacuum field equations.

2.4.1 Kasner’s particular solution In 1921, Kasner proposed an exact particular solution to the vacuum field equa- tions in normal form by assuming a homogenous but anisotropic space corre- sponding to the line element given by:

ds2 “ ´dt2 ` t2p1 dx2 ` t2p2 dy2 ` t2p3 dz2 (24) where pj satisfies the conditions:

2 2 2 p1 ` p2 ` p3 “ 1 and p1 ` p2 ` p3 “ 1 (25)

The first condition defines the Kasner plane, and the second describes the Kasner sphere. It follows that the metric determinant is (taking the condition (9) into account): g “ ´t2pp1`p2`p3q “ ´t2 (26) We can see that t “ 0 is a suspected singularity, we can confirm its coordinate independence by calculation of the Kretschman invariant (detailed derivation is provided in appendix B):

3 abcd ´4 ´2pi 2 2 R Rabcd “ t t pi ppi ´ 1q (27) i“1 ÿ 2 2 which goes to infinity at least as long as pi ppi ´ 1q ‰ 0. Note that if p1 “ p2 “ 2 2 0, p3 “ 1, we have pi ppi ´ 1q “ 0 and the system is one-dimensional. In this case t “ 0 is a coordinate singularity and we get a flat spacetime simply by the transformation ξ “ t sinh z, τ “ t cosh z. For pp1, p2, p3q ‰ p0, 0, 1q satisfying the 2 2 conditions (25) however, pi ppi ´ 1q ‰ 0 and we have a gravitational singularity at t “ 0.

Discussion on solution behaviour

The conditions (25) put some restraints on the values of pj. Only one of the parameters is independent, and by the ordering p1 ď p2 ď p3 one can show 1 2 2 that pp1, p2, p3q can take on values between p´ 3 , 3 , 3 q and p0, 0, 1q such that all parameters are distinct except for in the case of these specific triads. We shall restrict attention to the cases where the parameters are distinct so that:

1 2 ´ 3 ă p1 ă 0 ă p2 ă 3 ă p3 ă 1 (28)

6 Carin Jakobsson The BKL Phenomenon

The parameters may be represented in terms of a variable u, as:

´u 1`u up1`uq p1puq “ 1`u`u2 , p2puq “ 1`u`u2 , p3puq “ 1`u`u2 , u ě 1. (29)

1 The same values can be obtained by considering the parameters in terms of u subject to u ď 1:

1 1 1 p1p u q “ p1puq, p2p u q “ p3puq, p3p u q “ p2puq (30)

Plotting these curves (figure 1) we see that all the values of pj satisfying (28) are obtained by monotonically decreasing functions p1puq, p3puq and a mono- tonically increasing function p2puq. Note that the metric defined by (24) is anisotropic since p1, p2, p3 cannot take on the same values. Furthermore the linear distances in each space element decrease in two directions and increase in the third direction as the singularity is approached.

1 Figure 1: Parameteric behaviour in u [4]

2.4.2 Kasner’s generalized solution Kasner’s generalized solution to the gravitational vacuum field equations is ob- tained by imposing Gaussian gauge conditions on the metric such that (15)- (17) holds and assuming that in the limit of the singularity, Kasner’s generalized so- lution approaches Kasner’s particular solution (in principal terms in powers of t) such that the singularity at t Ñ 0 persist. In the synchronous reference frame, the spatial metric is given by:

2 2 2 gαβ “ a lαlβ ` b mαmβ ` c nαnβ (31) where a, b and c are given by:

a “ tpl , b “ tpm , c “ tpn (32)

7 Carin Jakobsson The BKL Phenomenon

and pl, pm, pn are functions of the spatial coordinates such that the require- ments (25) are satisfied. The vectors lα, mα, nα are also functions of the spatial coordinates and satisfy:

α α α α α α l lα “ n nα “ m mα “ 1, l nα “ l mα “ n mα “ 0 (33)

The metric determinant is given by (see equation (232) of appendix C):

g “ a2b2c2v2 “ t2v2 (34) where v “ l ¨ pn ˆ mq. Note that neither homogeneity nor isotropy is assumed by (31). Let’s investigate to what extent (31) satisfies the equations (15)-(17) in the vicinity of the singularity.

Investigation of (15) close to the singularity:

The inverse spatial metric is given by:

gαβ “ a´2lαlβ ` b´2mαmβ ` c´2nαnβ (35)

According to (10) we have: Bg κα “ gαγ γβ β Bt ´2 α γ ´2 α γ ´2 α γ “ pa l l ` b m m ` c n n qp2aal9 γ lβ ` 2bbm9 γ mβ ` 2ccn9 γ nβq The dot denoting a time derivative. Using the conditions (33) we get:

2a9 2b9 2c9 κα “ lαl ` mαm ` nαn (36) β a β b β c β The first term of (15) becomes:

1 B a: a9 2 :b b92 c: c92 κα “ ´ ` ´ ` ´ (37) 2 Bt α a a2 b b2 c c2 and the second term: 1 a9 2 b92 c92 κακβ “ ` ` . 4 β α a2 b2 c2 Thus, the first field equation (15) is given by:

a: :b c: pp ´ 1qp ` pp ´ 1qp ` pp ´ 1qp R0 “ ` ` “ l l m m n n “ 0 (38) 0 a b c t2 Note that the last equality is, according to the constraints (25), automatically fulfilled. Thus, (15) is satisfied close to the singularity.

Investigation of (17) close to the singularity:

8 Carin Jakobsson The BKL Phenomenon

Accordning to (36) and (34) we have

1 ? κα « , g « t (39) β t and therefore the time derivative term of (17) is identically equal to zero. In order to investigate under which conditions the remaining term (the spatial Ricci tensor) is zero, we introduce the notation:

l ¨ p∇ ˆ lq m ¨ p∇ ˆ mq n ¨ p∇ ˆ nq λ “ , µ “ , ν “ (40) v v v where v “ l¨pmˆnq and l, m, n are the three-dimensional vectors which specifies the directions for the power-law behaviour in t. Note that if λ “ µ “ ν “ 0, we obtain Kasner’s particular solution. We will now be interested in the projection of the components (17) onto the spatial vectors l, m, n i.e. the equation:

1 B ? Rl “ P l ` ? p ´gκlq (41) l l 2 ´g Bt l

m n and similarly for Rm and Rn. Thus, the third vacuum field equation (17) gives: 1 B ? Rl “ P l ` ? p ´gκlq “ 0,Rm “ ... “ 0,Rn “ ... “ 0 (42) l l 2 ´g Bt l m n where we define the projection of a tensor Tαβ onto the spatial vectors l, m, n by: α β m α β Tlm Tlm “ Tαβl m ,Tl “ Tβ mαl “ (43) gmm Thus, in the vicinity of the singularity we get:

2 2pl 2 2pm 2 2pn gll “ a “ t , gmm “ b “ t , gnn “ c “ t (44)

In this notation, the time derivative term of the expressions (41) becomes:

1 B ? l 1 1 B l ? B l ? p ´gκlq “ ? ? p´gqκl ` ´g κl 2 ´g Bt 2 ´g ˜2 ´g Bt Bt ¸

1 B l 1 B l 1 γδ B l 1 B l “ p´gqκ ` κ “ ´ p´gqg pgγδqκ ` κ 4g Bt l 2 Bt l 4g Bt l 2 Bt l

1 γδ l 1 B l 1 γ l 1 B l “ g κγδκ ` κ “ κ κ ` κ (45) 4 l 2 Bt l 4 γ l 2 Bt l According to (43) have:

2a9 B 2a: 2a9 2 κl “ , κl “ ´ (46) l a Bt l a a2

9 Carin Jakobsson The BKL Phenomenon

Thus, with use of (36), (17) is equivalent to:

pabc9 q¨ Rl “ ` P l “ 0 (47) l abc l

pabc9 q¨ Rm “ ´ P m “ 0 (48) m abc m pabc9q¨ Rn “ ´ P n “ 0 (49) n abc n

We can insert the expressions for a, b and c in terms of pl, pm, pn, and show that the time derivative terms will each vanish identically:

pabc9 q¨ p pp ` p ` p ´ 1q “ l l m n “ 0 (50) abc t2 (similarily for the m and n projections). Furthermore we can see that the time 1 1 derivative terms are of second order in t (note that t Ñ 8 at the singularity, 1 so the higher order in t , the larger the expression close to the singularity). Thus, if the remaining terms of (41), i.e. the spatial Ricci tensors projected 1 onto directions l, m and n, are of order ă 2 in t , the expressions (47)-(49) will be satisfied. Let us therefore investigate the order of the spatial Ricci tensors projected onto directions l, m and n. We have:

l Pll ´2 m Plm ´2 Pl “ “ a Pll,Pl “ “ b Plm (51) gll gmm with: γ γ γ σ γ σ α β Pll “ pBγ Γαβ ´ BβΓαγ ` ΓγσΓαβ ´ ΓβσΓαγ ql l (52) Inserting the spatial Christoffel symbols (14) gives [3]:

a2 “ 1 pal ¨ p∇ ˆ alqq2 ´ 1 pbm ¨ p∇ ˆ bmqq2 ´ 1 pcn ¨ p∇ ˆ cnqq2 ∆2 2 2 2 ˆ ´pcn ¨ p∇ ˆ bmqq2 ´ pbm ¨ p∇ ˆ cnqq2 ´ pbm ¨ p∇ ˆ alqq2 ´ pcn ¨ p∇ ˆ alqq2

`pcn¨p∇ˆcnqqpbm¨p∇ˆbmqq`pcn¨p∇ˆalqqpal¨p∇ˆcnqq`pal¨p∇ˆbmqqpbm¨p∇ˆalqq ˙ cn ¨ p∇ ˆ alq 1 cn ¨ p∇ ˆ bmq 1 bm ¨ p∇ ˆ cnq 1 bm ¨ p∇ ˆ alq `a2 1 ´ ` ´ b ∆ a ∆ a ∆ c ∆ # ˆ ˙,m ˆ ˙,l ˆ ˙,l ˆ ˙,n+ m n Similar expressions are obtainable for Pm and Pn . By assuming pl “ p1 ă 0, l m n one can show that the highest order terms of Pl ,Pm ,Pn are:

pl ¨ p∇ ˆ lqq2 λ2a2 P l “ ´P m “ ´P n “ t´2ppm`pn´plq “ (53) l m n 2v2 2b2c2

10 Carin Jakobsson The BKL Phenomenon

The conditions pl ă 0 and (25) gives that the parenthesis in the exponent is larger than 1, which makes this term, the spatial Ricci tensor of equations (41) of significant order. Thus, we may satisfy the conditions (47)-(49) by requiring:

l ¨ p∇ ˆ lq λ “ “ 0 (54) v A similar analysis of equation (16) gives no further information and will be of no importance for the derivation of the BKL phenomenon.

