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Maxwell-Klein-Gordon fields in black hole spacetimes Petryk, Roman J. W. 2006

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Maxwell-Klein-Gordon Fields in Black Hole Spacetimes by Roman J . W . Petryk Bachelor of Science, The University of Manitoba, 1996 Certificate of Advanced Study in Mathematics, The University of Cambridge, 1997 Master of Science, The University of British Columbia, 2000 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y T H E F A C U L T Y O F G R A D U A T E STUDIES (Physics) T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A November, 2005 © Roman J . W . Petryk, 2005 ABSTRACT 11 A B S T R A C T In this thesis I present results for the evolution and dynamics of massive electromagnetically cou-pled Maxwell-Klein-Gordon fields in black hole spacetimes. The first part of my investigation for gravitationally and electromagnetically self-interacting fields in spherical symmetry reveals two dis-tinct types of solution at the threshold of black hole formation. For fields with relatively small mass parameter I observe Type II discretely self-similar behaviour for the critical solutions and obtain the black hole mass and charge scaling relations. However, when the mass parameter is sufficiently large a different type of critical solution is obtained.. This new solution is periodic and resembles a perturbed charged boson star solution. This new solution exhibits Type I critical behaviour and its lifetime obeys a well-defined scaling law. The second aspect of investigation involves massive electromagnetically coupled scalar field perturbations in axial symmetry on a Kerr black hole spacetime. Here, results show that both the mass and charge coupling parameters play a significant role in the field dynamics on the spacetime background. For relatively weak parameter values the perturbations exhibit strong gravitational interaction through the phenomenon of orbiting resonances. In the case of pure electromagnetic perturbation there is also evidence of superradiant scattering when the black hole rotation is large. When the parameter values are large both the physics and complexity of the dynamics change. For intermediate values of the mass and charge parameter, the perturbations exhibit trapping and a preference for scattering along the axis of black hole rotation. Finally, all electromagnetically coupled solutions generically display charge separation and dynamo-like behaviour. C O N T E N T S iii CONTENTS Abstract i i Contents i i i List of Tables v i List of Figures v i i Acknowledgements x i i i 1 Introduction 1 1.1 Scalar Field Critical Collapse 1 1.2 Scattering and Scalar Accretion in Kerr Spacetime 3 1.3 Outline 5 1.4 Conventions and Units 5 1.5 Significant Figures and Uncertainties 6 1.6 Web References 6 2 Physical Formalism 7 2.1 The 3+1 or Arnowitt-Deser-Misner (ADM) Formalism 7 2.1.1 Foliations 7 2.1.2 Projections 7 2.1.3 Covariant Derivatives on the Spacelike Hypersurface 8 2.1.4 Intrinsic and Extrinsic Curvature 8 2.1.5 The Spacetime Constraint Equations 9 2.1.6 Time Derivatives 10 2.1.7 Spacetime Evolution Equations 10 2.1.8 Spherically Symmetric Spacetime in Polar Areal Coordinates 12 2.2 The Initial Value Problem 15 2.2.1 The Initial Value Problem for Electromagnetism 15 2.2.2 The Initial Value Problem for General Relativity 18 2.3 Conserved Quantities 19 2.3.1 Conservation of Charge 19 2.3.2 Conservation of Energy and Momentum 19 2.4 Regularity Conditions 20 2.4.1 Regularity at r = 0 in Polar-Areal Coordinates 20 2.4.2 Regularity at 9 = 0 and 9 = 7r in Kerr-Schild Coordinates 21 2.5 The Sommerfeld Condition 21 2.5.1 The Sommerfeld Condition in Polar Areal Coordinates 22 2.5.2 The Sommerfeld Condition in Kerr-Schild Coordinates 22 C O N T E N T S iv 3 C h a r g e d Scalar C r i t i c a l Co l l apse : 24 3.1 Preliminary Discussion : 24 3.1.1 Context of This Study 24 3.1.2 Issues Regarding Horizons 25 3.2 System of Equations 29 3.2.1 The Matter Lagrangian and Equations of Motion 29 3.2.2 The Einstein Lagrangian and Field Equations , 31 3.2.3 The Spacetime Metric and Equations 31 3.2.4 Matter Equations in First-Order Form 32 3.2.5 Physical and Conserved Quantities 34 3.2.6 Regularity, Boundary Conditions and Dissipation 34 3.3 Results 35 3.3.1 Initial Data Profiles 35 3.3.2 Results for e = 1.0 with = 0.0 38 3.3.3 Results for e = 1.0 with m$ = 1.0 43 3.3.4 Code Testing 53 4 C h a r g e d Scalar A c c r e t i o n 57 4.1 Scattering and the Physics of Black Hole Perturbations 59 4.1.1 The Effective Potential Formulation for Schwarzschild Spacetime 59 4.1.2 The Glory Effect and Orbiting Resonances 68 4.1.3 The Effective Potential Formulation for Kerr Spacetime 71 4.1.4 Polarization, Superradiance, and the Black Hole Bomb 72 4.1.5 Late Time Decay, and the Coupling of Electromagnetic and Gravitational Perturbations 73 4.1.6 Charged Scalar Perturbations in Static and Stationary Spacetimes . . . . . . 75 4.2 System of Equations for Charged Scalar Accretion 76 4.2.1 The Spacetime Metric 76 4.2.2 The Matter Lagrangian and Equations of Motion 79 4.2.3 Matter Equations in First-Order Form 79 4.2.4 Physical and Conserved Quantities ' . 85 4.2.5 Regularity, Boundary Conditions and Dissipation 89 4.2.6 Strategies for A M R and Parallel Computation 91 4.3 Results 94 4.3.1 Initial Data 94 4.3.2 Klein-Gordon Dynamics: Orbiting Resonances and Massive Mode Trapping for s = 0 Perturbations 96 4.3.3 Maxwell Equation Dynamics: Orbiting Resonances and Superradiance for s = 1 Perturbations 107 4.3.4 Massless Maxwell-Klein-Gordon Equation Dynamics I: Orbiting Resonances, Charge Separation and Dynamo-Like Behaviour for e = 0.6 Electromagneti-cally Coupled s = 0 and s = 1 Perturbations 125 4.3.5 Massless Maxwell-Klein-Gordon Equation Dynamics II: Comparing Electro-magnetically Coupled s = 0 and s = 1 Perturbations W i t h Different Values of Parameter e 152 4.3.6 Massive Charged Maxwell-Klein-Gordon Equation Dynamics: Collimation, Orbiting Resonances and Decay of m$ = 0.3, e = 0.6 Electromagnetically Coupled s = 0 and s = 1 Perturbations 163 4.3.7 Convergence of Conserved Quantities • . . 189 4.3.8 Independent Residual Testing Dynamical Variables 193 4.3.9 Testing The Sommerfeld Boundary Condition 194 C O N T E N T S v 5 Conc lus ions and Fu tu r e W o r k 200 B i b l i o g r a p h y • 202 A G los sa ry o f N o t a t i o n 211 B Difference and Disc re te Opera to r s 213 B . l Finite Difference Operators 213 B . 2 Other Discrete Operators .214 B.2.1 Quadratic F ix Operators 214 B.2.2 Interpolation and Extrapolation Operators 215 B.2.3 Kreiss-Oliger Dissipation Operators 216 B . 2.4 Integration Rules 217 C N u m e r i c s 218 C. l Finite Difference Techniques and Approximations 218 C. l . l Basic Definitions and Concepts 218 C. l .2 Convergence Testing and Independent Residual Evaluation 219 C . l . 3 Deriving Finite Difference Formulae • 221 C. l .4 The Leap-Frog and Crank-Nicholson Schemes 223 C. l .5 Stability • - 225 C. l .6 Kreiss-Oliger Dissipation Operators . 227 C . l . 7 Iterative Solution Scheme for Evolutionary Equations ' . . . 228 C.2 Constraint Equations 229 C.2.1 The Poisson Equation in Spherical Symmetry '. 230 C.2.2 The.Poisson Equation in Axia l Symmetry 233 C.3 Adaptive Mesh Refinement 238 C.3.1 The Gr id Hierarchy 238 C.3.2 The Time-Stepping Algorithm 239 C.3.3 Dynamical Regridding via Relative Solution Error 239 C. 4 Parallel Computing 241 C. 4.1 Parallel Program Design 242 D M I B Coord ina t e s • • . 244 D. l The Monotonically Increasingly Boosted (MIB) Coordinate System . 244 D.2 Equations of Motion 252 D. 2.1 Physical and Conserved Quantities . 254 D.2.2 Regularity, Boundary Conditions and Dissipation 257 D.3 Results 259 D.3.1 Initial Data 259 D.3.2 Testing the Boundary Conditions 260 D.3.3 Maxwell-Klein-Gordon Fields on a Flat Background 263 D.3.4 Convergence of Conserved Quantities 281 L I S T O F T A B L E S vi LIST OF TABLES 3.1 Classification hierarchy of critical solutions 25 3.2 Critical gravitational collapse initial data families A through F 37 3.3 Black hole properties from gravitational collapse of slightly supercritical Type I solutions for families A through F when e = 1.0 and m$ = 1.0. 50 4.1 Predicted and observed physics of the s = 0 and s = 1 perturbations in Kerr spacetime. 58 A . l Mathematical miscellanea 211 A.2 Tensor notation • • • 211 A.3 Derivative notation • • • • 212 A.4 Manifolds and Riemannian spaces 212 A . 5 Causal structure of spacetime ; 212 B . l Centered difference operators 213 B.2 Forward difference operators 214 B.3 Backward difference operators • • • 214 B.4 Quadratic fix operators 215 B.5 Adjacent quadratic fix operators. . 215 B.6 Interpolation and extrapolation operators 216 B.7 Kreiss-Oliger dissipation operators 216 B.8 Integration rules for equally spaced abscissae . • • • -'217 L I S T O F F I G U R E S vii LIST OF FIGURES 2.1 A schematic depiction of the 3+1 decomposition. 11 3.1 Penrose diagram showing causal structure of Reissner-Nordstrom spacetime 26 3.2 Conformal diagram for gravitational collapse and apparent horizon formation for a spherical shell of charged dust when the charge is too small to prevent spacelike singularity formation in its interior 27 3.3 Conformal diagram for gravitational collapse and apparent horizon formation for a spherical shell of charged dust when the charge is large enough to prevent spacelike singularity formation in its interior. 28 • 3.4 Conformal diagram for gravitational collapse and horizon formation for a spherically symmetric distribution of charged scalar field matter 29 3.5 Discrete self-similarity for the critical solution of A with e = 1.0, m$ = 0.0 '. 39 3.6 Power-law scaling of the black hole mass JTIBH for family A with e — 1.0 and m$ = 0.0. 40 3.7 Power-law scaling of the black hole mass QBH for family A with e = 1.0 and m$ = 0.0. 41 3.8 Charge to mass scaling of the black hole solutions for family A with e = 1.0 and m$ = 0.0. 42 3.9 max(2m(t,r)/r) of the critical periodic state for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields evolved from initial data family F 44 3.10 A closer view of max(2m(t,r)/r) showing oscillations within the critical periodic state for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields evolved from family F. . . . 45 3.11 A view of r(max(2m(£, r)/r)) within the critical periodic state for e = 1.0, ra$ = 1.0 Maxwell-Klein-Gordon fields evolved from family F 46 3.12 Modulus of the scalar field amplitude at r = 0 for the critical periodic state of e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields evolved from family F 47 3.13 Charge contained within spheres of radius r for the critical periodic state from e = 1.0, 771$ = 1.0 Maxwell-Klein-Gordon fields evolved from initial data family F. . 48 3.14 Lifetime of the critical periodic state for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields evolved from initial data family F 49 3.15 Mass gap for (supercritical) black hole solutions for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields evolved from initial data family F 51 3.16 Minimum charge of (supercritical) black hole solutions for e = 1.0, rn$ = 1.0 Maxwell-Klein-Gordon fields evolved from initial data family F 52 3.17 m&x(2m(t,r)/r) and |$(t, 0)| in coordinate time t for family F 54 3.18 max(2m(t,r)/r) and |$(£, 0)| in coordinate time t for family F 55 3.19 Lifetime of the critical periodic state for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields evolved from initial data family F on three successive grid resolutions 56 4.1 The effective (Regge-Wheeler) potential for massless scalar perturbations in M = 0 Minkowski spacetime 60 4.2 The effective (Regge-Wheeler) potential for massless scalar perturbations in Schwarzschild spacetime with M = 1 61 4.3 The effective potential for scalar massless perturbations in Schwarzschild spacetime with M = 5 62 LIST OF FIGURES viii 4.4 T h e effective potent ia l for electromagnetic per turbat ions i n Schwarzschi ld spacetime w i t h M = 5 63 4.5 T h e effective potent ia l for = 0.3 massive scalar per turbat ions i n Schwarzschi ld spacetime w i t h M = 5 64 4.6 T h e effective potent ia l for = 0.6 massive scalar per turbat ions i n Schwarzschi ld spacetime w i t h M = 5 65 4.7 T h e I = 5 mode effective potent ia l for my = 0 . 