Rochester Institute of Technology RIT Scholar Works Theses 8-14-2018 Binary Black Holes in the Inspiral Regime: The Effect of Spins and Eccentricity on Spacetime Dynamics and Gravitational Radiation Brennan M. Ireland [email protected] Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Ireland, Brennan M., "Binary Black Holes in the Inspiral Regime: The Effect of Spins and Eccentricity on Spacetime Dynamics and Gravitational Radiation" (2018). Thesis. Rochester Institute of Technology. Accessed from This Dissertation is brought to you for free and open access by 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]. Rochester Institute of Technology Ph.D. Dissertation Binary Black Holes in the Inspiral Regime: The Effect of Spins and Eccentricity on Spacetime Dynamics and Gravitational Radiation Author: Advisor: Brennan M. Ireland Manuela Campanelli A dissertation submitted in partial fulfillment of the requirements for the degree of Astrophysical Sciences and Technology in the School of Physics and Astronomy August 14, 2018 Rochester Institute of Technology Ph.D. Dissertation Binary Black Holes in the Inspiral Regime: The Effect of Spins and Eccentricity on Spacetime Dynamics and Gravitational Radiation Author: Advisor: Brennan M. Ireland Manuela Campanelli A dissertation submitted in partial fulfillment of the requirements for the degree of Astrophysical Sciences and Technology in the School of Physics and Astronomy Approved by Prof. Andrew Robinson Date Director, Astrophysical Sciences and Technology Certificate of Approval Astrophysical Sciences and Technologies R·I·T College of Science Rochester, NY, USA The Ph.D. Dissertation of Brennan M. Ireland has been approved by the undersigned members of the dissertation committee as satisfactory for the degree of Doctor of Philosophy in Astrophysical Sciences and Technology. Dr. Nathaniel Barlow, Committee Chair Date Dr. Manuela Campanelli, Dissertation Advisor Date Dr. Richard O'Shaughnessy Date Dr. Lawrence Kidder Date To the awesome people in my life who made this possible. You know who you are. ABSTRACT Observations of black hole binaries via the emission of gravitational waves are one of the most exciting discoveries in physics in the past 50 years. The most generic black holes in nature are ones with spin, which may be misaligned with the orbital angular momentum of the binary, and also orbital eccentricity. This demands computationally inexpensive and accurate models of spinning binary black holes for hundreds of orbits as the binary inspirals. This dissertation is divided into two projects, both of which focus on binary black holes with spin. In the first project, I construct and present a new global, fully analytic, approximate spacetime which accurately describes the dynamics of nonprecessing, spinning black hole binaries during the inspiral phase of the relativistic merger process. This approximate solution of the vacuum Einstein's equations can be obtained by asymptotically matching perturbed Kerr solutions near the two black holes to a post-Newtonian metric valid far from the two black holes. This metric is then matched to a post-Minkowskian metric even farther out in the wave zone. The procedure of asymptotic matching is generalized to be valid to all times, instead of a small group of initial hypersurfaces discussed in previous works. I then re-examine the asymptotic matching in the case of precession of the spins, allowing for generically spinning black hole binary metrics. This metric is well suited for long term dynamical simulations of spinning black hole binary spacetimes prior to merger, such as studies of circumbinary gas accretion which requires hundreds of binary orbits. In the second project, I present a method for developing and calculating the gravitational wave- forms from generically spinning, black hole binaries, with significant orbital eccentricity. I use the Lagrangian formulation of the post Newtonian equations of motion in the harmonic gauge for the generation of precessing, eccentric gravitational wave signatures. The equations of motion describ- ing the black hole binary system are important to our understanding of fundamental relativity, for both the context of supermassive black holes as well as stellar mass systems. If gravitational wave measurements are able to measure a non-negligible eccentricity from the binary, this may point to a unique formation model through relativistic 3-body interactions in dense stellar fields, which will impart occasionally significant eccentricity. This provides insight into the formation history of the binary, and explicitly the last dynamical effect the binary experienced before merging. i CONTENTS Abstract i Declaration v Acknowledgements vi List of published work vii List of Tables viii List of Figures ix 1 Introduction 1 1.1 Motivation.........................................1 1.2 Introduction to General Relativity............................3 1.3 Black Holes and Gravitational Waves...........................6 1.3.1 Black Holes.....................................6 1.3.2 Gravitational Waves................................8 1.4 Observations of Black Holes and Black Hole Binaries..................9 1.5 Projects Herein....................................... 