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Bachian Gravity in Three Dimensions a Thesis BACHIAN GRAVITY IN THREE DIMENSIONS A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY MUSTAFA TEK IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN PHYSICS SEPTEMBER 2019 Approval of the thesis: BACHIAN GRAVITY IN THREE DIMENSIONS submitted by MUSTAFA TEK in partial fulfillment of the requirements for the de- gree of Doctor of Philosophy in Physics Department, Middle East Technical University by, Prof. Dr. Halil Kalıpçılar Dean, Graduate School of Natural and Applied Sciences Prof. Dr. Altug˘ Özpineci Head of Department, Physics Prof. Dr. Bayram Tekin Supervisor, Physics, METU Examining Committee Members: Prof. Dr. Atalay Karasu Physics, METU Prof. Dr. Bayram Tekin Physics, METU Assoc. Prof. Dr. Tahsin Çagrı˘ ¸Si¸sman Astronautical Engineering, UTAA Prof. Dr. Ismail˙ Turan Physics, METU Assoc. Prof. Dr. Özgür Açık Physics, AU Date: I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Surname: Mustafa Tek Signature : iv ABSTRACT BACHIAN GRAVITY IN THREE DIMENSIONS Tek, Mustafa Ph.D., Department of Physics Supervisor: Prof. Dr. Bayram Tekin September 2019, 118 pages Modified theories in 3-dimensions such as the topologically massive gravity (TMG), new massive gravity (NMG) or Born-Infeld extension of NMG arise from the vari- ations of diffeomorphism invariant actions; hence the resulting field equations are divergence free. Namely, the rank two tensor defining the field equations satisfy a Bianchi identity for all smooth metrics. However there are some recently constructed theories that do not identically satisfy Bianchi identities for all metrics, but only for the solutions of the theory. These are called on-shell consistent theories of which examples are the minimal massive gravity (MMG) and the exotic massive gravity (EMG). We work out the generic on-shell consistent model in 3-dimensions as a modified Einstein gravity theory which is based on the analog of the Bach tensor, hence we name it as the Bachian gravity. Conserved charges are found by using the linearization about maximally symmetric backgrounds for the Bañados-Teitelboim- Zanelli (BTZ)-black hole metric. It is complicated to solve the field equations of the gravity theory and hence very few solutions with only maximal symmetry are known. We use the projection formalism to obtain a reduction of the some relevant 2-tensors defining the field equations with the help of the Geroch’s reduction method. v Keywords: 3-dimensional gravity, Bachian Gravity, Topologically Massive Gravity, New Massive Gravity, Exotic Massive Gravity,Symmetry reduction vi ÖZ ÜÇ BOYUTTA BACHIAN KÜTLE ÇEKIM˙ I˙ Tek, Mustafa Doktora, Fizik Bölümü Tez Yöneticisi: Prof. Dr. Bayram Tekin Eylül 2019 , 118 sayfa Topolojik kütleli kütleçekim, Yeni kütleli kütleçekim veya Born-Infeld geni¸sletilmi¸s Yeni kütleli kütleçekim gibi modifiye edilmi¸s3 boyutlu teoriler difeomorfizmler al- tında degi¸smezkalan˘ Etki’lerin varyasyonları sonucu elde edilirler ve ula¸sılanalan denklemlerinin diverjansı sıfırdır. Yani bütün düzgün metrikler için, alan denklem- lerini tanımlayan rank-2 tensörler Bianchi özde¸sligini˘ saglar.˘ Bununla birlikte son zamanlarda geli¸stirilenbazı teoriler bütün metrikler için Bianchi özde¸sligini˘ direkt saglamak˘ yerine, teorinin çözümü üzerinden saglamaktadır.˘ Bu tür teoriler "on-shell" tutarlı olarak adlandırılırlar. Örnek olarak Minimal kütleli kütleçekim ve Egzotik küt- leli kütleçekim teorilerini verebiliriz. 3-boyutta Bach tensörünü göz önüne alarak is- mini verdigimiz˘ Bachian kütleçekim teorisi, Einstein kütleçekim teorisinin bir mo- difikasyonu olarak 3-boyutlu kapsamlı on-shell tutarlı bir model olarak çözülmü¸stür. Korunumlu yükler, maksimum simetrik Bañados-Teitelboim-Zanelli BTZ kara de- ligi˘ metrigi˘ etrafında linerizasyon kullanılarak bulunmu¸stur. Kütleçekim teorilerinin alan denklemlerini çözmenin zorlugu˘ nedeniyle çok az sayıda ve sadece maksimum simetriye sahip çözümler bilinmektedir. Geroch indirgeme metodu yardımıyla alan denklemlerini tanımlayan bazı rank-2 tensörlerin indirgenmi¸shalleri projeksiyon for- vii mulasyonu kullanılarak bulunmu¸stur. Anahtar Kelimeler: 3-boyutlu Kütleçekim, Bachian Kütleçekim, Topolojik Kütleli Kütleçekim, Yeni Kütleli kütleçekim, Egzotik Kütleli Kütleçekim,Simetri indirge- mesi viii To my family and to people who unjustifiably prosecuted in Turkey ix ACKNOWLEDGEMENTS I would like to thank my supervisor Prof. Bayram Tekin with my sincere feelings. It is almost impossible to express my gratitude in words. If I had not worked with him, I would not have been able to get a PhD in a field I wanted since my childhood. I will never forget his endless support and help