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Wave Extraction in Numerical Relativity Doctoral Dissertation Wave Extraction in Numerical Relativity Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der Bayrischen Julius-Maximilians-Universitat¨ Wurzburg¨ vorgelegt von Oliver Elbracht aus Warendorf Institut fur¨ Theoretische Physik und Astrophysik Fakultat¨ fur¨ Physik und Astronomie Julius-Maximilians-Universitat¨ Wurzburg¨ Wurzburg,¨ August 2009 Eingereicht am: 27. August 2009 bei der Fakultat¨ fur¨ Physik und Astronomie 1. Gutachter:Prof.Dr.Karl Mannheim 2. Gutachter:Prof.Dr.Thomas Trefzger 3. Gutachter:- der Dissertation. 1. Prufer¨ :Prof.Dr.Karl Mannheim 2. Prufer¨ :Prof.Dr.Thomas Trefzger 3. Prufer¨ :Prof.Dr.Thorsten Ohl im Promotionskolloquium. Tag des Promotionskolloquiums: 26. November 2009 Doktorurkunde ausgehandigt¨ am: Gewidmet meinen Eltern, Gertrud und Peter, f¨urall ihre Liebe und Unterst¨utzung. To my parents Gertrud and Peter, for all their love, encouragement and support. Wave Extraction in Numerical Relativity Abstract This work focuses on a fundamental problem in modern numerical rela- tivity: Extracting gravitational waves in a coordinate and gauge independent way to nourish a unique and physically meaningful expression. We adopt a new procedure to extract the physically relevant quantities from the numerically evolved space-time. We introduce a general canonical form for the Weyl scalars in terms of fundamental space-time invariants, and demonstrate how this ap- proach supersedes the explicit definition of a particular null tetrad. As a second objective, we further characterize a particular sub-class of tetrads in the Newman-Penrose formalism: the transverse frames. We establish a new connection between the two major frames for wave extraction: namely the Gram-Schmidt frame, and the quasi-Kinnersley frame. Finally, we study how the expressions for the Weyl scalars depend on the tetrad we choose, in a space-time containing distorted black holes. We apply our newly developed method and demonstrate the advantage of our approach, compared with methods commonly used in numerical relativity. Abriss Diese Arbeit konzentriert sich auf eine fundamentale Problematik der nu- merischen Relativitatstheorie:¨ Die Extraktion von Gravitationswellen in einer eich- und koordinateninvarianten Formulierung, um ein physikalisch interpretierbares Ob- jekt zu erhalten. Es wird eine neue Methodik entwickelt, um die physikalisch relevanten Großen¨ aus einer numerisch erzeugten Raumzeit zu extrahieren. Wir prasentieren¨ eine allgemein- gultige¨ kanonische Formulierung der Weyl Skalare im Newman-Penrose Formalismus v als eine Funktion von fundamentalen Raumzeit-Invarianten. Dadurch zeigt sich, dass mit Hilfe dieser Methodik die explizite Konstruktion eines Vierbeins vollstandig¨ re- dundant ist. Als weiteren Schwerpunkt charakterisieren wir innerhalb des Newman-Penrose Formalismus eine spezielle Untergruppe von Tetraden, die transversen Frames. Es wird eine bisher unbekannte Verbindung zwischen den primar¨ genutzen Vierbeinen fur¨ die Extraktion der Wellenform abgeleitet, dem Gram-Schmidt Vierbein und dem quasi- Kinnersley Vierbein. Abschliessend studieren wir die Abhangigkeit¨ der Gravitations- wellen eines gestorten¨ Schwarzen Loches vom verwendeten Vierbein. Wir berechnen die Form der Gravitationswellen in dieser Raumzeit und demonstrieren inwieweit unsere neue Methodik robustere und exaktere Ergebnisse liefert, als die gewohnlich¨ verwendeten Ansatze¨ zur Extraktion des Signals. vi Wave Extraction in Numerical Relativity Full list of publications by the author This thesis is mainly based upon the following publications: • Nerozzi, Andrea; Elbracht, Oliver - Using curvature invariants for wave extrac- tion in numerical relativity, accepted by Physical Review D (2009). • Elbracht, Oliver; Nerozzi, Andrea - Using curvature invariants for wave extrac- tion in numerical relativity. II. Wave extraction in distorted black hole space- times, submitted to Physical Review D. • Elbracht, Oliver; Nerozzi, Andrea - A new approach to wave extraction in nu- merical relativity, submitted to Journal of Physics: Conference Series (refereed). Other publications by the author: • Elbracht, Oliver; Nerozzi, Andrea; Matzner, Richard - Wave extraction in nu- merical evolutions of distorted black holes, oclc/66137068 (2005). • Burkart, Thomas; Elbracht, Oliver; Spanier, Felix - Simulation results of our newly developed PIC codes, AN, Vol.328, Issue 7 (unrefereed). • Rodig,¨ Constanze; Burkart, Thomas; Elbracht, Oliver; Spanier, Felix - Multi- wavelength periodicity study of Markarian 501, Astronomy and Astrophysics, Vol- ume 501, Issue 3, 2009, pp.925-932. vii • Burkart, Thomas; Elbracht, Oliver; Ganse, Urs; Spanier, Felix - The influence of the mass-ratio on the acceleration of particles by lamentation instabilities, submitted to The Astrophysical Journal. The work has been supported through a research scholarship from the Elitenetz- werk Bayern (Elite Network of Bavaria). The work contained in this thesis is in part done within the International Research Training Group (GRK 1147/1) funded by the Deutsche Forschungsgesellschaft (DFG). Doctoral Dissertation Author: Oliver Elbracht email address: [email protected] viii Contents 1. Introduction1 1.1. Notation and Units...............................5 2. The 3+1 Split and Initial Data9 2.1. Initial Value Problem..............................9 2.2. The 3+1 Decomposition - Separating Space from Time.......... 12 2.3. The ADM Formalism............................. 