Visualizing, Approximating, and Understanding Black-Hole Binaries

424 Pages·2012·20.57 MB·English

Visualizing, Approximating, and Understanding Black-Hole
Binaries
Thesis by
David A. Nichols
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California
2012
(Defended April 30, 2012)
ii
c 2012
(cid:13)
David A. Nichols
All Rights Reserved
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To my parents and sister
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Acknowledgments
It is often said that it takes a village to raise a child, and it seems to take no fewer number of
peopletotakeanundergraduateandmakehimapostdoctoralresearcher. Forthemanypeoplewho
havehelped me asI workedtowardmy Ph.D., I wouldliketo acknowledgethem individually andin
(mostly) chronologicalorder from when they first contributed (knowingly or not) to my completing
this thesis.
I must, therefore, begin by thanking my parents. They brought me into this world and instilled
in me a value for education, integrity, and the pursuit of the truth; they supported my interests,
were encouraging,and always willing to listen; they even attempted to keep up on my latest aspect
of black-hole research—I do not think there is much else I could ask from them.
Iamalsoindebtedtomysister,Rebecca. Shehasbeenafriend,aninspiration,andarolemodel.
I do not think I would haveattempted as much, or workedas hard,had it notbeen for her example
leading the way since I was young.
I would like to acknowledge Kip Thorne. It was his teaching that drew me to relativity, and
his challenging homeworkassignments thatbroughtme into the theoreticalastrophysicsgroup,and
started me on research that would become the first chapters of this thesis. He introduced me to
Yanbei Chen, who would become my thesis adviser. He has an incredible understanding of general
relativity, and I have learned and benefitted much from my interactions with him.
I would like to express my gratitude to my adviser Yanbei Chen. He has been a wealth of ideas,
and was always happy to discuss a wide range of topics across many branches of physics with me.
He provided me with invaluable guidance on choosing research problems and presenting my results
in writing and in speech. He encouraged me to deepen my understanding of relativity and helped
me to develop as a scientist.
I also benefitted from working with many collaborators during this thesis: Jeff Kaplan and Kip
Thorne for Chapter 2; Yanbei Chen, Drew Keppel, and Kip Thorne for Chapter 3; Yanbei Chen,
MichaelCohen,DrewKeppel,GeoffreyLovelace,KeithMatthews,MarkScheel,andUlrichSperhake
forChapter 4;YanbeiChen forChapters5 and6; YanbeiChen, HuanYang,FanZhang,andAaron
Zimmerman for Chapter 7; and Jeandrew Brink, Yanbei Chen, Jeff Kaplan, Geoffrey Lovelace,
Keith Matthews, Robert Owen, Mark Scheel, Kip Thorne, Fan Zhang, and Aaron Zimmerman for
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Chapters 8–11.
MycollaboratorsandIhadmanyfruitfuldiscussionswithseveralpeoplewhilewritinganddoing
researchfor the papers that became the chapters of this thesis. They include Luc Blanchet, Yanbei
Chen, and Drew Keppel for Chapter 2; Jeff Kaplan and Geoffrey Lovelace for Chapter 3; Michael
Boyle, Jeandrew Brink, Lawrence Kidder, Robert Owen, Harald Pfeiffer, Saul Teukolsky, and Kip
ThorneforChapter4;LeeLindblom,GeoffreyLovelace,YasushiMino,UlrichSperhake,MarkScheel
and B´ela Szil´agyi, Kip Thorne, and Huan Yang for Chapter 5; Jeandrew Brink, Tanja Hinderer,
Lee Lindblom, Yasushi Mino, Mark Scheel, Bel´a Szil´agyi, Kip Thorne, Huan Yang, and Aaron
Zimmerman for Chapter 6; Emanuele Berti, Jeandrew Brink, and Zhongyang Zhang for Chapter 7;
Larry Kidder and Saul Teukolsky for Chapter 8; Yanbei Chen, Tanja Hinderer, Jeffrey D. Kaplan,
Geoffrey Lovelace, Charles Misner, Ezra Newman, Robert Owen, and Kip Thorne for Chapter 9;
John Belcher, Larry Kidder, Richard Price, and Saul Teukolsky for Chapter 10; and Larry Kidder
and Saul Teukolsky for Chapter 11.
Inworkingonthechaptersofthisthesis,Iwassupportedbyseveralpublicandprivatesourcesof
funding: the public grants were NSF Grants PHY-0601459, PHY-0653653,and PHY-1068881, and
NSF CAREERGrant PHY-0956189;the private grants came fromthe BrinsonFoundation and the
David and Barbara Groce Fund at the California Institute of Technology. My collaborators were
funded by these grants and several others, including: NSF Grants DMS-0553302, DMS-0553677,
PHY-0652929, PHY-0652952, PHY-0652995, PHY-0960291, PHY-0969111, PHY-1005426, PHY-
1005655, and PHY-1068881; NASA Grants NNX09AF96G and NNX09AF97G; and the Sherman
FairchildFoundation. Some calculationswere done usingthe SpECcode, andotherswere performed
on the Ranger cluster under NSF TeraGrid Grant PHY-090003.
