DOCSLIB.ORG

PDF (Thesis.Pdf)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
PDF (Thesis.Pdf) 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 .
Load more

Recommended publications
  • The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 53, Number 4, October 2016, Pages 513–554 http://dx.doi.org/10.1090/bull/1544 Article electronically published on August 2, 2016 THE EMERGENCE OF GRAVITATIONAL WAVE SCIENCE: 100 YEARS OF DEVELOPMENT OF MATHEMATICAL THEORY, DETECTORS, NUMERICAL ALGORITHMS, AND DATA ANALYSIS TOOLS MICHAEL HOLST, OLIVIER SARBACH, MANUEL TIGLIO, AND MICHELE VALLISNERI In memory of Sergio Dain Abstract. On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein’s prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detec- tion of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole bina- ries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science.
    [Show full text]
  • Generalizations of the Kerr-Newman Solution
    Generalizations of the Kerr-Newman solution Contents 1 Topics 1035 1.1 ICRANetParticipants. 1035 1.2 Ongoingcollaborations. 1035 1.3 Students ............................... 1035 2 Brief description 1037 3 Introduction 1039 4 Thegeneralstaticvacuumsolution 1041 4.1 Line element and field equations . 1041 4.2 Staticsolution ............................ 1043 5 Stationary generalization 1045 5.1 Ernst representation . 1045 5.2 Representation as a nonlinear sigma model . 1046 5.3 Representation as a generalized harmonic map . 1048 5.4 Dimensional extension . 1052 5.5 Thegeneralsolution . 1055 6 Tidal indicators in the spacetime of a rotating deformed mass 1059 6.1 Introduction ............................. 1059 6.2 The gravitational field of a rotating deformed mass . 1060 6.2.1 Limitingcases. 1062 6.3 Circularorbitsonthesymmetryplane . 1064 6.4 Tidalindicators ........................... 1065 6.4.1 Super-energy density and super-Poynting vector . 1067 6.4.2 Discussion.......................... 1068 6.4.3 Limit of slow rotation and small deformation . 1069 6.5 Multipole moments, tidal Love numbers and Post-Newtonian theory................................. 1076 6.6 Concludingremarks . 1077 7 Neutrino oscillations in the field of a rotating deformed mass 1081 7.1 Introduction ............................. 1081 7.2 Stationary axisymmetric spacetimes and neutrino oscillation . 1082 7.2.1 Geodesics .......................... 1083 1033 Contents 7.2.2 Neutrinooscillations . 1084 7.3 Neutrino oscillations in the Hartle-Thorne metric . 1085 7.4 Concludingremarks . 1088 8 Gravitational field of compact objects in general relativity 1091 8.1 Introduction ............................. 1091 8.2 The Hartle-Thorne metrics . 1094 8.2.1 The interior solution . 1094 8.2.2 The Exterior Solution . 1096 8.3 The Fock’s approach . 1097 8.3.1 The interior solution .
    [Show full text]
  • Arxiv:Gr-Qc/0112028V3 28 May 2002
    Sergiu Vacaru and Panayiotis Stavrinos SPINORS and SPACE–TIME ANISOTROPY arXiv:gr-qc/0112028v3 28 May 2002 University of Athens, 2002 ————————————————— c Sergiu Vacaru and Panyiotis Stavrinos ii ... iii ABOUT THE BOOK This is the first monograph on the geometry of anisotropic spinor spaces and its applications in modern physics. The main subjects are the theory of grav- ity and matter fields in spaces provided with off–diagonal metrics and asso- ciated anholonomic frames and nonlinear connection structures, the algebra and geometry of distinguished anisotropic Clifford and spinor spaces, their extension to spaces of higher order anisotropy and the geometry of gravity and gauge theories with anisotropic spinor variables. The book summarizes the authors’ results and can be also considered as a pedagogical survey on the mentioned subjects. iv - v ABOUT THE AUTHORS Sergiu Ion Vacaru was born in 1958 in the Republic of Moldova. He was educated at the Universities of the former URSS (in Tomsk, Moscow, Dubna and Kiev) and reveived his PhD in theoretical physics in 1994 at ”Al. I. Cuza” University, Ia¸si, Romania. He was employed as principal senior researcher, as- sociate and full professor and obtained a number of NATO/UNESCO grants and fellowships at various academic institutions in R. Moldova, Romania, Germany, United Kingdom, Italy, Portugal and USA. He has published in English two scientific monographs, a university text–book and more than hundred scientific works (in English, Russian and Romanian) on (super) gravity and string theories, extra–dimension and brane gravity, black hole physics and cosmolgy, exact solutions of Einstein equations, spinors and twistors, anistoropic stochastic and kinetic processes and thermodynamics in curved spaces, generalized Finsler (super) geometry and gauge gravity, quantum field and geometric methods in condensed matter physics.