2.5 The BKL phenomenon The Kasner-like metric (31) contain ten unknowns, namely nine spatial com- ponents belonging to the three vectors l, m, n and one w.r.t. the exponential parameters (the other two are given by the relations (25)). The relations (47)- (49) determine three of them, and the condition (54) determines one more. Furthermore the synchronous reference frame is invariant under arbitrary time- independent transformations which reduces the number of unknowns by three. Consequently, Kasner’s general solution has 10 ´ 4 ´ 3 “ 3 unknowns, which is one more than required for the solution to be completely general [4]. Note that the condition (54) is set only to arrive at the generalized Kasner solution, and is not a necessary property of a solution to the field equations (17) or for that matter (42). BKL were interested in the behaviour of the completely general solution to the field equations, for which the condition (54) is not necessarily satisfied. For this purpose, they took on the study of the equations (obtained by inserting (53) into (47)-(49) for λ ‰ 0):

pabc9 q¨ λ2a2 Rl “ ` “ 0 (55) l abc 2b2c2

pabc9 q¨ λ2a2 Rm “ ´ “ 0 (56) m abc 2b2c2 pabc9q¨ λ2a2 Rn “ ´ “ 0 (57) n abc 2b2c2 together with the requirement (38):

a: :b c: R0 “ ` ` “ 0 (58) 0 a b c

2.5.1 The BKL conjecture The system of differential equations (55)-(58) is non-linear, of second order in terms of the time-coordinate and of first order in terms of the spatial coordi- nates and thus difficult to solve. (Note that the spatial dependence is encoded in the directional vector l which depends on λ as (40)). In order to simplify the equations, BKL argued that:

11 Carin Jakobsson The BKL Phenomenon

”In the vicinity of the singularity, the terms containing time deriva- tives dominate over those containing spatial derivatives.”

This statement became known as the BKL conjecture. The BKL conjecture is sometimes expressed as ”the spatial gradients can be disregarded in the vicinity of the singularity”, which is not correct, since the BKL phenomenon is very much dependent on the presence of spatial gradients. In the early stages after the proposal, the conjecture was rejected [8], however, later on several studies (analytical and numerical) came to support, for example [24].

BKL further argued that the behaviour of the solution to the system of equa- tions (55)-(58), resembles the solution to the field equations of a homogeneous, anisotropic space. Homogeneous spaces can be classified according to Bianchi (table 3, appendix C) and in order to leave the metric as general as possible in the spatial directions, BKL decided to investigate the models of Bianchi type VIII and IX.

2.5.2 Derivation of oscillatory behaviour From table 3 (appendix C) we get the commutation relations of a homogeneous space of type IX as:

rX1,X2s “ ´X3, rX2,X3s “ ´X1, rX3,X1s “ ´X2 (59)

3 1 2 i.e. the non-zero structure constants are given by C21 “ C32 “ C13 “ 1 which by (239) coincides with the constants λ, µ, ν respectively. In the same way, for the homogeneous space of type VIII we get: λ “ ´µ “ ´ν “ 1. Einsteins equations for the homogeneous Bianchi models of type VIII and IX take the form [4]: pabc9 q¨ 1 Rl “ ` pλ2a4 ´ pµb2 ´ νc2q2q “ 0 l abc 2a2b2c2 pabc9 q¨ 1 Rm “ ` pµ2b4 ´ pλa2 ´ νc2q2q “ 0 (60) m abc 2a2b2c2 pabc9q¨ 1 Rn “ ` pν2c4 ´ pλa2 ´ µb2q2q “ 0 n abc 2a2b2c2 a: :b c: R0 “ ` ` “ 0 (61) 0 a b c 0 0 0 m n m The components Rl ,Rm,Rn,Rl ,Rl ,Rn vanish identically. Note that these equations, since λ, µ, ν are constant, contain only derivatives of the time coor- dinate, confirming the homogeneity of space. The time derivative terms of (60) can be written in a simpler form by the substitution:

a “ eα, b “ eβ, c “ eγ (62)

dt “ abcdτ (63)

12 Carin Jakobsson The BKL Phenomenon

We get: dτ α a9 “ α eα “ τ τ dt bc Thus, dτ α pabc9 q¨ “ α9 “ α “ ττ (64) τ ττ dt abc and α α α a: “ τ ¨ “ ττ ´ τ pbc9 ` bc9q bc ab2c2 b2c2 ˆ ˙ 1 ô a: “ pα ´ α β ´ α γ q (65) ab2c2 ττ τ τ τ τ Similar expressions are derived for b and c. Plugging in the expressions into (60) and (61), we get:

2 2 2 2 4 2αττ “ pµb ´ νc q ´ λ a 2 2 2 2 4 2βττ “ pλa ´ νc q ´ µ b (66) 2 2 2 2 4 2γττ “ pλa ´ µb q ´ ν c

1 2 pαττ ` βττ ` γττ q “ ατ βτ ` ατ γτ ` βτ γτ (67) Note that we can combine (66) and (67) to obtain restraints on the initial conditions ατ , βτ , γτ to the system (66) namely:

1 2 4 2 4 2 4 2 2 2 2 2 2 ατ βτ ` ατ γτ ` βτ γτ “ 4 pµ b ` ν c ` λ a ´ 2µb νc ´ 2λa νc ´ 2λa µb q As before, for Kasner’s stable generalised solution to solve (66), the right sides of the equations must be zero. Assuming pl “ p1 ă 0 we see that t Ñ 0 implies a Ñ 8. Thus, the condition λ “ 0 must apply in order to obtain Kasner’s generalised solution. We are however, looking for a solution to the equations (60) that doesn’t necessarily satisfy this requirement close to the singularity. The perturbations to the Kasner regime is, as stated, due to the terms λ2a4 “ a4, accordingly, close to the singularity (66) becomes:

1 4α αττ “ ´ 2 e 1 4α βττ “ 2 e (68) 1 4α γττ “ 2 e These equations are satisfied within a small neighbourhood of the singularity with initial time close to but not equal to zero, and we can move in the direction toward the singularity. Now, since the right hand sides of the system (68) is non-zero, we can’t assume the Kasner metric to solve it. Let’s instead try with the following ansatz: a „ tp1 , b „ tp2 , c „ tp3 (69) so that: abc “ Λt (70)

13 Carin Jakobsson The BKL Phenomenon where Λ is a constant. Furthermore define:

τ “ Λ´1 ln t ` const (71) so that (63) still holds. This gives:

p1 α a „ t “ e ñ α „ p1 ln t „ p1τΛ (72) and similarly for β and γ, so we get initial conditions:

ατ “ Λp1, βτ “ Λp2 γτ “ Λp3, as τ Ñ 8 (73) The system of equations (68) are nonlinear differential equations of second order, which are solved by [4]:

2|p |Λ a2 “ 1 coshp2|p1|Λτq

2 2 2Λpp2´|p1|qτ (74) b “ b0e coshp2|p1|Λτq

2 2 2Λpp3´|p1|qτ c “ c0e coshp2|p1|Λτq where b0 and c0 are constants. The asymptotic behaviour of (74) for τ Ñ 8 is in agreement with that of (69). If τ Ñ ´8 the asymptotic expressions for the solution (74) are given by:

a „ e´Λp1τ , b „ eΛpp2`2p1qτ , c „ eΛpp3`2p1qτ (75) and the function tpτq can in the same limit be expressed as:

t „ eΛp1`2p1qτ . (76)

Expressing a, b, c as functions of t, we obtain:

p1 p1 p1 a „ t l , b „ t m , a „ t n (77) where: 1 |p1| 1 2|p1| ´ p2 1 p3 ´ 2|p1| pl “ , pm “ , pn “ (78) 1 ´ 2|p1| 1 ´ 2|p1| 1 ´ 2|p1| and: 1 1 abc “ Λ t, Λ “ p1 ´ 2|p1|qΛ (79)

2.5.3 Discussion on oscillatory behaviour

1 1 1 Note that pl ą 0, pm ă 0 and pn ą 0. This is the core of the BKL phenomenon. As the singularity is approached, the regime with pl ă 0, pm ą 0, pn ą 0 is replaced by a new regime. In the transition, the negative exponent is transferred from direction l to direction m. The course of change of parameters pl, pm, pn can be represented in terms of the parameter u (see expression (29) and figure 1) such that if: pl “ p1puq, pm “ p2puq, pn “ p3puq (80)

14 Carin Jakobsson The BKL Phenomenon then 1 1 1 pl “ p2pu ´ 1q, pm “ p1pu ´ 1q, pn “ p3pu ´ 1q. (81) 1 If pm ă 0, the right sides of equations (68) would have expressions in terms of β instead of α and further approach to the singularity would again transfer the negative exponent from direction l to that direction for which the exponent is positive and smallest. At some point of the approach to the singularity, the parameter u goes from u ą 1 to u ă 1, up until this transition, the negative exponent is transferred between directions l and m and the solution spatially os- cillates in directions l and m and decrease monotonically in the direction n. The BKL phenomenon can thus be thought of as oscillations on the Kasner sphere (see 25). For u ă 0, pn is the smaller of the two positive exponents and a series of alterations where the negative exponent is transferred between directions l and n occur. A series of oscillations during which the monotonically decreasing direction is unchanged is referred to as a Kasner era/epoch. The alternation of Kasner eras continues without limit in the approach to the singularity. The sequence of directional changes of monotonic decrease, as well as the number of oscillations per era, takes the character of a chaotic process [4].

15 Carin Jakobsson Algebraic framework

3 Algebraic framework

In section 4 we will see that the oscillatory BKL behaviour can be interpreted as the motion of a billiard ball restricted to a specific area of the hyperbolic disk, which in turn can be associated with a hyperbolic KM algebra. KM algebras are Lie algebras, which can be defined in terms of so called Cartan matrices. This section gives some background on Lie and KM algebras, and provides the necessary tools to relate the billiard picture to the hyperbolic KM algebras. For the sake of understanding the importance of the connection of BKL oscillations to hyperbolic KM algebras (see section 5.4), we shall also include some theory on Lie groups and the connection to symmetries and Lie algebras.

3.1 Groups

A group G is a set of elements g1, g2, ..., gn such that • G is closed w.r.t. multiplication: gh “ f, g, h P G ñ f P G

• G is associative: pg1g2qg3 “ g1pg2g3q • There exists an identity element in G • There exist inverse elements for all elements in G. If n is finite, then G is called finite; if n is infinite, then G is called infinite.

3.2 Lie Groups A Lie group is a continuous group, with elements specified by one or more con- tinuous parameters θ1, θ2, ..., θn, which vary smoothly. For example the special orthogonal group of 2 dimensions, SOp2q, is the group of rotations in 2D. ”Spe- cial” implies that the determinant of the elements are unity. The elements of the group are the two by two matrices which rotate two-dimensional vectors an angle θ, i.e. the elements of the group SO(2) are given by:

cos θ ´ sin θ Rpθq “ (82) sin θ cos θ ˆ ˙ for all 0 ď θ ă 2π. The dimension of a Lie group is the number of free parameters of the group elements. In SOp2q we have only one free parameter namely θ, i.e. dimpSOp2qq “ 1. Some important examples of Lie groups and their dimensions is given in table 1.

16 Carin Jakobsson Algebraic framework

Groupname Notation Dimension General linear group GLpNq N 2 Special linear group SLpNq N 2 ´ 1 NpN´1q Special orthogonal group SOpNq 2 Unitary group UpNq N 2 Special unitary group SUpNq N 2 ´ 1

Table 1: Examples of Lie groups

3.2.1 Symmetries, Lie groups and Lie algebras In physics, a symmetry is a change that preserves key features of a physical system. A Lie group is a so called symmetry group, that is, the group elements are symmetries (changes which leave some system key features invariant). For example the orthogonal groups SOpNq preserve distances in Euclidean space Rn. A Lie group always gives rise to a Lie algebra, elements of which are tangent vectors to the Lie group elements close to the identity.

