Topics in Gravitational Physics: Tidal Coupling in Gravitational Wave Searches and Mach’s Principle Thesis by Hua Fang In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2007 (Defended January 9, 2007) ii © 2007 Hua Fang All Rights Reserved iii Acknowledgements Foremost, I would like to thank my advisor, Kip Thorne, for being a great mentor. He introduced me to the science of relativity and gravitational waves, and to a group of enthusiastic, talented Kiplings with whom I have a lot of valuable interactions. I thank Kip for suggesting interesting research topics to me, for inspiring discussions and constant encouragement, and for patiently demonstrating and urging me to a clear, well- organized presentation of ideas. I am also very thankful to my former advisor, Thomas Prince, for supporting my study and research for two years when I first came to Caltech, and for introducing me to the exciting field of gravitational-wave physics. I am thankful to Massimo Tinto, for introducing me to LIGO-related data analysis and for interesting discussions on numerical techniques such as Markov chain Monte Carlo methods. I am thankful to Jonathan Gair, who assisted me in coding up the Press formula for numerical kludge (NK) waveforms. He was also kind enough to educate me a lot on geodesic integrators and on practical concerns in template counting, waveform comparison, etc. In addition, I must thank my other collaborators on our NK paper, Stanislav Babak, Kostas Glampedakis, and Scott Hughes, for helpful discussions. I wish to thank Geoffrey Lovelace, with whom I studied the black hole perturbation theory and applied it to the tidal coupling problem we jointly worked on. The collaboration was based on our genuine quest to seek the truth, so there was never a lack of critical debates and earnest discussions. I thank Geoffrey for this very rewarding experience and for the finishing of our tidal coupling paper. I am thankful to everyone who participated in the LIGO-IMRI project: Duncan Brown, Jonathan Gair, Chao Li, Geoffrey Lovelace, Ilya Mandel, Kip Thorne, Jeandrew Brink, Yanbei Chen, Teviet Creighton, Curt Cutler, Steve Drasco, Yi Pan, Cole Miller, and Sterl Phinney, for helpful inputs and comments, and for the final submission of the IMRI paper. In addition to the above, I would like to thank Yasushi Mino, Shane Larson, Frans Pretorius, Rob Owen, Luisa Buchman, Harald Pfeiffer, Mark Scheel, Lee Lindblom, and all others who attend Kip and Lee’s group meetings, where a lot of interesing presentations and discussions are made. I am very grateful to Chris Mach, who has helped me solve many computer problems, and to JoAnn Boyd, Shirley Hampton, and Donna Driscoll for helping me with administrative matters. There are people outside my research group who have influenced me during my time as a graduate student. I thank Pengpeng Sui, Delores Bing, Qun Cao, Jian Wu, Zhimei Yan, Xin Liu, and Loredana Prisinzano, for iv their enlightenment and for sharing great moments. Last, I am deeply grateful to my family, who have been supportive all through the years. I owe them for their love and kindness. The research presented in this thesis was supported in part by NASA grants NAG5-12834, NAG5-10707, and NNG04GK98G, and by NSF grants PHY-0099568 and PHY-0601459. v Abstract The gravitational waves emitted by a compact object inspirling into a massive central body (e.g., a massive black hole) contain exquisite information about the spacetime geometry around that body and the tidal in- teraction (energy and angular momentum transfer) between the body and the inspiraling object’s orbit. The first part (chapters 2–4) of this thesis presents (i) an analysis of tidal coupling between a massive, nonrotat- ing black hole with circularly orbiting moon; (ii) an estimate of tidal-coupling measurement accuracy—in the context of using ground-based interferometers (e.g., Advanced LIGO) to detect gravitational waves from intermediate-mass-ratio inspirals (IMRIs); and (iii) an efficient and effective algorithm to generate “kludge” gravitational waveforms, which could be useful in real graviational-wave data analysis for the space-based LISA (Laser Interferometer Space Antenna) mission. More specifically: • In chapter 2 (research in collaboration with Geoffrey Lovelace), we use first-order perturbation theory to study a simplified model: a Schwarzschild black hole of mass M is tidally perturbed by a “moon” moving along a circular orbit with angular velocity Ω; the moon’s perturbing tidal field Ei j gives rise to an induced quadrupole moment I jk in the hole’s external gravitational field at large radii. Much to our surprise, we find that the induced quadrupole moment is proportional to the time derivative of the 6 moon’s tidal field, Ii j = (32/45)M