I am particularly grateful to Barbara and David Groce, who started a graduate fellowship for
my adviser’s students (of which I was the first recipient). I enjoyed our interactions over these past
years, and I was humbled by the interest they took in my work and my progress in my Ph.D.
I am also lucky to have had many interesting and informative discussions with members of the
theoretical astrophysics group, which, although they did not directly contribute to the chapters
of this thesis, were just as important to the Ph.D. process. In particular, I would like to bring
to attention the graduate-student-lunch and theoretical-astrophysics-meeting attendees (who are
too many to mention). I would also like to mention my office companions over the years, Kristen
Boydstun, Tony Chu, Drew Keppel, Keith Matthews, and Elena Murchikova, who helped to make
working in the office as pleasant as possible.
Iamalsopleasedtoacknowledgethemembersofmythesiscommittee,MarkScheel,Christopher
Hirata,andAlanWeinstein,whowereallobligingandgood-naturedaboutservingonmycommittee.
I am also thankful for JoAnn Boydand Shirley Hampton, who made the administrative aspects
of being a graduate student a breeze, and Chris Mach, who kept the computers, on which so many
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of the calculations in this thesis were performed, running smoothly.
TherearecertainlyotherswhomIamforgettingtomention,andtothosepeopleIapologize;my
memory may be short, but their contributions to this thesis are not.
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Abstract
Numerical-relativity simulations of black-hole binaries and advancements in gravitational-wave de-
tectors now make it possible to learnmore about the collisions of compact astrophysicalbodies. To
be able to infer more about the dynamicalbehavior of these objects requires a fuller analysis of the
connectionbetweenthe dynamics of pairsof black holes andtheir emitted gravitationalwaves. The
chapters of this thesis describe three approaches to learn more about the relationship between the
dynamics ofblack-holebinaries andtheir gravitationalwaves: modeling momentum flowin binaries
with the Landau-Lifshitz formalism, approximating binary dynamics near the time of merger with
post-Newtonianandblack-hole-perturbationtheories,andvisualizingspacetimecurvaturewithtidal
tendexes and frame-drag vortexes.
In Chapters 2–4,my collaboratorsand I present a method to quantify the flow of momentum in
black-holebinariesusingtheLandau-Lifshitzformalism. Chapter2reviewsanintuitiveversionofthe
formalism in the first-post-Newtonian approximation that bears a strong resemblance to Maxwell’s
theoryofelectromagnetism. Chapter3appliesthisapproximationtorelatethesimultaneousbobbing
motionofrotatingblackholesinthesuperkickconfiguration—equalmassblackholeswiththeirspins
anti-alignedandinthe orbitalplane—tothe flowofmomentuminthespacetime,priortotheblack-
holes’ merger. Chapter 4 then uses the Landau-Lifshitz formalism to explain the dynamics of a
head-on merger of spinning black holes, whose spins are anti-aligned and transverse to the infalling
motion. Before they merge, the black holes move with a large, transverse, velocity, which we can
explain using the post-Newtonian approximation; as the holes merge and form a single black hole,
wecanusetheLandau-Lifshitzformalismwithoutanyapproximationstoconnecttheslowingofthe
final black hole to its absorbing momentum density during the merger.
InChapters5–7,wediscussusing analyticalapproximations,suchaspost-Newtonianandblack-
hole-perturbationtheories,togainfurtherunderstandingintohowgravitationalwavesaregenerated
by black-hole binaries. Chapter 5 presents a way of combining post-Newtonian and black-hole-
perturbationtheories—whichwecallthehybridmethod—forhead-onmergersofblackholes. Itwas
able to produce gravitational waveforms and gravitational recoils that agreed well with compara-
ble results from numerical-relativity simulations. Chapter 6 discusses a development of the hybrid
model to include a radiation-reaction force, which is better suited for studying inspiralling black-
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hole binaries. The gravitational waveform from the hybrid method for inspiralling mergers agreed
qualitativelywiththatfromnumerical-relativitysimulations;whenappliedtothesuperkickconfigu-
ration,itgaveasimplified picture ofthe formationofthe largeblack-holekick. Chapter 7 describes
an approximate method of calculating the frequencies of the ringdown gravitational waveforms of
rotating black holes (quasinormal modes). The method generalizes a geometric interpretation of
black-hole quasinormal modes and explains a degeneracy in the spectrum of these modes.