3.3 Lie algebras A Lie algebra always exist in correspondence with a Lie group. The Lie algebra is the mathematical structure that underlies the infinitesimal structure of its Lie group. It is generated by matrix generators X1, ..., Xn such that the Lie group elements are obtained through:

i Rpθq “ eθ Xi . (83) As previously stated, SOp2q is the one-dimensional group of rotations in two dimensions defined by (82) with only parameter θ. According to (83), the Lie algebra generator can be defined simply by: 0 ´1 X “ (84) 1 0 ˆ ˙ so that the elements of the SO(2) group are obtained through: x1 cos θ ´ sin θ x “ . (85) y1 sin θ cos θ y ˆ ˙ ˆ ˙ ˆ ˙ For SOp3q (special rotation group in 3D, which includes all 3D matrices which rotate a 3D vector along any of the coordinate axes x, y, z by finite angles θ1, θ2, θ3 respectively) the Lie algebra generators can be expressed as: 0 0 0 0 0 1 0 1 0 X1 “ 0 0 1 ,X2 “ 0 0 0 ,X3 “ ´1 0 0 (86) ¨0 ´1 0˛ ¨´1 0 0˛ ¨ 0 0 0˛ ˝ ‚ ˝ ‚ ˝ ‚ So that the elements of the SO(3) group are obtained through:

1 2 3 i θ X1`θ X2`θ X3 ˆi ˆi 2 Rpθ q “ e “ ... “ I ` θ Xi sin θ ` θ Xi p1 ´ cos θq (87)

17 Carin Jakobsson Algebraic framework where θi θˆi “ , θ “ pθ1q2 ` pθ2q2 ` pθ3q2 θ and the Baker-Campell-Hausdorf formulaa has be used due to non-commutivity of the elements of SO(3).

3.3.1 Notation and classification A Lie algebra corresponding to a Lie group G is denoted as g. Some common examples of Lie algebras and the structure of their elements are (compare to table 1):

Algebra name Notation Description General linear algebra gln n ˆ n matrices Special linear algebra sln traceless n ˆ n Orthogonal algebra on antisymmetric n ˆ n matrices Unitary algebra un antihermitian n ˆ n matrices Special unitary algebra sun traceless antihermitian matrices

Table 2: Examples of Lie algebras

3.3.2 Rigorous definition of a Lie algebra A Lie algebra is a vector space g over some field F equipped with a binary operation, the so called Lie bracket: r¨, ¨s, x, y P g ñ rx, ys P g satisfying:

• Bilinearity in both arguments, i.e. a, b P R, x, y, z P g gives:

rax ` by, zs “ arx, zs ` bry, zs, rx, ay ` bzs “ arx, ys ` brx, zs (88)

• Alternativity: rx, xs “ 0 @ x P g (89)

• The Jacobi identity:

rx, ry, zss ` ry, rz, xss ` rz, rx, yss “ 0 @ x, y, z P g (90)

3.4 Kac-Moody algebras 3.4.1 Cartan matrix and Dynkin diagram A Kac-Moody (KM) algebra can be defined by a Cartan matrix A, that is, an n ˆ n matrix satisfying:

Aii “ 2 @i “ 1, ..., n (91)

Aij P Z´ i ‰ j (92)

18 Carin Jakobsson Algebraic framework

Aij “ 0 ñ Aji “ 0 (93) Any Cartan matrix can be related to a so called Dynkin diagram. For our purposes we should consider only Cartan matrices which satisfies AijAji ď 4, in which case its Dynkin diagram can be obtained by:

1. construct n nodes, labeled by i “ 1, ..., n

2. two nodes i and j are linked by a number of maxp|Aij|, |Aji|q lines.

3. if |Aij| ą |Aji| an arrow from j to i is added.

Figure 2: Dynkin diagrams of finite-dimensional Lie algebras A2, B2, G2 and p2q ` affine Kac-Moody algebras A2 , A1 [27]

Some examples of Dynkin diagrams and their corresponding Lie algebras are seen in figure 2, the related Cartan matrices are given by:

2 ´1 ApA q “ (94) 2 ´1 2 ˆ ˙ 2 ´2 ApB q “ (95) 2 ´1 2 ˆ ˙ 2 ´3 ApG q “ (96) 2 ´1 2 ˆ ˙ 2 2 ´4 ApAp qq “ (97) 2 ´1 2 ˆ ˙ 2 ´2 ApA`q “ (98) 1 ´2 2 ˆ ˙ We shall only be concerned with KM algebras related to connected (no loose roots) Dynkin diagrams. Furthermore we will (for most part) restrict our study to Cartan matrices A which are invertible (detpAq ‰ 0) and symmetrizable i.e. can be written as: A “ DS (99)

19 Carin Jakobsson Algebraic framework where D is diagonal with positive entries and S is symmetric. The matrix S is called a symmetrization of A and is unique up to an overall positive factor. One can characterize Cartan matrices fulfilling the above criteria by the signature of its symmetrization. A Cartan matrix is said to be finite if signpSq “ pn, 0q (Eu- clidean signature) and Lorentzian if signpSq “ pn ´ 1, 1q (Lorentzian signature). There is another important class of Cartan matrices which consist of matrices with zero determinant and positive semi-definite symmetrization S (only one zero eigenvalue), these are classified as affine KM algebras. One can verify from the Cartan matrices (94)-(132) that the finite-dimensional Lie algebras A2, B2 and G2, are of, or has symmetrizations of Euclidean signature, whereas p2q ` the affine algebras A2 and A1 are of or have symmetrizations of positive semi- definite (affine) signature.

3.4.2 Generators Assuming we have a symmetrizable, invertible Cartan matrix A, the correspond- ing KM algebra is defined as a Lie algebra gpAq ((88)-(90) satisfied) generated by 3n generators hi, ei, fi such that the Chevalley-Serre relations are satisfied, namely: rhi, hjs “ 0 (100)

rhi, ejs “ Aijej pno summation on jq (101)

rhi, fjs “ ´Aijfj pno summation on jq (102)

rei, fjs “ δijhj pno summation on jq (103)

rei, rei, ...rei, ejss...s “ 0 (104)

1´Aij commutators

rfi, rfi, ...rfi, fjss...s “ 0 (105)

1´Aij commutators The equations (104) and (105) are the Serre relations defined for i ‰ j. The relations (100)- (105), together with the Jacobi identity (90), allows one to split a KM algebra into a direct sum:

g “ n` ‘ h ‘ n´ (106) where n` includes all multi-commutators

rei1 , rei2 , ..., reik´1 , eik s, ...s (107) n´ includes all multi-commutators

rfi1 , rfi2 , ..., rfik´1 , fik s, ...s (108) and h is the abelian or Cartan subalgebra of g, containing the generators hi. The dimension n of the Cartan subalgebra is the rank r of g. The dimension d of the algebra g is instead related to the number of non-zero multi-commutators

20 Carin Jakobsson Algebraic framework

(107) and (108). The Serre relations, impose certain constraints among the multi-commutators, such that the KM algebra may be finite-dimensional or infinite-dimensional. A finite-dimensional KM algebra can also be classified as a finite-dimensional, semi-simple Lie algebra (see figure 13) and one can show that this is the case if and only if S has Euclidean signature. On the other hand, if S has Lorentzian signature the corresponding KM algebra is infinite- dimensional and the algebra is refereed to as a Lorentzian KM algebra.

In case the Cartan matrix of a KM algebra has zero determinant, and is sym- metrizable such that S is positive semi-definite (one zero eigenvalue) the al- gebra is said to be affine (see figures 14 and 15). A hyperbolic KM algebra is a Lorentzian KM algebra with the property that removing any node from its Dynkin diagram leaves one with a Dynkin diagram of finite-dimensional or affine type. Note that the figures 13, 14 and 15 include all finite-dimensional and affine Lie algebras. Thus, given a Dynkin diagram, one can determine whether or not the corresponding algebra is of the hyperbolic type from these figures.

3.4.3 Examples

Finite-dimensional simple Lie algebra A2

A2 (also denoted su3) has the symmetric Cartan matrix (94) of Euclidean sig- nature, i.e. a finite-dimensional simple Lie algebra. (94) gives rise to relations:

rh1, h2s “ 0,

rh1, e1s “ 2e1, rh1, e2s “ ´e2,

rh2, e1s “ ´e1, rh2, e2s “ 2e2

rh1, f1s “ ´2f1, rh1, f2s “ f2, (109)

rh2, f1s “ f1, rh2, f2s “ ´2f2

re1, re1, e2ss “ 0, re2, re2, e1ss “ 0,

rf1, rf1, f2ss “ 0, rf2, rf2, f1ss “ 0

By these relations all commutators with three or more e’s or f’s are killed. One may take as a basis of A2 the generators:

h1, h2, e1, e2, f1, f2, re1, e2s, rf1, f2s (110) thus, A2 is an eight-dimensional simple Lie algebra.

Finite-dimensional simple Lie algebra B2

B2 has Cartan matrix (95) which is symmetrizable with Euclidean signature, i.e. a finite-dimensional simple Lie algebra. (95) gives rise to commmutation

21 Carin Jakobsson Algebraic framework relations: rh1, h2s “ 0,

rh1, e1s “ 2e1, rh1, e2s “ ´2e2,

rh2, e1s “ ´e1, rh2, e2s “ 2e2

rh1, f1s “ ´2f1, rh1, f2s “ 2f2, (111)

rh2, f1s “ f1, rh2, f2s “ f2,

re1, re1re1, e2sss “ 0, re2, re2, e1ss “ 0,

rf1, rf1rf1, f2sss “ 0, rf2, rf2, f1ss “ 0 By these relations (and with use of the commutation rules (88)-(90)), all commu- tators rre1, re1...re1, e2ss...ss with four or more e’s vanishes and all commutators rre2, re2...re2, e1ss...ss with three or more e’s vanishes. Similarly for the f’s. One may take as a basis of B2 the generators:

h1, h2, e1, e2, f1, f2, re1, e2s, re1, re1, e2ss, rf1, f2s, rf1, rf1, f2ss (112) thus B2 is a ten-dimensional, simple Lie algebra.

` Infinite-dimensional affine Kac-Moody algebra A1

` A1 has Cartan matrix (98) which is symmetric with affine signature, that is one zero eigenvalue and the rest positive. (98) gives rise to commmutation re- lations similar to those of B2, with the exception that rh2, e1s “ ´2e1 and that re2, re2, e1ss and rf2, rf2, f1ss don’t vanish. This small change makes all the dif- ference, the commutation relations no longer kill the commutators of more than four e’s and f’s and the algebra becomes infinite-dimensional.

