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Integrable Nonlinear Relativistic Equations Integrable Nonlinear Relativistic Equations Item Type text; Electronic Dissertation Authors Hadad, Yaron Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 07/10/2021 09:43:36 Link to Item http://hdl.handle.net/10150/293490 Integrable Nonlinear Relativistic Equations by Yaron Hadad A Dissertation Submitted to the Faculty of the Department of Mathematics In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy In the Graduate College The University of Arizona 2 0 1 3 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the disser- tation prepared by Yaron Hadad entitled Integrable Nonlinear Relativistic Equations and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Date: April 9, 2013 Nicholas Ercolani Date: April 9, 2013 Leonid Friedlander Date: April 9, 2013 Andrei Lebed Date: April 9, 2013 Mikhail Stepanov Date: April 9, 2013 Vladimir Zakharov Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. Date: April 9, 2013 Vladimir Zakharov 3 Statement by Author This dissertation has been submitted in partial fulfillment of re- quirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to bor- rowers under rules of the Library. Brief quotations from this dissertation are allowable without spe- cial permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or repro- duction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. Signed: Yaron Hadad 4 Dedication For my grandmother Miriam, who foretold this moment three decades ago. 5 Acknowledgments The last six years were a fascinating journey for me with several life-changing events and perplexing crossroads. It is therefore impossible for me to express gratitude to all the people who contributed to this project. Nevertheless, some personal thanks. I would like to thank my wonderful family who supported me endlessly throughout this process. Their unfailing love is what allowed me to pursue my dream of studying the world around us. I wish to thank my advisor Vladimir Zakharov from whom I learned a great deal. He is the one who revealed to me the infinitude of science, an infinitude that is sometimes hidden underneath the surface of the smallest problem. His teaching goes well beyond mathematics, physics or even science in general. As the biggest lesson I learned from him is that even the greatest scientist should be first and foremost a modest person. Without his enlightening guidance on this path, I'd probably still be buried underneath the math books. There was one door I had to pass through dozens of times on my path. This was the entrance to the office of Lennie Friedlander who, with his endless patience and incorruptible rigor, helped me with every question or when in any doubt. Lennie's analysis classes inspired me to pursue research in this vast beautiful field. Thank you. It is a pleasure thanking Misha Stepanov for his direction in my RTG project. He was the first who opened my eyes to the inverse transform of Einstein's equation, a step, without which this work would never have come into existence. The numerical work described here is the result of our joint work at the beginning of my graduate studies. My gratitude to Nicholas Ercolani for his assistance during the last two years. Sev- eral sections of this work would not have been written without his help. His profound knowledge of so many distinct fields is a pool of inspiration and great support. 6 I thank Andrei Lebed for reviewing this work and taking the important role of my physics minor advisor. Finally, I would like to thank Tom Kennedy, Klaus Lux, Sandra Sutton and Alan Newell for their guidance and help. Alan assisted me when I was standing at the biggest crossroad, and introduced me to my mentor Vladimir Zakharov. Thank you all. 7 Table of Contents List of Figures ................................. 9 List of Tables .................................. 10 Abstract ..................................... 11 Chapter 1. Introduction .......................... 12 1.1. The special theory of relativity . 12 1.2. The general theory of relativity . 15 1.3. Integrability . 19 1.4. Chiral fields . 21 1.5. The Belinski-Zakharov transform . 24 1.6. Construction of N gravitational solitons . 33 1.7. An algorithm for construction of N gravitational solitons . 43 Chapter 2. The symmetric principal chiral field equation .... 46 2.1. Reduction to a single scalar equation . 47 2.2. Conservation laws . 