Using perturbation theory to understand the two body problem in general relativity by Pranesh Adhyam Sundararajan M. Sc. (Hons) Physics, B. E. (Hons) Electrical & Electronics Engineering Birla Institute of Technology & Science, Pilani, India (2005) Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2009 @ Massachusetts Institute of Technology 2009. All rights reserved. A uthor ........................................ ...................... Department of Physics -e 30, 2009 Certified by ............................... .S... Hu h ScttScottA. Hughes Associate Professor of Physics Thesis Supervisor Accepted by............. ............ omas J. Greytak MASSACHUSETTS INSTIJTE rofessor of Physics OF TECHNOLOGY Associate Departmen Head for Education AUG 0 4 2009 ARCHIVES LIBRARIES Using perturbation theory to understand the two body problem in general relativity by Pranesh Adhyam Sundararajan Submitted to the Department of Physics on June 30, 2009, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Abstract Binary systems composed of compact objects (neutron stars and black holes) radi- ate gravitational waves (GWs). The prospect of detecting these GWs using ground and space based experiments has made it imperative to understand the dynamics of such compact binaries. This work describes several advances in our ability to model compact binaries and extract the rich science they encode. A major part of this dissertation focuses on the subset of binaries composed of a massive, central black hole (105 - 10SM®) and a much smaller compact object (1 - 100M®). The emission of gravitational energy from such extreme mass ratio inspirals (EMRIs) forces the separation between the two components to shrink, leading to their merger. We treat the smaller object as a point-like particle on the stationary space-time of the larger black hole. The EMRI problem can be broken down into two related parts: (i) A determination of the inspiral trajectory followed by the smaller object, and (ii) A characterization of the gravitationalwaveforms that result from such an inspiral. The initial part of this work discusses the development of a numerical algorithm that solves for the GWs that result from the perturbations generated by the smaller object. It accepts any reasonable inspiral trajectory as an input and produces the resulting waveforms with an accuracy greater than 99%. Next, we present a technique to model the part of the inspiral trajectory that immediately precedes the final plunge of smaller object into the massive black hole. Along with earlier research, this enables us to compute the smaller object's complete inspiral trajectory. We now have a versatile toolkit that can model GWs from EMRIs. Finally, we present another application of this work. GWs carry linear momen- tum away from a binary. Integrating the lost momentum leaves an asymmetric binary with a non-zero recoil velocity after merger. We compute the recoils from EMRIs and extrapolate them to comparable mass binaries. We find that extrapolating pertur- bation theory gives results that agree well with those from numerical relativity, but require far less computation time. Thesis Supervisor: Scott A. Hughes Title: Associate Professor of Physics Acknowledgments There are several people who have offered invaluable guidance, inspiration, and sup- port leading up to this dissertation. I would like to thank Prof. Arun Kulkarni, Prof. Suresh Ramaswamy, and the rest of the Physics department at the Birla Institute of Technolgy and Science (Pilani, India). I learnt most of my undergraduate physics from them. During the course of my undergraduate studies, I was fortunate enough to have the opportunity to spend summers at the Tata Institute of Fundamental Re- search (Mumbai, India) and the Raman Research Institute (Bangalore, India). My work with Prof. T. P. Singh at TIFR was my first exposure to research. I published my first refereed paper under Prof. Bala R. Iyer's guidance at RRI. I am indebted to my advisor Prof. Scott A. Hughes (MIT) and my advisor-at- large Prof. Gaurav Khanna (UMass., Dartmouth). They have been an illimitable resource to rely on, and have shaped my ability to think scientifically, do research, write papers, and give talks. I also thank the Physics department at MIT and the entire MIT community for fostering an environment that is conducive to education and research. I thank my parents, family, friends, and fellow graduate students who are too numerous to mention here. In particular, I thank all those who encouraged me to pursue Physics, despite the typical mindset of the Indian middle-class that research in a purely scientific subject rarely leads to an economically viable career. Contents 1 Introduction and overview 21 1.1 Gravitational waves, the two-body problem, and general relativity . 