ANALYZING AND IMPROVING INITIAL DATA FOR BINARY BLACK HOLES BY JASON D. GRIGSBY A Dissertation Submitted to the Graduate Faculty of WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Physics August 2009 Winston-Salem, North Carolina Approved By: Greg Cook, Ph.D., Advisor Examining Committee: Jennifer Erway, Ph.D., Chairman Paul Anderson, Ph.D. Jed Macosko, Ph.D. Eric Carlson, Ph.D. This work is dedicated to my parents, Linda and Robert, and to my wife, Melanie, for all the support they’ve given me. ii Acknowledgments My efforts for the last six years alone would have landed me no-where if it were not for the efforts and assistance of so many — my family, friends, fellow graduates students, the faculty at the department of physics and, in particular, my advisor, Gregory B. Cook. I am also indebted to the other members of my research advisory committee, Paul Anderson, Jed Macosko, Eric Calrson and Jennifer Erway. I should also mention Matthew Caudill and Harald P. Pfeiffer with whom I’ve worked. Harald P. Pfeiffer has been particularly helpful in my recent work. I would also like to thank Fred Salsbury for taking notice of me in the first place. I am grateful for the support of Wake Forest University’s Department of Physics. The spectral solver Eric was developed by Harald P. Pfeiffer and Gregory B. Cook. SpEC was develoepd by Mark A. Scheel, Harald P. Pfeiffer and Lawrence E. Kidder. The vast majority of the numerical data created for this work was created on the Wake Forest University Distributed Environment for Academic Computing cluster. iii Contents 1 Introduction 1 2 3+1 (ADM) Decomposition 6 3 The Extended Conformal Thin-Sandwich Decomposition with Excision 18 4 The physics of binary black holes 33 4.1 Physical measurements . 34 4.2 Effective Potentials . 37 4.3 The Komar-mass ansatz . 39 4.4 Discussion . 40 5 Computational Tools 41 5.1 Parameters and freely specifiable data of the ECTS equations with excision 41 5.2 The core solver of Eric . 44 5.3 Domain choice . 46 5.4 Applying EP and Komar methods in Eric . 47 6 Testing methods for finding quasicircular orbits and testing notions of spin 49 iv 6.1 Comparing effective potentials to Komar-mass ansatz data . 50 6.1.1 Comparison between methods in the corotating case . 51 6.1.2 Comparison between methods in the nonspinning case . 54 6.2 Testing against the thermodynamic identity . 60 6.3 Discussion . 63 7 Measuring eccentricity in binary black-hole initial data 67 7.1 The 3PN equations . 67 7.2 Can numerical effective potentials be used to find eccentric orbits? . 72 7.3 Measuring eccentricity . 74 7.3.1 Measuring eccentricity on quasicircular orbit predictions . 75 7.3.2 Measuring eccentricities along effective potentials . 78 7.4 Parameterizing eccentric orbits with the Komar-mass difference . 85 7.5 Discussion . 87 8 Conformal Flatness 96 8.1 Deriving equations and boundary conditions for the conformal metric . 99 8.2 Implementation . 106 8.3 Tests and Results . 108 8.4 Discussion . 113 9 Conclusion 114 A Komar sequences for corotating and nonspinning binaries 117 v B Post-Newtonian Equations for Energy and Angular Momentum 121 vi List of Figures 6.1 effective-potential (EP) curves Eb/µ for corotating black holes vs. separation `/m. These curves are labeled by the orbital angular momentum J/µm which is kept constant along each curve. The thick red line connecting the minima of the EP-curves represents circular orbits; it terminates at the the inflection point in the EP-curve at J/µm = 3.38. Also plotted as a dashed blue line is the sequence of circular orbits determined by the Komar-mass ansatz. 52 6.2 Violation of the Komar-mass condition when the effective-potential method is used to determine the sequence of circular orbits. Here, corotating equal- mass binaries are considered. mΩ0 denotes the orbital angular frequency, so that large separations correspond to small values of mΩ0........... 54 6.3 EP curves Eb/µ for equal-mass “leading-order” non-spinning binaries (Ωr = Ω0) plotted vs. separation `/m. These curves are labeled by the value of J/µm along each curve. Also plotted are the line connecting the minima of the EP-curves, as well as the sequence of circular orbits as determined by the Komar-mass ansatz. The Komar-mass ansatz and the effective-potential method clearly disagree. 55 vii 6.4 Quasi-local spin of the “leading order” non-spinning (LN) binaries (as shown in Fig. 6.3) plotted vs. orbital angular velocity. The inset displays the 2 dimensionless spin value S/Mirr. The main figure displays the magnitude of the spin relative to the spin of a Kerr black hole with ΩB = Ω0. The residual spin is quite large. 