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Dynamics of Spinning Compact Binaries in General Relativity

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Dynamics of Spinning Compact Binaries in General Relativity Dynamics of Spinning Compact Binaries in General Relativity Thesis by Michael David Hartl In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2003 (Defended May 22, 2003) ii c 2003 Michael David Hartl All Rights Reserved iii Acknowledgements Earning a Ph.D. in physics from Caltech represents the fulfillment of a lifelong dream, and there are many who have contributed to its realization. First are my mother and father, whose encouragement, love, and support I could take for granted—if only all children were so lucky. It was they, more than anyone else, who were responsible for my early science education, especially through their indulgence of my voracious appetite for popular science (although my wonderful step-mother deserves considerable blame for this indulgence as well). The vacuum created by uninspiring science teachers was filled by books, from authors such as Timothy Ferris, John Casti, Douglas Hofstadter, and Stephen Hawking. My deepest gratitude in this vein is to Carl Sagan, whose Cosmos book and miniseries were a revelation. Cosmos sparked, among other things, an interest in amateur astronomy, but pursuing this interest might never have happened without the intervention of Lang Dana, whose generous and utterly unexpected gift of a first-rate telescope came—as he well knew—at a critical juncture in my life; for this I shall be eternally grateful. As an undergraduate at Harvard, it was my great privilege to learn physics from David Bear, who, though then a lowly graduate student, was the best teacher I could have asked for. He was the role model for my own teaching at Caltech, and I hope I have lived up to the lofty standard he set. My other best teachers were my friends, especially Ranjit Aiyagari, Long Nguyen, and Sumit Daftuar (who, among other things, was an ebullient and insightful physics study partner). Perhaps my most valued friend at Harvard was someone I actually met long before—my sister Kara, the best sister a brother could have, and soon to become a real doctor. My journey to Caltech started in the Solar and Stellar X-Ray group at the Harvard-Smithsonian Center for Astrophysics, where I had the good fortune of working with Leon Golub and Ed DeLuca (as well as fellow student and valued friend Meredith Wills). After the end of my sophomore year, I hoped to spend the summer doing research in my home state of California, so the question was obvious: I asked Leon, “Do you know anyone at Caltech?” The answer was yes, and Leon’s old friend Hal Zirin managed to get me a Summer Undergraduate Research Fellowship, even though the application deadline had already long passed. Though I lived far from campus that summer (in the idyllic mountain resort town of Big Bear), then-recent Caltech graduate Glenn Eychaner thoroughly indoctrinated me into the Way of Tech. When I returned to the SURF program for a iv second summer, this time living on campus, I already felt like an honorary Techer. I owe a debt of gratitude to Dale Gary for making that second summer possible, and for his patient guidance of the SURF project that led to my first scientific publication. Much of this thesis draws on the results and techniques of chaos theory, and it was my great privilege to learn this subject from one of its masters, Jim Yorke. I would like to thank the University of Maryland for the Distinguished Research Fellowship program that made that study possible. To Jim Yorke my gratitude is profound, not only for his generosity and wisdom, but also for writing me a strong letter of recommendation to Caltech—even though it meant helping me move to a rival institution. At Caltech, my friends have been a source of encouragement, support, inspiration, and occasion- ally beer. Many thanks go especially to Kashif Alvi, Sumit Daftuar, Samantha Edgington, Robert Forster, Scott Hughes, Keith Matthews, Shanti Rao, Kevin Scaldeferri, Luke Sollitt, Chip Sumner, and Shimpei Yamashita. Of my teachers, I especially want to thank Kip Thorne. I will always feel lucky that I had the opportunity to learn general relativity from the man who put the T in MTW—and refuses to be called anything other than Kip. Much thanks also to Alessandra Buonanno, who supplied the copious notes and references that made Chapter 4 possible. I look forward to collaborating with her on an expanded form of that project. My deepest appreciation goes to my advisor, Sterl Phinney, who was willing (and able!) to indulge my desire to find a project mixing chaos theory and general relativity. It was he who waved the magic wand, seemingly possessed by so many at Caltech, that created a connection between that project and gravitational waves. I am particularly grateful for Sterl’s careful reading of—and resulting