3 massive scalar per turbat ions i n Schwarzschi ld spacetime w i t h M = 5 66 4.8 T h e I — 10 mode effective potent ia l for = 0.6 massive scalar per turbat ions i n Schwarzschi ld spacetime w i t h M = 5 67 4.9 F rac t iona l t r ansmi t ted energy for spherical ly symmetr ic massless scalar per turba-t ions scat ter ing off a Schwarzschi ld black hole 69 4.10 T h e glory effect 70 4.11 O r b i t i n g resonances 71 4.12 Hor izons of K e r r spacetime 77 4.13 Penrose d iag ram showing causal s tructure of K e r r spacetime 78 4.14 A d a p t i v e mesh refinement (to a m a x i m u m of four levels) of the energy densi ty for massive electromagnet ical ly coupled per turbat ions i n a = M = 5 K e r r spacetime. . . 93 4.15 T h e g lory effect and o rb i t i ng resonances for a = M = 5, m $ = 0.0 scalar per turbat ions . 98 4.16 T h e glory effect and o rb i t ing resonances for a = M = 5, m $ = 0.0 scalar per turba-tions at late t ime 99 4.17 Integrated normal ized energy density for massless scalar per turbat ions 100 4.18 N o r m a l i z e d energy density at t = 53.98 for a = M = 5, m * = 0.0 scalar per turbat ions. 101 4.19 Integrated normal ized energy densi ty for massive scalar per turbat ions 103 4.20 M o d e t r app ing and o rb i t ing resonances for a = M = 5, m $ = 0.3 scalar per turbat ions . 104 4.21 M o d e t r app ing and orb i t ing resonances for a = M = 5, m $ = 0.3 scalar per turba-t ions at late t ime 105 4.22 Loca l i z a t i on , t i d a l d is tor t ion , and rap id decay of m<j> = 2.4 scalar per turbat ions i n a = M = 5 K e r r spacetime 106 4.23 Integrated normal ized energy density for m $ = 0.3 scalar per turbat ions 107 4.24 Integrated normal ized energy densi ty for = 2.4 scalar per turbat ions 108 4.25 In i t i a l magnet ic field d i s t r ibu t ion for a = M = 5, electromagnetic per turbat ions . . . 109 4.26 T h e glory effect and o rb i t ing resonances for a = M = 5, electromagnetic p e r t u r b a t i o n s . I l l 4.27 T h e g lory effect and o rb i t ing resonances for a = M — 5, e lectromagnetic per turba-tions at late t ime 112 4.28 Integrated normal ized energy density for electromagnetic per turbat ions i n M = 5 K e r r spacetime 113 4.29 Integrated normal ized energy density for electromagnetic per turbat ions i n M = 5 K e r r spacetime 114 4.30 N o r m a l i z e d energy densi ty at the superradiant scat ter ing peak t ime t = 53.98 for electromagnetic per turbat ions i n an a = M = 5 K e r r spacetime 115 4.31 BaBa for a = M = 5, electromagnetic per turbat ions 117 4.32 EaEa for a = M = 5, electromagnetic per turbat ions 118 4.33 Integrated squares of electric and magnet ic fields i n a = M — 5 K e r r spacetime. . . . 119 4.34 A n g u l a r m o m e n t u m density for electromagnetic per turbat ions i n a = M = 5 K e r r spacetime 121 4.35 A n g u l a r momen tum density for electromagnetic per turbat ions at late t ime i n a = M = 5 K e r r spacetime 122 4.36 A n g u l a r m o m e n t u m density for electromagnetic per turbat ions at superradiant scat-ter ing peak t ime t = 53.98 i n a = M = 5 K e r r spacetime 123 4.37 A n g u l a r m o m e n t u m density for electromagnetic per turbat ions i n a = 0, M = 5 K e r r spacetime 124 L I S T O F F I G U R E S ix 4.38 Angular momentum density for electromagnetic perturbations at late time in a = 0, M = 5 Kerr spacetime 125 4.39 Angular momentum density for electromagnetic perturbations at t = 53.98 in a = 0, M = 5 Kerr spacetime : 126 4.40 Integrated angular momentum density for electromagnetic perturbations in M - 5 Kerr spacetime 127 4.41 Integrated angular momentum density for electromagnetic perturbations in M = 5 Kerr spacetime 128 4.42 The glory effect and orbiting resonances for a — M = 5, m$ = 0.0, e = 0.6 coupled perturbations 130 4.43 The glory effect and orbiting resonances for a = M = 5, m$ =0.0, e = 0.6 coupled perturbations at late time 131 4.44 Integrated normalized energy density for e = 0.6 coupled massless perturbations in M = 5 Kerr spacetime 132 4.45 Evolution of the scalar contribution to the total energy for a = M = 5, m$ = 0.0, e = 0.6 coupled perturbations • • • 134 4.46 Evolution of the electromagnetic contribution to the total energy for a — M = 5, m$ = 0.0, e = 0.6 coupled perturbations 135 4.47 Evolution of the coupling terms contributing to the total energy for a = M = 5, m$ = 0.0, e = 0.6 perturbations • 136 4.48 Decomposition of integrated normalized energy density for e = 0.6 coupled massless perturbations in a = M — 5 Kerr spacetime 137 4.49 BaBa for a = M = 5, e = 0.6 coupled perturbations. 138 4.50 EaEa for a = M = 5, e = 0.6 coupled perturbations. 139 4.51 Integrated squares of electric and magnetic fields in a = M = 5 Kerr spacetime for e = 0.6 coupled perturbations. 140 4.52 Angular momentum density for a = M = 5, e = 0.6 coupled perturbations 141 4.53 Angular momentum density for a = 0, e = 0.6 coupled perturbations 142 4.54 Integrated angular momentum density for e = 0.6 coupled massless perturbations in M = 5 Kerr spacetime 143 4.55 Charge density showing separation in the a = M = 5 spacetime for e = 0.6 pertur-bations 144 4.56 Charge density showing separation in the a = 0, M = 5 spacetime for e = 0.6 perturbations 145 4.57 Integrated charge density for e = 0.6 coupled massless perturbations in M = 5 Kerr spacetime 146 4.58 Axia l current, J$, oscillating in a = M = 5 Kerr spacetime for e = 0.6 coupled perturbations 148 4.59 Axia l current, J ^ , oscillating in a = 0, M = 5 Kerr spacetime for e = 0.6 coupled perturbations 149 4.60 Global maxima of for e = 0.6 coupled massless perturbations in M = 5 Kerr spacetime 150 4.61 Global minima of for e = 0.6 coupled massless perturbations in M — 5 Kerr spacetime 151 4.62 Total normalized energy for massless electromagnetically coupled perturbations in a — M = 5 Kerr spacetime 154 4.63 Total angular momentum for massless electromagnetically coupled perturbations in a'= M = 5 Kerr spacetime 155 4.64 Total charge for massless electromagnetically coupled perturbations in a = M = 5 Kerr spacetime 156 4.65 Total normalized energy density in a = M = 5 Kerr spacetime for e .= 2.4 massless perturbations—early phase of evolution 158 LIST OF FIGURES 4.66 Total normalized energy density at t •= 16.06 in a = M = 5 Kerr spacetime for e = 2.4 massless perturbations. . . . ; 159 4.67 Total angular momentum density in a = M = 5 Kerr spacetime for e = 2.4 massless perturbations—early phase of evolution 160 4.68 Total angular momentum density at t = 16.06 in o = M = 5 Kerr spacetime for e = 2.4 massless perturbations 161 4.69 Charge density in o. = M = 5 Kerr spacetime for e = 2.4 massless perturbations— early phase of evolution 162 4.70 Charge density at t = 16.06 in a = M = 5 Kerr spacetime for e = 2.4 massless perturbations 163 4.71 Total normalized energy density in a = M = 5 Kerr spacetime for e = 0.6, m$ = 0.3 perturbations 165 4.72 Total normalized energy density at t = 77.11 and t = 154.22 in a = M = 5 Kerr spacetime for e = 0.6, m$ = 0.3 perturbations 166 4.73 Normalized scalar energy density in a = M = 5 Kerr spacetime for e = 0.6, m$ = 0.3 perturbations 167 4.74 Normalized scalar energy density at t = 77.11 and t = 154.22 in a = M = 5 Kerr spacetime for e = 0.6, m$ = 0.3 perturbations 168 4.75 Normalized electromagnetic energy density in a = M = 5 Kerr spacetime for e = 0.6, m<j> = 0.3 perturbations 169 4.76 Normalized electromagnetic energy density at t = 77.11 and t = 154.22 in a = M = 5 Kerr spacetime for e = 0.6, TO$ = 0.3 perturbations. . '. 170 4.77 Normalized coupling energy density in a = M = 5 Kerr spacetime for e = 0.6, m$ = 0.3 perturbations 171 4.78 Normalized coupling energy density at t = 77.11 in a = M = 5 Kerr spacetime for e = 0.6, m$ = 0.3 perturbations 173 4.79 Decomposition of integrated normalized energy density for m$ = 0.3, e = 0.6 per-turbations in a = M = 5 Kerr spacetime 174 4.80 Normalized total energy density at t = 77.11 in a = 0, M = 5 Kerr spacetime for e = 0.6, m$ = 0.3 perturbations 175 4.81 Integrated normalized energy density for m$ = 0.3, e = 0.6 perturbations in M = 5 Kerr spacetime 176 4.82 Angular momentum density at t = 0,77.11,154.22 for e = 0.6, m$ = 0.3 perturba-tions in a = M = 5 Kerr spacetime: 177 4.83 Angular momentum density at t = 77.11 for e = 0.6, TO$ = 0.3 perturbations in a = 0, M = 5 Kerr spacetime 178 4.84 Integrated angular momentum density for e = 0.6, m$ = 0.3 perturbations in M = 5 Kerr spacetime. . 179 4.85 Charge density at t = 0,77.11,154.22 for e — 0.6, m$ = 0.3 perturbations in a = M = 5 Kerr spacetime 180 4.86 Charge density at t = 77.11 for e = 0.6, m$ = 0.3 perturbations in a — 0, M = 5 Kerr spacetime 181 4.87 Integrated^ charge density for e = 0.6, m$ = 0.3 perturbations in M = 5 Kerr spacetime , 182 4.88 Axia l current component, J ^ , at t — 0,77.11,154.22 for e = 0.6, ra$ = 0.3 pertur-bations in a = M = 5 Kerr spacetime 183 4.89 Axia l current component, J^, at t = 77.11 for e = 0.6, m$ = 0.3 perturbations in a = 0, M = 5 Kerr spacetime 184 4.90 Global maxima of for e = 0.6, m$ = 0.3 perturbations in M = 5 Kerr spacetime. 185 4.91 Global minima of for e = 0.6, m$ = 0.3 perturbations in M = 5 Kerr spacetime. . 186 4.92 BaBa at t = 0,77.11,154.22 for e = 0.6, ?n$ = 0.3 perturbations in a = M = 5 Kerr spacetime 187 LIST OF FIGURES x i 4.93 EaEa at t = 0,77.11,154.22 for e = 0.6, m* = 0.3 perturbations in a = M = 5 Kerr spacetime .188 4.94 Integrated squares of electric and magnetic fields for e = 0.6, m$ = 0.3 perturbations in a = M = 5 Kerr spacetime ' 189 4.95 Convergence of total normalized energy for e = 0.6, m$ = 0.3 perturbations in a = M = 5 Kerr spacetime 190 4.96 Convergence of total angular momentum for e = 0.6, ra$ = 0.3 perturbations in a = M = 5 Kerr spacetime 191 4.97 Convergence of total charge for e = 0.6, m$ = 0.3 perturbations in a = M. = 5 Kerr spacetime 192 4.98 Convergence factors, C, for energy density, angular momentum density and charge density of e = 0.6, ?n$ = 0.3 perturbations in a = M = 5 Kerr spacetime. 193 4.99 Independent residual norms \\rh\\2 from the equations Arr, Agr and A$r 195 4.100Independent residual norms \\rh\\i from the equation $0 196 4.101Relative errors for energy density, angular momentum density and charge density for e = 0.6, m<j> = 0.3 perturbations in a — M = 5 Kerr spacetime . 198 4.102Errors for normalized integrated energy, integrated angular momentum and inte-grated charge of e = 0.6, m$ = 0.3 perturbations in o = M = 5 Kerr spacetime. . . . 199 C . l A schematic representation of the finite difference grid in one spatial and one tem-poral dimension . 222 C.2 The four-point stencil of the leap-frog scheme for the advection equation 224 C.3 The four-point stencil of the Crank-Nicholson scheme for the advection equation. . . 225 C.4 The CFL condition for a stable leap-frog scheme 227 C.5 The five-point stencil of the second order FDA for the Poisson equation off boundary in the two dimensions r and 6 235 C.6 The two-point stencil of the second order FDA for the Poisson equation in the two dimensions r and 9 at boundary 9 = 0 236 C.7 The two-point stencil of the second order FDA for the Poisson equation in the two dimensions r and 9 at boundary r = 0 237 C.8 A schematic representation of the A M R grid hierarchy in two spatial dimensions. . . 240 C. 9 Schematic representation of domain decomposition 243 D. l MIB interpolating function f(p) 246 D.2 MIB interpolating function g(z) . 247, D.3 Radially outgoing geodesic \+(t, p) in the MIB coordinates 248 D.4 Radially ingoing geodesic (t, p) in the MIB coordinates 249 D.5 Axially outgoing geodesic Afj_(£, z) in the MIB coordinates 250 D.6 Axially ingoing geodesic \z_(t,z) in the MIB coordinates 251 D.7 MIB coordinates and dissipation acting on PQ for m$ = 1.0, e = 1.0 Maxwell-Klein-Gordon fields V 262 D.8 Error in total normalized energy for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields. 263 D.9 Error in angular momentum for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields. . . 264 D.10 Error in charge for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields 265 D . l l m$ = 0.0, e = 0.0 Klein-Gordon field dynamics in flat spacetime 267 D.12 m$ = 0.3, e = 0.0 Klein-Gordon field dynamics in flat spacetime 268 D.13 m$ = 2.4, e = 0.0 Klein-Gordon field dynamics in flat spacetime .' 269 D.14 Maxwell field dynamics in flat spacetime 271 D.15 Angular momentum of the Maxwell fields in flat spacetime 272 D.16 Massless e = 0.6 Maxwell-Klein-Gordon field dynamics in flat spacetime 274 D.17 Angular momentum of the massless e = 0.6 Maxwell-Klein-Gordon fields in flat spacetime 275 LIST OF FIGURES xii D. 18 Angular momentum of the massless e = 0.6 Maxwell-Klein-Gordon fields in flat spacetime 276 D.19 m$ = 0.3, e = 0.6 Maxwell-Klein-Gordon field dynamics in flat spacetime 278 D.20 Angular momentum of the m$ = 0.3, e = 0.6 Maxwell-Klein-Gordon fields in flat spacetime. . 