56 6.5 EP curves Eb/µ for “true” non-spinning black holes vs. separation `/m. The curves are labeled by the orbital angular momentum J/µm which is kept constant along each curve. The thick red line connecting the minima of the EP-curves represents circular orbits, and terminates at the innermost stable circular orbit at the inflection point in the EP-curve at J/µm = 3.12. Also plotted as a dashed blue line is the sequence of circular orbits as determined by the Komar-mass ansatz. 57 6.6 Violation of the Komar-mass condition when the effective-potential method is used to determine the sequence of circular orbits. Here, non-spinning (NS) equal-mass binaries are considered, and the sequence is parametrized by the orbital angular momentum. 58 6.7 Quasi-local spin for a “true” non-spinning (NS) black hole in an equal-mass binary along the sequence of circular orbits, parametrized by the orbital angular velocity. The same six cases are plotted as in the inset of Fig. (6.4). Note that here SK = 0 by construction. 59 6.8 Rotation parameter fr = Ωr/Ω0 along the sequence of non-spinning (NS) black holes, parametrized by the 2/3-power of the orbital angular velocity. 2/3 The dotted line plots the fit to fr through the term of order (mΩ0) .... 60 viii 7.1 Parameter space of J(, ζ) (Eq. (7.2)) for non-spinning equal-mass black holes. Short-dashed (green or black) lines of constant are constructed using the 3PN equations. The solid (red) line represents all 3PN models with a specific value of mΩ = 0.0614 corresponding to a particular numerical model. The horizontal dotted (black) line represents the corresponding orbital an- gular momentum of that numerical model. 70 7.2 Parameter space of E(, ζ) (Eq. (7.1)) for non-spinning equal-mass black holes. Lines as in Fig. 7.1 except the horizontal dotted(black) line represents the corresponding binding energy of the numerical model. 71 7.3 EP curves for non-spinning equal-mass black holes from both numerical and 3PN data. Numerical EP curves are plotted as short-dashed (blue, green, and red) lines. 3PN EP curves are plotted as solid (black) lines. The Komar sequence through the numerical data is plotted as a dash-dot (purple) line. The EP sequence through the numerical data is plotted as a long-dash (black) line. The EP sequence through the 3PN data is plotted as a light-solid (orange) line. A boundary of the allowable region for the 3PN equations is shown as a light-solid (brown) line. 73 7.4 The 3PN eccentricity measures Eb and J applied to both Komar and EP sequences of non-spinning equal-mass black holes. Note that the EP minima yield quasi-circular data with a smaller eccentricity than is obtained from the Komar-mass ansatz. 76 ix 7.5 The 3PN eccentricity measures Eb and J applied to both Komar and EP sequences of corotating equal-mass black holes. The dot-dashed (blue) lines correspond to quasi-circular data defined via the Komar-mass ansatz and where the Newtonian notion of corotation is used in the 3PN equations. The dashed (red) lines show the improvement obtained by including the 1PN correction to the notion of corotation. The solid (black) lines show the added improvement of using quasi-circular data based on the EP method. 77 7.6 Eccentricity of non-spinning numerical effective potentials using d plotted against (mΩ)(2/3). Dashed (red) lines are used to denote the smallest values of angular momentum, solid (green) lines represent medium values of angular momentum and dotted-dashed (blue) lines represent the largest values of angular momentum. 79 7.7 Eccentricity of non-spinning numerical effective potentials using Ω plotted against (mΩ)(2/3). Lines as in Fig. 7.6. 80 7.8 The relative difference between the Ω and d definitions of eccentricity for the non-spinning equal-mass black-hole numerical EP curves. Lines as in Fig. 7.6. 81 7.9 Eccentricity of non-spinning equal-mass black holes computed along numer- ical EP curves. The 3PN Eb definition of eccentricity is plotted against the orbital angular velocity. Lines are as in Fig. 7.7. The × symbols mark the minima of each EP curve. 82 7.10 Eccentricity of non-spinning equal-mass black holes computed along numer- ical EP curves. The 3PN J definition of eccentricity is plotted against the orbital angular velocity. Lines are as in Fig. 7.7. The × symbols mark the minima of each EP curve. 83 x 7.11 Difference between Ω and Eb plotted against Ω for non-spinning equal-mass black holes computed along numerical EP curves.