insightful comments on—the papers composing this thesis, which always led to major improvements in both content and presentation. Though technically single-author works, the chapters in this thesis owe much to his guidance and advice. This thesis was supported financially by Caltech teaching assistantships (including the Richard Feynman Book Fund Teaching Assistantship) and by NASA grant NAG5-10707. v Abstract This thesis investigates the dynamics of binary systems composed of spinning compact objects (such as white dwarfs, neutron stars, and black holes) in the context of general relativity. In particular, we use the method of Lyapunov exponents to determine whether such systems are chaotic. Compact binaries are promising sources of gravitational radiation for both ground- and space-based gravitational-wave detectors, and radiation from chaotic orbits would be difficult to detect and analyze. For chaotic orbits, the number of waveform templates needed to match a given gravitational- wave signal would grow exponentially with increasing detection sensitivity, rendering the preferred matched filter detection method computationally impractical. It is therefore urgent to understand whether the binary dynamics can be chaotic, and, if so, how prevalent this chaos is. We first consider the dynamics of a spinning compact object orbiting a much more massive rotating black hole, as modeled by the Papapetrou equations in Kerr spacetime. We find that many initial conditions lead to positive Lyapunov exponents, indicating chaotic dynamics. The Lyapunov exponents come in positive/negative pairs, a characteristic of Hamiltonian dynamical systems. Despite the formal existence of chaotic solutions, we find that chaos occurs only for physically unrealistic values of the small body’s spin. As a result, chaos will not affect theoretical templates in the extreme mass-ratio limit for which the Papapetrou equations are valid. Chaos will therefore not affect the ability of space-based gravitational-wave detectors (such as LISA, the Laser Interferometer Space Antenna) to perform precision tests of general relativity using extreme mass-ratio inspirals. We next consider the dynamics of spinning black-hole binaries, as modeled by the post-Newtonian (PN) equations, which are valid for orbital velocities much smaller than the speed of light. We study thoroughly the special case of quasi-circular orbits with comparable mass ratios, which are particularly relevant from the perspective of gravitational wave generation for LIGO (the Laser Interferometer Gravitational-wave Observatory) and other ground-based interferometers. In this case, unlike the extreme mass-ratio case, we find chaotic solutions for physically realistic values of the spin. On the other hand, our survey shows that chaos occurs in a negligible fraction of possible configurations, and only for such small radii that the PN approximation is likely to be invalid. As a result, at least in the case of comparable mass black-hole binaries, theoretical templates will not vi be significantly affected by chaos. In a final, self-contained chapter, we discuss various methods for the calculation of Lyapunov exponents in systems of ordinary differential equations. We introduce several new techniques ap- plicable to constrained dynamical systems, developed in the course of studying the dynamics of spinning compact binaries. Considering the Papapetrou and post-Newtonian systems together, our most important general conclusion is that we find no chaos in any relativistic binary system for orbits that clearly satisfy the approximations required for the equations of motion to be physically valid. vii Contents Acknowledgements iii Abstract v 1 Introduction 1 1.1 The Papapetrou and post-Newtonian equations . 4 1.1.1 Papapetrou (or Dixon) . 4 1.1.2 Post-Newtonian (with spin) . 6 1.1.3 Normalized units and µM ............................. 7 1.1.4 Normalization constraints . 8 1.1.5 Spin supplementary conditions and the center of mass . 9 1.1.6 Degrees of freedom . 11 1.2 Lyapunov exponents and chaos . 12 1.2.1 Deviation vector method . 12 1.2.2 Jacobian method . 13 1.2.3 The origin of exponential separation . 13 1.2.4 Constrained systems . 14 1.2.5 The meaning of zero . 14 1.3 Chaos in general relativity: a review and critique . 15 1.3.1 Chaos detection . 16 1.3.2 Time coordinate subtleties . 16 1.3.3 Genericity and the choice of initial conditions . 17 1.3.4 Papapetrou spin parameter confusion . 17 1.3.4.1 Chaos in this thesis . 18 2 Dynamics of spinning test particles in Kerr spacetime 21 Abstract . 21 2.1 Introduction . 21 2.2 Spinning test particles . 24 viii 2.2.1 Papapetrou equations . 24 2.2.1.1 Spin supplementary conditions . 25 2.2.1.2 A reformulation of the equations . 25 2.2.1.3 Range of validity . 27 2.2.2 Comments on the spin parameter . 27 2.2.2.1 Bounds on S for stellar objects . 27 2.2.2.2 Tidal disruption .
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