50 2.3. Solitons on diagonal backgrounds . 55 Chapter 3. The 1 + 1 Einstein equation ................. 63 3.1. Lagrangian formulation . 65 3.2. The geometric representation . 67 3.3. Hamiltonian formulation . 69 3.4. Conservation laws . 72 3.5. Synchronous reference frames . 75 3.6. Gravitational solitons . 76 3.6.A. List of seeds studied in the literature . 77 3.6.B. Diagonal backgrounds . 78 3.7. Numerical study . 100 Chapter 4. The 1 + 2 Einstein equation ................. 104 4.1. General diagonal spacetime metrics . 105 4.2. Diagonal spacetime metrics in 1 + 2 coordinates . 107 4.2.A. Introduction . 107 4.2.B. Plane gravitational waves . 111 4.2.C. Stability . 115 4.3. Weak gravitational waves . 124 8 Table of Contents|Continued 4.3.A. The general case . 124 4.3.B. Waves in 1 + 2 coordinates . 125 4.3.C. Higher order analysis . 128 4.3.D. Axial symmetry . 133 4.4. The nonlinear Schr¨odinger-Einsteinequations . 135 4.4.A. Derivation . 136 4.4.B. Solutions . 139 Appendix A. Notations and conventions ................ 144 Appendix B. Curvature tensors for the block diagonal metric . 146 Appendix C. The integrability condition for Ai ........... 148 Appendix D. The commutation relation for Di ............ 149 Appendix E. Solving the pole trajectory equations ......... 151 Appendix F. Derivatives with respect to the metric ........ 152 References .................................... 153 9 List of Figures Figure 1.1. The trajectory of a light ray in flat spacetime . 13 Figure 1.2. The light-cone . 15 Figure 1.3. Trajectories of the dressing matrix poles (non-stationary metrics) 36 Figure 1.4. Trajectories of the dressing matrix poles (stationary metrics) . 38 (1) Figure 2.1. g12 for one soliton of the symmetric Chiral field equation . 59 (1) Figure 2.2. g11 for one soliton of the symmetric Chiral field equation . 59 (1) Figure 2.3. g11 for one soliton of the symmetric Chiral field equation . 60 (2) Figure 2.4. g12 for two solitons of the symmetric Chiral field equation . 61 (2) Figure 2.5. g11 for two solitons of the symmetric Chiral field equation . 62 Figure 3.1. Spacetime diagram for a single gravitational soliton . 82 Figure 3.2. Spacetime diagram for two gravitational solitons . 83 (1) Figure 3.3. g11 for one gravitational soliton on flat spacetime . 87 (1) Figure 3.4. g22 for one gravitational soliton on flat spacetime . 88 (1) Figure 3.5. g12 for one gravitational soliton on flat spacetime . 89 Figure 3.6. f (1) for one gravitational soliton on flat spacetime . 90 (2) Figure 3.7. g22 for two (real) gravitational solitons on flat spacetime . 91 (2) Figure 3.8. g22 for soliton & anti-soliton interaction on flat spacetime . 92 (2) Figure 3.9. g12 for soliton & anti-soliton interaction on flat spacetime . 93 (2) Figure 3.10. g11 for two (complex) gravitational solitons on flat spacetime . 94 (2) Figure 3.11. g22 for two (complex) gravitational solitons on flat spacetime . 94 (2) Figure 3.12. g12 for two (complex) gravitational solitons on flat spacetime . 95 Figure 3.13. The interpretation of "s ...................... 96 (1) Figure 3.14. g11 for one gravitational soliton on Einstein-Rosen background 99 (1) Figure 3.15. g22 for one gravitational soliton on Einstein-Rosen background 100 (1) Figure 3.16. g12 for one gravitational soliton on Einstein-Rosen background 101 Figure 3.17. The 'triangle of validity' of the numerical scheme . 103 Figure 4.1. Classes of solutions of the vacuum Einstein equation . 111 Figure 4.2. The profile of the Bondi-Pirani-Robinson gravitational wave . 116 Figure 4.3. Reflectionless Bondi-Pirani-Robinson wave . 123 Figure 4.4. Collapse of one gravitational wave into three . 132 Figure 4.5. Interaction of two gravitational waves . 133 10 List of Tables Table 3.1. Seeds studied in the literature using the Belinski-Zakharov trans- form . 77 Table 3.2. Error in numerical solutions of the reduced Einstein equation . 102 Table F.1. Derivatives with respect to the metric coefficients gab . 152 11 Abstract This work focuses on three nonlinear relativistic equations: the symmetric Chiral field equation, Einstein's field equation for metrics with two commuting Killing vectors and Einstein's field equation for diagonal metrics that depend on three variables. The symmetric Chiral field equation is studied using the Zakharov-Mikhailov transform, with which its infinitely many local conservation laws are derived and its solitons on diagonal backgrounds are studied.
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