E˙ i j, instead of being proportional to the field itself (as one might expect from a Newtonian analogy)! This time-varying induced quadrupole moment is gauge-invariant, and its gravitational force acting back on the moon is responsible for the orbital energy and angular momentum loss to the hole’s horizon. We discuss the implication of this result in the static limit and conclude that the static induced quadrupole moment for a black hole is inherently ambiguous. We also show that the orbiting moon’s tidal field induces a tidal bulge on the hole’s horizon, and that the horizon shear (i.e., the rate of change of the horizon shape) leads the perturbing tidal field at the horizon by an angle 4MΩ. • In chapter 3 (research largely by the author alone), we give a survey of initial explorations of the prospects for using Advanced LIGO to detect gravitational waves from intermediate-mass-ratio inspi- rals (IMRIs)—analogous to the extreme-mass-ratio inspirals (EMRIs) targeted by LISA. We describe initial estimates of the detection range and the number of IMRI wave cycles in the Advanced LIGO band. Motivated by the possibility of using IMRI waves to probe the properties of its central body, e.g, to constrain its deviation from a Kerr black hole, we give a detailed analysis of Advanced LIGO’s vi accuracy for measuring the tide-induced energy transfer E˙H between the central black hole and the orbit. Our study shows that at signal-to-noise ratio 10, the accuracy of measuring tidal coupling E˙H is roughly a few percent of the gravitational energy flow to infinity. This suggests that Advanced LIGO, working together with a network of detectors, will be able to put a modest but interesting constraint on a source’s deviation from a Kerr black hole in terms of its tidal response. • In chapter 4 (research in collaboration with Stanislav Babak, Jonathan Gair, Kostas Glampedakis, and Scott Hughes), we describe a new waveform-generating scheme in the context of LISA’s data analysis for EMRI waves. Our scheme involves combining flat-spacetime wave-emission formulas (i.e., quadrupole, quadrupole-octupole, and the Press formula) with a full relativistic treatment of the orbiting object’s motion. The result is a family of “Numerical Kludge” (NK) waveforms. We compare our NK waveforms with the corresponding canonical, but more computational-intensive Teukolsky- based waveforms, and find remarkable agreement between the two. The second part (chapter 5) of this thesis (research in collaboration with Kip Thorne) discusses another pre- diction from general relativity, the dragging of inertial frames, in connection with Mach’s principle. Specifi- cally: • We first explain frame dragging outside the rotating Earth by two physical descriptions, gravitomag- netism and a fluid like space-drag, and demonstrate the equivalence between the two. Then we study frame dragging inside a rotating universe, i.e., Mach’s principle, by a simple model. We idealize our universe as a homogeneous, isotropic expanding, and slowly rotating sphere, surrounded by vacuum. We find that as the universe expands, the frame dragging weakens at its center; and that at later times inertia at the center completely breaks free of the grip of the universe’s rotating matter. vii Contents Acknowledgements iii Abstract v 1 Introduction 1 1.1 Tidal interaction between compact objects and massive black holes . 3 1.1.1 Tidal coupling between Earth and Moon . 3 1.1.2 Black hole tidal coupling . 4 1.2 IMRI waves for Advanced LIGO . 6 1.3 Template waveforms for EMRI waves . 8 1.4 General relativity and Mach’s Principle . 9 Bibliography . 10 2 Tidal Coupling of a Schwarzschild Black Hole and Circularly Orbiting Moon 14 2.1 Introduction and summary . 15 2.1.1 Motivations . 15 2.1.2 Framework and results . 16 2.2 Problem setup . 19 2.3 Time-dependent part of the perturbation . 20 2.3.1 The perturbed metric . 20 2.3.2 Induced quadrupole moment in the LARF . 22 2.4 The static, axisymmetric part of the perturbation . 23 2.4.1 Static induced quadrupole moment . 23 2.4.2 Ambiguity of the static induced quadrupole moment . 25 2.5 The tidal phase shift . 28 2.5.1 Phase of the tidal bulge on the horizon . 29 2.5.2 Phase shift between the tidal bulge and the moon . 31 2.5.2.1 Tidal phase shift between a rotating horizon and stationary moon . 32 2.5.2.2 Tidal phase shift between a non-rotating horizon and rotating moon . 34 viii 2.6 Concluding discussion . 35 Appendices . 36 2.A Symmetric trace-free tensor notation for spherical harmonics . 36 2.B Time-dependent perturbation equations . 36 2.C Time-independent perturbation equations . 38 2.D Newman-Penrose formalism . 39 2.D.1 Newman-Penrose quantities for Schwarzschild spacetimes . 39 2.D.2 Newman-Penrose quantities for Kerr spacetimes .