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Regge Calculus As a Numerical Approach to General Relativity By Regge Calculus as a Numerical Approach to General Relativity by Parandis Khavari A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Astronomy and Astrophysics University of Toronto Copyright c 2009 by Parandis Khavari Abstract Regge Calculus as a Numerical Approach to General Relativity Parandis Khavari Doctor of Philosophy Department of Astronomy and Astrophysics University of Toronto 2009 A (3+1)-evolutionary method in the framework of Regge Calculus, known as “Paral- lelisable Implicit Evolutionary Scheme”, is analysed and revised so that it accounts for causality. Furthermore, the ambiguities associated with the notion of time in this evolu- tionary scheme are addressed and a solution to resolving such ambiguities is presented. The revised algorithm is then numerically tested and shown to produce the desirable results and indeed to resolve a problem previously faced upon implementing this scheme. An important issue that has been overlooked in “Parallelisable Implicit Evolutionary Scheme” was the restrictions on the choice of edge lengths used to build the space-time lattice as it evolves in time. It is essential to know what inequalities must hold between the edges of a 4-dimensional simplex, used to construct a space-time, so that the geom- etry inside the simplex is Minkowskian. The only known inequality on the Minkowski plane is the “Reverse Triangle Inequality” which holds between the edges of a triangle constructed only from space-like edges. However, a triangle, on the Minkowski plane, can be built from a combination of time-like, space-like or null edges. Part of this thesis is concerned with deriving a number of inequalities that must hold between the edges of mixed triangles. Finally, the Raychaudhuri equation is considered from the point of view of Regge Cal- ii culus. The Raychaudhuri equation plays an important role in many areas of relativistic Physics and Astrophysics, most importantly in the proof of singularity theorems. An analogue to the Raychaudhuri equation in the framework of Regge Calculus is derived. Both (2+1)-dimensional and (3+1)-dimensional cases are considered and analogues for average expansion and shear scalar are found. iii Dedication To my beloved Husband and my Dear Parents iv Acknowledgements The acknowledgement section was the very last section that I had to complete to formally submit my thesis. At first, I intended to write a ”standard acknowledgement” similar to what is seen in most Ph.D. theses. On a second thought however, I decided to write a document that reflects upon my characteristics as a person not as a Ph.D. candidate. Fifteen years ago, I could not decide whether I wanted to become an astrophysicist or a musician. Just the thought of achieving either of these two goals made me very happy. When I entered university to study engineering, I thought that my dream would never come true. If I have learned one thing during the past few years, it is that: “Men can do all things if they will.” After obtaining my B.Sc., I realised that the passion for Astrophysics in me is too strong to be killed. So I decided to start studying Astrophysics despite the fact that changing my field of study appeared to be quite challenging. All I had was the passion for Astro- physics and it turned out that it was all that I needed. Indeed, writing this document brings extreme joy to me as I now know that despite all hardships one of my dreams has indeed come true. I shall confess that I have learned many things during the completion of this thesis that might appear utterly irrelevant to this work. Most importantly, I have come to a deep understanding of Sir Isaac Newton’s statement that: “To explain all nature is too difficult a task for any one man or even for any one age.” Over the past four years, I have learned not to get disappointed with failure and not to be overjoyed with success. I have learned not to believe anyone’s scientific statements unless I follow in their footsteps and come to the same conclusions. Finally, I have learned that v even those in whom I have much faith can make mistakes. During these years, I was blessed with having a companion without whose support and encouragement entering the Ph.D. program in Astrophysics and completing it was not possible. My greatest gratitude goes to my beloved husband Mahdi to whom I dedicate this thesis. He always listened to me when I needed to and always encouraged me when I was most disappointed. He gently endured me when I was upset or stressed out and always tried his best to calm me. The completion of this thesis was concurrent with the birth of my precious son, Yusuf whose arrival has brighten up my life. I wish that he will respect science and more im- portantly the quest for the truth in his future life. I would like to sincerely thank my