3.4.4 Roots

A KM algebra can be defined in terms of its simple roots αi, given by:

rh, eis “ αiphqei pno summation on iq (113) such that αiphjq “ Aji so that (101) is satisfied. αiphq is an r-dimensional vector and two simple roots are by definition linearly independent. A relation similar to (113) applies to the elements of n`, namely:

rh, rei1 , rei2 , ..., reik´1 , eik s, ...ss

“ pαi1 ` αi2 ` ... ` αik qphqrei1, rei2, ..., reik´1, eiks, ...s (114) and for the elements of n´:

rh, rfi1 , rfi2 , ..., rfik´1 , fik s, ...ss

“ ´pαi1 ` αi2 ` ... ` αik qphqrfi1 , rfi2 , ..., rfik´1 , fik s, ...s (115) If the commutator

rei1 , rei2 , ..., reik´1 , eik s, ...ss (116)

22 Carin Jakobsson Algebraic framework

is non-zero, αphq ” pαi1 ` αi2 ` ... ` αik qphq is a positive root (defined as a linear combination of the simple roots αi with integer non-negative coefficients) of g. Likewise, if

rfi1 , rfi2 , ..., rfik´1 , fik s, ...ss (117) is non-zero, ´αphq ” ´pαi1 ` αi2 ` ... ` αik qphq is a negative root (defined as a linear combination of the simple roots with integer non-positive coefficients) of g. In general, a root can be thought of as an eigenvalue-vector of the adjoint map adhpxq “ rh, xs where x ‰ 0 P g, i.e.

rh, xs “ αphqx (118)

One can show that all roots are positive or negative. The simple roots are all positive roots and form a basis to the dual Cartan subalgebra or rootspace h˚ which contain all roots. The roots in h˚ can be obtained by Weyl reflections of the simple roots, that is, reflections in a hyperplane perpendicular to the simple roots. The Weyl group W of an algebra g is generated by repeated Weyl reflections on the set of simple roots. If the simple roots can be repeatedly Weyl reflected such that the reflections eventually coincides with the simple roots, the algebra is finite-dimensional, if not, it is infinite-dimensional. The roots can also be obtained by summation of simple roots for non-zero commutators according to (114) and (115). A diagram (in Rn, n “ r being the rank of g) with all roots located, is called a root diagram of g. According to (118) and the relations (114) and (115), a finite-dimensional algebra has a finite number of roots, whereas an infinite-dimensional algebra has an infinite number of roots. A Weyl chamber of an algebra g is a space in the root diagram onto which all roots can be reflected by Weyl reflections.

3.4.5 The bilinear form We shall assume the Cartan matrix is invertible and symmetrizable (as required for a hyperbolic KM algebra). We have:

A “ DS, D “ diagp1, ..., nq (119) and we define a bilinear form in the root space h˚ by setting:

pαi, αjq “ Sij (120) for simple roots αi and αj. The length or norm of a simple root is defined by:

|αi| “ pαi, αiq (121) a The off-diagonal entries of S encodes an angle θij between αi and αj. In the case of a Euclidean metric, the bilinear form is just the scalar product

pαi, αjq “ |αi||αj| cos θij (122)

23 Carin Jakobsson Algebraic framework

The fundamental Weyl chamber of an algebra g is the Weyl chamber such that the scalar product between a vector α located inside the chamber, and any of the simple roots αi, is positive i.e.

pα, αiq ą 0 (123)

Note that a Cartan matrix must have Aii “ 2 (according to (91)), so we get: 2 2 “ iSii ñ i “ 2 (124) |αi| and the Cartan matrix can be obtained from the simple roots as (no summation on i): pαi, αjq Aij “ iSij “ ipαi, αjq ñ Aij “ 2 2 (125) |αi| We see that the scaling of the roots is irrelevant w.r.t. the Cartan matrix (which fully determines? the algebra), a standard convention is to set the length of the longest roots to 2 so that in case all simple roots have same length, we get:

Aij “ pαi, αjq (126) and Aij must be symmetric.

3.4.6 Fundamental weights In order to find the fundamental Weyl chamber of an algebra, one may use the concept of fundamental weights, namely, vectors Λi P h such that:

pΛi, αjq “ δij (127)

The fundamental weights are thus orthogonal to all simple roots w.r.t. the bilinear product. The condition (123) is then equivalent to the requirement:

i i α “ λ Λi, λ ą 0 (128) i.e. the fundamental weights lie on the intersections of the boundary walls of the fundamental Weyl chamber.

3.4.7 Examples

Finite-dimensional simple Lie algebra A2 ? The Cartan matrix (94) is symmetric so the simple roots are of equal length 2 and (126) is applicable. Furthermore, (94) has two positive eigenvalues i.e. has Euclidean signature, so the bilinear form can be defined as the Euclidean scalar product (122). Thus, the angle between the two simple roots is given by:

A12 “ ´1 “ pα1, α2q “ |α1||α2| cos θ12 “ 2 cos θ12 (129)

24 Carin Jakobsson Algebraic framework

2π i.e. θ12 “ 3 . A root diagram of A2 can be obtained by aligning α1 with the x-axis of the Euclidean two-dimensional coordinate system with metric g “ diagp1, 1q. The simple roots are then represented by the blue arrows in figure 3. The rest of the roots (represented by dotted arrows in the same figure) can be located in the diagram by Weyl reflections w1 and w2 corresponding to the simple roots α1 and α2 respectively. We see that:

w1pα2q “ α1 ` α2

w2pα1q “ ´pα1 ` α2q

w2pw1pα2qqq “ α1 (130) w1pw2pα1qqq “ α2

w2pw1pw2pα1qqqq “ ´α2

w1pw2pw1pα2qqqq “ ´α1

Using the notation wipwjpαkqq “ αk ô wiwj “ 1 we also have the relations:

2 3 2 wi “ 1, pw1w2q “ 1, pw1w2w1q “ 1 (131) indicating that there are no further roots, which corresponds to the fact that the commutators with three or more e’s and f’s vanishes (see (109)).

Figure 3: Root diagram of A2 “ su3

Thus, the roots of A2 may be represented? in a Euclidean two-dimensional co- ordinate system, all roots have length 2 and are equally distributed at the corners of a hexagon. The directions of the fundamental weights are given by (127) and are represented by the green arrows in figure 3, and the fundamental Weyl chamber is by the requirement (127) given by the yellow area, an infinitely large area, partly bounded by the fundamental weights.

25 Carin Jakobsson Algebraic framework

`` Hyperbolic Kac-Moody algebra A1

`` A1 “ AE3 “ H3 is the hyperbolic KM algebra that can be associated to the BKL oscillations for pure gravity. It has Cartan matrix:

2 ´2 0 `` ApA1 q “ ´2 2 ´1 (132) ¨ 0 ´1 2 ˛ ˝ ‚ By use of the rules of section 3.4.1 we can construct the corresponding Dynkin diagram:

`` Figure 4: Dynkin diagram of the hyperbolic Kac-Moody algebra A1

Note that by removing the right, left and middle node from the Dynkin diagram `` ` of A1 one obtains the affine algebra A1 , finite-dimensional algebra A2 and the finite-dimensional A1 ‘ A1. Thus, removing any of the nodes, one obtains an affine or a finite-dimensional algebra, as required for a? hyperbolic algebra. The Cartan matrix has eigenvalues λ1 “ 2, λ2,3 “ 1 ˘ 2 i.e. has Lorentzian signature. The Cartan matrix is symmetric and provides a natural metric for `` ? the root space of A1 where the length of the simple roots is 2 and (126) apply. `` The Weyl chamber of A1 can be visualized in different ways. One may for example express the simple roots in Minkowski space (metric η “ diagp´1, 1, 1q) as: ? ? ? ? α 2 0, 1, 0 , α 2 ?1 , 1, ?1 , α 2 0, 1 , 3 (133) 1 “ p q 2 “ p 3 ´ ´ 3 q 3 “ p ´ 2 2 q

`` since that will give us the correct scalar products. The rest of the roots of A1 are obtained by Weyl reflections w1, w2, w3 in the simple roots. Since all the simple roots are space-like, the roots are generated by reflections in time-like `` planes. The following relations apply for the Weyl reflections of A1 :

3 2 8 pw2w3q “ 1, pw1, w3q “ 1, pw1w2q “ 1 (134) where the last equation means that we always get a new root by the reflec- tion w1w2 (compare to the relations (131)). The fundamental weights can be calculated by use of (127) giving:

? Λ 2 2 , ?1 , ?1 , Λ 3 , 0, 0 , Λ 2 1, 0, 1 (135) 1 “ p´ 3 2 6 q 2 “ p´ 2 q 3 “ 3 p´ q b b We see that two of the fundamental weights are time-like, whereas the third one is light-like. In figure 5 the fundamental weights are represented by the red arrows (some of the weights are reversed in order to visualize the fundamental

26 Carin Jakobsson Algebraic framework

Weyl chamber) and the light-cone by the transparent surface. Note that in Lorentzian space with coordinates pt, x, yq equal length time-like vectors are defined by a hyperbolic surface:

´t2 ` x2 ` y2 “ const ă 0 (136)

The hyperbolic surface (blue surface in figure 5), is an analog to the circle of the root space of the finite-dimensional A2 algebra (figure 3). The fundamen- `` tal Weyl chamber of A1 is the bounded by the three infinitely large planes, each plane containing two of the three fundamental weights. These planes will intersect the hyperboloid on hyperbolic lines. One can equally well represent the Weyl chamber boundaries on a hyperbolic disk, in which case one obtains the purple area in figure 6. The corner on the boundary of the disc represents the unreachable Weyl chamber corner specified by the light-like fundamental `` weight. A1 has infinitley many Weyl chambers, some of which are the remain- ing bounded areas within the simplex on the hyperbolic disk in figure 6. More Weyl chambers can be constructed by moving the simplex corners along the boundary of the hyperbolic disc, one can do this infinitely many times, some of `` the infinitely many Weyl Chambers of A1 can be seen in figure 7. Note that all Weyl chambers has a corner at the boundary of the hyperbolic disk.

Figure 5: Light cone and hyperbolic surface with fundamental weights defined by (135)

27 Carin Jakobsson Algebraic framework

`` Figure 6: Fundamental Weyl chamber of A1 [27]

`` Figure 7: Weyl chambers of A1 [28]

28 Carin Jakobsson Algebraic framework

4 The billiard picture

The following is a derivation of the so called billiard picture based on the BKL- oscillations and the corresponding connection to hyperbolic KM algebras, for the sake of investigating symmetries of gravitation.

