UNIVERSITY OF CINCINNATI Consequences of quantum mechanics in general relativity by Souvik Sarkar A thesis submitted to the graduate school in partial fulfillment for the degree of Doctor of Philosophy in the Department of Physics Cenalo Vaz, L.C.R. Wijewardhana August 2018 UNIVERSITY OF CINCINNATI Abstract Cenalo Vaz, L.C.R. Wijewardhana Department of Physics Doctor of Philosophy by Souvik Sarkar We have presented a gauge theoretic approach for gravity in 2 + 1-D, starting from the Chern-Simons action with the SO(2; 2) and ISO(2; 1) Lie group structures. With these groups, we are able to construct some known results from general relativity namely the Banados, Teteilboim and Zanelli (BTZ) solution and the spinning point particle solution respectively. We obtain the canonical structure in each case. To get a more physi- cal picture we probe the canonical structure by embedding the Arnowitt-Deser-Misner (ADM) metric into the corresponding spacetime. After several canonical transformations we found in each case that the spacetime is described by a two dimensional phase space with two degrees of freedom, the mass and the angular momentum. These are stationary solutions. In the second part of this thesis, we have extended this approach to non- rotating dynamical collapse. During a spherical collapse the end state does not always lead to a black hole. Instead, due to quantum effects, collapsing shells in the exterior of the apparent horizon are accompanied by outgoing Unruh radiation in its interior. Both collapsing shells and the outgoing Unruh radiation appear to stop at the apparent hori- zon. This solution is obtained by solving the Wheeler-DeWitt equation (WDW) which is the Hamiltonian constraint, elevated to operator form by applying Dirac's quantization procedure. As a consequence, we can expect that the collapse process would end in a quasi-stable, static compact object without forming a black hole. We have shown that, using Einstein equations, such a stable configuration is possible where the BTZ horizon radius is 87% of the boundary radius, so the BTZ horizon lies inside the boundary. In the last part, we consider a scalar field model of dark matter(DM) which forms a Bose Einstein condensate. We coupled a non-relativistic scalar field with φ4 interactions to linearized gravity in the Gravitoelectromagnetic formulation. This model predicts that halo sizes vary significantly depending on the mass of the scalar field particles and the nature of the self-interaction. Dark matter halos ranging from asteroid size to galaxy size are possible within this model. This was one of the major challenges in the Cold Dark Matter (CDM) paradigm. We examined the stability of the halos by studying small os- cillations about the equilibrium using a collective coordinates approach with both types of self-interaction. We found oscillations about the minima of the energy with a time period of 10 billion years for attractive interactions and 1:3 billion years for repulsive interactions. ii iii Acknowledgements Firstly, I would like to express my sincere gratitude to my advisors Dr. Cenalo Vaz and Dr. L. C. R. Wijewardhana for the continuous support during my PhD study and research and for their patience, motivation, enthusiasm and knowledge. Their persistent guidance helped me in every way in my reasearch and writing of this thesis. It has been an honour to be PhD student under them. Without their help, I could not have imagined this day. I would like to use this opportunity to thank my thesis committee members Dr. Rostislav A. Serota, Dr. Philip Argyres, Dr. Kay Kinoshita for their time, interest and helpful comments in my research. I would like to convey my heartiest gratitude to Dr. Paul Esposito, Dr. Peter Suranyi, Dr. Michael Ma for their Constructive discussions and sharing their thoughts. It has considerably improved the environment to carry on the research during my PhD. All the group sessions with my other PhD colleagues have added another polished layer in my research journey through my PhD career. I thank all of them from the core for their interest and helping me in every way possible. Last but not the least, I would mention my parents , Sreetama and my friends for their constant support, love and encouragement to reach my goal. It has been a wonderful journey through my research career and many more people are responsible for making it memorable. I would like to thank all of them. Their contributions in my life is unforgettable and will be cherished through the rest of my life. Thank you... iv Contents Abstract i Acknowledgements iv List of Figures vii Physical Constants viii 1 Introduction1 2 Chern-Simons theory and quantization of BTZ black hole8 2.1 Introduction..................................8 2.2 The Connection................................ 10 2.3 Chern-Simons gravity............................. 14 2.4 Equations of motion............................. 19 2.4.1 Static case............................... 20 2.5 Fall-off conditions and boundary action................... 23 2.6 Embedding.................................. 