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Event Horizons, Gravitational Waves and Astrophysical Kicks in Black-Hole Spacetimes J Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 10-1-2011 Event horizons, gravitational waves and astrophysical kicks in black-hole spacetimes J. Marcelo Ponce Castro Follow this and additional works at: http://scholarworks.rit.edu/theses Recommended Citation Ponce Castro, J. Marcelo, "Event horizons, gravitational waves and astrophysical kicks in black-hole spacetimes" (2011). Thesis. Rochester Institute of Technology. Accessed from This Dissertation is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. Event Horizons, Gravitational Waves and Astrophysical Kicks in Black-Hole Spacetimes Ph.D. Doctor of Philosophy in Astrophysical Sciences and Technology M.Sc. Marcelo Ponce C. Rochester Institute of Technology Rochester, New York October 2011 ASTROPHYSICAL SCIENCES AND TECHNOLOGY COLLEGE OF SCIENCE ROCHESTER INSTITUTE OF TECHNOLOGY ROCHESTER, NEW YORK CERTIFICATE OF APPROVAL Ph.D. DEGREE DISSERTATION The Ph.D. Degree Dissertation of J. Marcelo Ponce Castro has been examined and approved by the dissertation committee as satisfactory for the dissertation requirement for the Ph.D. degree in Astrophysical Sciences and Technology. Dr. Douglas Meadows, Committee Chair Dr. Manuela Campanelli, Committee Member Dr. Joshua A. Faber, Committee Member Dr. Joel Kastner, Committee Member Dr. Carlos Lousto, Committee Member Dr. Jorge Pullin, Committee Member Professor Dr. Yosef Zlochower, Thesis Advisor Date EVENT HORIZONS, GRAVITATIONAL WAVES AND ASTROPHYSICAL KICKS IN BLACK HOLES SPACETIMES By J. Marcelo Ponce Castro A dissertation submitted in partial fulfillment of the requirements for the degree of Ph.D. in Astrophysical Sciences and Technology, in the College of Science, Rochester Institute of Technology. October, 2011 Approved by Prof. Andrew Robinson Date Director, Astrophysical Sciences and Technology Para ALI Acknowledgments I want to express my most profound gratitude and admiration to my advisor, Prof. Yosef Zlochower, who has been not only and incredible mentor, but who also has expended large amounts of time teaching me the science behind the numerical simulations. I have learnt a lot from him not only in our research field but also in the motivation, dedication and passion that he has for his work. Many thanks for his scientific guidance that generated many interesting discussions, questions, and pushing the goals of our research. Thanks so much for reading this thesis “thousands of times” and the editorial assistance in writing it. It has been an honor and a enormous pleasure for me to work with him. Very special thanks to Prof. Manuela Campanelli and Prof. Carlos Lousto for giving me the chance and honor of belonging to the CCRG, for enlightening discussions and for the high quality science developed within the CCRG. Many thanks to all the members from CCRG: faculties, postdocs and students. Special thanks to Prof. Hans-Peter Bischof, for his help with the visualizations, Imagine RIT and for being always so nice with me. Many thanks to Prof. Joshua Faber and Prof. Jamie Lombardi. It has been a pleasure to work with them. Thanks for many interesting discussions and forthcoming projects... Thanks to the faculty of the AST program at RIT, from who I have learnt most of the astrophysics I known. To the members of this committee, for accepting the hard task of reading and evaluating this thesis. I gratefully acknowledge NSF for financial support from grants PHY-0722315, PHY-0653303, PHY-0714388, PHY-0722703, DMS-0820923, PHY-0929114, PHY-0969855, PHY-0903782, and CDI-1028087. and NASA for financial support from grants NASA 08-ATFP-0093, 07- ATFP07-0158 and HST-AR-11763. This research was supported in part by the National Science Foundation through TeraGrid resources provided by NCSA and TACC under grant numbers TG-PHY080040N, TG-PHY060027N and TG-AST100048. Computational resources were provided by Ranger cluster at TACC, Lincoln cluster at NCSA and by NewHorizons at iii RIT. I gratefully acknowledge financial support from the Astrophysical Sciences and Technology (AST) program at RIT, and the Graduate Student Office at RIT. I gratefully acknowledge travel awards from the American Physical Society (APS): Division of Astrophysics (DAP) and Topical Group in Gravitation (GGR). To all my fellows graduate student for sharing a really enjoyable ride. I found amazing people and friends that I hope to keep close. To RIT, for allowing me to pursue my dreams, to live in an amazing place, to enjoy the snow one more time, to play rugby again... go TIGERS!!! I want to thank my family in Uruguay for their support and unconditional love: my aunt Alicia, my parents Hector and Griselda, my cousins Sebi