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Black Holes, Singularity Theorems & the Global Structure of Spacetime

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Black Holes, Singularity Theorems & the Global Structure of Spacetime Black Holes, Singularity Theorems & The Global Structure of Spacetime in General Relativity Rohan Kulkarni Matriculation number : 3753615 Group : Elementary Particle Theory, Institute of Theoretical Physics, University of Leipzig i+ i+ J + J + i0 i0 J − J − i− Author's Declaration I wrote this thesis on my own and it is primarily an extended literature review with my own comments all over the document. In order to achieve 10 ECTS for the course 12-PHY- BIPPP : Project Oriented Course, the sections 1.1, 1.2, 1.3, 1.4 and chapter 3 in a draft format were successfully submitted along with a presentation at this very same institution under Prof. Dr. Stefan Hollands. The goal of this course was to prepare for my Bachelor's thesis in which I self-studied the rst seven chapters from [1] and submitted a document with the sections and chapters mentioned above. Other than that I hereby declare that I have not used any sources other than those listed in the bibliography and identied as references. I further declare that I have not submitted this thesis at any other institution in order to obtain a degree. Thesis submitted at the University of Leipzig in partial fulllment of the requirements for the degree of Bachelor of Physics IPSP (International Physics Studies Program). Supervisor : Prof. Dr. Stefan Hollands Month and Year of Submission April 2019 3 4 Author's Declaration Acknowledgments First and foremost, I thank Prof. Dr. Stefan Hollands for accepting to be my supervisor and for suggesting me this topic for my Bachelor's thesis. I also thank him for his guidance and helpful advice during this work. Apart from being the supervisor for my thesis, Prof. Hollands was also my lecturer for theoretical physics lectures in Quantum mechanics and Statistical mechanics at the University of Leipzig. I learned a great deal of Physics and Mathematics under him in these courses, project, and my thesis. I would also like to thank Dr. Zahn who agreed to be the second person to asses this thesis. Finally, I would like to thank my family for all their support during my studies. They have always encouraged me to reach my dreams. They kept this task up even after the fact that most of them live on a dierent continents and I couldn't be more grateful about it. 5 List of figures Closing of light cones near the Schwarzschild radius . 23 Eddington-Finkelstein diagram . 29 Spacetime diagram of the Kruskal coordinates. 30 Einstein-Rosen bridge (Figure 17.2, Page 233, [7]) . 32 Time evolution of Einstein-Rosen bridge (Figure 17.3, Page 233, [7]) . 32 A wormhole (Figure 17.3, Page 233, [7]) . 32 Penrose Diagram of Schwarzschild spacetime. 34 Sub-extremal RN metric diagram . 45 Example of a lightcone . 48 Example of an achronal set . 49 Cylindrical spacetime . 50 Domain of dependence . 53 Cauchy Horizons . 54 Diagram showing the idea of the least upper bound . 72 Step 1: (i) . 72 ! Step 2 . 73 Step 3 . 73 Step 4 . 74 Step 5 . 74 Misner Space . 84 Identied points in Misner space (Denoted by Greek symbols Greek symbols primed) . 86 Geodesics in Misner space. Taken from Pg.172,[8] . 88 Constant x geodesics plotted in (T ; ) coordinates. 88 Symmetry of a single x0 > 0 geodesic around z = 0 . 89 Closed timelike curves in Gödel Universe . 93 Warping of Spacetime in the Alcubierre drive . 99 https://de.wikipedia.org/wiki/Datei:Alcubierre.png Pullback of a function . 103 Pushforward of a vector . 104 Minkowski spacetime (v; w) . 108 Minkowski spacetime (T ; X) . 110 Timelike and Spacelike geodesics in Minkowski spacetime. 111 The picture on the front page represents a Black hole. It was redrawn but inspired from a diagram in these lecture notes https://web.math.princeton.edu/~aretakis/ columbiaGR.pdf. 7 Table of contents Author's Declaration . 3 Acknowledgments . 5 List of figures . 7 Introduction . 13 Notations and Conventions . 15 Mathematical and Physical . 15 Textual . 15 Prerequisites . 16 A few remarks by the author . 16 1. Schwarzschild Black Holes . 17 1.1. Derivation of the Schwarzschild metric. 18 1.2. Singularities in Schwarzschild metric . 22 1.2.1. Radial null curves of Schwarzschild metric . 23 1.3. Kruskal and Eddington Finkelstein coordinates. (Using Tortoise coordinates). 24 1.3.1. Kruskal Coordinates for r > 2m . 24 1.3.2. Kruskal Coordinates for r < 2m . 27 1.3.3. Metric for the exterior and interior of a Schwarzschild black hole. 28 1.4. Eddington Finkelstein coordinates . 28 1.4.1. The Finkelstein diagram . 29 1.5. Kruskal Extension . 30 1.6. Einstein-Rosen Bridge . 31 1.7. Penrose Diagram for the Kruskal Solution . 33 1.8. Bibliographical Notes . 34 2. Reissner-Nordström (RN) Black Holes . 37 2.1. Derivation of the RN metric . 37 2.2. Few Properties of Reissner-Nordström Black holes . 44 2.3. Bibliographical notes . 46 3. Causal structure . 47 3.1. Denitions of Future and Pasts . 47 3.2. Denitions of Causal relations between events . 48 3.3. Denitions of Causality, Initial conditions. 50 3.4. Denitions/Theorems of Global dependence and Cauchy surfaces. 52 3.5. Bibliographical Notes . 55 4. Singularity Theorems : Congruences and Raychaudhuri equa- tion. 57 4.1. What do we mean by singularity? . 57 9 10 Table of contents Fundamental problem . 58 4.2. Design of Singularity theorems . 59 4.2.1. Strategy for Singularity theorems . 59 Algorithm for singularity theorems . 59 4.2.2. Singularity lemma I. (Timelike geodesic congruence) . 60 4.2.2.1. Tool I : Extrinsic curvature . 60 4.2.2.2. Tool II : Strong energy condition . 61 4.2.2.3. Tool III : Congruences and their properties. 62 4.2.2.4. Tool IV : Dynamics of these congruences. (Raychaudhuri's equation.) . 63 4.2.2.5. Proof . 66 4.2.3. Singularity lemma II (Null geodesic congruence) . 67 4.3. Conjugate Points . 67 4.3.1. Why are we interested in conjugate points? . 67 4.3.2. Existence of Conjugate points. 68 4.4. Topology of Causal Curves . 68 4.5. Singularity Theorems . 69 4.5.1. First Theorem / Timelike case . 70 4.5.2. Hawking's Theorem . 71 4.6. Bibliographical Notes . ..
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