13 1.5.1 Hybrid Metric Construction for spinning black hole binaries in the inspiral regime........................................ 13 1.5.2 Evolution equations for generic black hole binaries............... 14 1.6 Organization........................................ 15 2 Analytic Techniques in General Relativity 17 2.1 The Mathematics of General Relativity......................... 17 2.1.1 Manifolds, Vectors, One-forms, and Tensors................... 17 2.1.2 The Covariant Derivative............................. 19 ii CONTENTS 2.1.3 Geodesics...................................... 19 2.1.4 Curvature...................................... 20 2.1.5 Einstein's Equations................................ 23 2.1.6 Conserved Quantities............................... 24 2.2 Mathematical Properties of Black Holes......................... 25 2.2.1 The Schwarzschild Solution............................ 26 2.2.2 The Kerr Solution................................. 28 2.2.3 Coordinate and Gauge Choices.......................... 30 2.3 Approximate Solutions to the Einstein Equations.................... 31 2.3.1 Post-Minkowski Theory.............................. 31 2.3.2 Post-Newtonian Theory.............................. 36 2.3.3 Black Hole Perturbation Theory......................... 40 2.4 Formulation of Gravitational Radiation......................... 43 3 Construction of an Approximate Global Spacetime with Spin Via Asymptotic Matching 46 3.1 Introduction......................................... 46 3.2 Construction of Approximate Global Metric....................... 49 3.2.1 Subdividing Spacetime............................... 50 3.2.2 Inner Zone..................................... 51 3.2.3 Near Zone...................................... 55 3.2.4 Asymptotic Matching............................... 56 3.2.5 Expansion of the Nonspinning Part of the IZ and NZ Metrics......... 58 3.2.6 Matching Calculation............................... 59 3.2.7 Global Metric.................................... 63 3.3 Numerical Analysis..................................... 64 3.3.1 Ricci Scalar..................................... 65 3.3.2 Relative Kretschmann............................... 66 3.3.3 Accuracy of the Global Metric: the Ricci Scalar................. 68 3.3.4 Accuracy of the Global Metric: the Hamiltonian Constraint.......... 75 3.3.5 Aligned Spin.................................... 76 3.3.6 Accuracy of the Global Metric: the Relative Kretschmann........... 78 3.3.7 Numerical Methods................................ 84 3.3.8 Orbital Hang-up Effect, and Long Time Evolutions of the BHB........ 87 3.4 Discussion.......................................... 88 3.5 Final Remarks for Non-Precessing Matching Calculations............... 90 3.5.1 Transition functions................................ 90 3.5.2 Ingoing Kerr Coordinates to Cook-Scheel Harmonic Coordinates....... 92 3.5.3 Details about the Horizon and the Innermost Stable Circular Orbit...... 96 3.5.4 Computationally Effective IZ Metric....................... 98 3.6 Asymptotic Matching in the Precessing Case...................... 99 3.6.1 Coordinate rotation................................ 100 3.6.2 Matching calculation................................ 101 iii CONTENTS 3.6.3 Concluding Remarks................................ 104 4 Post-Newtonian Dynamics of Eccentric, Spinning Binaries 105 4.1 Introduction......................................... 105 4.2 Formulation of PN Equations of Motion......................... 107 4.2.1 Orbital Position Equations of Motion...................... 107 4.2.2 Spin Precession Equations............................. 111 4.2.3 Quasi-Circular Orbits............................... 113 4.2.4 Eccentricity Definitions.............................. 113 4.2.5 The Taylor T4 Method.............................. 116 4.3 Gravitational Radiation: T4 and Direct EOM...................... 117 4.4 Comparing to T4...................................... 119 4.4.1 Eccentricity Reduction in the Direct Integration................ 120 4.4.2 Consistent PN Order in Taylor T4........................ 122 4.4.3 Quantifying the Comparison: Orbital Frequencies...............