4.1 Iwasawa decomposition of metric BKL define the line element as:

2 2 2pl 2pm 2pn α β ds “ ´dt ` pt lαlβ ` t mαmβ ` t nαnβqdx dx (137) and suggest the signs of the parameters pl, pm, pn undergoes chaotic oscillations between the spatial directions defined by vectors l, m, n [4]. For the sake of deriving the billiard picture, we rewrite the expression of the line element: 2 m n 2 µ ν ds “ gmndx dx “ ´gsdt ` gµν dx dx (138) where gs “ |gµν | is the spatial metric determinant. The spatial metric is further- more expressed as a diagonalization or in a so called Iwasawa decomposition: T 2 gµν “ N A N (139) subject to:

1 N12 N13 1 2 3 A “ exp ´diagpβ , β , β q , N “ 0 1 N23 (140) ¨0 0 1 ˛ ` ˘ One may thus express the line element as: ˝ ‚

d k dl2 “ e´2β pwkq2 (141) k“1 ÿ where: k i w “ Nikdx (142) i ÿ Comparing to the BKL-parameters pl, pm, pn of (137) we get:

1 2 3 e´β “ tpl , e´β “ tpm , e´β “ tpn (143) So that as one approaches the singularity t “ 0 we get: 1 pl ă 0 β Ñ `8 2 pm ą 0 ñ β Ñ ´8 (144) 3 pn ą 0 β Ñ ´8 Thus, the formulation of the metric (139) implies an oscillatory exchange of signs of the parameters βi in the limit of the singularity, according to section 2.5.3. Note that the Greek parameters of section 2.4.2 is related to βi by α “ ´β1, β “ ´β2, γ “ ´β3 (145)

29 Carin Jakobsson Algebraic framework

4.2 System potential A system of pure gravity corresponds to the Einstein Lagrangian:

? ? ab LGpgmnq ” K ´ V “ ´gR ” ´gg Rab (146) For Kasner’s particular solution, we have only time dependence, i.e. only kinetic energy K is present, and we get, by inserting the metric of the line element (24) with (143) into (146):

2 µ ν 9i 2 9i LG “ Gµν β9 β9 “ pβ q ´ β (147) i i ÿ ´ ÿ ¯ where the dot now (for the sake of simplifying notation) denotes derivative w.r.t. τ which is related to t by (63). The sums of (147) indices from 1 to 3 and:

0 ´1 ´1 Gµν “ ´1 0 ´1 (148) ¨´1 ´1 0 ˛ ˝ ‚ may be interpreted as a metric in β-space, in which a particle move along a parametrized curve βpτq. A detailed derivation is provided in appendix D. Investigating the signature of the β-space metric, we get:

´λ ´1 ´1 ´1 ´λ ´1 “ ´λ3 ` 3λ ´ 2 “ 0 (149) ˇ ˇ ˇ´1 ´1 ´λˇ ˇ ˇ ˇ ˇ ˇ λ ˇ 2, λ λ 1 (150) ˇ ñ 1 “ˇ ´ 2 “ 3 “ I.e. two positive eigenvalues and one negative, the three-dimensional β-space is thus of Lorentzian signature (just as the Cartan matrix of the hyperbolic KM `` algebra A1 ). As stated in section 2.4.2, the particular solution of Kasner has equations of motion (take use of (145)):

β:i “ 0 (151) On the other hand, for Kasners generalized solution, the system is not only time dependent, but corresponds to a potential V pβq, represented by the terms of the right sides of the equations (66), namely:

:1 1 2 2 2 2 4 β “ 2 pµb ´ νc q ´ λ a :2 1 2 2 2 2 4 β “ 2 pλa ´ νc q ´ µ b (152) :3 1 2 2 2 2 4 β “ 2 pλa ´ µb q ´ ν c A particle following the curve βpτq defined by this system is subject to a La- grangian obtained by inserting the expression of the metric corresponding to (141) into (146) as:

µ ν LG “ K ´ V “ Gµν β9 β9 ´ V pβq (153)

30 Carin Jakobsson Algebraic framework

The potential of the system has one part connected to gravitation VG and one part connected to symmetry VS. These are given by:

V “ VG ` VS (154)

1 ´2αijkpβq i 2 1 ´2mj pβq i k VGpβq “ 4 e pCjkq ` 2 e pCjkCji ` ”more”q i‰j,i‰k,j‰k j ÿ ÿ (155) 2 1 ´2pβj ´βiq VSpβq “ e PimNjm (156) 2 iăj m ÿ ˆ ÿ ˙ Where ”more” stands for the terms in the first sum that arise upon taking i “ j or i “ k. Furthermore we have:

m mjpβq “ β , (157) m‰j ÿ i m αijkpβq “ 2β ` β (158) m‰i,j,k ÿ Pij is defined by an extension of the transformation equation (139) in configu- ration space to a canonical transformation in phase space by the formula:

ij i π dgij “ π dβi ` PijdNij (159) iăj ÿ and: i i1 ´1 ´1 ´1 Cjk “ f j1k1 Nii1 Njj1 Nkk1 (160) i1,j1,k1 ÿ i where fj1k1 are structure constants of the spatial homogeneity group (see ap- pendix C).

4.3 The limit t Ñ 0 4.3.1 Potential In the limit of the singularity, the parameters βi approaches ˘8 so that the exponential functions become infinitely steep, and we may replace them by the sharp-wall function: 0, x ă 0 Θpxq “ (161) #`8 x ą 0 Furthermore aΘpxq “ Θpxq for all a ą 0 so we get:

j i V “ Θp´2αijkpβqq ` Θp´2mjpβqq ` Θp´2pβ ´ β qq (162) i‰j,i‰k,j‰k j iăj ÿ ÿ ÿ More specifically (with use of (157) and(158)):

V pβq “ Θp´2p2β1 ` β2 ` β3qq ` Θp´2p2β2 ` β1 ` β3qq ` Θp´2p2β3 ` β1 ` β2qq

31 Carin Jakobsson Algebraic framework

`Θp´2pβ2 ` β3qq ` Θp´2pβ3 ` β1qq ` Θp´2pβ1 ` β2qq `Θp´2pβ2 ´ β1qq ` Θp´2pβ3 ´ β1qq ` Θp´2pβ3 ´ β2qq (163) So that the first term of (163) defines a plane in beta space 2β1 ` β2 ` β3 “ 0 such that the potential is zero on one side and infinite on the other (a potential wall), and likewise for the other terms. The whole sum specifies a volume in β-space and it turns out that some of the terms of (163) define walls that are not needed in order to define this volume, i.e. subdominant walls. Singling out only the dominant walls of the potential and simplifying we get:

V pβq “ Θp´2β1q ` Θp´pβ2 ´ β1qq ` Θp´pβ3 ´ β2qq (164)

The first term originates from VG and is therefore associated with a gravita- tional wall: 1 α1pβq “ 2β “ 0 (165)

The last two terms originates from VS and are therefore associated with the symmetry walls:

2 1 3 2 α2pβq “ β ´ β “ 0, α3pβq “ β ´ β “ 0 (166)

4.3.2 Interpretation of potential

In the three-dimensional β-space, with Lorentzian metric Gµν , a system with potential (164) corresponds to a particle following a trajectory βipτq which is always light-like, but the trajectory direction is varied according to the signature oscillations of the components βi. The directional changes are the particle’s response to the symmetry and gravitation wall-potentials. It is possible to project the reflections in the wall-potentials in three-dimensional β-space into reflections in lines on a hyperbolic disc, in which case we get the picture in figure 8. The straight lines represent gravitational walls 2β1 “ 0, 2β2 “ 0 and 2β3 “ 0), whereas the hyperbolic lines are due to the symmetry walls β2 ´β1 “ 0, β3 ´β2 “ 0 and β3 ´β1 “ 0. The particle following a path βipτq is contained by, and reflected by the dominant wall potentials α1pβq “ 0, α2pβq “ 0, α3pβq “ 0 (noted by 1, 2, 3 respectively in figure 8) giving rise to the chaotic, oscillatory behaviour described in section 2.

4.3.3 Spatial decoupling Since the direction of the trajectories of the system particle is always light- like, the particle will always remain inside a light-cone in the beta-spacetime. Consequently, neighbouring points in β-space decouple as one moves in direction towards the singularity at t “ 0 see figure 9. This means that the spatial derivatives of the field equations must vanish in the vicinity of the singularity [27], and the solution becomes homogeneous (compare to the BKL conjecture in section 2.5.1). Note that this statement of spatial decoupling, as well as the BKL conjecture are not rigorously derived, and should be regarded as hand-waving, heuristic physical arguments.

32 Carin Jakobsson Algebraic framework

Figure 8: Billiard picture of the pure gravity case [27]

Figure 9: Decoupling of spatial points as t Ñ 0

`` 4.4 Relation to hyperbolic Kac-Moody algebra A1 Note that the β-space metric:

0 ´1 ´1 Gµν “ ´1 0 ´1 (167) ¨´1 ´1 0 ˛ ˝ ‚ (see derivation in appendix D) defines a bilinear form:

µ ν K “ Gµν β9 β9 . (168) (kinetic energy). For vectors F and G in β-space, defined by:

µ µ F “ Fµβ ,G “ Gµβ (169)

˚ such that Fµ and Gµ belong to the dual β -space, we define the inverse bilinear form: µν pF,Gq “ G FµGν (170)

33 Carin Jakobsson Generalizations and symmetries such that pF |Gq encodes the angle between two vectors F and G and pF |F q gives the norm of a vector F in β-space. Note that:

1 ´1 ´1 µν 1 G “ 2 ´1 1 ´1 (171) ¨´1 ´1 1 ˛ ˝ ‚ so we may equivalently express the bilinear form as:

1 pF,Gq “ FiGi ´ 2 Fi Gi (172) i i i ÿ ´ ÿ ¯´ ÿ ¯ When applied on the wall potentials α1, α2, α3 we get norms of 2 and:

pα1, α2q “ tF1 “ 2,G2 “ 1,G1 “ ´1u “ ´2 (173)

pα1, α3q “ tF1 “ 2,G3 “ 1,G2 “ ´1u “ 0 (174)

pα2, α3q “ tF2 “ 1,F1 “ ´1,G3 “ 1,G2 “ ´1u “ ´1 (175) Define a matrix A by:

Aij “ pαi, αjq, i, j “ 1, 2, 3 (176) i.e. 2 ´2 0 A “ ´2 2 ´1 (177) ¨ 0 ´1 2 ˛

˝ ‚ `` we see that A is the Cartan matrix of the hyperbolic KM algebra A1 (132). Thus, the dominant potential walls defining the system of the BKL oscillations `` in β-space, in fact corresponds to the simple roots of A1 (up to a norma- tive constant) and the volume defined by the potential walls defines the Weyl `` chamber of A1 .

34 Carin Jakobsson Generalizations and symmetries

5 Generalizations and symmetries

At the time of discovery of the BKL phenomenon researchers were fascinated by the fact that the general solution indeed has a physical singularity close to which it exhibits oscillations of chaotic nature. Naturally, they were motivated to find generalizations to systems of higher dimension and to more general cosmological systems. The discovery of the connection to Lie algebras and Lie groups fueled the interest further as they imply that symmetric properties are inherent in cosmological systems (see section 3.2.1). This section provides a discussion on the connection between the BKL phenomenon and hyperbolic KM algebras for more general cases as well as some insight on the symmetry-aspect.