279 D.21 Angular momentum of the m$ = 0.3, e = 0.6 Maxwell-Klein-Gordon fields in flat spacetime. 280 D.22 Convergence of total normalized energy for e = 0.6, m$ =.0.3 fields in the M I B flat spacetime. 281 D.23 Convergence of total angular momentum for e. = 0.6, m$ = 0.3 fields in the M I B flat spacetime 282 D.24 Convergence of total charge for e = 0.6, m$ = 0.3 fields in the M I B flat spacetime. . 283 A C K N O W L E D G E M E N T S Xl l l A C K N O W L E D G E M E N T S I would like to express gratitude to my advisor Matthew Choptuik, and thesis committee members Kris t in Schleich, Douglas Scott and Wil l iam Unruh for their efforts and assistance. Thanks are also forwarded to Frans Pretorius for use of his parallel adaptive mesh refinement routine library. And I especially thank Andrea and Anastazia Petryk, along with family and friends for their encouragement arid support. Results presented in this thesis were computed primarily on the vn, vnp4 and WestGrid Glacier clusters located at the University of British Columbia. Substantial use was also made of the WestGrid Network Storage Facility at Simon Fraser University. The vn and vnp4 clusters are funded and supported by the British Columbia Knowledge Development Fund, the Canada Foundation for Innovation, the Canadian Institute of Advanced Research, the Natural Sciences and Engineering Research Council of Canada and the University of British Columbia. Funding and support for Westgrid Glacier and the Westgrid Network Storage Facility is provided by the Alberta Science and Research Authority, the British Columbia Knowledge Development Fund, the Canada Foundation for Innovation, the University of British Columbia, Simon Fraser University and T R I U M F . I also thank the Natural Sciences and Engineering Research Council of Canada, the R. Howard Webster Foundation and the University of British Columbia for their support. C H A P T E R 1. I N T R O D U C T I O N 1 CHAPTER 1 INTRODUCTION Of all conceptions of the human mind perhaps the most fantastic is the black hole. Black holes are neither bodies nor radiation. They are clots of gravity. 1 This dissertation concerns the numerical study of black hole formation from charged, massive scalar fields in spherical symmetry, and the axially symmetric scattering and accretion of such matter onto rotating black holes. In the gravitational collapse to a black hole, and in accretion onto black holes, astrophysically realistic matter will likely be magnetohydrodynamic ( M H D ) — fluid-like, and ionized. Only in very few highly symmetric cases has the fully general relativistic problem (where matter sources dynamically influence the curvature of spacetime) been completely studied. Even then, computational techniques are required. Shock formation in fluid systems further complicates the details. Numerical simulation of general relativistic M H D systems is in its infancy, and insight gained through the study of M H D analogues like the charged, massive scalar field.is required for progress. Furthermore, scalar field models can represent the pions 7 T ° , 7r + and 7 T ~ , and have been proposed as candidates for early universe and dark matter. The study of charged, massive scalar fields is therefore interesting in and of itself. Finally, there is the connection to matter models in which critical behaviour has been studied at the threshold of gravitational collapse and open questions in that regard. Essential to a complete understanding of the physics are both: (1) spherically symmetric in-vestigation of the general relativistic problem; and (2) analysis of the dynamics in axial symmetry. While the spherically symmetric investigation provides information on how the matter dynamically influences spacetime curvature, the effects of magnetic fields and electromagnetic radiation can not be extracted until the imposed symmetry is (at most) axial. The charged, massive scalar field system has not been studied in much detail in these regards, so any additional insight regarding its physics wil l certainly be useful. 1.1 Scalar Field Critical Collapse In 1993 Choptuik published the results of an interesting phenomenon in gravitational collapse [23]. There he gave answers to the questions "What happens at the threshold of gravitational collapse?" and "What happens in the limit of increasingly small mass black hole formation?" The system in which he first observed these critical phenomena was the massless scalar field minimally coupled to gravity. Since that time, critical phenomena have been observed in a variety of different matter models. Those I will mention are but a small subset of those studied, but they do form the set most important to the discussions which follow. Specifically, I will mention works by Hod and Piran [66], [68], Brady, Chambers and Goncalves [12], [16], Hawley and Choptuik [63], [64], [65], as well as an analysis by Gundlach and Martin-Garcia [57]. Finally, relatively complete overviews of critical phenomena in gravitational collapse have also been provided by Gundlach in [59] and [60]. In brief, for the minimally coupled massless Klein-Gordon field in spherical symmetry, Choptuik considered families of solutions S\p) with the property that the critical value of the parameter p = p* separates solutions which form black holes from those which do not [23]. He found evidence that supported two conjectures. First, he obtained numerical evidence supporting the universality of the solution p —> p* and that it forms structure on arbitrarily small spatiotemporal scales. That 1 Quoted from [116]. C H A P T E R 1. INTRODUCTION 2 is, not only were all solutions approaching criticality "the same", but they also exhibited a type of self-similarity in space and time as the evolution proceeded. Second, he observed that there exists • a mass-relation for the solutions which formed black holes. Specifically, that the mass of the black hole obeyed a power law m B H oc | p - p * | 7 m , . (1.1) where jm « 0.37, is a universal exponent. In principle, the mass of the black hole can be made arbitrarily small (i.e., there is no mass gap in the black hole spectrum). In analogy to phase transitions in condensed matter systems, this continuous scaling of the black hole mass has since been identified as Type II critical behaviour in black hole formation. Key to the discussion of his results was the observation that, expressed in the usual (t, r, 9, <fi) spherical coordinates, his system was invariant under the rescalings t -> kt , . (1.2) r —> kr , (1.3) k being an arbitrary positive constant. Expressing his variables in terms of the form-invariant quantities X(t, r) and Y(t, r) , he found there exists a unique sequence of this set of variables [X, Y] in the limit of the exactly critical solution [X, Y] —> [X*,Y*]. Expressing this sequence in terms of the logarithmic spatial and temporal variables r = r(t) and p = p(r), Choptuik found that for [X,Y\-*[X*,Y*] [X*(T-AT,P-AP),Y*(T-AT,P-AP)}~[X*(T,P),Y*(T,P)} , (1.4) where A T ~ A ~ A p is a universal constant with approximate value A ss 3.4. Specifically, Choptuik found A T « 3.43 , A p « 3 . 4 4 . •' (1.5) This means that every time the profiles X, Y are re-examined after evolution of geometric time interval STo, To = To(t) being the proper time of a stationary observer at r = 0, the solutions essentially appear rescaled by a factor of exp(A) « 30. Therefore, there exists a discrete self-similarity in the solution in the form of a series of ever-decreasing echoes as the scalar field evolves on the threshold of black hole formation. The universality of critical phenomena can be understood in terms of perturbation theory. Consider linear perturbations of the exactly critical solution. The perturbed critical solution will have a number of modes, both stable and unstable. The stable modes decay, thus driving the perturbed solution to the critical solution. The unstable modes, however, drive the system away from the exactly critical solution. In the case at hand, the unstable modes would drive the system to either black hole formation or dispersal. The exactly critical solution has only one unstable mode. It is through the existence of this single mode that the universality of the solution can be explained—tuning similar families of initial data to the critical solution, the system is driven to the same critical point. Furthermore, perturbation theory analysis explains how the eigenvalue of the unstable mode is directly correlated with the black hole scaling exponent 7m, and correctly predicts its value. In subsequent research, Brady, Chambers and Goncalves examined critical phenomena in the case of a minimally coupled massive scalar field [12], [16]. Their results showed that, for the massive case, when the radial extent of the initial pulse A is less than the Compton wavelength of the scalar field m _ 1 , then Type II critical behaviour is observed. The critical solution observed in this case is exactly that which was found by Choptuik. However, they also observed that there exists a minimum mass black hole that can be formed when the radial extent of the initial pulse is much greater than the Compton wavelength (that is, a mass gap exists in the black, hole spectrum when A ^> m~1). Again, in analogy to condensed matter systems, such behaviour has come to C H A P T E R 1. I N T R O D U C T I O N 3 be known as Type I critical phenomena.2 In the regime Of Type I solutions, the exactly critical solutions are unstable soliton stars rather than the naked singularities of the Type II case. In fact, it is the presence of the mass parameter m which destroys the scaling of the system under transformations (1.3) and (1.2). Hence, a different type of critical solution is expected and observed. Noting similarities between the soliton stars examined by Brady, Chambers and Goncalves, and boson stars composed of massive complex scalar fields, Hawley and Choptuik investigated the behaviour of boson stars driven to the brink of black hole formation [63], [64], [65]. Hawley and Choptuik thereby found Type I critical behaviour for massive complex scalar fields, the exactly critical solutions being perturbed and periodically oscillating boson stars, their lifetimes obeying a well-defined scaling relation as p —> p*. Finally, Gundlach and Martin-Garcia [57] discussed the possibility that, as in the case of the massive minimally coupled scalar field, the presence of a parameter breaking the invariance under transformations (1.3) and (1.2) could lead to behaviour in the electromagnetically coupled complex scalar field different from that first found by Choptuik. The parameter in the electromagnetically coupled complex scalar field system which breaks this invariance is the charge coupling parameter e. However, Gundlach and Martin-Garcia performed an analysis which showed that, as the system evolves, the effects of charge will play a decreasingly significant role. Specifically, assuming a self-similar solution and also that field equations in this system always contain dominant contributions from e in the combination er, a rescaling r —> exp(—A)r effectively scales e as e —> exp(—A)e. Therefore, after each echo, the effects of the charge coupling will decrease exponentially with A. In fact, they showed that in critical collapse the black hole mass should scale as r r i B H oc \p-p*\lm , (1.6) where 7 m « 0.37 (just as found by Choptuik [23]), and that the charge of the black hole should scale as Q B H O C | P - P * P , « , . (1.7) where ~JQ W 0.88. In fact, they predicted that for other charged'matter models the relationship between -ym and JQ should obey 7Q > 2jm. They also predicted the solutions would be universal. Furthermore, they postulated that if a second dimensionful parameter such as scalar field mass m<j> is added to the system, then universality will not hold generically, but might be recovered in the limit when e/m$ —> 0 or m$/e —> 0. In conclusion, they also suggested the value of |e|/m$ could determine whether the system displays a Type I mass scaling typical of perturbed boson star critical solutions. Soon after Gundlach and Martin-Garcia published their results, Hod and Piran performed the numerical study of the charged massless minimally coupled scalar, field [66], [68]. As predicted by Gundlach and Martin-Garcia, they found mass and charge scaling relations with exponents of about 7 m w 0.37 and JQ > 2jm. They also verified that the infinitesimal mass black holes formed were neutral or obeyed the relationship QBH <C