Numerical Simulations of Gravitational Collapse by Frans Pretorius B.Eng., The University of Victoria, 1996 M.Sc, The University of Victoria, 1999 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 2, 2002 © Frans Pretorius, 2002 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University Of British Columbia Vancouver, Canada ABSTRACT 11 ABSTRACT In this thesis we present a numerical study of gravitational collapse, within the framework of Einstein's theory of general relativity. We restrict our attention to spacetimes possessing axial symmetry, and incorporate a massless scalar field as the matter source. Our primary objectives are the study of critical phenomena at the threshold of black hole formation, and the stable simulation of black hole spacetimes. An integral part of the thesis is concerned with developing the necessary numerical tools and techniques to successfully solve these problems. To that end, we have implemented a variant of Berger and Oliger's adaptive mesh refinement (AMR) algorithm, with enhancements that allow us to incorporate elliptic equations into the AMR framework. Using this code, we simulate critical collapse of axisymmetric distributions of scalar field energy, which is the first time this has been done in the non-perturbative regime. For several classes of initial data, our results are consistent with a hypothesized universal critical solution. However, from the collapse of prolate initial data, we find indications that there may be an additional, non-spherical unstable mode. This putative instability eventually causes a near- critical echoing solution to bifurcate into two, causally disconnected solutions that each resemble the spherical critical solution. Furthermore, we speculate that this bifurcation process would continue indefinitely at threshold, resulting in an infinite cascade of near-spherical solutions. However, the evidence for this second unstable mode is not conclusive, and more work will be needed, possibly with an enhanced code, to answer this question. To numerically study spacetimes containing black holes, one needs to avoid the singularities that occur inside of the holes. The technique that we have implemented to accomplish this is called black hole excision. This aspect of the code is still work-in-progress, for we have not yet incorporated excision into the AMR-based code, and the class of excision boundary conditions we currently employ are inconsistent with the complete set of field equations. However, we are able to obtain stable simulations using a constrained evolution scheme, and we present two preliminary examples, one showing black hole formation, the other a head-on black hole collision. CONTENTS iii CONTENTS Abstract ii Contents • iii List of Tables v List of Figures vi Acknowledgements viii 1 Introduction 1 1.0.1 Notation 3 1.1 The Field Equations of General Relativity 3 1.2 Gravitational Collapse 5 1.3 Critical Phenomena 6 1.4 Black Hole Collisions and Gravitational Waves 7 1.5 Numerical Solution of the Field Equations 9 1.5.1 The ADM formalism 10 1.5.2 Coordinate and Slicing Conditions 12 1.5.3 Black Hole Excision . 14 1.5.4 Solution of Equations via Finite Difference Techniques 15 1.5.5 Adaptive Mesh Refinement 23 1.5.6 Solution of Elliptic Equations via Multigrid 28 2 Numerical Evolution in Axisymmetry 34 2.1 The 2+1+1 Formalism ' 34 2.1.1 A Scalar Field Matter Source 37 2.1.2 Incorporating Angular Momentum using a Complex Scalar Field 37 2.2 Structure of the Unigrid Numerical Code 38 2.2.1 Coordinate System and Variables 38 2.2.2 Regularity and Outer Boundary Conditions 40 2.2.3 Initial Data 41 2.2.4 Discretization and Solution Scheme 41 2.2.5 Finding Apparent Horizons 43 2.2.6 Black Hole Excision Technique 44 2.2.7 Convergence, Consistency and Mass Conservation Results . 47 2.3 Structure of the Adaptive Code 49 2.3.1 Modifications to Berger and Oliger's Algorithm 52 2.3.2 Multigrid on an Adaptive Hierarchy 56 2.3.3 Clustering 58 2.3.4 Truncation Error Estimation 58 2.3.5 Interpolation and Restriction Operators 59 2.3.6 Initializing the Grid Hierarchy 59 2.3.7 Controlling High-Frequency Grid-Boundary Noise 60 2.3.8 Dealing with Multigrid Failures 62 CONTENTS iv 2.3.9 Convergence, Consistency and Mass Conservation Results 63 2.4 Probing the Geometry of a Numerical Solution 67 2.4.1 Geodesies 70 3 Scalar Field Critical Collapse 72 3.1 Spherically Symmetric Initial Data 73 3.2 Small Deviations from Spherical Symmetry 77 3.3 Large Deviations from Spherical Symmetry 82 3.3.1 Equatorial-Plane Antisymmetric Scalar Field Collapse . 