5.1 The general case We have analysed the case of pure gravity in four dimenions, where the La- grangian is defined by: ? LGpgmnq “ ´gR (178) We found that the system potential was given by (154), (155) and (156). Taking the limit of the singularity t Ñ 0 we arrived at the association with the Weyl `` chamber of a hyperbolic KM algebra A1 “ AE3 “ H3 with Cartan matrix (177) and Dynkin diagram in figure 4. In the general case of dimension D, the system Lagrangian is a function of not only the metric, but also of p-form fields Appq and a scalar dilaton φ. The system Lagrangian is given by:

p eλp qφ ppq D m 1 ppq ppqm1..mp`1 Lpgmn, φ, A q “ ´ g R´BmφB φ´ F F 2 p 1 ! m1...mp`1 ˜ p p ` q ¸ a ÿ (179) where the notation Dg has been used for the total metric in order to save g ppq for the spatial part. The field strengths Fm1...mp`1 are related to the various p-forms Appq as:

F ppq Appq p permutations (180) m1...mp`1 “Bm1 m2...mp`1 ˘ For more detailed definitions see [27] section 2.1. By rewriting the action (179) on canonical form and performing an Iwasawa decomposition of the metric (139), one obtains the kinetic energy (compare to (147)):

µ ν 92 9 2 92 K “ Gµν βτ βτ “ Σβ ´ pΣβq ` φ (181) and the system potential:

el magn V “ VS ` VG ` Vppq ` Vppq ` Vφ (182) p ÿ ` ˘ Where we have, in addition to symmetry and gravitation, contribution from the p-form fields and the dilaton, some of which depend on the β-parameters expo- nentially. In the limit of the singularity, the exponentials can be approximated

35 Carin Jakobsson Generalizations and symmetries by sharp wall potentials, and the potential reduces to a sum of dominant sharp wall potentials, given by:

Symmetry walls: βj`1 ´ βj “ 0 j “ 1, 2, ..., d (183) Gravitational wall: 2β1 ` β2 ` ... ` βd´2 “ 0 (184) Electric p-form wall: λppq β1 ` ... ` βp ` φ “ 0 (185) 2 Magnetic p-form wall:

λppq β1 ` ... ` βd´p´1 ´ φ “ 0 (186) 2 where d “ D ´ 1. It is possible to derive the billiard picture for any system, regardless of space-time dimension, p-forms and dilaton couplings. However, only some of these cases can be associated with the Weyl chamber of a hyperbolic KM algebra. Depending on the form of the potential, one arrives at different walls on the hyperbolic disk (billiard table). If the billiard ball is bounded by potential walls in every direction on the disk (see left side of figure 10), the area corresponds to the fundamental Weyl chamber of some hyperbolic KM algebra and the behaviour towards the singularity becomes chaotically oscillatory. On the other hand, if the ball is not in every direction bounded by potential walls, it will instead exhibit a stable motion according to Kasner’s particular solution toward the boundary of the billiard table, never actually reaching the boundary which represents infinity (see right side of figure 10). In this case the billiard motion is no longer restricted to the fundamental Weyl chamber of a hyperbolic KM algebra, it can however be associated with other algebras [23].

36 Carin Jakobsson Generalizations and symmetries

Figure 10: Billiard motion projected onto the hyperbolic disk [26]

5.2 Pure gravity of higher dimension The BKL phenomenon was discovered for pure gravity in four dimensions in 1970, in 1986, it was shown to persist in generalizations up ten dimensions of pure gravity, but to vanish for higher dimensions [11], [13]. We shall see that this has a simple explanation in terms of the association with KM algebras. Deriving the billiard picture for pure gravity of arbitrary dimension gives kinetic energy (compare to derivation in appendix D):

2 µ ν i 2 i K “ Gµν β9 β9 “ pβ9 q ´ β9 (187) i i ÿ ´ ÿ ¯ and potential energy with contributions from symmetry and gravitation, corre- sponding to dominant walls (183) and (184) (see [21] section 2 for derivation). Note that the bilinear form, defined by the inverse metric Gµν can in the general case be derived as:

37 Carin Jakobsson Generalizations and symmetries

Let 1 1 ... 1 1 1 ... 1 Md “ ¨. . . .˛ (188) . . .. . ˚ ‹ ˚1 1 ... 1‹ ˚ ‹d so that: ˝ ‚ Gµν “ Id ´ Md. (189) where Id is the d-dimensional identity matrix. The inverse metric is given by

µν 1 G “ Id ´ Md (190) d ´ 1 since then: µν 1 Gµν G “ pId ´ MdqpId ´ Mdq d ´ 1

1 1 2 “ Id ´ Md ´ Md ` M (191) d ´ 1 d ´ 1 d and we have: 2 Md “ dM (192) so we get: µν 1 d Gµν G “ Id ´ Md ´ Md ` Md “ Id. (193) d ´ 1 d ´ 1 Taking the bilinear form:

µν A “ pαµ, αν q “ G αµαν (194) where αµ are the dominant walls (183) and (184), one arrives at the d-dimensional Cartan matrix: 2 ´1 0 0 ... 0 0 0 ´1 2 ´1 0 ... 0 0 ´1 ¨ 0 ´1 2 ´1 ... 0 0 0 ˛ ˚ . ‹ A “ ˚ . ‹ (195) ˚ ‹ ˚ 0 0 0 0 ... 2 ´1 0 ‹ ˚ ‹ ˚ 0 0 0 0 ... ´1 2 ´1‹ ˚ ‹ ˚ 0 ´1 0 0 ... 0 ´1 2 ‹ ˚ ‹ ˝ ‚ `` representing the algebra AEd “ Ad´2 with Dynkin diagram depicted in figure 11 (note that figure 4 is the special case d “ 3). In order for the BKL phenomenon to occur, the corresponding KM algebra must be of the hyperbolic type, that is, the algebra must be reducible to a finite-dimensional or affine KM algebra by the removal of any of its nodes. All finite-dimensional algebras are listed in figure 13 and all affine algebras are listed in figure 14 and 15. Going through the Dynkin diagrams for 4 ď D ă 11 one can see that removing any node, generates a finite-dimensional or an affine KM algebra. Take for example the

38 Carin Jakobsson Generalizations and symmetries

`` case D “ 10 corresponding to the algebra AE9 “ A7 , which has d ´ 1 “ 8 nodes in a connected ring and one node which is connected to only one of the nodes on the ring. Taking away the node with only one connection, we ` obtain the affine twisted A7 , taking away the node with three connections, we obtain the finite-dimensional A7 (plus A1, which doesn’t affect the algebraic structure in this context). Removing any of the remaining nodes on the ring, ` gives finite-dimensional algebras A8,D8,E8 or the twisted affine algebra E7 . For dimensions D ě 11 however, a finite-dimensional or affine algebra is not always obtained by the removal of a node, and the time-like planes which specify the shape of the billiard table, are not contained on the hyperbolic disk (see left side of figure 10).

Figure 11: Dynkin diagram related to BKL oscillations in spacetime dimension D “ d ` 1 (β-space dimension d).

5.3 11D SUGRA and other systems of interest One can show that pure gravity systems which include dilaton φ doesn’t exhibit the BKL oscillations in any dimension [6]. The general solutions relevant to su- perstring and M-theory however, can be associated with BKL oscillations thanks to the presence of p-forms [18]. The BKL oscillations close to the singularity may thus have some physical relevance and it has been speculated that they would be a contradiction w.r.t. the pre-big-bang scenario [18]. It was furthermore proved that the BKL oscillatory behaviour is a feature of p-form systems of arbitrary dimension ě 4 [22]. An important example is eleven-dimensional supergrav- ity (11D SUGRA), the low-energy limit of M-theory, which corresponds to an eleven-dimensional system including three-form fields. This system indeed ex- hibit the BKL oscillations with associated hyperbolic algebra E10 corresponding

39 Carin Jakobsson Generalizations and symmetries

Figure 12: Dynkin diagram of hyperbolic KM algebra E10 to the Cartan matrix: 2 ´1 0 0 0 0 0 0 0 0 ´1 2 ´1 0 0 0 0 0 0 0 ¨ 0 ´1 2 ´1 0 0 0 0 0 ´1˛ ˚ 0 0 ´1 2 ´1 0 0 0 0 0 ‹ ˚ ‹ ˚ 0 0 0 ´1 2 ´1 0 0 0 0 ‹ A “ ˚ ‹ (196) ˚ 0 0 0 0 ´1 2 ´1 0 0 0 ‹ ˚ ‹ ˚ 0 0 0 0 0 ´1 2 ´1 0 0 ‹ ˚ ‹ ˚ 0 0 0 0 0 0 ´1 2 ´1 0 ‹ ˚ ‹ ˚ 0 0 0 0 0 0 0 ´1 2 0 ‹ ˚ ‹ ˚ 0 0 ´1 0 0 0 0 0 0 2 ‹ ˚ ‹ ˝ ‚ and Dynkin diagram depicted in figure 12.

5.4 Dualities, hidden symmetries, dimensional reduction and extended geometries A duality is a special type of symmetry, where two equivalent descriptions of a particular physical situation exist, using different physical concepts. For exam- ple, Maxwell’s equations in the absence of electrical charges are invariant under an exchange of fields E Ñ ´B. The state of the system could be described in terms of an electric field, or equivalently, in terms of a magnetic field. The exchange of fields E Ñ ´B defines a duality of Maxwell’s equations in the absence of electric charges [15]. In his doctoral thesis, Ehlers found a duality between different components of the metric of a stationary (metric independent of the time-coordinate) vacuum spacetime, which maps solutions of Einstein’s field equations to other solutions [2]. Ehlers duality was expanded to a larger symmetry corresponding to the finite-dimensional special linear group SLp2q and the corresponding finite-dimensional KM algebra A1. Further generaliza- tions give rise to the infinite-dimensional Geroch Lie group associated with the affine KM algebra E9 [5]. Ehlers and Geroch duality symmetries appear af- ter dimensional reduction of certain gravitational systems, and are therefore referred to as hidden symmetries of these gravitational systems [12] [17]. For example, reduction of four-dimensional pure gravity to three dimensions gives rise to Ehlers symmetry SLp2q, and pure gravity systems of 3 ` n dimen- sions reduced to three dimensions gives as symmetry groups SLpn ` 1q “ An (extensions of Ehlers symmetry) [19]. The reduction of 11D SUGRA to two dimensions requires a larger symmetry group, namely the infinite-dimensional

40 Carin Jakobsson Generalizations and symmetries

Geroch symmetry [16]. The BKL phenomenon resembles dimensional reduc- tion to a one-dimensional (time-dependent) system in the sense that the spatial gradients are regarded of less importance than the time-gradients, and that the system solution in the vicinity of the singularity can be obtained by analysis of completely homogeneous models. The BKL conjecture is only valid locally, but there is reason to believe that the theory could apply in a more general manner, leading up to the question: Is it possible to achieve a hyperbolic KM algebra as a result of BKL-like dimensional reduction of a gravitational system, such that the association is valid non-locally [27]? An argument in favour of this inquiry refers to the size of the related hyperbolic KM algebras. BKL-like reduction of 11D SUGRA to a one-dimensional system would require a larger symmetry group than the Geroch group and it turns out that the hyperbolic KM algebra `` A1 corresponding to E10 is indeed large enough [25].

Hyperbolic KM algebras are of interest also in extended geometry, [29], [30], [31], [32], [33], [34], which provides a way to “geometrise” duality symmetries. In extended geometry, the gauge symmetry containing diffeomorphisms and gauge transformations of tensor fields, is promoted to “generalized diffeomor- phisms” where the duality group plays the role of a structure group, analogous Bx1m to GLpdq in gravity ( Bxn P GLpdq). Just like certain solutions of gravity may show symmetry in the form of isometries, flat solutions of extended geometry `` show enhanced symmetries, dualities. If extended geometry with A1 or E10 as structure group is constructed, it is likely to provide the correct sense in which these hyperbolic algebras are present already in gravity or supergravity, and may also give the concrete realization of how spatial gradients are encoded algebraically in infinite-dimensional algebras.

41 Carin Jakobsson Conclusion

6 Conclusion

The BKL phenomenon is an oscillatory behaviour of chaotic nature, exhibited by an inhomogeneous and anisotropic Kasner-like solution to the gravitational field equations in the vicinity of a space-like singularity. The inhomogeneous properties vanish in the limit of the singularity as the spatial points decouple and cannot interact. BKL oscillations are in essence derived by perturbation of the generalized solution of Kasner, identification of similarities of the perturbed equations to some homogeneous Bianchi models and analysis of the solution to these homogeneous models close to the singularity. The solution oscillates in two of three spatial directions during each so called Kasner era. Further analysis of the BKL phenomenon can be found in [4].