21 1.2 Gravitational waves, the two-body problem, and Astronomy .... 24 1.3 The simplest relativistic binary . .............. 27 1.4 Gravitational wave detection . .................. .. 30 1.5 Black hole perturbation theory . ................ .. .. 32 1.6 This dissertation . ................... ....... 35 1.6.1 Building gravitational waveforms . ............... 36 1.6.2 Generating inspiral trajectories . ................ 37 1.6.3 Recoil velocities from black hole mergers . ........... 38 2 Towards adiabatic waveforms for inspiral into Kerr black holes: I. A new model of the source for the time domain perturbation equation 41 2.1 Introduction ................ .. ... ... ....... 41 2.1.1 Background ................... .. .. 41 2.1.2 Our approach to adiabatic inspiral . ............. 43 2.1.3 This chapter ............ ..................... 45 2.1.4 Organization of this chapter . ............ ....... .. 49 2.2 Numerical implementation of the Teukolsky equation in the time domain 51 2.2.1 Homogeneous Teukolsky equation . ............. .. 52 2.2.2 The source term ........... ........ ....... 56 2.3 The discrete delta function and its derivatives . ........... 60 2.3.1 A simple numerical delta function .......... .. ... 62 2.3.2 A multiple point delta function . ................ 2.3.3 Higher order interpolation for smoothness ........... 2.3.4 Convergence with the discrete delta function .......... 2.4 Numerical Implementation and Evaluation of the Discrete Delta Function 2.4.1 Comparison of different discrete delta functions ........ 2.4.2 Convergence of our code . .................... 2.4.3 Comparison of discrete and Gaussian approximations for the numerical delta ................. .. .. 78 2.5 Accounting for finite extraction radius ........ 2.6 Summary and Future work . .............. 2.7 Acknowledgments . ................... 3 Towards adiabatic waveforms for inspiral into Kerr black holes: II. Dynamical sources and generic orbits 89 3.1 Introduction......................... ...... 89 3.1.1 Background . .................. .. 89 3.1.2 Time-domain black hole perturbation theory . ... ... 9 1 3.1.3 This chapter .. .................. .. 92 3.2 Dynamically varying discrete delta functions . .. .. .. .. ... 93 3.2.1 Higher order delta functions .. .. .. ... .. .. 95 3.2.2 Wider stencils at a given interpolation order . .. .. 98 3.2.3 Smoothing the source with a Gaussian filter .. .. .. .. 100 3.2.4 Order of convergence of the filtered delta . .. .. .. .. 101 3.3 Waveforms and comparisons for generic geodesic Kerr orbits 104 3.3.1 Geodesics in Kerr spacetime . ............. 104 3.3.2 Comparison with frequency-domain waveforms . .. 106 3.4 Inspiral waveforms . ...................... 109 3.5 Summary and future work ................... 112 3.6 Appendix: Waveform extrapolation . ............. 116 3.7 Acknowledgments . .......... ............. 118 4 The transition from adiabatic inspiral to geodesic plunge for a com- pact object around a massive Kerr black hole: Generic orbits 121 4.1 Introduction and motivation . ............... ...... 121 4.2 The transition trajectory for circular orbits ........ ...... 123 4.2.1 Kerr Geodesics .. ..... ............. ...... 124 4.2.2 The last stable orbit . ............... ....... 125 4.2.3 The constants in the transition regime .. .. ...... 126 4.2.4 Reparametrization of the 0-equation .. .. .. ...... 128 4.2.5 The prescription ................ .... ....... 128 4.2.6 Initial conditions . ................. ..... 130 4.2.7 Code algorithm and numerical results ....... ...... 131 4.2.8 Comparison with Ref. [74] . ............ ...... 133 4.3 Eccentric orbits . .. ...................... ..... 135 4.3.1 The last stable orbit . ...... ......... ....... 136 4.3.2 The constants during the transition ........ ..... 136 4.3.3 The prescription for eccentric orbits ........ ....... 137 4.3.4 Initial conditions . ................. ...... 138 4.3.5 Code implementation and numerical results . .. ...... 139 4.3.6 Comparison with Ref. [75] . ............ ..... 141 4.4 Summary and Future work . ................ ...... 144 4.5 Appendix A: The LSO for eccentric orbits . ....... ..... 145 4.6 Appendix B: Numerical integration across turning points ...... 146 4.7 Acknowledgments .......... ...... ....... 147 5 Recoil velocities from black hole mergers 149 5.1 Introduction and background . .............. ....... 149 5.1.1 Background ....... .. .... ....... ....... 150 5.1.2 This chapter ......... ... .. .... ....... 151 5.2 Summary of our approach to inspirals ........... ...... 153 5.2.1 Geodesics in a Kerr spacetime