82 3.3.2 Highly Prolate Initial Scalar Field Profiles 84 3.4 Sample AMR Mesh Structure . 89 4 Black Hole Excision 101 4.1 Scalar Field Collapse 101 4.2 Black Hole Collisions . 102 4.2.1 Boosted Scalar Field Initial Conditions 108 4.2.2 Excision Boundary Conditions 108 4.2.3 A Sample Merger 109 5 Conclusions and Future Work 117 Bibliography 118 A Equations and Finite Difference Operators 124 A.l Analytic Equations 124 A. 2 Finite Difference Operators 126 B Data Analysis and Visualization 129 B. l The Data-Vault Virtual Machine Specification . 129 B.l.l Register Structure 129 B.l.2 Instruction Set , 130 B.1.3 Execution Model 134 B.2 Current Implementation 135 B.2.1 Generalized Index Vector format 135 LIST OF TABLES v LIST OF TABLES 2.1 AMR code test: £2 and £00 norms of the data shown in Figures 2.11 and 2.12 .... 63 3.1 Extremes of the central value of $ in near-critical spherically symmetric collapse . 74 3.2 Extremes of the central value of $ in near-critical, near-spherically symmetric collapse 81 3.3 Extremes of the local central values of $ in near-critical antisymmetric collapse ... 86 3.4 Extremes of the central values of $ in near-critical prolate collapse 89 B.1 DV register structure definition 130 B.2 DV time structure definition 130 B.3 DV level structure definition 132 B.4 DV grid structure definition 132 LIST OF FIGURES vi LIST OF FIGURES 1.1 A schematic representation of the ADM space+time decomposition 11 1.2 A schematic demonstration of the CFL condition 21 1.3 An example of a Berger and Oliger mesh hierarchy 25 1.4 A pseudo-code representation of the Berger and Oliger time stepping algorithm ... 26 1.5 A pseudo-code representation of the FAS, V-cycle multigrid algorithm 32 2.1 The effect of an excision surface on the definition of grid points and FD stencils ... 45 2.2 A pseudo-code representation of the reconstruction function and smoothing filter used to smooth grid functions during excision 48 2.3 Test of the unigrid code for Brill wave evolution 50 2.4 Test of the unigrid code for scalar field evolution 51 2.5 A pseudo-code representation of the Berger and Oliger time stepping algorithm, as modified and implemented in our code 54 2.6 An illustration of the technique we use to extrapolate elliptic variables within the AMR framework 55 2.7 A hypothetical MG grid configuration that will adversely affect execution speed ... 57 2.8 An illustration of the interpolation method used for a and ft during AMR evolution 60 2.9 A pseudo-code description of part of the interpolation method used for a and ft during AMR evolution 61 2.10 AMR code test: a at t = 2.34, showing the AMR hierarchy 64 2.11 AMR code test: comparison of a to a unigrid evolution, at t = 2.34 65 2.12 AMR code test: comparison of $ to a unigrid evolution, at t = 2.34 66 2.13 AMR code test: depth of the hierarchy as a function of time 68 2.14 AMR code test: MADM 68 2.15 AMR code test: consistency 69 2.16 AMR code test: convergence factors 69 3.1 Initial profile of <£, for spherically symmetric critical collapse 73 3.2 ln(i?max) vs. \n(A* - A) from sub-critical evolution of spherically symmetric initial 3 data, with eT = 10~ . 75 3.3 ln(i?max) vs. ln(A* - A) from sub-critical evolution of spherically symmetric initial 4 data, with eT = 10~ 76 3.4 Central value of $ as a function of - ln(r* - r), from spherically symmetric critical collapse 77 3.5 Evolution of $ from spherically symmetric critical collapse 78 3.6 ln(i?max) vs. \n(A* - A) from sub-critical evolution of near-spherically symmetric -3 initial data, with er = IO 79 3.7 ln(i?max) vs. ln(^4* - A) from sub-critical evolution of near-spherically symmetric 4 initial data, with er = 10~ 80 3.8 Comparison of the central value of $ versus — ln(r* — r), for near-spherically sym• metric initial data 81 3.9 Apparent horizon shapes from super-critical, antisymmetric collapse 83 3.10 Initial profile of $ for antisymmetric critical collapse 84 3.11 Evolution of $ from antisymmetric initial data 85 LIST OF FIGURES vii 3.12 Late-time profile of 4>, evolved from near-critical antisymmetric initial data 86 3.13 ln(jRmax) vs.