Numerical Simulations of Black-Hole Spacetimes Thesis by Tony Chu In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2012 (Defended July 8, 2011) ii c 2012 Tony Chu All Rights Reserved iii Acknowledgements This thesis would not have been possible without the wisdom and guidance of many people. First, I thank my thesis advisor Mark Scheel, for allowing me to take part in the ex- citement of numerical relativity, and for teaching me valuable lessons along the way. I am indebted to him for his kindness and encouragement throughout my graduate studies. His door was always open, and he was never too busy to entertain my many naive questions and whimsical ideas. Each time I left his office, physics made a lot more sense. I thank Harald Pfeiffer, who was my mentor when I first came to Caltech and was still clueless about numerical relativity. He had no hesitations in helping me to get started, and was infinitely patient in providing detailed explanations of important concepts, supple- mented by his sharp insight. His resourcefulness never ceases to amaze me, and I am truly fortunate to have crossed paths with him. I thank Rana Adhikari and my formal advisor Yanbei Chen for serving on my candicacy and thesis committees, Lee Lindblom for serving on my candidacy committee, and Christian Ott and B´elaSzil´agyifor serving on my thesis committee. The time and care they have given me are deeply appreciated. I also thank Kip Thorne, my advisor before his retirement, who nudged me in the direction of numerical relativity upon my arrival at Caltech, and conveyed to me the wonder of gravitation. I thank my fellow relativity students for brightening and enlightening my days: Kristen Boydstun, Jeffrey Kaplan, Keith D. Matthews, David Nichols, Huan Yang, Fan Zhang, and Aaron Zimmerman. I am especially grateful to my officemates David and Keith for frequent and helpful interactions, and to Keith for being my travel companion to physics conferences. I also thank the many others whose knowledge I have benefited from in the last several years: Michael Boyle, Jeandrew Brink, Alessandra Buonanno, Michael Cohen, Marc Favata, Chad Galley, Tanja Hinderer, Nathan Johnson-McDaniel, Geoffrey Lovelace, Haixing Miao, Samaya Nissanke, Robert Owen, Yi Pan, Christian Reisswig, Ulrich Sperhake, Sherry Suyu, Heywood Tam, Nicholas Taylor, Saul Teukolsky, Anil Zenginoglu, and anyone else I may have inadvertently left out. I thank Yanbei Chen, Harald Pfeiffer, Mark Scheel, and Kip Thorne for writing recom- mendation letters for my postdoctoral applications, and JoAnn Boyd for making sure that I did not miss any deadlines. I thank JoAnn Boyd and Shirley Hampton for sorting out all sorts of administrative stuff, so that I never had to worry about them. I also thank Chris Mach for setting up new computers for me, and for fixing them whenever they misbehaved. iv I thank Alan Weinstein for having enough faith to admit me into the physics graduate program. I thank the physics department for supporting me as a teaching assistant during my first year at Caltech, and the Brinson Foundation and Sherman Fairchild Foundation for supporting me as a research assistant in subsequent years. I also acknowledge sup- port from NSF grants PHY-0601459, PHY-0652995, and DMS-0553302 and NASA grant NNX09AF97G. I thank Kenneth Chang and Geoffrey Ho, who are my best friends, comrades, and partners in crime, for the fun-filled times we have had together. They are thoroughly absurd, in the best possible way. I cannot recount all of the numerous occasions on which they have compelled me to burst out in laughter. And of course, I am eternally thankful for my mother, for giving me a reason to always be able to smile, and for blessing me with more than I deserve. Somehow she has maintained an inexhaustible confidence in me and has supported every one of my endeavors, even when they were far-fetched and she clearly knew so. Among the things I have dreamed of doing, perhaps studying black holes was not so crazy after all. v Abstract This thesis covers various aspects of the numerical simulation of black-hole spacetimes ac- cording to Einstein's general theory of relativity, using the Spectral Einstein Code (SpEC) developed by the Caltech-Cornell-CITA collaboration. The first topic is improvement of binary-black-hole initial data. Chapter 2 is primarily concerned with the construction of binary-black-hole initial data with nearly extremal spins. It is shown that by superposing two Kerr metrics for the conformal metric, the spins remain nearly constant during the initial relaxation in an evolution, even for large spins. Other results include non-unique conformally-flat black hole solutions to the extended-conformal- thin-sandwich