    [Show full text]
  • A Festschrift for Thomas Erber
    Physics:Doing A Festschrift For Thomas Erber Edited by Porter Wear Johnson GC(2251)_PhysicsBookCvr10.indd 1 9/15/10 10:33 AM Doing Physics: A Festschrift for Tom Erber Edited by Porter Wear Johnson Illinois Institute of Technology IIT Press Doing Physics: A Festschrift For Thomas Erber edited by Porter Wear Johnson Published by: IIT Press 3300 S. Federal St., 301MB Chicago, IL 60616 Copyright °c 2010 IIT Press All rights reserved. No part of this book, including interior design, cover design, and icons, may be reproduced or transmitted in any form, by any means (electronic, photo- copying, recording, or otherwise) without the prior written permission of the publisher. ISBN: 1-61597-000-2 Series Editor: Technical Editor: Sudhakar Nair Julia Chase Editorial Board David Arditi Roya Ayman Krishna Erramilli Porter Wear Johnson Harry Francis Mallgrave Mickie Piatt Katherine Riley Keiicho Sato Vincent Turitto Geo®rey Williamson Doing Physics: A Festschrift for Tom Erber Preface \Au¼ertenÄ wir oben, da¼ die Geschichte des Menschen den Menschen darstelle, so lÄa¼tsich hier auch wohl behaupten, da¼ die Geschichte der Wissenschaft die Wissenschaft selbst sei." Goethe, Zur Farbenlehre: Vorwort (1808) It is widely asserted that the great physicists who grasped the full unity of physics are all dead, having been replaced in this age of specialization by sci- entists who have a deep understanding only for issues of rather limited scope. Indeed, it is di±cult to refute such a viewpoint today, at the end of the ¯rst decade of the twenty-¯rst century. The unifying principles of the quantum the- ory and relativity are part of the ethos of physics, but fragmented development in various disjointed areas has characterized the past several decades of progress.
    [Show full text]
  • Axisymmetric Spacetimes in Relativity
    9, tr' Axisymmetric Spacetimes in Relativity S. P. Drake Department of Physics and Mathematical Physics îåäi,î31 tÏfåi" Australia 22 July 1998 ACKNO\MTEDGEMENTS It is an unfortunate fact that only my name Soes on this thesis' This is not the work of one person, it is an accumulation of the labor of many' AII those who have helped me in ways that none of us could image should be acknowledged, but how. None of this would have been possible were it not for the wisdom and guidance of my supervisor Peter Szekeres. I have often pondered over the difficulty of supervising students. It must be heart-bleaking to watch as students make necessary mistakes. Patience I'm sure must be the key' So for his wisdom, patience and kindness I am deeply indebted to Peter Szekeres. Without the love and support of my family this thesis would never have begun, much less been completed, to them I owe my life, and all that comes with it. It would take too long to write the names of all those I wish to thank. Those who are special to me, have helped me through so much over the years will receive the thanks in Person. I would like to the department of physics here at Adelaide where most of my thesis work was done. I would like to thank the department of physics at Melbourne university, where I did my undergraduate degree and began my PhD. I would like to thank the university of Padova for there hospitality dur- ing my stay there.
    [Show full text]
  • GR15 Workshop A3---Mathematical Studies of Field Equations---A Report
    GR15 Workshop A3 Mathematical Studies of Field Equations A report Piotr T. Chru´sciel D´epartement de Math´ematiques, Facult´edes Sciences, Parc de Grandmont, F–37200 Tours, France September 25, 2018 1 Introduction In this report I have tried to describe the main streams of research presented at the workshop. Occasionally I have described some contributions which were presented in poster form, whenever they were directly related to the subjects presented in the talks. This paper is, essentially, a patchwork of various people description of their work, which I have edited in an attempt to obtain some uniformity in style, and to produce a report of finite length. Much to my regret, I had to cut or shorten a lot of material because of length constraints. I wish to thank all those who have sent me a contribution. arXiv:gr-qc/9803008v1 2 Mar 1998 2 The Riemannian Penrose Inequality In 1972, Penrose [89] gave a heuristic argument to establish the geometric in- equality mADM (M) ≥ |N|/16π (2.1) q for a time-reversible initial data set (M,g), where |N| is the area of a marginally trapped surface bounding M. We refer to (2.1) as the Riemannian Penrose inequality, because throughout this section it is assumed that the initial data have vanishing extrinsic curvature, Kij = 0. Penrose’s argument relies on knowledge of the future evolution of M, including the Hawking area monotonicity of the (suitably differentiable) event horizon. Therefore, Penrose suggested that (2.1) can be seen as a Riemannian test for the standard picture of black hole formation, particularly the Weak Cosmic Censorship Hypothesis.