25 2.6.1 Spinning particle........................... 25 2.6.2 BTZ black hole............................ 27 2.7 New canonical variables........................... 28 2.8 The boundary action............................. 30 2.9 Summary................................... 34 3 Classical dust collapse model 36 3.1 Collapse wave functionals.......................... 39 3.2 A quasi-classical static configuration.................... 46 3.3 Energy extraction............................... 48 3.4 Summary................................... 49 v Contents 4 Dark matter as Bose-Einstein condensate 51 4.1 Introduction.................................. 51 4.2 Dark matter and BEC............................ 54 4.3 The action................................... 56 4.3.1 The GPP Action........................... 56 4.3.2 The Gravitational Action...................... 59 4.3.3 Equations of Motion......................... 61 4.3.4 Stable Configuration......................... 64 4.3.4.1 Total energy........................ 64 4.3.4.2 Minimum energy configuration.............. 67 4.3.4.3 Single vortex ansatz.................... 70 4.3.4.4 Size of condensates(halos)................. 72 4.3.5 Vortex Oscillations.......................... 74 4.4 Summary................................... 78 5 Conclusion 81 A Simplification of the Hamiltonian constraint 92 B Derivation of unique solution of WDW equation 94 C Calculations of Israel's junction conditions 97 D The Gross-Pitaevskii equation 99 vi List of Figures 4.1 Gravitational acceleration on the equatorial plane of the BEC for our ansatz 74 4.2 Gravitational acceleration on the equatorial plane of the BEC for our ansatz 77 4.3 Gravitational acceleration on equatorial plane of the BEC for our ansatz 78 vii Physical Constants Speed of light c = 2:997 924 58 × 108 ms−1 Reduced Planck constant ~ = 1:054 571 800 × 10−34 J.s Gravitational constant G = 6:674 08 × 10−11 Nm2kg−2 Boltzmann constant k = 1:380 650 5 × 10−23 JK−1 −35 Planck length lP = 1:616 229 × 10 m −8 Planck mass mP = 2:176 470 × 10 kg −44 Planck time tP = 5:391 16 × 10 s viii To my parents. ix Chapter 1 Introduction There are four fundamental forces in nature based on its properties e.g. strong, weak, electromagnetic, and gravitational force. While the description of the first three forces is well-built on the principles of quantum mechanics on the microscopic level with a very high precision, general relativity takes into account the gravitational force on the macroscopic (galactic) scales. So far, quantum mechanics and general relativity are proved to be two of the most successful theories in modern physics to account for these fundamental forces. More specifically, general relativity discards the concept of force and introduces a new way to look at gravity as the curvature of spacetime itself, which in turn, dictates the objects to move in the shortest path in the presence of matter. On the other hand, according to quantum mechanics, the microscopic forces are mediated by quantum particles which are the quanta of a force field which pervades all over the universe. This generation of quanta out of the field is known as `quantization' of force. Since gravity is simply the curvature of spacetime, quantization of gravity means quantizing spacetime itself which is difficult to realize in our usual understanding. The task of quantizing gravity is one of the most outstanding problems of modern theo- retical physics. Many attempts have been made to reconcile these two theories starting in 1930s. Decades of hard work has enabled us to develop insights into quantum field theory and quantization of constrained systems. Despite the progress in these areas, there is still 1 Quantization of gravity no self-consistent quantum theory of gravity. There are some unavoidable reasons that make quantum gravity very challenging to formulate. They are the following: • General relativity is a highly non-linear theory and standard power counting argu- ments indicate that the theory is nonrenormalizable. This was finally confirmed in 1986 by explicit computations. • Observables in quantum field theory are local, wherein, quantum gravity physical observables are nonlocal. • Quantum field theory respects causality but for quantum gravity, light cones and causal structure are themselves subject to quantum fluctuations. • It turns out, that quantum gravity is in itself a constrained system. Hamiltonian is identically zero when acting on physical states. Therefore time evolution in quantum gravity does not have any obvious meaning. It has to be handled carefully. Our aim is to understand the quantum theory of gravity which is far more complicated in 3 + 1-D. In consideration of these problems, it is reasonable to look at a simpler theory namely 2 + 1-D gravity and develop our insight based on it. We might hope to generalize that understanding to 3 + 1 case. We could have looked at 1 + 1 case, but in one spatial dimension, the concept of Bekenstein-Hawking entropy of black holes does not make sense. This is because the entropy is proportional to the area or the circumference of the black hole.