in the diffucult times I have had. He is not only a supervisor but a wise person enlightened me about the dark roads of the life. I believe that he is a polymath, not just a scholar, who is very much needed for the uncivilized countries such as Turkey. Special thanks to Dr. Gökhan Alkaç for all the help he gave me and for the patient an- swers to my questions, even if the questions are silly sometimes. Other lovely friends of mine, Dr. Merve Demirta¸s,Özgür Durmu¸sve Hikmet Öz¸sahinalso deserves thank- fulness for the enormous contribution to the "goygoy" during my PhD. Without them, it was almost impossible to concentrate on the study at so many times. I am very grateful to my parents for their endless support. I thank my brother Mehmet Tek, and his lovely wife Sevgi Kümü¸sTek, and also their beatiful daughter Ne¸seTek for the happiness that they bring to my life. x TABLE OF CONTENTS ABSTRACT . .v ÖZ......................................... vii ACKNOWLEDGEMENTS . .x TABLE OF CONTENTS . xi LIST OF ABBREVIATIONS . xv CHAPTERS 1 INTRODUCTION: A BRIEF REVIEW OF GENERAL RELATIVITY . .1 2 MATHEMATICAL PRELIMINARIES . .5 2.1 Basics of Riemannian Geometry . .5 2.1.1 Affine Connection . .6 2.1.2 Parallel Transport . .7 2.1.3 The Covariant Derivative of Tensor Fields . .8 2.1.4 The Metric Compatible Connection . 10 2.1.5 Curvature and Torsion . 11 2.1.6 The Ricci Tensor and the Scalar Curvature . 13 2.2 Hypersurfaces . 14 2.2.1 Gaussian Normal Coordinates: . 15 2.2.2 Projection Tensor: . 17 xi 2.3 Stokes’ Theorem . 23 3 THREE DIMENSIONAL GRAVITY THEORIES . 27 4 GENERIC EXOTIC MASSIVE GRAVITY THEORIES . 39 4.1 3D Bach Tensor and On-shell Consistency . 40 4.2 Generalization of 3D Bach Tensor . 44 4.3 Ψµν from Quadratic Gravity . 49 4.4 Bachian Gravity . 51 4.5 Conserved Charges . 59 4.6 Further Developments in Exotic Massive Gravity . 61 5 SYMMETRY REDUCTION VIA THE GEROCH METHOD . 63 5.1 Reduction of the Various tensors under a Killing Symmetry . 63 5.1.1 The Stationary Metric . 64 5.1.2 Coordinate or Gauge Transformation . 65 5.1.3 Scalar Twist . 66 5.1.4 Ricci Tensor and The Scalar Curvature . 68 5.1.5 Reductions of the Cotton Tensor Cµν, and the Jµν and Hµν tensors . 70 6 CONCLUSIONS . 75 REFERENCES . 77 APPENDICES A MAPS AND TOPOLOGICAL SPACES . 83 A.1 Maps . 83 A.1.1 Properties of the Maps . 84 xii A.1.2 Equivalence Class . 84 A.2 Topological Spaces . 85 A.2.1 Compactness and Paracompactness . 87 A.2.2 Connectedness and Path-connectedness . 87 A.2.3 Homeomorphism . 87 B MANIFOLDS AND TENSOR FIELDS . 89 B.1 Manifolds: . 89 B.1.1 Curves and Functions: . 92 B.1.2 Vectors: . 93 B.1.3 One-forms: . 94 B.1.4 Tensors: . 96 B.1.5 Tensor Field: . 96 B.1.6 Push-forward and Pull-back: . 96 B.1.7 Lie Derivatives: . 98 B.1.8 Differential Forms: . 103 B.1.9 Integration of Differential Forms: . 105 B.1.10 Lie Groups and Lie Algebras: . 107 B.1.11 The one parameter subgroup: . 110 B.1.12 Frames and Structure Equation: . 111 B.1.13 The action of Lie groups on manifolds: . 113 B.1.14 Orbits and Isotropy groups: . 114 B.1.15 Induced Vector Fields: . 114 B.1.16 The adjoint representation: . 115 xiii CURRICULUM VITAE . 117 xiv LIST OF ABBREVIATIONS ABBREVIATIONS GR General Relativity dS de Sitter AdS Anti-de Sitter TMG Topologically Massive Gravity NMG New Massive Gravity EMG Exotic Massive Gravity xv xvi CHAPTER 1 INTRODUCTION: A BRIEF REVIEW OF GENERAL RELATIVITY General relativity models gravity as a four dimensional manifold M 1 with certain desired properties that we shall discuss, whose metric g is determined from the Ein- stein’s equations 1 8πG G := R − g R + Λg = N T ; (1.1) µν µν 2 µν µν c4 µν where Rµν is the Ricci tensor, R is the scalar curvature, Λ is the cosmological con- stant, Tµν is the energy-momentum tensor and GN is the Newton’s constant while c is the speed of light. The left hand side is purely related to geometry, while the right hand side represents all possible matter distribution. The cosmological constant Λ was observed to be tiny but positive: in SI units Λ = 10−52m−2 hence it plays its ma- jor role in the global dynamics of the universe [1]. On the other hand, the numerical 8πGN −43 value of the "coupling constant" κ := c4 is κ = 2:1 × 10 m=J which is again small but when it gets multiplied by possibly large Tµν (as in the interior of a neutron star) whose unit is J=m3 one gets a large effect. The fact that the left hand side of µν Eq.(1.1) satisfies the so called Bianchi identity rνG = 0 for all smooth metrics, re- µν quires the so called the covariant conservation law rνT = 0. The matter content of the Universe in small scales is very complicated, therefore there is no hope of solving Eq.(1.1) exactly.
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