    [Show full text]
  • Arxiv:1608.03021V1 [Gr-Qc]
    Ernst formulation of axisymmetric fields in f(R) gravity: applications to neutron stars and gravitational waves Arthur George Suvorov∗ and Andrew Melatos† School of Physics, University of Melbourne, Parkville VIC 3010, Australia (Dated: October 30, 2018) The Ernst formulation of the Einstein equations is generalised to accommodate f(R) theories of gravity. It is shown that, as in general relativity, the axisymmetric f(R) field equations for a vacuum spacetime that is either stationary or cylindrically symmetric reduce to a single, non-linear differential equation for a complex-valued scalar function. As a worked example, we apply the generalised Ernst equations to derive a f(R) generalisation of the Zipoy-Voorhees metric, which may be used to describe the gravitational field outside of an ellipsoidal neutron star. We also apply the theory to investigate the phase speed of large-amplitude gravitational waves in f(R) gravity in the context of soliton-like solutions that display shock-wave behaviour across the causal boundary. PACS numbers: 04.20.Jb, 04.30.Nk, 04.50.Kd, 04.70.Bw I. INTRODUCTION function f and its derivatives, as well as the Ricci scalar R and its derivatives. Since their initial presentation, the Einstein equations There are two flavours of Ernst equation that we gen- have been rewritten in many different ways [1, 2]. For sta- eralise here to the f(R) theory of gravity: stationary, tionary and axisymmetric spacetimes, the Ernst formula- and cylindrically symmetric. Stationary spacetimes arise tion has proved to be an especially useful representation in numerous physically important contexts; for example, [3].
    [Show full text]
  • Gravitational and Electromagnetic Potentials of the Stationary Einstein
    Gravitational and electromagnetic potentials of the stationary Einstein-Maxwell field equations by Thaddeus Charles Jones A thesis submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Physics Montana State University © Copyright by Thaddeus Charles Jones (1979) Abstract: Associated with the stationary Einstein-Maxwell field equations is an infinite hierarchy of potentials. The basic characteristics of these potentials are examined in general and then in greater detail for the particular case of the Reissner-Nordstrom metric. Their essential utility in the process of solution generation is elucidated and the necessary equations for solution generation are developed. Appropriate generating functions, which contain the complete infinite hierarchy of potentials, are developed and analyzed. Particular attention is paid to the inherent gauge freedom of these generating functions. Two methods of solution generation, which yield asymptotically flat solutions in vacuum, are generalized to include electromagnetism. One method, using potentials consistent with the Harrison transformation and the Reissner-Nordstrom metric, is discussed in detail and its resultant difficulties explored. GRAVITATIONAL AND ELECTROMAGNETIC POTENTIALS OF THE STATIONARY EINSTEIN-MAXWELL FIELD EQUATIONS by THADDEUS CHARLES JONES A thesis submitted in partial fulfillment of the requirements for the degree . o* DOCTOR OF PHILOSOPHY in Physics Approved: MONTANA STATE UNIVERSITY Bozeman, Montana June, 1979 iii ACKNOWLEDGMENTS I have but one person to thank for the most chal­ lenging and exciting intellectual period of my life. Dr. William Kinnersley. I express my gratitude for his continued faith in me during times when I had almost lost my own. I also gratefully acknowledge the many hours of clarifying discussion with Dr.
    [Show full text]
  • Fundamental Equations for Hypersurfaces
    Appendix A Fundamental Equations for Hypersurfaces In this appendix we consider submanifolds with codimension 1 of a pseudo- Riemannian manifold and derive, among other things, the important equations of Gauss and Codazzi–Mainardi. These have many useful applications, in particular for 3 + 1 splittings of Einstein’s equations (see Sect. 3.9). For definiteness, we consider a spacelike hypersurface Σ of a Lorentz manifold (M, g). Other situations, e.g., timelike hypersurfaces can be treated in the same way, with sign changes here and there. The formulation can also be generalized to sub- manifolds of codimension larger than 1 of arbitrary pseudo-Riemannian manifolds (see, for instance, [51]), a situation that occurs for instance in string theory. Let ι : Σ −→ M denote the embedding and let g¯ = ι∗g be the induced metric on Σ. The pair (Σ, g)¯ is a three-dimensional Riemannian manifold. Other quanti- ties which belong to Σ will often be indicated by a bar. The basic results of this appendix are most efficiently derived with the help of adapted moving frames and the structure equations of Cartan. So let {eμ} be an orthonormal frame on an open region of M, with the property that the ei ,fori = 1, 2, 3, at points of Σ are tangent μ to Σ. The dual basis of {eμ} is denoted by {θ }.OnΣ the {ei} can be regarded as a triad of Σ that is dual to the restrictions θ¯i := θ j |TΣ(i.e., the restrictions to tangent vectors of Σ). Indices for objects on Σ always refer to this pair of dual basis.