TOBH-To date, nothing has been published in the way of massive electromagnetically coupled scalar field critical phenomena, and only modest work has been presented in hypothesis of the details. The numerical results presented in Chapter 3 from my study of black hole critical phenomena of charged massive scalar fields are the first of their kind. 1.2 Scattering and Scalar Accretion in Kerr Spacetime From the Rutherford scattering of alpha particles off atomic nuclei, to modern experiments like those at the Large Hadron Collider and proposed TeV electron-positron experiments at the Next Linear Collider, particle scattering has revealed, and will continue to reveal, the physical nature of 2lt should be noted that, even before the results of Brady, Chambers and Goncalves, Type I critical phenomena was known to exist for the minimally coupled Yang-Mills field [26]. C H A P T E R 1. I N T R O D U C T I O N 4 the subatomic world. Photon scattering in its various forms has also provided much information on the physics of matter—Thomson scattering has shown that atomic constituents can have both positive and negative charge, Rayleigh scattering explains why the sky appears blue, Compton scat-tering has revealed the corpuscular nature of the photon, Raman scattering provides information on the excited energy states of a target material, and Bragg scattering has determined the lattice spacing of crystalline substances. Particle and photon scattering experiments could, in principle, also be performed on black holes. Under the hypothesis that the details would reveal interest-ing physics, there has been rather detailed theoretical investigation of black hole wave scattering. While much of the work has involved obtaining closed-form solutions, there recently has been much effort in solving the black hole scattering problem using numerical techniques. Rather than provide a complete overview of the developments I will simply address previous work in the field when it pertains directly to the discussion of my results, and refer the reader to [49] and [51] for general reference. There are essentially three phases in the dynamics of any matter scattering in a black hole spacetime: (1) an early phase which in many ways resembles the initial dynamics of the same matter in a Minkowski spacetime; (2) an intermediate phase in which curvature effects become apparent; and (3) an asymptotic late phase which can in many cases be described by a power law decay in time. The late phase evolution has been studied in some detail in the context of quasi-normal (i.e., damped, complex frequency) black hole modes.3 Since certain frequencies of the quasi-normal modes should be independent of the process that generates them, these frequencies are characteristic of, and directly connected to, the black hole parameters (mass, charge, and angular momentum). They thereby provide a means of identifying the black hole spacetime. While, by definition, scattering involves dynamics governed predominantly by interactions over a limited range of space and a relatively short interval of time, the physics of interest in accretion is also over a relatively small region of space but now persists over a prolonged period of time. In black hole accretion, there are basically two modes of flow: (1) Bondi-like flow where angular momenta are low and centrifugal forces are too weak to balance gravitational attraction and provide an equilibrium state; and (2) thin-disc where angular momenta are sufficient to generate a stationary state. Bondi-like flow qualitatively resembles spherical accretion, while thin-disk is typified by matter in Keplerian and super-Keplerian orbits. Consequently, locally dissipated energy, entropy and angular momentum are advected or carried along by the flow and lost to the black hole in Bondi-like accretion. Meanwhile, in thin-disc accretion, matter exists at a fixed radius for many orbits allowing for the opportunity to locally transfer substantial energy, entropy and angular momentum. In black hole accretion, astronomical evidence shows that a portion of inspiralling plasma and generated electromagnetic radiation collimate into bipolar jets from the black hole. Only modest progress has been made through the study of magnetohydrodynamical systems, and the details are not well understood. It is assumed electromagnetic fields play a significant role in the collimation process, but the details are again vague. Some notable computational simulations of this accretion and collimation process have been presented in [42], [82], [107], [83] and [84]. Since the knowledge is incomplete, there is clearly interest to investigate possible electromagnetic effects in black hole accretion and scattering phenomena through matter models simpler than magnetohydrodynamics. As a step towards this goal, and in order to distinguish, between matter and gravitational effects— essential to discussion of the fully general relativistic system—Chapter 4 of this dissertation presents results from my axially symmetric study of charged, massive scalar field (i.e., the Maxwell-Klein-Gordon system) accretion and scattering in rotating black hole spacetimes. In this situation, I treat the scalar field as test matter, (it evolves about the black hole, but does not itself contribute to the curvature of spacetime) and observe its dynamics. To my knowledge, this study is the first of its kind. 3 [85] provides a good summary of the research before 1999. C H A P T E R 1. I N T R O D U C T I O N 5 1.3 Outline In Chapter 2, I present methodology in terms of the Arnowitt-Deser-Misner (ADM) formalism,, focusing specifically on the polar areal coordinate system. I then discuss the initial value formalism for electromagnetism and general relativity, and close with brief discussion of regularity conditions. Chapter 3 presents the study of critical solutions for massive charged scalar fields in spherical symmetry, beginning with a presentation of the system of equations. Among the results are both Type I and Type II solutions and the scaling relations associated with each. By far the most voluminous is Chapter 4 where I present the investigation of massive electro-magnetically coupled scalar field dynamics in Kerr spacetime. I provide discussion of the effective potential formalism and its implications. Following that, I provide the system of equations, and then proceed with presentation of my results. Included in the results are electromagnetically uncoupled (i.e., e = 0) Klein-Gordon field dynamics, Maxwell field dynamics, and finally, electro-magnetically coupled Maxwell-Klein-Gordon equation dynamics in Kerr spacetime. These results cover a range of scalar field mass, electromagnetic coupling, and black hole angular momentum pa-rameters. Among the results shown are orbiting resonances, superradiant scattering and trapping of fields in the black hole spacetime. Also included are solutions demonstrating the preference for Maxwell-Klein-Gordon fields to scatter along the axis of black hole rotation under a certain set of conditions, arid also the generic emergence of charge separation and dynamo-like behaviour when, the Maxwell and Klein-Gordon fields are coupled electromagnetically through charge. Chapter 5 provides a summary and outlook, Appendix A provides a glossary of notation, and Appendix B provides a summary of finite difference and discrete operators used throughout the dissertation. Appendix C provides discussion of the numerical and computational techniques used to obtain all the presented results. Finally, Appendix D provides a detailed account of Maxwell-Klein-Gordon dynamics in Minkowski spacetime. There I discuss the Monotonically Increasingly Boosted (MIB) coordinate system and its utility in treating outer boundary conditions. Along with a presentation of the equations, I provide results to compare and contrast flatspace dynamics to those for the black hole background. These results include Klein-Gordon fields, Maxwell fields, and the Maxwell-Klein-Gordon system for a variety of field mass and charge coupling parameter values. 1.4 Conventions and Units Throughout this dissertation I employ the index notation, where, for example, 'it;:;: ' (1.8) designates the abc... contravariant components and def... covariant components.of tensor T. Indices a, b, c, d, e, / , . . . take on values 0,1,2,3, where 0 typically designates the temporal component and 1,2,3 designate the spatial components in a given coordinate system. In a situation where indices are repeated, the Einstein summation convention is implied as per 3 - -nbat\ = Y,Tdat-- (1-9) a=0 I also employ the metric signature (—, +, +, +), and use Planckian units where h = G = c = 1. A quantity with dimension length7" x mass™ x timep in ordinary units is therefore dimensionless in Planckian units. To retrieve an equation valid in ordinary units one simply multiplies each quantity C H A P T E R 1. I N T R O D U C T I O N 6 in Planckian units by its conversion factor n ( c3 \ (m+n+p)/2 (1.10) Gh In SI units the fundamental constants have values c = 3.00x 10 8m s 1,G = 6.67x 10 n m 3 kg 1 s 2 , and h = 6.626 x 1 0 " 3 4 J s. 1.5 Significant Figures and Uncertainties Finally, where other's numerical results are quoted, all significant figures are assumed accurate. For example, Choptuik's original result, A p w 3.44, is accurate in all three digits. M y numerically computed results, however, are usually accurate in all figures but the last significant digit. Thus, my A « 3.47 is accurate in the first two digits but uncertain in the last. The exception is in cases where my result has but one significant digit. The quantity is then accurate in that single digit. For example, in my k ss 1 x 1 0 - 2 , uncertainties would enter at the level of subsequent figures. 1.6 Web References In the body of this thesis there are a number of references to files found in the World Wide Web. While the availability of all files cannot be guaranteed in the future, efforts wil l be made to maintain access to those whose Uniform Resource Locator (URL) currently has l a p l a c e . p h y s i c s . u b c . c a for host. While there wil l be a change in file U R L in the event of a host change, the path should be preserved. Thus, file U R L s like http:/ /$WWW/People/petryk/animations /9.mpeg wil l still be valid in the future, but $WWW may differ from its current value l a p l a c e .phys ics .ubc. ca . C H A P T E R 2. PHYSICAL F O R M A L I S M 7 C H A P T E R 2 P H Y S I C A L F O R M A L I S M 2.1 The 3+1 or Arnowitt-Deser-Misner (ADM) Formalism In this section I introduce the so-called 3+1 or Arnowitt-Deser-Misner (ADM) formulation of relativity. Relevant historical references are [47], [33], [4], and [148]. While the 3+1 formalism was originally developed for use in quantum gravity (see [4]), this approach has perhaps seen its greatest successes in numerical relativity. I employ the 3+1 approach in my investigations and now just briefly cover some of its key points. The following exposition is'based upon [27] which is itself a reworking of the results in [148]. 2.1.1 Foliations Start with a spacetime manifold M. endowed with a metric gab- Assume the metric is expressed in ' coordinates, where t is the temporal parameter. The spacetime can always be foliated as a sequence { E t } which are isosurfaces to t. These spacelike hypersurfaces are locally described by the dual vector field Qa defined fia = V 0 t , (2.1) and has norm gabnanb = -a-2, (2.2) where a is known as the lapse function (or simply the lapse). Given this, I can construct a unit-norm dual vector, na, as per na = -aQa. = -oiVat, (2.3) where the sign is chosen so that the related unit-norm vector, na, na = gabnb (2.4) is future-directed. na can be viewed as the four-velocity of an observer moving orthogonally to the hypersurfaces of S j . Finally, such an observer wil l have a four-acceleration, aa, according to aa = nbVbna. (2.5) 2.1.2 Projections In the 3+1 formalism I aim to decompose tensors into their single temporal (hypersurface-normal) and three spatial (hypersurface-tangential) parts. To extract the temporal part of vector Wa I simply contract with na as per W* = -Wana. (2.6) For a covector Wa I contract with na Wh = +Wana. (2.7) There are two items to note in the preceding pair of equations. First, the superscripts and subscripts ii designate the original tensor index has been contracted with na or na. Second, the choice of sign — or + is a convention adopted directly from that of [148]. Meanwhile, partial components are C H A P T E R 2. PHYSICAL F O R M A L I S M extracted with the rank-two projection tensor ± a b = 6% + nanb. (2.8) Then any tensor T^f" which has had all its free indices projected out with ± a b in the manner ••itt: = - V - V - r , • • • • • • 7^!::. (2-9) is referred to as a spatial tensor. Above and onwards _L (with no indices) is shorthand to designate that spatial projection is applied to all free indices in the trailing tensor. When applied to all free indices of the metric tensor, gab, projection yields the spatial metric labi •jab = -i-9ab = 9ab + nanb. (2.10) By extension, the contravariant spatial metric is given by 7 a b = ±gab = gab + n a n 6 _ ^.11) Note that tensor indices of continue to be raised and lowered with spacetime metric gab—not with 7 a(,—as jab and 7 a b are not inverses in the four-dimensional spacetime manifold. The only time jab can be used to raise or lower indices is when the tensor acted upon is spatial (i.e., it has no temporal piece). 