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 ¥ 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 ¥ 4 is associated with algebras AEd and that pure gravity systems of dimension D ¥ 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 ¥ 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.