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Toroidal Event Horizons and Other Relativistic Ray Tracing Adventures Toroidal event horizons and other relativistic ray tracing adventures ADissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Andrew Douglas Bohn August 2016 c 2016 Andrew Douglas Bohn This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. http://creativecommons.org/licenses/by-sa/4.0/ TOROIDAL EVENT HORIZONS AND OTHER RELATIVISTIC RAY TRACING ADVENTURES Andrew Douglas Bohn, Ph.D. Cornell University 2016 This thesis deals with the appearance of colliding black holes from two vantage points, dividing the thesis into two parts. Part I (Chapters 2 to 4)investigatesthetopologyofmergingblackholeeventhorizons. Chapter 2 introduces a new code for locating event horizons in numerical simulations. The code can automatically refine arbitrary regions of the event horizon surface to find features such as the hole in a toroidal event horizon. With these tools, Chapter 3 shows the first binary black hole event horizon with a toroidal topology. It had been predicted that generically the event horizons of merging black holes should briefly have a toroidal topology, but such a phase has not been seen prior to this work. In all previous binary black hole simulations, in the coordinate slicing used to evolve the black holes, the topology of the event horizon transitions directly from two spheres during the inspiral to a single sphere as the black holes merge. Chapter 3 presents a transformation of the time coordinate that results in a toroidal event horizon. A torus could potentially provide a mechanism for violating the so-called topological censorship theorem; however, since these toroidal event horizons arise from a coordinate choice, they can be removed by the inverse coordinate transformation and do not violate the theorem. Chapter 4 presents work toward reslicing the event horizon on slices of constant affine parameter in an e↵ort to find an event horizon on a toroidal topology. Part II (Chapters 5 and 6) considers what a binary black hole merger actually looks like for anearbyobserver.Chapter 5 presents both demonstrative images illustrating details of the spatial distortion and realistic images of collections of stars taking both lensing amplification and redshift into account. On large scales, the lensing from inspiraling binaries resembles that of single black holes, but on small scales the resulting images show complex and in some cases self-similar structure across di↵erent angular scales. Chapter 6 presents miscellaneous gravitational lensing projects including producing content for the LIGO press conference for the first ever detection of gravitational waves. We also present a project visualizing physically accurately an accretion disk around a black hole, similar to the system from the movie Interstellar. Biographical Sketch Andy was born in Wyandotte, Michigan in September 1987 to Michael and Kimberly Bohn. Shortly after, he and his family moved to northwest Indiana (neither to be confused with nor associated with the rest of Indiana), where he was raised. He went to high school at Lake Central High School in Saint John, Indiana from 2002-2006, where he spent most of this time drumming. He auditioned and was selected for the first Bands of America honor band snare line in 2004, to march in the Tournament of Roses Parade in Pasadena, California. He also composed and performed in a bass drum ensemble winning the Indiana Percussion state championship in 2005. In 2006, he began collegiate studies at Purdue University in West Lafayette, Indiana, where he majored in both Honors Physics and Mathematics. At Purdue, he investigated how to focus the Large Synoptic Survey Telescope using intra-focal and extra-focal images, under the supervision of Ian Shipsey. He spent the summer of 2008 at Caltech studying the strength and thermal noise of laser pulled and polished fused silica glass fibers to hold the 40kg mirrors for the Advanced LIGO detectors. In the summer of 2010, he began research at Cornell University under Julia Thom studying AC coupling in Silicon on insulator devices for the Large Hadron Collider. The following summer, he joined the Simulating eXtreme Spacetimes collaboration under Saul Teukolsky and Lawrence Kidder studying event horizons in merging black hole systems, where he carried out the work for this thesis. He received his Master’s degree in 2013. He accepted a position at SpaceX in early 2016, where he will begin as a flight software engineer for low Earth orbit satellites in late July 2016. iii In the immortal words of my Dad, “now hear this!” iv Acknowledgments This thesis would not have been possible without the help and support of a large number of people. I would like to begin by thanking my advisors, starting