2.1.3 Covariant Derivatives on the Spacelike Hypersurface The spatial projection tensor can then be applied to the operator V a to define the natural derivative operator Da for spatial tensors, Da = ±Va. (2.12) The action of Da on a spatial tensor UT^f" would then be Dj±T$;;; = . V , . 7^;;;. . • (2.13) where shorthand _L again indicates I apply the projection operator to all free tensor indices. Since Da^bc = J - V 0 7 6 c = -LV a(c/6c"+ nbnc) = ASIa(nbnc) = _ L ( n c V a n 6 + n 6 V a n c ) = ••••= 0, (2.14) and similarly for Dajbc, Da is clearly the natural derivative operator for the spatial metric jab-2.1.4 Intrinsic and Ex t r ins i c Curva ture Given the covariant derivative on the spacelike hypersurface Da, it is natural to define the curvature of the spacelike hypersurface, 7£ a b (f, via (DaDb- DbDa) ±WC = TZabcd±Wd, (2.15) where Wa is some covariant vector. 7c a b c d is referred to as the intrinsic curvature. ~R-abd is itself a spatial tensor and possesses the expected symmetries TZ(ab)cd = TZab(cd) = 0, (2.16) T^abcd = T^cdab, (2-17) and Klabc]d = 0. . (2.18) C H A P T E R 2. PHYSICAL F O R M A L I S M 9 In continuation, the spatial Ricci tensor , lZab, and scalar, TZ, are defined via •Rab = nacbc (2.19) and K = Kaa. (2.20) While the intrinsic curvature describes the geometry of the spacelike hypersurfaces, it is the extrinsic curvature that describes how spacelike hypersurfaces are embedded in the spacetime. The extrinsic curvature, Kab, is defined either through covariant differentiation as per Kab = -±Vanb, (2.21) or equivalently via Lie differentiation along the normal vector field as in Kab = - ^ ± £ n 9 a b . (2.22) Using the (easily derived) property - L V a n b = V 0 n 6 + naab • (2.23) in the first form of Kab, I obtain Kab = - V a n 6 - naab, (2.24) which makes clear a relationship between the spatial extrinsic curvature tensor, and the four-velocities and four-accelerations of observers moving orthogonally to the spacelike hypersurfaces. 2.1.5 The Spacetime Constraint Equations Having introduced the necessary elements of the 3+1 formalism, I are now in a position to derive ' the evolution of spacetime from Einstein's field equations, Gab = Rab - \ g a b R = 8nTab. (2.25) I begin by denning three projections of the stress-energy tensor, Tab, p = T m = Tabnanb, (2.26) •a = ±Tan = _± ( T a b n 6 ) , (2.27) gab = ^rpab^ (2.28) p, ja, and Sab, can respectively be interpreted as the local energy density, momentum density and spatial stress tensor as seen by an observer with four-velocity na. Omitting the detailed algebra, I simply state that contracting Einstein's equations twice with na obtains the Hamiltonian constraint TZ + K2 - K\Kba = 167T/9 , (2.29) where K = T r X = Kaa. Likewise, contracting Einstein's equations once with na and then project-ing onto the spacelike hypersurface, I obtain the momentum constraint DbKab — DaK — 8"7rj°. (2.30) Notice the preceding two equations contain only spatial tensors and derivatives, and no explicit C H A P T E R 2. P H Y S I C A L F O R M A L I S M 10 temporal derivatives, as they are truly equations of constraint. Being constraints, they must be satisfied at every time or hypersurface in the foliation {E t } by the values of •jab and Kab (the fundamental 3+1 variables). 2.1.6 Time Derivatives While the spacetime history could, in principle, be obtained from the constraint equations alone, it is often useful to obtain the solutions from evolutionary equations instead. To this end, I must first define the appropriate time derivatives. It turns out that vector field Na defined Na = ana (2.31) is a natural vector along which I could Lie differentiate. Since na = -a€ia, (2.32) and contracting i V a with Q,a is a unit norm, £lana = 1, (2.33) Nana = 1. (2.34) However, there is no reason to be so restrictive, and I could, in principle, construct a more general vector field, ta, appropriate for Lie differentiation, ta = Na + j3a. (2.35) Providing the shift vector pa is spatial Pana = 0, (2.36) I prove ta = is an equally valid choice, since t»n„ = {Na + pa) na = Nana + pana = Nana = 1 (2.37) also has the desired property. The vector pa I just introduced can be interpreted as follows. Four-velocity ta represents a congruence of observers moving in the future direction. In moving from one spacelike hypersurface to the next in foliation {£*}> the observers move into the future an amount na perpendicular to the original hypersurface E t . But on the future hypersurface Y^+du the position of the observers has been spatially shifted by an amount pa. So, only if (3a = 0 is the propagation purely perpendicular to Et- This scenario is depicted in Figure 2.1 below. 2.1.7 Spacetime Evolution Equations W i t h all the preliminaries developed, I are now in a position to state the equations of spacetime evolution. The derivation of evolution equations for the spacetime begins with the Lie derivative definition of the extrinsic curvature, Kab, Kab = - \ £ t 9 a b , (2-38) where for full generality I Lie differentiate along vector ta- rather than na. I thus obtain £tlab = -2aKab + £f3lab, (2.39) C H A P T E R 2. P H Y S I C A L F O R M A L I S M 11 S i Figure 2.1: A schematic depiction of the 3+1 decomposition. na is the unit normal to E t at coordinate (t,xa). Moving in the future direction amount dt, an observer with four-velocity (J^ J = ta = Na+pa = ana+f3a ends up on hypersurface T^t+dt at coordinate (t + dt, xa). a is the lapse function and measures movement into the future. The shift vector f3a measures the spatial displacement of coordinates i n going from E t to Et+dt-CHAPTER 2. PHYSICAL FORMALISM 12 which are the evolution equations of the spatial metric. The procedure to obtain the evolution of the extrinsic curvature is somewhat more lengthy and involves, among other things, contractions of Einstein's field equations and projections of the curvature tensor Rabcd (and its contractions) onto the spacelike hypersurfaces. For brevity I omit the details and simply state the results.1 Evolution equations for the extrinsic curvature thus obtained are £tKab = £pKab - DaDba + a (nab + KKab - 2KacKcb - 8n (sab - ^ a b (S "/»))). (2-40) where S = TrS = Saa. Finally, raising an index produces the often more useful form £tK% = £0Kab - DaDba + a {llab + KK\ + 8n Q i _ ° f c (S - p) - Sab^ ) . (2.41) 2.1.8 Spherically Symmetric Spacetime in Polar Areal Coordinates In a following chapter I study the evolution of spherically symmetric spacetime in the context of critical gravitational collapse of a massive electromagnetically coupled complex scalar field. Since the study employs the 3+1 formalism in the polar areal coordinate system, it is natural now to present the spacetime constraints and evolution equations for that choice of metric. The most general spherically symmetric spacetime metric in (t,r,9, <f>) coordinates is given by line element ds2 = ( - a 2 + a 2/? 2) dt2 + 2a2[3dtdr + a2dr2 + r2b2dfl2, (2.42) where a = a(t,r), (3 = f3(t,r), a = a(t,r), b = b(t,r), and where dn2 = d92 + s in 2 9dcp2 (2.43) is the usual metric on the unit sphere. In the special case 3 = 0, b = 1 (2.44) the line element reduces to that of the polar areal coordinate system ds2 = -a2dt2 + a2dr2+r2d{l2. (2.45)' The future-directed unit-norm spatial hypersurface orthogonal vector is na = (1/a, 0,0,0). The term polar is used because of the usual association of r, 9, and cf> with the polar coordinate system in flat spacetime, while areal is used because the proper area of a two-sphere of radius r, centered at r = 0, is 47rr2. r is thus a geometric quantity. This polar slicing condition places a restriction on the trace of the extrinsic curvature tensor, Kab, K = TrK E E Kaa = K\. (2.46) It is straightforward to show (algebra omitted) this in turn implies K%=K\=Q. (2.47) Finally, since a generalized Birkhoff's theorem tells us that spherically symmetric electrovac solu-tions are uniquely Reissner-Nordstrom, I can identify polar areal metric functions a and a outside I refer the reader to [148] or [27] for the full procedure. C H A P T E R 2. P H Y S I C A L F O R M A L I S M 13 a 2 = ( l - - + T^V (2-50) sources through 2 a ( r 47rr2 J ' where parameters m = m(t, r) and q = q(t, r) respectively represent the gravitational mass and electric charge of the source. Then, rewriting (2.51) I obtain 2\ a? + 47rr2 ) ' the so-called mass aspect function [28]. In an evolutionary problem, the divergence of a2 signals the formation of an apparent horizon—and therefore existence of a black hole—for this choice of coordinates.3 In the limit r —> co the mass aspect function is then interpreted as the total effective mass of the black hole, consisting of the usual A D M mass (see [4]) less the difference of total ingoing and outgoing matter flux at the time of apparent horizon formation [56] . 4 As noted in [21], this total effective mass is the natural measure in the signal of black hole formation for time-dependent problems. Once a2 has diverged, I can integrate the charge density to obtain q(t,r). Knowing all the functions on the right hand side of (2.52) I can immediately deduce m(t,r) for the solution. I now state (without derivation) the values of DaDba, Kab, K, DbKab, DaK, TZab and TZ required by the equations of spacetime constraint and evolution. The non-vanishing contributions are / DrDra = - (-) , D6Dea = D^D^a = (2.57) K \ = - A (2.58) aa 2The last term employs a common convention different from those of [62], [145], [40], [49], and [17]. The factor has been inherited by the current term since I use the Maxwell equation convention VaFab = -Jb, (2.48) rather than VaFab = - 4 T T J 6 . (2.49) 3 The equation for a trapped surface is [44] ^ab _ sagb^ (DaSb _ K a b ) = 0 t ( 2 5 3 ) where sa is a spacelike, outward-pointing, unit-norm to the surface. Using the polar areal coordinate system in spherical symmetry sa is given by s° = ^0, ^>0,0^ . (2.54) In polar areal coordinates, expression (2.53) then evaluates to — = 0 . (2.55) ar Since a becomes arbitrarily large, but never reaches the limit a —> oo, apparent horizons never actually form in this coordinate system. However, apparent horizon formation is imminent as a increases to arbitrarily large values. With some confidence, the location at which a diverges can then be used to estimate the eventual location of the apparent horizon. Finally, if an apparent horizon forms, an event horizon is known to exist at or outside the location of the apparent horizon. It is in this sense that I refer to the formation of a black hole. 4 As can be seen from results that follow, choosing polar areal coordinates allows the Hamiltonian constraint in the q = 0 limit to be rewritten in the suggestive form (2.56) C H A P T E R 2. PHYSICAL F O R M A L I S M 14 K = - — , ' (2.59) aa D°K*a = - ' ( - i - \ ( 2 . 