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Singularity Resolution in Anisotropic and Black Hole Spacetimes in Loop Quantum Cosmology Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 12-18-2018 Singularity Resolution in Anisotropic and Black Hole Spacetimes in Loop Quantum Cosmology Sahil Saini Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Elementary Particles and Fields and String Theory Commons, and the Quantum Physics Commons Recommended Citation Saini, Sahil, "Singularity Resolution in Anisotropic and Black Hole Spacetimes in Loop Quantum Cosmology" (2018). LSU Doctoral Dissertations. 4775. https://digitalcommons.lsu.edu/gradschool_dissertations/4775 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. SINGULARITY RESOLUTION IN ANISOTROPIC AND BLACK HOLE SPACETIMES IN LOOP QUANTUM COSMOLOGY A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Physics and Astronomy by Sahil Saini M.S., Indian Institute of Technology, Delhi, 2010 May 2019 I dedicate this dissertation to my parents - Surya Kanta Saini and Jai Prakash Saini, who sowed the seeds of curiosity in me. ii Acknowledgments First of all, I would like to thank my advisor Parampreet Singh for his continuous support, guidance and encouragement through out my PhD. I feel deeply indebted to him for showing me how to do research. He has shown immense patience in guiding me through various research problems, helping me along the way and showing me how to pen down and present my ideas. He has also been kind enough to help me navigate my life outside of work. I feel very fortunate to have worked with him and will forever be indebted for his support and guidance. I would specially like to thank Jorge Pullin for devoting his valuable time and patience in helping me prepare this dissertation. It has been my privilege to know him, and I am very grateful for his guidance and support, and the warm and welcoming environment he has extended to all the members at the quantum gravity group at LSU. I have specially enjoyed many of the courses delivered by him. His clarity of thought and grasp of fundamental ideas will always be an inspiration for me. I have immensely benefited from numerous discussions with Ivan Agullo. His incisive lectures on a wide range of topics, openness to discuss new ideas and endless enthusiasm have been a constant source of inspiration. I hope some of his enthusiasm and vibrancy has rubbed off on me as well. I would specially like to thank Javier Olmedo for his collaboration. His support and discus- sions have also been crucial in the completion of this dissertation. I am grateful to him for many stimulating discussions, and for patiently explaining various intricacies of quantum cosmology to me. I would like to thank Juhan Frank, A. Ravi P. Rau and Tryfon T. Charalampopoulos for agreeing to be a part of my PhD committee, and for their valuable feedback. I specially thank A. Ravi P. Rau for stimulating courses, discussions and informal lunch seminars which have enriched me in numerous ways. My sincere thanks to all my colleagues and friends at LSU, especially, Alison Dreyfuss, Anthony Brady, David Kekejian, David Alspaugh, Eklavya Thareja, Emily Safron, Eneet Kaur, iii Grigor Sargsyan, Karunya Shirali, Kelsie Krafton, Kevin Valson Jacob, Kunal Sharma, Samuel Cupp, Sergio Lopez Caceres, Siddhartha Das, Siddharth Soni, Sudarsan Balakrishnan, Sumeet Khatri, Tyler Ellis and Vishal Katariya. I thank all of them for making my time memorable here. I would also like to thank the administrative staff members at the Department of Physics and Astronomy especially Arnell Nelson, Carol Duran, Claire Bullock, Stephanie Jones, Laurie Rea, Shanan Schatzle and Shemeka Law for their continuous help and support. Finally, I would like to thank my relatives and friends in India who have been a constant pillar of support. I would specially like to thank my cousins Ajay and Kamal for constant support, emotional and otherwise. I would also like to thank my friends Lucky, Nandu and Yogi for their warmth and support. iv Table of Contents ACKNOWLEDGMENTS . iv ILLUSTRATIONS . vii ABSTRACT ........................................................................... ix CHAPTER 1 INTRODUCTION . .1 1.1 The canonical approach to quantum gravity . .6 1.2 Loop quantum gravity . 