12 2.4. BSSN - An Alternative............................. 16 3. Gravitational Waves 21 3.1. The Linearized Theory of Gravity...................... 21 3.2. A Wave Solution and the Transverse-Traceless Gauge........... 26 3.2.1. Interaction of Gravitational Waves with Test-Particles....... 28 3.2.2. Polarization of a Plane Wave..................... 30 3.3. Interaction of Gravitational Waves with Detectors............. 33 3.4. The Energy of Gravitational Radiation.................... 37 3.5. Gravitational Waves from Perturbed Black Holes.............. 38 3.5.1. Perturbation Theory and Quasi-Normal Modes.......... 38 3.5.2. The Regge-Wheeler and Zerilli Equation.............. 40 4. The Newman-Penrose Formalism 47 4.1. Mathematical Preliminaries.......................... 48 4.1.1. Directional Derivatives and Ricci Rotation Coefficients...... 50 4.1.2. The Commutation Relation and Structure Constants....... 51 4.1.3. The Ricci Identities........................... 51 ix Contents 4.1.4. The Bianchi Identities......................... 52 4.2. Null Tetrads and Null Frames......................... 53 4.3. Spin Coefficients................................ 55 4.4. Weyl Tensor and Weyl Scalars......................... 56 4.5. Curvature Invariants.............................. 58 4.6. Tetrad Transformations............................. 59 4.6.1. Type I Rotations............................ 60 4.6.2. Type II Rotations............................ 61 4.6.3. Type III Rotations............................ 62 4.7. Petrov Classification.............................. 63 4.8. Physical Interpretation of the Weyl Scalars & Peeling-off Theorem... 65 4.8.1. Petrov Type N.............................. 67 4.8.2. Petrov Type III............................. 67 4.8.3. Petrov Type D.............................. 68 4.8.4. Petrov Type II.............................. 68 4.8.5. Petrov Type I.............................. 69 4.9. Goldberg-Sachs Theorem........................... 69 4.10. Bondi Frame................................... 70 4.11. Kinnersley Tetrad................................ 72 4.12. Black Hole Space-Times in the NP Formalism............... 73 4.13. Perturbation Approach in the NP Formalism................ 74 4.13.1. The Perturbation Equations...................... 74 4.13.2. Teukolsky Master equation...................... 75 4.13.3. Asymptotic Behavior.......................... 77 4.14. An Energy Measurement........................... 78 4.15. Selecting the Proper Frame for Wave Extraction.............. 79 4.15.1. Transverse Frames & Quasi-Kinnersley Frame........... 80 4.15.2. Finding the Quasi-Kinnersley Frame................ 81 5. Non-Perturbative Approach for Wave Extraction 87 5.1. A new Formalism for Wave Extraction.................... 88 x Contents 5.2. Redefining the Weyl Scalars in Transverse Frames............. 89 5.3. The Bianchi Identities............................. 91 5.4. The Ricci Identities............................... 92 5.5. Directional Derivatives............................. 94 5.6. The Type D Spin Relation........................... 96 5.7. Spin Coefficients as Directional Derivatives................. 99 5.8. The Function H ................................ 101 5.9. Weyl Scalars in Terms of Curvature Invariants............... 103 5.9.1. Final Expressions for the Weyl Scalars and Peeling Behavior... 105 5.9.2. Conclusion............................... 107 6. Distorted Black Hole Space-Times in the NP Formalism 109 6.1. Brill Wave Initial Data............................. 110 6.1.1. Pure Gravitational Waves....................... 110 6.1.2. Distorted Black Hole Initial Data................... 113 6.2. Weyl Scalars and Spin Coefficients on the Initial Slice........... 114 6.3. Finding the Transverse Frames........................ 118 6.3.1. The First Transverse Frame...................... 118 6.3.2. The Quasi-Kinnersley Frame..................... 122 6.4. Connecting the Tetrads in Type D...................... 126 6.5. Spin-Boost Degree of Freedom........................ 128 6.6. Perturbation Theory for the
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