6 0 ) \aaJ raa DrK = --%-, ' (2.61) 2a' -iHy+*(^ )y where prime, ', designates partial differentiation with respect to radial coordinate r, and overdot, ', designates partial differentiation with respect to time coordinate t. Substituting these quantities into the results of subsections 2.1.5 and 2.1.7 I obtain the Hamiltonian constraint - + - Anra2p = 0, (2.64) a 2r and the momentum constraint a — inraajr = 0. (2.65) Rather than directly using the evolution equations for the spatial metric and extrinsic curvature, I find the so-called slicing condition more useful in the calculations. This equation can be derived from the evolution equations for K^. First note that, since the spacetime metric is torsion-free, Lie derivatives of Kab with respect to Nc reduce to covariant derivatives as per £ N K \ = (DCK\) Nc - K% (DcNa) + Kac (DbNc). (2.66) For polar areal metric (2.45) this reduces (2.41) to DtK% = -DaDba + a(n% + KK% + 8 w ( ^ ± a b ( S - p ) - S a b ^ . (2.67) Then, using Kg = Kg •= 0, and substituting Hamiltonian constraint (2.64) where appropriate, (2.67) becomes ^ - ^ l - 4 7 r r a 2 ( 5 - 2 S | ) = 0 . (2.68) However, using the Einstein field equations, it is possible to show that here from which it follows reducing the slicing condition to (2.69) (2.70) - - - 4irra2Srr = 0 . (2.71) a 2r " The set of equations (2.64), (2.65), (2.71) is sufficient to solve for the entire evolution of spacetime geometry. The method by which to choose appropriate data from which to begin evolution is the initial value problem of general relativity. It' is the topic of the next section. CHAPTER 2. PHYSICAL FORMALISM 15 2.2 The Initial Value Problem The Cauchy problem of a theory amounts to (1) the determination of a boundary and conditions on that boundary along with (2) equations which yield unique solutions for these boundary conditions. The boundary of the Cauchy problem is known as a Cauchy surface. In an evolutionary theory like general relativity or electromagnetism where a 3 + 1 formulation has been adopted, the Cauchy surface is the spatial hypersurface Et=o on which initial conditions are specified, from which the system is deterministically evolved. 5 In fact, the Cauchy problems of electromagnetism and gen-eral relativity are remarkably similar. Both the Einstein equations and the Maxwell equations with sources form sets of coupled nonlinear partial differential equations, both sets containing evolution-ary equations of hyperbolic character as well as constraint equations of elliptic type. And not only are the same solution methods applicable to the hyperbolic equations of both electromagnetism and general relativity, but'also similar are the methods for solving the elliptic constraints and initial value problems of both theories. The next chapter provides a rather detailed account of the numerical techniques used to solve the initial value and evolutionary equations. I now demonstrate well-posedness of the Maxwell equations as discussed in [145] and outline the solution of the initial value problem as developed by [149]. Since this thesis involves only the solutions of Einstein's equations in spherical symmetry, but the Maxwell equations in both spherical symmetry and axial symmetry, I focus discussion primarily on the initial value problem of electromagnetism. 2.2.1 The Initial Value Problem for Electromagnetism In covariant form the Maxwell's equations for non-polarizable, non-magnetizable media are VaFab = -Jb, (2.72) VaFbc + VcFab + VbFca = 0 , . • (2.73) where Fab is the electromagnetic field-strength tensor and Ja is the four-current. (2.72) are the inhomogeneous equations and (2.73) are the homogeneous Maxwell equations. Since Fab is anti-symmetric, contracting (2.72) with the covariant derivative obtains 0 = VbVaFab = -VbJb , (2.74) demonstrating Ja is conserved current. I can use Fab = yaAb - VbAa (2.75) to write them in terms of the four-vector potential Aa as Va(WaAb - VbAa) = -Jb, (2.76) V a ( V ( A - VcAb) + V c ( V a A h - VbAa) + Vb(VcAa - V a A c ) = 0 . (2,77) The inhomogeneous Maxwell equations are exactly what I derive for Aa using the Euler-Lagrange equations. In addition, it is clear that by substituting k i (VaVb-VbVa)T^-X..dl=-Y,RabeC*T^^ (2-78) i=l i=i 5 The one caveat is occurrence of a Cauchy horizon marking the boundary of the domain of dependence beyond which the evolution will not be determined by data supplied on the Cauchy surface. While Cauchy horizons occur in black hole spacetimes, such as Reissner-Nordstrom and Kerr, they are typically shrouded by event horizons. Furthermore, as discussed in Chapter 3, the Cauchy horizon is known to be unstable in the case of Reissner-Nordstrom, and it is also debatable whether one will form during the gravitational collapse of a realistic matter distribution. C H A P T E R 2. P H Y S I C A L F O R M A L I S M 16 into the left hand side of (2.77) I obtain V a ( V 6 A c - V c A f c ) + V c ( V a A h - VfcAa) + V b ( V c A a - VaAc) = — {Rbdac + Radcb + Rcdba) Ad = {Rdbac + Rdacb + R-dcba) Ad = 0 , (2.79) and so the homogeneous Maxwell's equations hold by construction, by virtue of the vector potential formulation of the theory.6 However, I have still not discussed the character of these equations. Because the homogeneous equations hold by construction, there is no need to examine them in any more detail. However, since the inhomogeneous equations must be solved explicitly, it is useful to examine their character, as it will determine how I must solve for the dynamics of the system. First, notice that, because of antisymmetry, the inhomogeneous equations VaFab = Va(VaAb-VbAa) =-Jb (2.84) contain no nonvanishing terms V * V i A i , where summation over repeated indices is in this instance not implied, and index i designates a specific component living in the n-dimensional spacetime. Because of this, (2.84) can only provide evolution equations for TI — 1 of the TI components Aa. Specifically, if At is the temporal component, then the inhomogeneous equation V a F a t = Va(VaAt - VtAa) = -Jt (2.85) is an elliptic equation of constraint for the electromagnetically coupled system. 7 (2.85) must be satisfied at all times. The question which naturally arises, is, "If (2.85) is satisfied at the initial time, will the evolution equations preserve the solution of (2.85) for all time?" The answer to this question is "yes", and it is reasonably straight-forward to demonstrate. To obtain the answer, I first choose a gauge. For the current purposes, Lorentz gauge VaAa = V M a = 0 (2.87) 6In electromagnetism texts this proof is typically presented for stationary observers in flat spacetime as follows. In (spatial) three-vector notation the homogeneous Maxwell's equations terms of the magnetic field, B, and electric field, E, are written V - B = 0 , (2.80) SB VxE+ — = 0. (2.81) dt In terms of the three-vector potential A and scalar potential <p, the B and E are B = V x A , (2.82) 3A E = -—-Vip. (2.83) dt Then, since the divergence of the curl of any vector field is identically zero (i.e., V • x A) =0) and since the curl of the gradient of any scalar field is also identically zero (i.e., V x V</> = 0) equations (2.80) and (2.81) hold by construction. 7Using the (spatial) three-vector nomenclature of flat space, this constraint equation is simply Gauss' law of electrodynamics, V • E = pQ , (2.86) where pQ is the charge density. C H A P T E R 2. PHYSICAL F O R M A L I S M 17 is the most convenient. Then, substituting VaFab = V Q ( V a A b - VbAa) = VaVaAb - RcacbAa = V a V a A 6 - R\Aa , (2.88) (which has incorporated in it the Lorentz gauge condition) into (2.76) I obtain the form VaVaAb-.RabAa + Jb = 0, (2.89) a wave equation for the field Ab. Since I are generally allowing gravitational back-reaction, the Ricci tensor Rab will be dependent on the stress-energy tensor. Now, covariantly differentiating (2.89) I obtain V 6 V a V a A f c - VaRaeAe = 0 , (2.90) where I have used the symmetries of the Ricci tensor and V a Ja = 0 to get the result. Next, using (2.78) I find that V 6 V ° V a A 6 = VbVaVaAb = V a V f c V M 6 - RbaeaVeAb - RbaebVaA* = V a V b V a A b - RbaeaS7eAb - RbaebVaAe = V a V a V b A 6 - VaRbabAe - RbaeaVeAb = V a V a V 6 A 6 + VaRaeAe . RbaeVaAe (2.91) Then, substituting (2.91) into (2.90) I find V a V a (VbAb) = 0 . (2.92) Clearly, the Euler-Lagrange equations for vector potential Aa (i.e., the inhomogeneous Maxwell equations) can be written as a linear, diagonal second-order hyperbolic system. As such, it has a well-posed initial value formulation as proven by theorems (see [145] for details). Then, given both an initial data set (Aa, satisfying (2.85) and the Lorentz gauge condition on a Cauchy surface E t = o , this system is guaranteed to satisfy the Maxwell equations and Lorentz gauge condition everywhere for all time. To solve constraint equation (2.85) I implement the procedure outlined by York and Piran [149], described in terms of the electric field8 Ea = Fabnb (2.94) as follows. Since the constraint equation is essentially Gauss' law for the electric field, I start by decomposing the electric field into its longitudinal (divergenceful) and transverse (divergenceless) parts, La and Ta respectively,' Ea=La+Ta . (2.95) Now, I introduce a scalar field U and define it to be the potential for the electric field, La = DaU , (2.96) 8 Here the magnetic field would be expressed Ba = --eabcdFcdnb , (2.93) . where eabcd is the totally antisymmetric tensor of positive orientation. C H A P T E R 2. PHYSICAL F O R M A L I S M 18 where DA is again the spatial covariant derivative operator. Then, clearly, D A E A = D A ( L A + TA) = D A L A = DADAU , (2.97) where D A D A is the (spatial) 3-dimensional Laplacian operator. 9 Then, since DADAU = D A E A = PQ , (2.99) where PQ = —nAJA is the electric charge density, 1 0 I recognize this constraint equation as the usual Poisson equation. As expected, it is of elliptic type. Once I have solved elliptic differential equation (2.99) for U, I can determine EA up to a transverse component TA as The electromagnetic initial value problem thus has been reduced to a single equation (2.99). Now, while the longitudinal component L A is explicitly involved in. the solution of U, the transverse component TA (which represents the radiative degrees of freedom) is freely specifiable. I could just as well choose it to be vanishing at initial time. Thus, given charge density P Q , I can solve for U via (2.99). Having found U, I can then determine L A via (2.96). Meanwhile, I are free to specify any reasonable (i.e., smooth, differentiable, and satisfying regularity conditions discussed in section 2.4) initial values for any remaining variables not determined by gravitational constraints. 2.2.2 The Initial Value Problem for General Relativity The Einstein, Einstein-Klein-Gordon and Einstein-Maxwell equations are known to possess well-posed initial value formulations. 1 1 The initial value problem of the Einstein equations is in many ways analogous to that of electromagnetism. Where in electromagnetism a choice of gauge con-dition had to be made for four-vector potential Aa, the gauge choice for the Einstein equations amounts to a choice of coordinates along with a slicing condition that determines the embedding of hypersurfaces E t . Furthermore, while the well-posedness of electromagnetism was dependent on the ability, to cast the inhomogeneous Maxwell equations as a linear, diagonal second-order hyper-bolic system, well-posedness for general relativity requires the reformulation of Einstein's equations as a quasilinear, diagonal second-order hyperbolic system (see [145] for details). Well-posedness of the Einstein equations was first demonstrated in [33]. Further developments of special significance were produced in [34], and [35]. While the inhomogeneous Maxwell equations (2.85) form the initial value constraints of elec-tromagnetism, for Einstein's equations they are furnished by the Hamiltonian constraint (2.29) and momentum constrains (2.30). Then, by theorem, providing an initial data set ( E ( = o , hao, Kao) satisfying (2.29) and (2.30) is supplied on a Cauchy surface diffeomorphic to Et=o, the system is guaranteed to satisfy the Einstein equations on subsequent E t . Here jab is the spatial three-metric as defined in subsection 2.1.2 and Kab is the extrinsic curvature as defined in subsection 2.1.4. 9The 3-dimensional Laplacian in polar areal coordinates acting on U(t, r.) is E A = L A + T A = DAU + TA . (2.100) (2.98) 1 0In polar-areal coordinates, PQ = —naJa = —^Jt-1 1 The equations of Chapter 3 form an Einstein-Maxwell system where the sources are given by an electromagnet-ically coupled complex scalar field. C H A P T E R 2. PHYSICAL F O R M A L I S M 19 The solution of (2.29) and (2.30) can be carried out by decomposing extrinsic curvature, Kab, into longitudinal and transverse components, and then freely specifying the divergenceless (trans-verse) component while solving the required elliptic for the divergenceful (longitudinal), part. This is closely analogous to the preceding decomposition and solution of the electric field where the transverse component represented the radiative degrees of freedom for the electromagnetic field. In the case of Einstein's equations, the transverse component of the extrinsic curvature encodes gravitational radiation content of the spacetime. 