Rochester Institute of Technology RIT Scholar Works Theses 8-2015 Puncture Initial Data and Evolution of Black Hole Binaries with High Speed and High Spin Ian Ruchlin Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Ruchlin, Ian, "Puncture Initial Data and Evolution of Black Hole Binaries with High Speed and High Spin" (2015). 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]. Puncture Initial Data and Evolution of Black Hole Binaries with High Speed and High Spin Ph.D. Doctor of Philosophy in Astrophysical Sciences and Technology Ian Ruchlin School of Physics and Astronomy Rochester Institute of Technology Rochester, New York August 2015 ASTROPHYSICAL SCIENCES AND TECHNOLOGY COLLEGE OF SCIENCE ROCHESTER INSTITUTE OF TECHNOLOGY ROCHESTER, NEW YORK Ph.D. DISSERTATION DEFENSE Candidate: . Dissertation Title: . Adviser: . Date of defense: . The candidate’s Ph.D. Dissertation has been reviewed by the undersigned. The Dissertation (a) is acceptable, as presented. (b) is acceptable, subject to minor amendments. (c) is not acceptable in its current form. Written details of required amendments or improvements have been provided to the candidate. Committee: Dr. Mihail Barbosu, Committee Chair Dr. Manuela Campanelli, Committee Member Dr. Yosef Zlochower, Committee Member Dr. Lawrence E. Kidder, Committee Member Dr. Carlos O. Lousto, Thesis Advisor Please submit form to AST Graduate Program Coordinator 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 Ian Ruchlin 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. Mihail Barbosu, Committee Chair Dr. Manuela Campanelli, Committee Member Dr. Yosef Zlochower, Committee Member Dr. Lawrence E. Kidder, Committee Member Dr. Carlos O. Lousto, Thesis Advisor Date PUNCTURE INITIAL DATA AND EVOLUTION OF BLACK HOLE BINARIES WITH HIGH SPEED AND HIGH SPIN By Ian Ruchlin 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. August, 2015 Approved by Dr. Andrew Robinson Date Director, Astrophysical Sciences and Technology Abstract This dissertation explores numerical models of the orbit, inspiral, and merger phases of black hole binaries. We focus on the astrophysically realistic case of black holes with nearly ex- tremal spins, and on high energy black hole collisions. To study the evolution of such systems, we form puncture initial data by solving the four general relativity constraint equations us- ing pseudospectral methods on a compactified collocation point domain. The solutions to these coupled, nonlinear, elliptic differential equations represent the desired configuration at an initial moment. They are then propagated forward through time using a set of hyper- bolic evolution equations with the moving punctures approach in the BSSNOK and CCZ4 formalisms. To generate realistic initial data with reduced spurious gravitational wave con- tent, the background ansatz is taken to be a conformal superposition of Schwarzschild or Kerr spatial metrics. We track the punctures during evolution, measure their horizon properties, extract the gravitational waveforms, and examine the merger remnant. These new initial data are compared with the well known Bowen-York solutions, producing up to an order of magnitude reduction in the initial unphysical gravitational radiation signature. We perform a collision from rest of two black holes with spins near to the extremal value, in a region of parameter space inaccessible to Bowen-York initial data. We simulate nonspinning black holes in quasi-circular orbits, and perform high energy head-on collisions of nonspinning black holes to estimate the magnitude of the radiated gravitational energy in the limit of infinite momen- tum. We also evolve spinning black holes in quasi-circular orbits with unequal masses and different spin orientations. These models provide insight into the dynamics and signals gener- ated by compact binary systems. This is crucial to our understanding of many astrophysical phenomena, especially to the interpretation of gravitational waves, which are expected to be detected directly for the first time within the next few years. i ii Contents Abstract i Contents iii List of Figures vii List of Tables xv 1 Introduction 1 1.1 Gravitation . .1 1.1.1 The Newtonian Theory of Gravitation . .2 1.1.2 The Theory of General Relativity . .5 1.2 Motivation . .7 1.2.1 Black Holes . .7 1.2.2 Observations of Black Hole Systems . .8 1.2.3 Gravitational Waves . 