and Andre, my grand-mother Mamaela, and to the rest of all my family that in our very own and particular way we keep our ties strong! To all my pets: I miss you a lot guys! The most beautiful and sweet white kitty in the world ’enana’, this little and sweetie buddy Newton and the greatest dog in the whole world Ernita who will be always in my thoughts. I want to specially thank to my lovely and beautiful fiancée, friend and future wife, Luciana, that has decide to come and join me in what will be an amazing and incredible journey for the rest of our lives, not only to reach our most challenging goals but also to begin our own family. Without her all this effort will not make any sense at all. Thanks darling for being there and being here for me! I love you! iv Abstract In this thesis we use computational techniques (numerical simulations) to study different stages of black hole mergers. A first project describes topological properties of the main performer of this play, the black hole and its event horizon. We investigate three configurations: a continuum ring singularity, a “discretized” ring (black holes arranged on a ring), and a linear distribution of black holes. We evolve each of the corresponding spacetimes forward and then backwards in time, searching for the respective event horizons. We find some evidence, based on configurations of multiple BHs arranged in a ring, that this configuration leads to singular limit where the horizon width has zero size, possibly indicating the presence of a naked singularity, when the radius of the ring is sufficiently large. In a second project, we study the dynamics of a hydrodynamical accretion disk around a recoiling black hole, which models the behavior of an accretion disk around a binary just after the merger, using “smoothed-particle hydrodynamics” techniques. We simulated different recoil angles between the accretion disk and the recoil velocity of the black hole. We find that ◦ for more vertical kicks (θ . 30 ), a gap remains present in the inner disk, while for more ◦ oblique kicks (θ & 45 ), matter rapidly accretes toward the black hole. There is a systematic trend for higher potential luminosities for more oblique kick angles for a given black hole mass, disk mass and kick velocity, and we find large amplitude oscillations in time in the case of a kick oriented 60◦ from the vertical. Key-words: Computational Astrophysics, Numerical Relativity, Smooth Particle Hydro- dynamics, Event Horizons, Ring Singularity, Black Holes, Gravitational Recoils, Accretion Disk. v vi Contents Abstract v Table of Contents vii List of Figures xi List of Tables xix 1 Introduction 1 1.1 General Relativity . .1 1.1.1 Einstein Field Equations . .2 1.2 Gravitational Radiation . .5 1.3 Astrophysical Applications . .9 1.3.1 Gravitational Recoils . 10 1.4 Numerical Relativity . 11 1.4.1 3+1 Decomposition . 14 1.5 Black Holes . 15 1.5.1 Definition of Horizons . 21 1.5.2 Gas Dynamics around BHs . 25 1.6 Research Projects . 28 1.6.1 Event Horizons and Topological Censorship . 29 1.6.2 Newtonian Gas Dynamics around Kicked BHs . 30 1.6.3 Organization . 32 vii 2 Numerical Relativity 33 2.1 General Relativity Formalism . 34 2.1.1 Mathematical Foundations of General Relativity . 34 2.1.2 Black Holes . 40 2.1.3 Asymptotically Flatness . 43 2.1.4 Einstein Field Equations . 47 2.2 The 3+1 formalism . 48 2.2.1 3+1 split of spacetime . 48 2.2.2 Extrinsic Curvature . 51 2.2.3 Einstein Equations . 53 2.2.4 Constraints . 53 2.2.5 The ADM Evolution Equations . 54 2.2.6 Numerical Evolutions: free vs constrained evolutions . 56 2.2.7 Hamiltonian Formulation . 57 2.3 The NOK-BSSN Formulation . 59 2.3.1 General Considerations: BSSN//NOK-BSSN//conformal Γ ....... 59 2.3.2 Conformal factor & Conformal metric . 60 2.3.3 Conformal Connection Functions . 61 2.3.4 Evolution Equations . 61 2.3.5 Final Set of Evolution Equations & Constraints . 62 2.4 Moving Puncture Approach . 63 2.5 Initial Data . 65 2.5.1 Black Hole Initial Data . 66 2.6 Horizons . 68 2.6.1 Apparent Horizons . 68 2.6.2 Events Horizons . 69 2.6.3 Applications: Irreducible Mass . 72 2.6.4 Isolated and Dynamical Horizons . 72 viii 2.7 Finite Differencing . 74 2.7.1 Classification of Partial Differential Equations . 74 2.7.2 Finite Difference Methods . 77 2.7.3 Uniform Grids, Fixed Mesh Refinement, and Adaptive Mesh Refinement 80 2.7.4 Other Approaches . 81 2.8 Simulations Framework . 82 2.8.1 Cactus . 82 2.8.2 LazEv . 83 3 Event Horizons for Nontrivial Black-Hole Topologies 85 3.1 Introduction . 85 3.2 Topology theorems and their consequences . 88 3.2.1 Application to the topology of black holes . 90 3.2.2 Higher-Dimensions and non-trivial Event Horizons . 91 3.3 Toroidal Event Horizons and Black Hole Rings .
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