with my professorial advisor, Saul Teukolsky. In addition to providing many useful suggestions on the direction of my research and frequently recalling relevant page numbers from Numerical Recipes on demand, he treated me like an adult instead of a subordinate. I am very grateful for everything I have learned from my time with Saul, from approaching problems with first principles and simple base cases, to the proper way to write delicate emails by “always starting with a compliment.” My work was additionally mentored by Larry Kidder, with whom I grew a strong rapport. His door was always open for me to bug him with arcane questions regarding the inner workings of scary portions of SpEC,ortochataboutboardgamesanddebateifMinnesota or Detroit had worse sports outings the previous night. He managed to simultaneously be a friend and mentor. I could not have asked for a better pair of advisors. Thanks also go to my additional defense committee members Eanna´ Flanagan who miraculously managed to teach me some General Relativity and Cosmology, and Julia Thom for graciously agreeing to sit through my thesis and defense. I am obliged to thank some of my officemates: Dan Hemberger for surviving DotA sessions with me, Kate Henriksson for making Debatable less debatable, and my psuedo-officemate Curran Muhlberger for indulging random computer science discussions and opening the channel from SXS to SpaceX. I would have been unable to complete this work without Fran¸cois H´ebert and Will Throwe for being an absolute joy to work with for five years, engaging with me in often poorly founded research conversations, and enabling the successes of the gravitational lensing project. I also need to thank the latecomer to the office Nils Deppe for being an irreplaceable and unyielding Strahlkorper to me in the office every day. I would like to thank the whiskey group of Cody Long, Ethan Kassner, John Stout, and Brian Leahy for fantastic tastings that were brief, civil, and classy. Special thanks go to v Ethan for living with me for almost 20% of my life, playing Mario Kart 64, and watching “Always Sunny” with the finest whiskey Canada has to o↵er. A song for Matt: Matt Bierbaum, Matt Bierbaum, opening up reddit.com; He is aplomb, and he is the bomb, Matt Bierbaum. Thanks to Corky “Robert” Wharton for being the Sir Donald Bradman of entertainment, but at last it is time for me to cash out of the Wharton money-friendship uncertainty principle σmσf ~/2. ≥ Finally, and most of all, I would like to thank my family, to whom I dedicate my thesis. My sister Emily, who I have always admired for her ability to battle through incredible adversities not limited to having to deal her brothers while growing up; e hale Space !; My brother ⇥ ! ⇥ Mike, for getting me started with programming and fostering my curiosity through (mostly) playful competitiveness; My brother-in-law Alex, for indulging me with puzzles, generally nerdy maths, and being a fantastic addition to the family; My mom and dad, who taught me how to be inquisitive and empathetic, and provided me with everything I could have possibly needed and more; And to my wife ,maywehave more year(s) of and !Iloveyouall. I thank not only those who I mentioned here, but everyone who has had a positive impact on my life. Table of Contents Biographical Sketch ................................... iii Dedication ........................................ iv Acknowledgments .................................... v Table of Contents .................................... vi List of Figures ...................................... vii List of Tables ...................................... viii 1 Introduction 1 1.1 Brief history of gravitational theory ....................... 2 1.2 General Relativity ................................ 5 1.3 Thesis overview .................................. 8 I Binary black hole event horizon topologies 19 2 A Parallel Adaptive Event Horizon Finder for Numerical Relativity 20 2.1 Introduction .................................... 20 2.2 Backwards geodesic method overview ...................... 22 2.3 Event horizon representation ........................... 24 2.4 Generator evolution ................................ 33 2.5 Handling metric data ............................... 35 2.6 Initial data .................................... 36 2.7 Identifying future generators ........................... 38 2.8 Conclusions .................................... 43 Chapter Appendices 45 2.A Null geodesic evolution equations in the 3 + 1 decomposition ......... 45 2.B Spacetime interpolations ............................. 49 2.C Removing triangles from the collision detection algorithm ........... 55 3 Toroidal Horizons in Binary Black Hole Mergers 63 3.1 Introduction
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