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Exploring Black Hole Coalesce

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Exploring Black Hole Coalesce The Pennsylvania State University The Graduate School Eberly College of Science RINGING IN UNISON: EXPLORING BLACK HOLE COALESCENCE WITH QUASINORMAL MODES A Dissertation in Physics by Eloisa Bentivegna c 2008 Eloisa Bentivegna Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2008 The dissertation of Eloisa Bentivegna was reviewed and approved* by the following: Deirdre Shoemaker Assistant Professor of Physics Dissertation Adviser Chair of Committee Pablo Laguna Professor of Astronomy & Astrophysics Professor of Physics Lee Samuel Finn Professor of Physics Professor of Astronomy & Astrophysics Yousry Azmy Professor of Nuclear Engineering Jayanth Banavar Professor of Physics Head of the Department of Physics *Signatures are on file in the Graduate School. ii Abstract The computational modeling of systems in the strong-gravity regime of General Relativity and the extraction of a coherent physical picture from the numerical data is a crucial step in the process of detecting and recognizing the theory’s imprint on our universe. Obtained and consolidated over the past two years, full 3D simulations of binary black hole systems in vacuum constitute one of the first successful steps in this field, based on the synergy of theoretical modeling, numerical analysis and computer science efforts. This dissertation seeks to model the merger of two coalescing black holes, employing a novel technique consisting of the propagation of a massless scalar field on the spacetime where the coalescence is taking place: the field is evolved on a set of fixed backgrounds, each provided by a spatial hypersurface generated numerically during a binary black hole merger. The scalar field scattered from the merger region exhibits quasinormal ringing once a common apparent horizon surrounds the two black holes. This occurs earlier than the onset of the perturbative regime as measured by the start of the quasinormal ringing in the gravitational waveforms, indicating that previous semianalytical evidence on the early validity of perturbative methods during a black hole merger is indeed correct. The scalar quasinormal frequencies are also used to associate a mass and a spin with each hypersurface: this measure is, within our error bars, compatible with the horizon mass and spin computed from the dynamical horizon framework. The emerging physical picture indicates that the behavior of a scalar field propagating on the spacetime of two merged (i.e., surrounded by a common apparent horizon) black holes is very close to that expected on a Kerr spacetime with mass and spin parameters equal to Mf and jf , the mass and spin of the common apparent horizon at the end of the coalescence. Some of the results discussed in this dissertation have been published as: E. Bentivegna, P. Laguna, and D. Shoemaker, The effect of gauge conditions on waveforms from binary black hole coalescence, AIP Conf. Proc., 873:9498 (2006). E. Bentivegna, D. Shoemaker, I. Hinder, and F. Herrmann, Probing the binary black hole merger regime with scalar perturbations, arXiv:0801.3478 (2008). iii Contents List of Tables vii List of Figures viii Acknowledgements xv 1 Introduction 1 1.1 The black hole concept in mathematical, numerical, astrophysical and quan- tum Relativity ................................... 3 1.2 Two-black-hole solutions ............................. 5 2 General Relativity and the three-plus-one decomposition 7 2.1 Gravitation according to General Relativity .................. 8 2.1.1 Exact solutions in vacuum: static and stationary black holes ..... 9 2.2 The dynamics of General Relativity ....................... 10 2.3 The ADM formulation .............................. 11 2.3.1 The 3 + 1 decomposition ......................... 12 2.3.2 Projections of the Riemann tensor .................... 14 2.3.3 Projections of Einstein’s equation .................... 15 2.4 Decomposition of the scalar wave equation ................... 18 iv 3 Scalar field evolution on a black hole background 19 3.1 Quasinormal ringing: damped oscillatory modes in open systems ....... 21 3.2 The scalar wave equation on a static, spherically symmetric spacetime .... 22 3.2.1 Green’s function analysis ......................... 25 3.3 The scalar wave equation on a stationary, axisymmetric spacetime ...... 28 3.3.1 Green’s function analysis ......................... 31 4 General Relativity and numerical methods 33 4.1 Scientific computing in General Relativity ................... 34 4.2 Modelling ..................................... 34 4.2.1 Evolution schemes and constraint preservation ............. 35 4.2.2 Coordinate conditions .......................... 39 4.2.3 Initial data ................................ 