parents for igniting the passion for learning in me. I am grateful to them for providing me with an outstanding education. I am also much indebted to them for taking excellent care of my son and thus providing me with the opportunity to finish my work. I am much grateful to my supervisor, Professor Charles C. Dyer, for teaching me not only General Relativity but more importantly life lessons. He patiently taught me many things and supported me whenever I faced a barrier in my work. I will always be indebted to him. I am also thankful to my supervisory committee for supporting me through my Ph.D. program and providing me with constructive comments. My sincere appreciation goes to Professors Mochnacki, McCann, Abraham, and Hobill for reading my thesis and making excellent comments. Finally, I would like to thank all my friends in the Department of Astronomy and Astro- physics. In particular, I would like to thank Ivana Damjanov, Marzieh Farhang, Preethi Nair, Maria Stankovic, Samaya Nissanke, Brian Lee, Duy Nguyen, Marc Goodman, and vi Lee Robbins. The material presented in this thesis is based on the work supported by Walter John Helm Graduate Scholarship, Canadian Institute for Advanced Research Graduate Scholarship, Helen Hogg Scholarship, Reinhardt Scholarship, and University of Toronto Fellowship. vii Contents 1 Introduction 1 1.1 NumericalRelativity ............................. 3 1.2 Objectives of this Thesis and its Contributions . 4 2 Regge Calculus 7 2.1 Introduction.................................. 7 2.1.1 Descretised Action and Regge’s Equation . 11 2.2 The Action must always be real . 14 2.2.1 Calculating the Deficit Angle . 17 2.3 Binachi Identities in Regge Calculus . 20 2.4 (3+1)-Evolutionary Methods in Regge Calculus . 21 2.5 Parallelisable Implicit Evolution Scheme for Regge Calculus . 22 2.6 Previous Applications of the Parallelisable Implicit Evolution Scheme for ReggeCalculus ................................ 26 3 Inclusion of Causality in the PIES 27 3.1 Introduction.................................. 27 3.2 Area of a Bone and the Issue of Causality . 28 3.2.1 Variation of a Time-Like Bone with respect to a Time-Like edge . 32 3.2.2 Variation of a Space-Like Bone with respect to a Space-Like edge 33 3.3 Conclusion................................... 35 viii 4 A Skeletonised Model of the FLRW Universe 37 4.1 Introduction.................................. 37 4.2 The Friedmann-Lemaˆıtre-Robertson-Walker Universe . 38 4.3 Standard Triangulations of a 3-sphere . 42 4.3.1 5-Cell Triangulation of S3 ...................... 43 4.3.2 16-Cell Triangulation of S3 ...................... 44 4.4 The Time-Function in the Parallelisable Implicit Evolutionary Method . 44 4.5 Construction of the Skeletonised FLRW Universe . 50 4.5.1 Construction of the Initial Hypersurface at the Moment of Time- Symmtry................................ 52 4.5.2 Evolution of the Initial Hypersurface . 53 4.5.3 Lattice Action and the Relevant Regge Equations . 56 4.6 Calculation of the Required Parameters . 57 4.6.1 The 5-Cell Universe . 57 4.6.2 The 16-Cell Universe . 58 4.7 Discussion of the General Space of Solutions . 58 4.8 Conclusion................................... 61 5 Triangle Inequalities in the Minkowski Plane 62 5.1 Introduction.................................. 62 5.2 Preliminaries ................................. 63 5.3 Triangle Inequalities for a SST Triangle . 65 5.4 Triangle Inequalities for a NSS Triangle . 71 5.5 Conclusion................................... 74 6 Raychaudhuri’s Equation in Regge Calculus 75 6.1 Introduction.................................. 75 6.2 Raychaudhuri’s equation in the Continuum . 76 ix 6.3 Geodesics in Regge Calculus . 78 6.4 Expansion of Two non-Parallel Geodesics in Flat Space-Time . 79 6.5 Raychauduri’s Equation in (2+1)-Dimensional Skeletonised Space-Times 80 6.5.1 Distance between two geodesics in (2+1) Skeletonised Space-Times 82 6.5.2 Expansion and Shear in (2+1)-dimensional Skeletonised Space-Times 84 6.6 Raychaudhuri’s Equation in (3+1)-Dimensional Skeletonised Space-Times 86 6.6.1 Expansion in (3+1)-dimensional Skeletonised Space-Times . 88 6.6.2 Shear in (3+1)-dimensinal Skeletonised Space-times . 90 6.7 Conclusion................................... 91 7 Conclusion and Future Work 92 A A Note about the Minkowski Plane 95 B Areas of Triangles on the Minkowski Plane 98 B.0.1 AreaofaSSSTriangle ........................ 98 B.0.2 AreaofaSSTTriangle........................ 100 B.0.3 Area of a NSS Triangle . 103 B.0.4 Area of a NST Triangle . 106 C Calculation of Hyperbolic Functions for angle ξ 108 D Flowchart of the Numerical Example 109 E Mathematica Code of the Numerical Example 111 Bibliography 116 x Chapter 1 Introduction The beginning of the twentieth century witnessed the development of a revolutionary new theory called the Special Theory of Relativity.
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