10 1.3 Projects Herein . 11 1.3.1 Collisions from Rest of Highly Spinning Black Hole Binaries . 11 1.3.2 Quasi-circular Orbits and High Speed Collisions Without Spin . 12 1.3.3 Highly Spinning Black Hole Binaries in Quasi-circular Orbits . 13 1.3.4 Organization . 14 iii 2 Numerical Techniques in Relativity 15 2.1 General Relativity Formalism . 15 2.1.1 Geometric Units . 16 2.1.2 Manifolds . 16 2.1.3 Tensors . 17 2.1.4 Differentiation . 23 2.1.5 Curvature . 29 2.1.6 The Einstein Field Equation . 31 2.1.7 Geodesics . 31 2.1.8 Causal Structure and Asymptotic Flatness . 32 2.1.9 Black Holes . 34 2.2 3 + 1 Formalism . 42 2.2.1 The 3 + 1 Decomposition . 43 2.2.2 Conformal Decomposition . 48 2.2.3 Transverse-traceless Decomposition . 49 2.2.4 Thin-sandwich Decomposition . 52 2.2.5 Mass, Momentum, and Angular Momentum . 53 2.2.6 Black Hole Initial Data . 54 2.2.7 Gauge Choices and Evolution Formalisms . 58 2.3 Pseudospectral Methods . 61 2.4 Cactus and the Einstein Toolkit ........................ 64 2.4.1 The TwoPunctures Thorn . 64 2.4.2 The HiSpID Thorn . 66 3 Collisions from Rest of Highly Spinning Black Hole Binaries 71 3.1 Stationary Kerr Initial Data . 74 3.1.1 Conformal Kerr in Quasi-isotropic Coordinates . 74 3.1.2 Punctures . 78 3.1.3 Attenuation and Superposition . 79 iv 3.2 Numerical Techniques . 80 3.2.1 Initial Data Solver with Spectral Methods . 80 3.2.2 Evolution . 81 3.3 Collisions from Rest with Nearly Extremal Spin . 82 3.4 Gauge Conditions . 86 3.5 Fisheye Transformations . 90 3.6 Summary . 93 4 Quasi-circular Orbits and High Speed Collisions Without Spin 95 4.1 Lorentz-boosted Schwarzschild Initial Data . 97 4.1.1 Conformal Boosted Schwarzschild in Isotropic Coordinates . 97 4.1.2 Punctures . 102 4.1.3 Attenuation and Superposition . 103 4.2 Numerical Techniques . 105 4.2.1 Initial Data Solver with Spectral Methods . 105 4.2.2 Evolution . 105 4.3 Quasi-circular Orbits . 107 4.4 Gauge Conditions . 112 4.5 Energy Radiated in High Energy Head-on Collisions . 115 4.6 Summary . 122 5 Highly Spinning Black Hole Binaries in Quasi-circular Orbits 125 5.1 Lorentz-boosted Kerr Initial Data . 128 5.1.1 Conformal Boosted Kerr in Isotropic Coordinates . 129 5.1.2 Punctures . 133 5.1.3 Attenuation and Superposition . 135 5.2 Numerical Techniques . 137 5.2.1 Initial Data Solver with Spectral Methods . 137 5.2.2 Unequal Masses . 138 v 5.2.3 Evolution and Gauge Choices . 140 5.3 Quasi-circular Orbits . 141 5.4 Summary . 145 6 Discussion 147 6.1 Summary of Results . 147 6.2 Future Research Directions . 149 6.2.1 Comparison with Excision Initial Data . 149 6.2.2 Conformal Thin-sandwich Initial Data . 150 6.2.3 Ultimate Kicks . 150 6.2.4 Trumpet Initial Data . 151 6.2.5 Multiple Black Hole Systems . 151 6.3 Acknowledgements . 151 Bibliography 153 vi List of Figures 2.1 A Penrose diagram representation of Minkowski spacetime in compactified spher- ical coordinates. The locations of conformal infinity are labeled. Time runs vertically, from the infinite past at i− and I −, to the infinite future at i+ and I +. Space extends horizontally from the origin (the vertical line labeled r = 0) out to spatial infinity at i0. A point on the diagram represents a two-sphere at that particular time and radius. Light rays move along 45◦ lines, parallel to I − and I +. Generic surfaces of constant time (dt = 0) and constant radius (dr = 0) are shown. 34 2.2 A Penrose diagram of Schwarzschild spacetime. The locations of conformal infinity are labeled. Time runs vertically, from the infinite past at i− and I −, to the infinite future at i+ and I +. Light rays move along 45◦ lines, parallel to I − and I +. A point on the diagram represents a two-sphere at that particular time and radius. Region I is the ordinary space outside of the black hole, extending from the event horizon at r = 2M to spatial infinity i0. Region II is the interior of the black hole, in which all timelike and lightlike curves end at r = 0, the singularity. Region III is another asymptotically flat region of space. Since no physical signal can move from I to III, or vice versa, they are causally disconnected, and represent “separate universes.”.