43 4.2.4 Boundary conditions ........................... 45 4.3 Numerical analysis ................................ 46 4.3.1 Integration of partial differential equations ............... 46 4.3.1.1 Finite differencing schemes .................. 46 4.3.1.2 Convergence to the continuum solution ............ 48 4.3.2 Function minimization .......................... 50 4.4 Computer science ................................. 52 5 Gravitational and scalar field evolution codes 55 5.1 BSSN evolution code ............................... 56 5.1.1 Scope ................................... 56 5.1.2 Structure ................................. 56 5.2 Scalar field evolution code ............................ 59 v 5.2.1 Scope ................................... 59 5.2.2 Structure ................................. 59 5.2.3 Code verification ............................. 61 5.2.3.1 Flat space tests ......................... 62 5.2.3.2 Single black hole tests ..................... 64 6 Probing the binary black bole merger regime with scalar perturbations 66 6.1 Binary black hole background .......................... 68 6.1.1 Energy and angular momentum ..................... 72 6.2 Scalar field evolution ............................... 72 6.2.1 Initial data ................................ 78 6.2.2 Frequency extraction ........................... 78 6.2.3 Mass and spin extraction ......................... 100 6.2.4 Scalar field evolution before the first common apparent horizon . 109 6.3 Error analysis ................................... 109 6.4 Discussion and conclusions ............................ 115 References 118 A Gauge conditions and physical observables 130 A.1 Zerilli waveforms ................................. 131 A.2 Error budget ................................... 131 B Extraction of quasinormal frequencies via an evolutionary strategy 134 B.1 Algorithm ..................................... 135 B.2 Application to quasinormal ringing waveforms ................. 135 vi List of Tables 6.1 The residual q and the best-fit values of ω20, α20, A20 and φ20, along with the corresponding mass and dimensionless spin estimates. ............. 107 6.2 The residual q and the best-fit values of ω40, α40, A40 and φ40, along with the corresponding mass and dimensionless spin estimates. ............. 107 6.3 The residual q and the best-fit values of ω60, α60, A60 and φ60, along with the corresponding mass and dimensionless spin estimates. ............. 108 vii List of Figures 3.1 The contour integral used to calculate G(x, y; t). The contribution from the poles of G(x, y; ω) in the lower half plane are responsible for the quasinormal mode components of the field, while the half circle as ω -if present at | |→∞ all- generates the prompt response to the initial data and the portion of the contour around the branch cut results in the late-time power-law tails (picture reproduced from [1]). ............................... 27 3.2 The spectrum of quasinormal modes for a Schwarzschild black hole, for the modes ℓ = 2 (diamonds) and ℓ = 3 (crosses). The absolute value of the imaginary part grows in a roughly linear fashion with n, so that higher and higher overtones tend to be more and more suppressed as the field evolves in time (plot reproduced from [2]). ........................ 29 4.1 Evolution of the spatial grid (here collapsed to a 2D plane) during the numer- ical evolution. At each time coordinate, the lapse and the shift are calculated to determine where each grid point is going to be positioned in the next it- eration. In binary black hole simulations, the gauge choices proposed, for instance, in [3] guarantee that the grid points remain far enough from the puncture singularities even when the puncture locations are advected through the grid. ...................................... 40 viii 4.2 Lapse function for four gauge types, on the x-y plane, for a single boosted puncture. The λ and γ values refer to the constants in equations (4.2.36) and (4.2.37), while µ = 0 and η = 4. In the case when λ = 1 and γ = 0, the collapsed-lapse region does not track the puncture location efficiently, lagging behind in the evolution and generating the stretched contour shown in the top right panel. .................................. 44 5.1 Structure and interdependencies of the BSSN evolution code used in the runs of chapter 6 .................................... 58 5.2 Structure and interdependencies of the scalar evolution code used in the runs of chapter 6 .................................... 60 5.3 Convergence factor for the plane wave. After a rapid initial transient, the value of Q settles, as expected, to 15. ...................... 62 5.4 Pointwise convergence for the plane wave, at t/λ = 0, 0.8, 1.6, 2.4, 3.2, 4.0. The difference between the two highest resolution runs is multiplied by the appropriate factor for a
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