1 2 2.3 Conserved Quantities If a given tensor T^[[[ is divergenceless, for example v ar$::: = o, . (2.101) it has associated with it zero or more conserved quantities, depending on the symmetries (and thus Ki l l ing vectors) of the spacetime. If the divergenceless tensor is the stress-energy, Tab, then the conserved quantities will be energy and a component of the momentum. If the tensor is instead the current four-vector Ja, the conserved quantity wil l be a charge. 2.3.1 Conservation of Charge Consider first the case of a conserved current four-vector, J Q , V a J ° = 0,. (2.102) an example being the electromagnetic current. Then, contracting this vector with the hypersurface orthonormal na yields charge density, PQ , PQ = ^naJa, as measured by an observer with four-velocity na. Using Gauss' theorem,'the total charge, Q, can be obtained by integrating (2.103) over one ofthe spacelike hypersurfaces E t , •Q = - I VWKJad3x, (2.104) where | —y | is the determinant of the spatial metric j a b , and o9x is the invariant volume element on E t . This integrated charge will then be a constant, regardless of any dynamics. Observe how conservation is manifest for divergenceless four-vectors regardless of spacetime symmetry.' The situation is different for tensors of rank-two or more—conservation there depends on the existence of Ki l l ing vectors. . 2.3.2 Conservation of Energy and Momentum The (rank-two) stress-energy tensor Tab is by construction divergenceless, VaTab = VaTba = 0. (2.105) 1 2For the spherically symmetric spacetime in polar-areal coordinates, the constraint equations and slicing condition have been reduced to ordinary differential equations (2.64), (2.65) and (2.71) which can easily be integrated. In this case, the system also possesses no radiative degrees of freedom. (2.103) C H A P T E R 2. PHYSICAL F O R M A L I S M 20 In situations where the spacetime possesses a timelike Ki l l ing vector field ta, the contraction pT = natbTab (2.106) is the energy density measured by an observer with four-velocity na. And when the spacetime possesses a spacelike Ki l l ing vector field sa, the quantity Ps=_-naSbTab (2.107) represents the momentum density in the direction of sa. Then integrals T= ( VW\natbTabd3x (2.108) and S = - / VWKsbTabd3x (2.109) respectively represent the total conserved energy and total conserved component of momentum. For instance, S could be the radial, angular, or other-directional momentum, the specifics depending on Ki l l ing vector sa. 2.4 Regularity Conditions In simulations presented later in this dissertation I employ coordinate systems which become de-generate over finite sets of spatial coordinate values. Specifically, in the spherically symmetric simulation of charged scalar critical collapse in Chapter 3, I use the (t,r,8,(f>) polar-areal coordi-nate system over a range which includes r = 0. Similarly, in Chapter 4, I study axially symmetric charged scalar accretion in (t,r,9,(p) Kerr-Schild coordinates over a range which includes 9 = 0 and 9 = ir. The system of equations to be evolved sometimes become singular at r = 0 for the polar-areal coordinates, and at 9 = 0 or 8 = IT for Kerr-Schild. In cases where the equations do not explicitly diverge, the evolution often becomes numerically unstable at the degenerate points unless the equations are recast in a more appropriate form. The form they take on is determined by so-called regularity conditions at the degenerate coordinate values [6], [45]. 2 . 4 . 1 Regular i ty at r = 0 i n Po l a r -Area l Coordinates Following [6], I say a tensorial quantity is regular at r = 0 of the (t,r, 9, cf>) polar-areal coordinate system if and only if its components can there be expanded in non-negative integer powers of the (x,y,z) spatial Cartesian components defined" x = rsin9cos(f> , (2.110) y = rsin.0sin</> , (2.111) z = rcos6>. (2.112) Clearly, assuming neither the spacetime nor matter are anywhere singular, I should ensure that all physical quantities are in this sense regular. Once the regularity conditions have been deter-mined for the physical quantities, their consequences should be consistently propagated through the equations of any unphysical auxiliary variables. For example, under this definition, a physical rank-zero tensor quantity such as electric charge density, PQ, or scalar field amplitude, 4> = $(£, r), would to leading order have to be an even function of radial coordinate at r = 0. This would in C H A P T E R 2. P H Y S I C A L F O R M A L I S M 21 turn imply that an auxiliary variable such as $ r 0 , r ) = « 9 r $ (2.113) would by consequence have to be an odd function of r and therefore vanishing about r = 0. Furthermore, as the electric field is a rank-one tensor, the radial component Er would have to be odd in r in order to be regular there. Indeed, if I were to integrate charge density PQ over space, I would find that Er would have to be odd about r = 0 by consequence of PQ being even there (cf. Appendix C) . This line of argument is similarly applied to determine the nature of each function about r = 0 in the evolution leading to critical collapse of charged scalar fields in Chapter 3. 2.4.2 Regularity at 9 — 0 and 0 = -K in Kerr-Schild Coordinates Extending the definition, I say a tensorial quantity is regular at 9 = 0 and 9 = ir of the (£, r, 9, <f>) Kerr-Schild coordinate system if and only if its components can be expanded in non-negative integer powers of its (x, y, z) spatial Cartesian components at these coordinate values. As a consequence, physical rank-zero tensors such as the scalar field, $ = Q(t,r,9), or charge density, P Q , are to leading order in 9 even functions about 9 = 0 and 9 = n. B y definition, auxiliary quantity $eXt,r,6) = de$ (2.114) would then have to be odd at these degenerate coordinate values. The components of a physical rank-one tensor can similarly be determined. For example, regularity requires the radial and axial components, Er and E^, of the electric field vector be even to leading order in 9, and the angular component, Ee, be odd in 9 about 9 = 0 and 9 = ir. The regularity conditions are obtained and enforced for all functions in Chapter 4. 2.5 The Sommerfeld Condition When studying a dynamical system—or any system, for that matter—on a finite spatial domain, properties of the system on the domain boundary must be considered. The exact properties may or may not be known. If not known, the only choice is to apply approximate boundary conditions and examine whether they are sufficient for the purpose of current investigation. As an example of a system where exact boundary conditions can be determined, consider the spherically symmetric massless Klein-Gordon equation V ° V a $ ( * , r ) = 0 (2.115) in Minkowski spacetime. In the usual (t, r, 9, (f>) spherical polar coordinates, the spacetime is de-scribed by line element ds2 = -dt2 + dr2 + r2d£l2 , (2.116) where dQ2 = d92 + s in 2 9dcj>2 . (2.117) The spherically symmetric massless Klein-Gordon equation (2.115) can then be written 9 „ ( r $ ) = 5 r r ( r $ ) , . (2.118) where du denotes the second partial derivative with respect to temporal coordinate t, and drr designates the second partial derivative with respect to radial coordinate r. There are both ingoing and outgoing solutions to (2.118), the ingoing being given by (r*)(t,r) = f(c-t + r) , (2.119) C H A P T E R 2. P H Y S I C A L F O R M A L I S M 22 f(c-t + r) being an arbitrary function of its single variable, while the outgoing solution is given by ( r * ) ( i , r ) =g(c+t-r) , (2.120) g(c+t — r) again being an arbitrary function of just one variable, c_ being the ingoing wave speed and c + being the outgoing wave speed. The wave speeds are given by solutions of the radial null geodesic equation, here (2.121) From (2.119) it follows that ingoing solutions can be written 9 t ( r$) - (c_)9 r (r$) = 0 , (2.122) while from (2.120) it can be shown that outgoing solutions have the form «9 t(r$) + (c+)d P(r$) = 0 . (2.123) A radial inflow boundary condition for (2.115) then isgiven exactly by (2.122), while radial outflow is given exactly by (2.123). When applied at the domain boundary, equation (2.123) is commonly known as an outgoing radiation boundary condition or the Sommerfeld condition (cf. [135], [136]). 2.5.1 The Sommerfeld Condition in Polar Areal Coordinates In the spacetime described by polar areal metric ds2 = -a2dt2 + a2dr2 + r2dQ.2 , (2.124) where a = a(t,r), a = a(t,r), and dfl2 again is given by (2.117), the Sommerfeld condition (2.123) wil l exactly describe outflow conditions of the massless Klein-Gordon equation when a = 1 and a = 1 on the boundary. The Sommerfeld condition can still be applied when the spacetime is curved, but the quality of its approximation to perfect outflow will have to be tested. In the case of curved spacetime, radial null geodesies satisfy = c± = ± ( ^ ) , (2.125) and so the (now approximate) condition (2.123) becomes 9 t ( r$) + ( ^ ) 9 r ( r$ ) = 0 : (2.126) In Chapter 3 (2.126) is used to study the critical gravitational collapse of charged massive complex scalar fields in spherical symmetry. While only approximate, the condition is sufficient for the choices-of scalar field parameters and coordinate range, as demonstrably little inflow or reflection is observed from the domain boundary. . 2.5.2 The Sommerfeld Condition in Kerr-Schild Coordinates Suppose, instead, the massless Klein-Gordon equation is evolving in the spacetime of a rotating black hole described by the Kerr metric. In (t,r,6,(j>) Kerr-Schild coordinates, this spacetime is CHAPTER 2. PHYSICAL FORMALISM 23 described by the line element 2 , 9 ( 2Mr\ , 9 / 2Mr\ „ . i n ( i - 2 2Mro? s i n2 0 d s 2 = - ( l - ^ ^ ^ J d t 2 + . [ l + - ^ - J d r 2 + E ^ 2 + s i n 2 6 l f r 2 + a 2 + — — + ( J cttdr - f j dtd(f> - 2 a s i n 2 6> ( 1 + — - J drd<j> , (2.127) where E = r 2 + a 2 cos 2 9 , (2.128) J M £ is the t empora l coordinate, and r, 9 and </> are respectively the r ad ia l and angular spat ia l coordi -nates, M represents the to ta l mass of the black hole spacetime and J i ts to ta l angular momentum. In this case, the r ad ia l nu l l geodesies satisfy equat ion 0 = — f 1 — J dt2 + ( - = - J rftrfr + ( 1 + J <*r2 , (2.130) (2.129) and so the wave speeds are dr\ / - 2 M r 7 E ± l \ , „ - , O - , N d t y ± V 1 + 2 M r / E A s s u m i n g the scalar field wavefronts are nearly spherical at the doma in boundary, the approximate Sommerfeld condi t ion can then be wr i t t en ^+(TTM^)d^ = °- <2132> T h i s approximate boundary condi t ion is used i n Chap te r 4 when s tudy ing the dynamics of electro-magnetic rad ia t ion and charged massive scalar fields on a K e r r spacetime background. A s discussed above, the implementa t ion must be tested for inflow and reflections off the outer boundary. In doing so (2.132) is found sufficient for the purposes of the study. To conclude, the Sommerfeld condi t ion is often acceptable i n practice. T h e computa t iona l boundaries can always be moved sufficiently far from regions of phys ica l interest that pa r t i a l re-flections at late t ime w i l l add on ly a to lerably smal l error to the solut ion. In si tuations where more precisely outgoing condit ions are required on the boundary of a finite domain , a technique using boosted coordinates i n conjunct ion w i t h numer ica l diss ipat ion can be appl ied . However , as the Sommerfeld condi t ion is sufficient for the investigations presented i n Chapte rs 3 and 4, discussion of this boosted coordinate me thod is reserved for A p p e n d i x D . C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 24 C H A P T E R 3 C H A R G E D S C A L A R C R I T I C A L C O L L A P S E 3.1 Preliminary Discussion 3.1.1 Context of This Study The near-critical collapse of a charged massive scalar field in spherical symmetry is studied in this chapter. Critical phenomena in gravitational collapse have been briefly discussed in the Chapter 1. More important, the interested reader can access a significant amount of review work on this subject, most notably due to Gundlach [59], [60], along with other detailed accounts of previous studies of critical phenomena in spherical symmetry. The latter include Ph . D . theses of many of Choptuik's former students (specifically, Liebling [93], Neilsen [106], Hawley [64], Honda [77], Pretorius [127], Ventrella [144], Noble [108], Olabarrieta [117] and La i [92]). Briefly, the reader is reminded that, in traditional critical behaviour, a critical solution has come to be identified as a type of separatrix (in solution space) between solutions modelling collapse to a black hole from those