In Chapters 8–11, we describe a new way of visualizing spacetime curvature using tools called
tidaltendexesandframe-dragvortexes. Thisreliesuponatime-spacesplitofspacetime,whichallows
one to break the vacuum Riemann curvature tensor into electric and magnetic parts (symmetric,
trace-free tensors that have simple physical interpretations). The regions where the eigenvalues
of these tensors are large form the tendexes and vortexes of a spacetime, and the integral curves
of their eigenvectors are its tendex and vortex lines, for the electric and magnetic parts, respec-
tively. Chapter 8 provides an overview of these visualization tools and presents initial results from
numerical-relativity simulations. Chapter 9 uses topological properties of vortex and tendex lines
to classify properties of gravitational waves far from a source. Chapter 10 describes the formalism
in more detail, and discusses the vortexes and tendexes of multipolar spacetimes in linearized grav-
ity about flat space. The chapter helps to explain how near-zone vortexes and tendexes become
gravitationalwaves far from a weakly gravitating,time-varying source. Chapter 11 is a detailed in-
vestigationofthevortexesandtendexesofstationaryandperturbedblackholes. Itdevelopsinsight
intohowperturbationsof(stronglygravitating)blackholesextendfromnearthe horizontobecome
gravitationalwaves.
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Contents
Acknowledgments iv
Abstract vii
List of Tables xviii
List of Figures xix
1 Introduction 1
1.1 The Two-Body Problem in General Relativity and Its Importance for Gravitational-
Wave Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 A Brief Overview of the Two-Body Problem. . . . . . . . . . . . . . . . . . . 1
1.1.2 Gravitational-Wave Measurements from Compact Binary Sources . . . . . . . 6
1.2 Methods to Understand the Dynamics of Compact Binaries in This Thesis . . . . . . 7
1.2.1 Understanding Momentum Flow in Black-Hole Binaries . . . . . . . . . . . . 7
1.2.1.1 Black-Hole Kicks as the Motivating Factor for Studying Momentum
Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1.2 The Challenge of Studying Momentum Flow . . . . . . . . . . . . . 8
1.2.1.3 Momentum Flow in the Post-Newtonian Approximation . . . . . . . 10
1.2.1.4 Momentum Flow in Numerical-Relativity Simulations . . . . . . . . 11
1.2.2 Approximating the Inspiral, Merger, and Ringdown in Black-Hole Binaries . 12
1.2.2.1 Approximation Methods in General Relativity . . . . . . . . . . . . 13
1.2.2.2 The Hybrid Method for Head-on Black-Hole Mergers . . . . . . . . 16
1.2.2.3 The Hybrid Method for Inspiralling Black-Hole Mergers . . . . . . . 17
1.2.2.4 A Geometric Approximation for the Ringdown of Black Holes . . . 19
1.2.3 Visualizing Spacetime Curvature with Vortex and Tendex Lines. . . . . . . . 20
1.2.3.1 The Challenges of Spacetime Visualization and the Method of This
Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.3.2 Overview of Spacetimes Visualized with this Method . . . . . . . . 21
1.3 Summarizing the Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 25
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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
I Understanding Momentum Flow in Black-Hole Binaries 31
2 Post-Newtonian Approximation in Maxwell-Like Form 32
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 The DSX Maxwell-Like Formulation of 1PN Theory . . . . . . . . . . . . . . . . . . 34
2.3 Specialization to a Perfect Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Momentum Density, Flux, and Conservation. . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 Gravitational Potentials in the Vacuum of a System of Compact, Spinning Bodies. . 40
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.A Appendix: Derivation of the 1PN Gravitational Potentials . . . . . . . . . . . . . . . 41
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Momentum Flow in Black Hole Binaries: I. Post-Newtonian Analysis of the
Inspiral and Spin-Induced Bobbing 44
3.1 Introduction: Motivation and Overview . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.2 Momentum Flow in Black-Hole Binaries . . . . . . . . . . . . . . . . . . . . . 45
3.1.3 Gauge-Dependence of Momentum Flow and the Landau-Lifshitz Formalism . 47
3.1.4 Overview of This Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Bobbing and Momentum Flow in the Extreme-Kick Configuration . . . . . . . . . . 49
3.2.1 Field Momentum in the Extreme-Kick Configuration . . . . . . . . . . . . . . 51
3.2.2 The Holes’ Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.3 Momentum Conservation for the Extreme-Kick Configuration . . . . . . . . . 57
3.3 The Landau-Lifshitz Formalism in Brief . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4 Four-Momentum Conservation for Fully Nonlinear Compact Binaries . . . . . . . . . 60
3.5 Post-Newtonian Momentum Flow in Generic Compact Binaries . . . . . . . . . . . . 62
3.5.1 Field Momentum Outside the Bodies . . . . . . . . . . . . . . . . . . . . . . . 62
3.5.2 Centers of Mass and Equation of Motion for the Binary’s Compact Bodies . . 63
3.5.3 The Momenta of the Binary’s Bodies . . . . . . . . . . . . . . . . . . . . . . . 65
3.5.4 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.A Appendix: The Total PN Momentum Density . . . . . . . . . . . . . . . . . . . . . . 67
3.B Appendix: Momentum of a Black Hole Computed via a Surface Integral of Superpo-
tential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68