The oscillatory BKL behaviour can, by a reparametrisation of the metric and an analysis of the Lagrangian, be linked to, and encoded in a billiard motion where the billiard ball chaotically switches direction according to the BKL os- cillations. The shape of the billiard table which gives rise to these directional changes can be identified with the Weyl chamber of a hyperbolic KM algebra. The association of the BKL phenomenon and hyperbolic algebras can be gen- eralized from pure gravity to several systems of interest [21], [20], [22], [26], including the general case of supergravity. It is fascinating that the association with a hyperbolic algebra applies to one of the most general theories, namely 11D SUGRA, especially in the context of dimensional reduction and extended geometry.

42 Carin Jakobsson Appendix

A Derivation of gravitational field equations on normal form

Combining (3) and (4) we get:

a ac ac d ac d d d e d e Rb “ g Rcb “ g Rcdb “ g pBdΓcb ´ BbΓcd ` ΓdeΓcb ´ ΓbeΓcdq (197) So the Ricci Tensor components become:

0 0c d d d e d e R0 “ g pBdΓc0 ´ B0Γcd ` ΓdeΓc0 ´ Γ0eΓcdq

d d d e d e “ ´pBdΓ00 ´ B0Γ0d ` ΓdeΓ00 ´ Γ0eΓ0dq α α β 1 α 1 α β “ ´p´B0Γ0α ´ Γ0βΓ0αq “ 2 κα ` 4 κβ κα where we have used the values of the inverse metric components (6) and Christof- fel symbols (254). Similarily:

0 0c d d d e d e Rα “ g pBdΓcα ´ BαΓcd ` ΓdeΓcα ´ ΓαeΓcdq

d d d e d e “ ´pBdΓ0α ´ BαΓ0d ` ΓdeΓ0α ´ ΓαeΓ0dq β β β γ β γ “ ´pBβΓ0α ´ BαΓ0β ` Γβγ Γ0α ´ Γαγ Γ0βq

1 β 1 β β 1 γ β 1 γ “ ´p 2 Bβκα ´ 2 Bακβ ` Γβγ 2 κα ´ Γαγ 2 κβq 1 β β β γ β γ “ 2 p´κα,β ` κβ,α ´ Γβγ κα ` Γαγ κβq Thus: 0 1 β β γ β β γ Rα “ 2 pκβ,α ` Γαγ κβ ´ κα,β ´ Γβγ καq γ β γ β Adding Γβακγ ´ Γαβκγ “ 0 to the right side we can identify the expression as an anti-symmetrization of covariant derivatives i.e.:

0 1 β β Rα “ 2 pκβ;α ´ κα;βq

α Furthermore the spatial components Rβ are given by:

α αc d d d e d e Rβ “ g pBdΓcβ ´ BβΓcd ` ΓdeΓcβ ´ ΓβeΓcdq

αγ d d d e d e “ g pBdΓγβ ´ BβΓγd ` ΓdeΓγβ ´ ΓβeΓγdq Each of the terms evaluated in their own parenthesis:

αγ 0 δ δ “ g ppB0Γγβ ` BδΓγβq ´ pBβΓγδq

δ 0 δ σ 0 δ δ 0 δ σ `pΓδ0Γγβ ` ΓδσΓγβq ´ pΓβδΓγ0 ` Γβ0Γγδ ` ΓβσΓγδqq Now, the sum of the terms containing only spatial indices is the spatial Ricci tensor namely: α αγ αγ δ Pβ “ g Pγβ “ g Pγδβ

43 Carin Jakobsson Appendix where α α α α σ α σ Pβγδ “Bγ Γβδ ´ BδΓβγ ` ΓγσΓβδ ´ ΓδσΓβγ (198) Thus, we get:

α α αγ 0 δ 0 0 δ δ 0 Rβ “ Pβ ` g pB0Γγβ ` Γδ0Γγβ ´ ΓβδΓγ0 ´ Γβ0Γγδq

α αγ 1 1 δ 1 δ 1 δ “ Pβ ` g p 2 B0κγβ ` 4 κδκγβ ´ 4 κβδκγ ´ 4 κβκγδq α 1 αγ 1 δ α 1 δα 1 δ α “ Pβ ` 2 g B0κγβ ` 4 κδκβ ´ 4 κβδκ ´ 4 κβκδ Note that the fourth and fifth terms are equivalent since:

δα δα αβ αβ δα α δ κβδκ “ pκβδκ qpg gαβq “ pκβδg qpκ gαβq “ κδ κβ (199) So we have: α α 1 αγ 1 δ α 1 δα Rβ “ Pβ ` 2 g B0κγβ ` 4 κδκβ ´ 2 κβδκ (200) In order to simplify this further, we consider the expression:

1 ? α ? B0p ´gκ q (201) 2 ´g β

1 α ? ? α “ ? pκ B0 ´g ` ´gB0κ q 2 ´g β β

1 α 1 ? αγ ? αγ “ ? κ ? B0p´gq ` ´gg B0κγβ ` ´gκγβB0g 2 ´g β 2 ´g ˆ ˙ 1 α 1 αγ 1 αγ “ κ B0p´gq ` g B0κγβ ` κγβB0g (202) 4p´gq β 2 2 Where we have used (10) and the product rule. Comparing expressions (200) and (202) we find that the second term of (202) is the second term of (200). The first term of (202) may be simplified with use of the relation between a matrix and its determinant, namely, for a matrix A:

B BA |A| “ |A|A´1 (203) Bx Bx

When applied to the metric determinant |gαβ| “ ´g the first term of the ex- pression (202) is given by:

1 α γδ 1 α γδ 1 α δ “ κ p´gqg B0gγδ “ κ p´gqg κγδ “ κ κ 4p´gq β 4 β 4 β δ which is the third term of (200). It remains to compare the last term of (202) with the last term of (200). Note that:

αβ B0pg gαβq “ 0

αβ αβ ô g B0pgαβq ` gαβB0pg q “ 0

44 Carin Jakobsson Appendix

αβ αβ ô g καβ “ ´gαβB0pg q αβ αβ Which indicates that κ “ ´B0g . With this in mind, it is clear that the last terms of expressions (200) and (202) are equivalent. Thus, the three last terms of (200) can be replaced by the expression (201).

Thus, the vacuum field equations for a synchronous reference frame are given by: 0 1 α 1 α β R0 “ 2 B0κα ` 4 κβ κα “ 0 (204) 0 1 β β Rα “ 2 pκβ;α ´ κα;βq “ 0 (205) ? Rα P α ?1 gκα 0 (206) β “ β ` 2 ´g B0p ´ β q “

45 Carin Jakobsson Appendix

B Derivation of Kretschmann scalar for Kas- ner’s particular solution

The Kretschmann scalar is given by:

abcd R Rabcd (207)

The Kasner metric: 2p gmn “ diagp´1, t q (208) Where the notation t2p ” pt2p1 , t2p2 , t2p3 q is used. Furthermore we will denote 2p 2p1 2p2 2p3 2p 2p Σt ” t `t `t and gµν “ diagpt q ” t δµν for the spatial metric (greek indices labels spatial coordinates and latin indices label spacetime coordinates). The inverse Kasner metric is given by:

gmn “ diagp´1, t´2pq (209)

The Kasner determinant: 2 g “ |gmn| “ ´t (210) The Riemann tensor components:

a a a a e a e Rbcd “BcΓbd ´ BdΓbc ` ΓceΓbd ´ ΓdeΓbc (211) we get f abcd bh ci dj a Rabcd “ gaf Rbcd,R “ g g g Rhij (212) so that the Kretschmann scalar is given by:

abcd f bh ci dj a bh ci dj f a RabcdR “ gaf Rbcdg g g Rhij “ gaf g g g RbcdRhij (213) The Christoffel symbols:

a 1 ad Γbc “ 2 g pBbgcd ` Bcgbd ´ Bdgbcq (214)

We note that Bµgνγ =0, g0µ “ 0 and Bag00 “ 0 so we get:

0 µ 0 µ Γ00 “ Γ00 “ Γµ0 “ Γνγ “ 0 (215)

0 1 00 2p´1 Γµν “ 2 g p´B0gµν q “ pt δµν (216) µ 1 µγ ´1 µ Γν0 “ 2 g B0gγν “ t pδν (217) where we denote:

2p´1 2p1´1 2p2´1 2p1´1 pδµν ” diagppq, pt δµν ” diagpp1t , p2t , p3t q. (218)

We must have at least two spatial indices to get non-zero components of the Riemann tensor, so we have:

0 µ 0 0 0 R000 “ R000 “ Rµ00 “ R0µ0 “ R00µ “ 0 (219)

46 Carin Jakobsson Appendix and furthermore get:

0 µ µ µ µ 0 µ R0µν “ Rν00 “ Rνγδ “ Rνγ0 “ Rν0γ “ Rµνγ “ R0νγ “ 0 (220)

0 0 0 0 0 e 0 e 2p´2 Rµν0 “ ´Rµ0ν “Bν Γµ0 ´ B0Γµν ` ΓνeΓµ0 ´ Γ0eΓµν “ ´ppp ´ 1qt δµν (221) µ µ µ µ µ e µ e ´2 µ R0ν0 “ ´R00ν “Bν Γ00 ´ B0Γ0ν ` ΓνeΓ00 ´ Γ0eΓ0ν “ ´ppp ´ 1qt δν (222) and similar notation to (218) applies. So we get the Kretschmann scalar as (only sum on higher and lower indices):

abcd µµ νν 00 0 0 µµ 00 νν 0 0 RabcdR “ g00g g g Rµν0Rµν0 ` g00g g g Rµ0ν Rµ0ν (223)

00 νν 00 µ µ 00 00 νν µ µ `gµµg g g R0ν0R0ν0 ` gµµg g g R00ν R00ν (224) The first term is given by:

2p1´2 p1pp1 ´ 1qt 0 0 ´2p1 ´2p2 ´2p3 2p2´2 pt , t , t q 0 p2pp2 ´ 1qt 0 ¨ 2p3´2˛ 0 0 p3pp3 ´ 1qt ˝ ‚ ´2p1 p1pp1 ´ 1q 0 0 t ´2 ´2p2 t 0 p2pp2 ´ 1q 0 t ¨ ˛ ¨ ´2p3 ˛ 0 0 p3pp3 ´ 1q t

˝ ´‚2˝p1 ‚ t p1pp1 ´ 1q ´2 ´2 ´2p2 “ t pp1pp1 ´ 1q, p2pp2 ´ 1q, p3pp3 ´ 1qqt t p2pp2 ´ 1q ¨ ´2p3 ˛ t p3pp3 ´ 1q ˝ ‚ ´4 ´2p1 2 2 ´2p2 2 2 ´2p3 2 2 “ t t p1pp1 ´ 1q ` t p2pp2 ´ 1q ` t p3pp3 ´ 1q And we get same´ results for the remaining terms, thus: ¯

abcd ´4 ´2p1 2 2 ´2p2 2 2 ´2p3 2 2 RabcdR “ 4t t p1pp1 ´ 1q ` t p2pp2 ´ 1q ` t p3pp3 ´ 1q ´ ¯ 3 ´4 ´2pi 2 2 “ 4t t pi ppi ´ 1q (225) i“1 ÿ

47 Carin Jakobsson Appendix

C Homogeneous spaces

A space is said to be homogeneous if the spatial line element is invariant under a transformation xα Ñ x1α in the sense that:

2 1 2 3 α β 11 12 13 1α 1β dl “ γαβpx , x , x qdx dx “ γαβpx , x , x qdx dx (226)

α 1α so that the functional dependence of γab is same for the coordinates x and x .