10 1.3 Loop quantum cosmology . 14 1.4 Motivation . 23 1.5 Overview of the dissertation . 26 2 A REVIEW OF CLASSICAL AND QUANTUM GRAVITY . 31 2.1 Strong and weak singularities in GR . 31 2.2 Hamiltonian formulation of GR . 37 2.3 Loop quantum cosmology . 44 2.4 Effective dynamics of loop quantized spatially flat FLRW model . 51 2.5 Anisotropic models . 53 3 SINGULARITY AVOIDANCE IN EFFECTIVE LOOP QUAN- TIZED BIANCHI-II SPACETIME . 57 3.1 Introduction . 57 3.2 Classical dynamics of diagonal Bianchi-II spacetime . 64 3.3 Effective dynamics: `A' quantization . 68 3.4 Effective dynamics: `K' quantization . 79 3.5 Potential divergences in curvature invariants and θ_ ......................... 82 3.6 Geodesic completeness . 84 3.7 Lack of strong singularities . 86 3.8 Conclusions . 88 4 SINGULARITY AVOIDANCE IN EFFECTIVE LOOP QUAN- TIZED BIANCHI-IX SPACETIME . 92 4.1 Introduction . 92 4.2 Classical aspects of Bianchi-IX spacetime in connection-triad variables .................................................................. 94 4.3 Effective loop quantum cosmological dynamics: `A' quantization . 97 4.4 Effective loop quantum cosmological dynamics: `K' quantization . 110 4.5 Possible divergences in curvature invariants . 112 4.6 Geodesic completeness . 115 4.7 Lack of strong singularities . 117 4.8 Conclusion . 119 v 5 SINGULARITY AVOIDANCE IN EFFECTIVE LOOP QUAN- TIZED KANTOWSKI{SACHS COSMOLOGY . 122 5.1 Introduction . 122 5.2 Classical dynamics of Kantowski{Sachs spacetime . 125 5.3 Effective dynamics of the Kantowski{Sachs spacetime . 128 5.4 Geodesics . 132 5.5 Lack of strong singularities . 134 5.6 Conclusions . 136 6 SYMMETRIC BOUNCE IN EFFECTIVE LOOP QUANTIZED BLACK HOLE SPACETIMES . 139 6.1 Introduction . 139 6.2 Classical setting . 145 6.3 Effective loop quantum cosmology . 151 6.4 Dynamical prescriptions for symmetric bounces . 156 6.5 Conclusions . 176 7 CONCLUSIONS . 180 APPENDIX A PERMISSIONS FROM IOP PUBLISHING LTD. 190 VITA .................................................................................. 203 vi Illustrations Table 4.1 Different possible fates for the triad variables upon evolution. 104 List of Figures 2.1 3 + 1 split of spacetime. 38 6.1 Choice 1: the evolution of the spatial volume as a function of the normalized time T= Tb . The curves ranging from thickest to the thinnest correspondj tojM = 0:1; 0:3; 1; 10; 1000 (in Planck units), respectively. 159 6.2 Choice 1: the volume at the bounce as a function of the mass. 159 6.3 Choice 1: the value of the Kretschmann scalar at the bounce as a function of mass. 161 6.4 Choice 2: we plot the LHS and RHS of Eq. (6.43) as a function of α for different values of the mass M. The horizontal lines correspond to the RHS of (6.43). The curves ranging from the thickest to the thinnest correspond to M = 0:1; 0:3; 1; 10; 1000 in Planck units, respectively. 162 6.5 Choice 2: α as a function of M................................................... 163 6.6 Choice 2: the logarithmic of pb as a function of the logarithm of pc for different values of the mass M. The curves ranging from the thickest to the thinnest correspond to M = 0:05; 0:1; 0:3; 1; 10; 1000 in Planck units respectively. 164 6.7 Choice 2: the evolution of the spatial volume as a function of the normalized time T= Tb.
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