equations. Chapter 3 presents work in progress, and builds on the method of the previous chapter to add physically realistic tidal deformations in binary-black-hole initial data, by superposing two tidally deformed black holes to compute the freely specifiable data. The aim of this is to reduce the junk radiation content, and represents a first step in incorporating post-Newtonian results in constraint-satisfying initial data. The next topic is the evolution of black-hole binaries and the gravitational waves they emit. Chapter 4 presents the first spectral simulation of two inspiralling black holes through merger and ringdown. The black holes are nonspinning and have equal masses. Gravita- tional waveforms are computed from the simulation, and have the lowest numerical error to date for this configuration. Gauge errors in these waveforms are also estimated. Chapter 5 extends this work to perform the first spectral simulations of two inspiralling black holes with moderate spins and equal masses, including the merger and ringdown. Two configura- tions are considered, in which both spins are either anti-aligned or aligned with the orbital angular momentum. The high accuracy of the gravitational waveforms for the nonspinning case is found to carry over to these configurations. Chapter 6 uses the waveforms from these simulations to make the first attempt to calibrate waveforms in the effective-one-body model for spinning, non-precessing black-hole binaries. The final topic is the behavior of quasilocal black-hole horizons in highly dynamical situations. Chapter 7 discusses simulations of a rotating black hole that is distorted by a pulse of ingoing gravitational radiation. For large distortions, multiple marginally outer trapped surfaces appear at a single time, and the world tubes they trace out are all dynamical horizons. The dynamical horizon and angular momentum flux laws are evaluated in this context, and the dynamical horizons are contrasted with the event horizon. The formation of multiple marginally outer trapped surfaces in the Vaidya spacetime is also treated. vi Contents Acknowledgements iii Abstract v List of Figures xi List of Tables xxiii 1 Introduction 1 1.1 Binary-black-hole initial data . .3 1.2 Binary-black-hole evolutions . .5 1.3 Effective-one-body formalism . .7 1.4 Quasilocal black hole horizons . .9 Bibliography . 11 2 Binary-black-hole initial data with nearly extremal spins 14 2.1 Introduction . 15 2.2 Initial data formalism . 19 2.2.1 Extrinsic curvature decomposition . 19 2.2.2 Bowen-York initial data . 21 2.2.3 Quasi-equilibrium extended-conformal-thin-sandwich initial data . 24 2.3 Single-black-hole initial data with nearly extremal spins . 29 2.3.1 Bowen-York (puncture) initial data . 29 2.3.2 Quasi-equilibrium extended-conformal-thin-sandwich data . 37 2.4 Binary-black-hole initial data with nearly extremal spins . 42 2.4.1 Conformally flat, maximal slicing data . 43 2.4.2 Superposed-Kerr-Schild data . 43 2.4.3 Suitability for evolutions . 49 vii 2.5 Exploratory evolutions of superposed Kerr-Schild initial data . 51 2.5.1 Description of evolution code . 51 2.5.2 Eccentricity removal . 52 2.5.3 Low-eccentricity inspiral with χ ≈ 0:93 ................ 54 2.5.4 Head-on plunge with χ ≈ 0:97 ...................... 56 2.6 Discussion . 58 2.6.1 Maximal possible spin . 58 2.6.2 Additional results . 60 2.7 Appendix A: Approximate-Killing-vector spin . 62 2.7.1 Zero expansion, minimal shear . 62 2.7.2 Normalization . 65 2.8 Appendix B: Scalar-curvature spin . 67 Bibliography . 69 3 Including realistic tidal deformations in binary-black-hole initial data 74 3.1 Introduction . 74 3.2 Initial data formalism . 76 3.2.1 Extended-conformal-thin-sandwich equations . 76 3.2.2 Freely specifiable data . 78 3.2.3 Boundary conditions . 83 3.3 Initial data . 84 3.3.1 Parameters . 84 3.3.2 Constraints . 85 3.3.3 Tidal deformations . 86 3.4 Evolutions . 87 3.4.1 Constraints . 88 3.4.2 Junk radiation . 89 3.4.3 Eccentricity . 90 3.5 Future work . 90 Bibliography . 92 4 High-accuracy waveforms for binary black hole inspiral, merger, and ring- down 95 4.1 Introduction . 95 4.2 Solution of Einstein's Equations . 97 viii 4.2.1 Initial data . 97 4.2.2 Evolution of the inspiral phase . 97 4.2.3 Extending inspiral runs through merger . 99 4.2.4 Evolution from merger through ringdown . 103 4.2.5 Properties of the numerical solution . 106 4.3 Computation of the waveform . 109 4.3.1 Waveform extraction . 110 4.3.2 Convergence of extracted waveforms . 111 4.3.3 Extrapolation of waveforms to infinity . 113 4.4 Discussion . 119 4.5 Appendix: Estimating the gauge error of numerical waveforms . 122 4.5.1 Damped harmonic gauge evolutions . 122 4.5.2 Gauge and numerical errors .