In an Euclidean space the transformation is just a translation x1α “ xα ` dxα which leaves invariant the differentials pdx, dy, dzq. In a non-Euclidean, coor- dinate dependent space however, the spatial line element is invariant under a more general translation, namely x1α “ xα ` dxαpxq, which corresponds to a transformation involving coordinates. This general translation (which can be thought of as a group of motion) leaves invariant three (spatially coordinate de- 1 α 2 α 3 α pendent) so called differential forms peαdx , eαdx , eαdx q or more compactly a α written eαdx , i.e.: a α a 1 1α eαpxqdx “ eαpx qdx (227) A spatial line element for a homogeneous non-euclidean space, invariant under the general translation, can be specified with use of these differential forms as:

2 a α b β dl “ γabpeαdx qpeβdx q (228) where γab is symmetric and depends on time. The spatial metric for a homoge- neous space can thus be written as:

a b γαβ “ γabeαeβ (229) The inverse metric is defined by:

αβ ab α β γ “ γ ea eb (230) cb b a a α α b b so that γacγ “ δa and eαeβ “ δβ , ea eα “ δa. One can show that: re2, e3s re1, e3s re1, e2s e “ , e “ , e “ (231) 1 v 2 v 3 v 1 2 3 a where v “ e ¨ e ˆ e and e , ea can be understood as cartesian vectors with a α components eα, ea (all spatially coordinate dependent). The determinant of the metric (229) is given by:

a 2 2 γ “ |γab||eα| “ |γab|v (232)

C.1 Structure constants of homogeneous spaces One can manipulate equation (227) into a constant relation between the differ- ential forms and the coordinate differentials by requiring the coordinates x1α to be integrable (see [4] appendix C). The relation is given by:

α β c c c ea eb pBβeα ´ Bαeβq “ Cab (233)

48 Carin Jakobsson Appendix

c and determines the so called structure constants Cab of the group of transfor- mations under which the differential forms are invariant. Clearly, the structure constants are anti-symmetric in their lower indices, i.e:

c c Cab “ ´Cba (234)

γ By multiplying (233) by ec we can obtain one more condition on the structure constants, we get: α γ β γ c γ ea Bαeb ´ eb Bβea “ Cabec (235) α Defining the linear differential operators as Xa “ ea Bα we get commutation relations: c rXa,Xbs “ CabXc (236) which can be manipulated into the Jacobi identity, namely:

rXa, rXb,Xcss ` rXb, rXc,Xass ` rXc, rXa,Xbss “ 0 (237) or in terms of the structure constants:

f d f d f d CabCfc ` CbcCfa ` CcaCfb “ 0 (238) The relations (234) and (238) are required conditions for structure constants belonging to a homogeneous space. The structure constants can be expressed in terms of the cartesian vectors e1, e2, e3 by noting that (233) can be written as: c c rea, ebs ¨ ∇ ˆ e “ ´Cab (239) More specifically we get, by (231):

e1 ¨ ∇ ˆ e1 e2 ¨ ∇ ˆ e1 e3 ¨ ∇ ˆ e1 C1 “ ,C1 “ ,C1 “ (240) 32 v 13 v 21 v and six more relations which can be obtained by cyclic permutation of indices 1,2 and 3.

C.2 Classification of homogeneous spaces a The choice of differential forms eα are not unique but can be subjected to any linear transformation: 1α b 1α ea “ Aaeb (241) under which the structure constants transform like a mixed tensor. From all structure constants satisfying the conditions (234) and (238), some only differ w.r.t. the linear transformation (241). It is thus possible to separate sets of structure constants from one another, by requiring the elements of two different sets to be non-equivalent w.r.t. the transformation (241), leading to a classifi- cation of homogeneous spaces.

49 Carin Jakobsson Appendix

One may express the 9 structure constant components in terms of six com- ab ponents of a symmetric tensor n and three components of a vector ac:

c dc c c Cab “ eabdn ` δb aa ´ δaab (242) where eabc is a unit anti-symmetric tensor. By this definition (234) is satisfied, and (238) is satisfied by requiring:

ab n ab “ 0 (243)

The requirement (241) causes the symmetric tensor be on the form nab “ 1 2 3 pn , n , n q. Without loss of generality, we can choose ab “ pa, 0, 0q and by (236) arrive at the commutation relations:

3 1 2 rX1,X2s “ aX2 ` n X3, rX2,X3s “ n X1, rX3,X1s “ n X2 ´ aX3, (244) An analysis of to what extent the values n1, n2, n3, a gives degenerate structure constants results in the conclusion that one can differentiate between 9 different types of homogeneous spaces by table 3 (using the notation of Bianchi [1]):

Space a n1 n2 n3 I 0 0 0 0 II 0 1 0 0 III 1 0 1 -1 IV 1 0 0 1 V 1 0 0 0 VI ‰ 1, ą 0 1 -1 0 VII ą 0 1 1 0 VIII 0 1 1 -1 IX 0 1 1 1

Table 3: Bianchi categorization of homogeneous spaces

50 Carin Jakobsson Appendix

D Derivation of metric in β-space for Kasner’s particular solution

The Einstein Lagrangian is given by:

? ? mn LGpτ, βpτq, βτ pτqq “ ´gR “ ´gg Rmn (245) where Latin indices label spacetime coordinates and Greek indices label spatial coordinates. The line element of Kasner’s particular solution is given by (24). By the substitution (143) we can write the Kasner metric components in terms of βµpτq as: ´e2A 0 0 0 1 0 e´2β 0 0 gab “ ¨ 2 ˛ (246) 0 0 e´2β 0 3 ˚ 0 0 0 e´2β ‹ ˚ ‹ ˝ ‚ where we assume A “ Apτq. The metric inverse is given by

´e´2A 0 0 0 2β1 ab 0 e 0 0 g “ ¨ 2 ˛ (247) 0 0 e2β 0 3 ˚ 0 0 0 e2β ‹ ˚ ‹ ˝ ‚ and the Kasner determinant by:

2A´2 β g “ |gmn| “ ´e (248) ř and we shall use notations:

1 2 3 1 2 3 1 2 3 β “ pβ , β , β q, Σβ “ β ` β ` β , diagpβ , β , β q “ βδµν . (249) It remains to calculate the Riemann tensor components:

a a a a e a e Rbac “BaΓbc ´ BcΓba ` ΓaeΓbc ´ ΓceΓab (250) where: a 1 ad Γbc “ 2 g pBbgcd ` Bcgbd ´ Bdgbcq. (251)

We note that Bµgab “ 0 and g0µ “ 0. We get:

0 1 00 1 ´2A 9 2A 9 Γ00 “ 2 g pB0g00q “ 2 p´e qp2Ap´e qq “ A (252)

µ 0 µ Γ00 “ Γµ0 “ Γνγ “ 0 (253) 0 1 00 1 ´2A 9 ´2β 9 ´2A´2β Γµν “ 2 g p´B0gµν q “ 2 p´e q2βe δµν “ ´βe δµν (254) µ 1 µγ 1 2β µγ 9 ´2β 9 µ Γν0 “ 2 g B0gγν “ 2 pe δ qp´2βe δγν q “ ´βδν (255) The non-zero Ricci tensor components are then given as:

a a a a e a e R00 “ R0a0 “BaΓ00 ´ B0Γ0a ` ΓaeΓ00 ´ Γ0eΓ0a

51 Carin Jakobsson Appendix

µ µ 0 µ ν “B0Γ0µ ` Γµ0Γ00 ´ Γ0ν Γ0µ 9 9 9 9 µ 9 ν “B0p´Σβq ` p´ΣβqA ´ p´βδν qp´βδµq “ ´Σβ: ´ A9Σβ9 ´ Σβ92 Assuming no summation on µ:

a a a a e a e Rµµ “ Rµaµ “BaΓµµ ´ BµΓµa ` ΓaeΓµµ ´ ΓµeΓµa

0 0 0 ν 0 0 ν “B0Γµµ ` Γ00Γµµ ` Γν0Γµµ ´ 2Γµν Γµ0 9 ´2A´2β 9 9 ´2A´2β 9 9 ´2A´2β 9 ´2A´2β 9 ν “B0p´βe q`Ap´βe q`p´Σβqp´βe q´2p´βe δµν qp´βδµq ´2A´2β 2 “ e p´β: ´ p´2A9 ´ 2β9qβ9 ´ A9β9 ` β9Σβ9 ´ 2β9 qδµµ ´2A´2β “ e p´β: ` A9β9 ` β9Σβ9qδµµ So we get the Ricci scalar as:

µν ´2A 2β µµ R “ g Rmn “ e R00 ` e δ Rµµ

“ e´2A pΣβ:`A9Σβ9`Σβ92q`p´Σβ:`A9Σβ9`pΣβ9q2 “ e´2A 2A9Σβ9`Σβ92`pΣβ9q2q and the` Einstein Lagrangian becomes: ˘ ` ˘ ´A´ β 2 2 LG “ e 2A9Σβ9 ` Σβ9 ` pΣβ9q ř Setting A “ ´Σβ we get A9 “ ´Σβ9`i.e.: ˘

2 2 2 2 2 1 2 1 3 2 3 LG “ ´pΣβ9q ` Σβ9 ` pΣβ9q “ Σβ9 ´ pΣβ9q “ ´2pβ9 β9 ` β9 β9 ` β9 β9 q

Σβ Note that the tt-component of the metric is e “ gs, so the metric:

´gs 0 0 0 1 0 e´2β 0 0 gmn “ ¨ 2 ˛ (256) 0 0 e´2β 0 3 ˚ 0 0 0 e´2β ‹ ˚ ‹ ˝ ‚ corresponding to Kasner’s particular solution gives a β-space Lagrangian:

µ ν LG “ Gµν β9 β9 where 0 ´1 ´1 Gµν “ ´1 0 ´1 (257) ¨´1 ´1 0 ˛ since: ˝ ‚ 0 ´1 ´1 β1 µ ν 1 2 3 2 Gµν β9 β9 “ pβ , β , β q ´1 0 ´1 β ¨´1 ´1 0 ˛ ¨β3˛ ˝ ‚˝ ‚ “ ´2pβ91β92 ` β91β93 ` β92β93q (258)

52 Carin Jakobsson Appendix

E Classification of Lie algebras

Figure 13: Finite-dimensional Lie algebras [27]

53 Carin Jakobsson Appendix

Figure 14: Untwisted affine Lie algebras [27]

54 Carin Jakobsson Appendix

Figure 15: Twisted affine Lie algebras [27]

55 Carin Jakobsson Acknowledgements

Acknowledgements

I would like to thank my supervisor Martin Cederwall who has been supporting my work from start to finish and who introduced me to this many-faceted and well-documented subject. I am furthermore grateful to Daniel Persson for dis- cussing his thesis with me and to Edwin Langmann for reviewing and enhancing my work.

56 Carin Jakobsson References

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