    [Show full text]
  • Towards Gravitating Discs Around Stationary Black Holes
    Towards gravitating discs around stationary black holes Oldrichˇ Semerak´ Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V Holeˇsoviˇck´ach 2, CZ-180 00 Praha 8, Czech Republic E-mail: oldrich.semerak@mff.cuni.cz Abstract This article outlines the search for an exact general relativistic description of the exterior (vacuum) gravitational field of a rotating spheroidal black hole surrounded by a realistic axially symmetric disc of matter. The problem of multi-body stationary spacetimes is first exposed from the perspective of the relativity theory (section 1) and astrophysics (section 2), listing the basic methods employed and results obtained. Then (in section 3) basic formulas for stationary axisymmetric solutions are summarized. Sections 4 and 5 review what we have learnt with Miroslav Z´aˇcekˇ and Tom´aˇsZellerin about certain static and stationary situations recently. Concluding remarks are given in section 6. Although the survey part is quite general, the list of references cannot be complete. Our main desideratum was the informative value rather than originality — novelties have been preferred, mainly reviews and those with detailed introductions. 1 The fields of multi-body systems involving black holes The subject of self-gravitating sources around rotating black holes is interesting in several re- spects, relevant from the point of view of the relativity theory itself as well as in the astrophysical arXiv:gr-qc/0204025v1 5 Apr 2002 context. First, due to the non-linearity of Einstein’s equations,
    [Show full text]
  • The Kerr Black Hole Hypothesis: a Review of Methods and Results
    The Kerr black hole hypothesis: a review of methods and results João Miguel Arsénio Rico Thesis to obtain the Master of Science Degree in Engineering Physics Examination Committee Chairperson: Doctor José Pizarro de Sande e Lemos Supervisor: Doctor Vitor Manuel dos Santos Cardoso Members of the Committee: Doctor Jorge Miguel Cruz Pereira Varelas da Rocha Doctor Paolo Pani September 2013 ii Acknowledgements Life and physics are hard. It is often easy to forget how much there is beyond the formulas we understand and that some of our basic assumptions are wrong. Experiment rules regardless of pet theories, and we are fortunate for the opportunity to at least refine our intuitions. It is essential to keep questioning, keep working, keep humble. Life and physics are a pleasure, and for the most part people are the reason why. I want to start by thanking Vitor Cardoso, my thesis supervisor, for all the help, patience and encouragement throughout this project. An excellent researcher and an excellent supervisor are not the same thing, and I have been extremely fortunate because Vitor really is both. His sheer energy, constant good spirits and relentless enthusiasm for Physics are an inspiring example for me, and I have yet to measure all that I have learned from him. I also want to thank Paolo Pani for being such a great co- supervisor! Thank you for all the guidance and fun through this initiation to research and black hole physics. It has been a real pleasure and privilege to work with both of you. I thank Jorge Rocha, my thesis examiner, for a very careful reading of the manuscript and valuable comments and suggestions.
    [Show full text]
  • Exact Solution of a Neumann Boundary Value Problem for the Stationary Axisymmetric Einstein Equations
    Journal of Nonlinear Science https://doi.org/10.1007/s00332-018-9527-1 Exact Solution of a Neumann Boundary Value Problem for the Stationary Axisymmetric Einstein Equations Jonatan Lenells1 · Long Pei1 Received: 4 October 2018 / Accepted: 11 December 2018 © The Author(s) 2019 Abstract For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By solving this equation with a Dirichlet boundary condition imposed along the disk, Neugebauer and Meinel in the 1990s famously derived an explicit expression for the spacetime metric corresponding to the Bardeen–Wagoner uniformly rotating disk of dust. In this paper, we consider a similar boundary value problem for a rotating disk in which a Neumann boundary condition is imposed along the disk instead of a Dirichlet condition. Using the integrable structure of the Ernst equation, we are able to reduce the problem to a Riemann–Hilbert problem on a genus one Riemann surface. By solving this Riemann–Hilbert problem in terms of theta functions, we obtain an explicit expression for the Ernst potential. Finally, a Riemann surface degeneration argument leads to an expression for the associated spacetime metric. Keywords Ernst equation · Einstein equations · Boundary value problem · Unified transform method · Fokas method · Riemann–Hilbert problem · Theta function Mathematics Subject Classification 83C15 · 37K15 · 35Q15 · 35Q76 1 Introduction Half a century ago, Bardeen and Wagoner studied the structure and gravitational field of a uniformly rotating, infinitesimally thin disk of dust in Einstein’s theory of relativity Communicated by Anthony Bloch. B Jonatan Lenells [email protected] Long Pei [email protected] 1 Department of Mathematics, KTH Royal Institute of Technology, 10044 Stockholm, Sweden 123 Journal of Nonlinear Science ζ D ρ0 ρ Fig.
    [Show full text]