which do not. This separatrix has a small number of unstable modes— indeed, often only a single unstable mode—in perturbation theory. Even within the context of spherical symmetry, in which the bulk of critical phenomena computations—including those in this chapter—have been carried out, critical solutions are characterized by additional symmetry relative to generic solutions admitted for the model. To date, the additional symmetry has been of one of two types, each of which is further sub-typed as to wheither the symmetry is discrete or continuous. This has lead to the following classification hierarchy of critical solutions. In the top of this hierarchy, the phase transition between black hole spacetimes and those that are not are classified first or second order (Type I or Type II respectively) depending on whether or not the minimum black hole mass above threshold is finite or infinitesimal. Historically, this motivation through observation that the mass could be viewed as an order parameter traces back to Christodoulou [36] and Evans [46]. Now, in the second level of this hierarchy, the solutions are further sub-classified as either discrete or continuous, depending on their symmetry. As discussed in Chapter 1, one of the additional symmetries of Type II solutions is spatiotemporal discrete self-similarity (DSS). Continuously self-similar (CSS) and static solutions have also been found for Type II case in spherical symmetry. On the other hand, Type I solutions in spherical symmetry have been found to additionally possess a time-translational symmetry, which manifests as either (discrete) periodic or (continuous) static evolution. This classification.hierarchy is presented with examples in Table 3.1. • One of the more novel features of the model studied in this chapter—shared with those of [24], [12] and [16] (for the massive real scalar field), [26] and [29] (for SU(2) Yang-Mills), [9] and [10] (for the SU(2) Skyrme model), [63] j [64], [65] and [92] (for massive complex scalar fields), and [114] and [108] (for perfect fluids)—is the existence of both Type I and Type II behaviour. This arises from dimensionful terms in the matter Lagrangian that set a length scale which factors into the type critical solution obtained. In the limit of small scalar field mass m$, and for an appropriate range of electric coupling parameter e, the massive charged scalar field solutions at black hole threshold should observe the Type II scaling relations noted in [66], [68] and [58]. The existence and properties of charged boson star, solutions (see [79], [80], [81]), together with the current understanding of massive scalar critical phenomena, suggest that Type I critical solutions resembling perturbed charged boson stars should exist for an appropriate set of initial conditions and range of parameter values e and m$ [31]. However, details regarding the emergence and C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 25 critical solution additional symmetry example (from spherical symmetry) Type I continuous (static) SU(2) Yang-Mills [26], [29] SU(2) Skyrme [9] perfect fluids [108] discrete (periodic) massive real scalar field [12], [16] Type II continuous (static) SU(2) Skyrme [10] continuous (CSS) perfect fluids [114], [108] discrete (DSS) massive real scalar field [24] SU(2) Yang-Mills [26], [29] Table 3.1: Classification hierarchy of critical solutions. At the top of the hierarchy, the solutions are classified as either Type I or Type II. In Type I, near-critical black hole masses are finite. The spherically symmetric Type I solutions are further sub-typed by additional symmetry as either continuous (static) or discrete (periodic). In Type II, near-critical black, hole masses are infinitesimal. The spherically symmetric Type II solutions are further sub-typed by additional symmetry as either continuously self-similar (CSS) or static, or discretely self-similar (DSS). Examples are given from spherically symmet-ric computations for each of these sub-types. A l l of the tabulated example models allow both Type I and Type II solutions, depending on parameter values and initial conditions. properties of such solutions have not been documented elsewhere. Note that the study described below has much in common with those of Brady, Chambers and Goncalves [12], [16], Hod and Piran [66], [68], Gundlach and Martin-Garcia [57], Hawley and Choptuik [63], [64], [65] and La i [92]. Also note that, for the case of the Type II behaviour described herein, the results of Hod and Piran [66], [68] are expected to be significantly more accurate, and possessive of better convergence properties due to the fact that they were computed using a high-order unigrid technique, originally due to Garfinkle [52], tailored to the critical solution. The value of this particular study then, has been largely one of providing further confirmation of the view described by Gundlach and Martin-Garcia [57], and Hod and Piran [66], [68] with regards to charge scaling in the model, as well as completeness, as yet another case of critical collapse has been examined and the basic theoretical understanding of the phenomenology found solid. Perhaps most important, this study has provided valuable experience and insight into the nature of critical phenomena. This experience is likely to be vital in the studies of non-spherical charged collapse planned for the future. 3.1.2 Issues Regarding Horizons Before solving the Einstein-Maxwell-Klein-Gordon equations with electromagnetically coupled com-plex scalar field sources, it is prudent to first consider what global structure one could expect for the spacetime. Even the spherically symmetric gravitational collapse of simple charged matter is nontrivial. The spacetime that results is one of a spherically symmetric charge-carrying black hole with some dynamic spherically symmetric charged matter distribution in its exterior. How-ever, a generalization of Birkhoff's theorem guarantees the spacetime external to all matter is represented by the static Reissner-Nordstrom solution. While the maximal analytic extension of Reissner-Nordstrom spacetime is described by the Penrose diagram 3.1, it does not actually depict the spacetime generated by the realistic collapse of charged matter in spherical symmetry. The dynamical evolution of a charged spherical shell of dust has been described in detail by CHAPTER 3. CHARGED SCALAR CRITICAL COLLAPSE 26 F i g u r e 3.1: Penrose diagram showing causal structure of Reissner-Nordstrom spacetime (cf. [62]). I and I' depict asymptotically flat regions of the spacetime outside the black hole. II and IF are interior to the event horizon at r = r+, while interior regions III and III' are separated from iFand II' by the Cauchy horizons at r = r_ . 3+ represents future null infinity and 3~ past null infinity, while %~ represents past timelike infinity, i+ future timelike infinity and i° spacelike infinity. The singularity at r = 0 is timelike, and can thus be avoided by a timelike trajectory. C H A P T E R 3. C H A R G E D S C A L A R C R I T I C A L C O L L A P S E 27 r = 0 r = r. Figure 3.2: Conformal diagram for gravitational collapse and apparent horizon formation for a spherical shell of charged dust when the charge is too small to prevent spacelike singularity formation in its interior (cf. [11]). 3 + represents future null infinity, J~ is past null infinity, i° is spacelike infinity, and i+ represents future timelike infinity. An apparent horizon forms at radial coordinate r = r+, and r = r_ represents a Cauchy horizon. The spacetime singularity is spacelike and appears in the interior at r = 0. Boulware [11] for various values of total mass, M, total charge, Q, and total rest mass M.1 While the spacetime interior to the spherically symmetric distribution is fiat, the complete structure of spacetime is generally complicated and dependent on the values M, Q and M. For example, if the charge carried by the matter is sufficiently small, it will not be able to prevent the formation of a spacelike spacetime singularity in its interior by electrostatic repulsion. In this case, the collapse will follow the evolution shown in Figure 3.2, which is a conformal diagram showing formation of an apparent horizon at radial coordinate value r = r+ during evolution. The spacetime singularity appears in the interior at r = 0, and there exists a Cauchy horizon at r = r_ beyond which spacetime is not determined by Einstein's equations with the given initial data. In the case where charge Q is large enough to prevent spacelike singularity formation, the evolution proceeds as depicted in Figure 3.3. An apparent horizon still forms at r = r + , and a Cauchy horizon at r = r_. A singularity again forms within the apparent horizon, but outside the charged matter. This singularity is now timelike and therefore can be avoided by the evolving matter. In fact, once inside r = r+, the matter can pass through r = r_, and, avoiding the singularity, re-expand into a second asymptotically flat region of space. However, the case where a charged shell collapses to leave a black hole spacetime with timelike singularity is thought to be unrealistic. In fact, Penrose was first to notice that an observer crossing Cauchy horizon r = r_ would see the entire history of asymptotically flat regions in a finite time [122], implying that r = r_ should be unstable to small perturbations as later demonstrated in [100]. Meanwhile, by studying generic self-gravitating charged scalar field collapse, Hod and Piran were the first to demonstrate that in addition to the spacelike singularity at r = 0, a null singularity forms along the Cauchy horizon which then contracts and meets the spacelike singularity 1Novikov similarly studied the evolution of a shell of charged matter in [115]. C H A P T E R 3. C H A R G E D S C A L A R C R I T I C A L C O L L A P S E 28 Figure 3.3: Conformal diagram for gravitational collapse and apparent horizon formation for a spherical shell of charged dust when the charge is large enough to prevent spacelike singularity formation in its interior (cf. [11]). 3 + represents future null infinity, J~ is past null infinity, z° is spacelike infinity, and i+ represents future timelike infinity. An apparent horizon forms at radial coordinate r = r+, and r = r_ represents a Cauchy horizon. A spacetime singularity again appears at r = 0, but it is now timelike and exterior to the matter. C H A P T E R 3 . C H A R G E D S C A L A R C R I T I C A L C O L L A P S E 20 r = 0 r = r_ Figure 3 . 4 : Conformal diagram for gravitational collapse and horizon formation for a spherically symmetric distribution of charged scalar field matter (cf. [72], [73]). 3+ represents future null infinity, 3~ is past null infinity, i° is spacelike infinity, and i+ represents future timelike infinity. An apparent horizon forms at radial coordinate r = r+, and r = r_ represents a Cauchy horizon. A spacelike singularity forms in the interior at r = 0. Meanwhile, a weak null singularity forms along the Cauchy horizon and then contacts to meet the spacelike singularity at r = 0. at r = 0 [72], [73]. This scenario is depicted in Figure 3.4. These results were confirmed and analyzed in somewhat greater detail by Oren and Piran [119]. Furthermore, their investigations demonstrated the contracting null singularity is weak, in that an observer crossing it will experience only finite tidal forces (cf. [120], [121]). So while their future is unpredictable after crossing r = r _ , observers can ultimately still escape to a causally disconnected region of spacetime which may also be asymptotically flat. The salient feature of all these charged matter collapse scenarios is the appearance of the apparent horizon, outside which the spacetime is still predictable, regardless of any details within the Cauchy horizon. This property is what allows the use of any of a large number of coordinates to model processes leading up to the formation of a black hole by charged matter. Providing I am not interested in evolution beyond apparent horizon formation, non-penetrating coordinates are a completely suitable choice. The system of equations used in simulations of critical gravitational collapse of charged massive scalar fields now follows. 3.2 System of Equations 3.2.1 The Matter Lagrangian and Equations of Motion The Lagrangian for a massive electromagnetically coupled complex scalar field is given by [62] (3-1) CHAPTER 3. CHARGED SCALAR CRITICAL COLLAPSE 30 where m$ is the scalar field mass parameter, e sets the strength of electromagnetic coupling be-tween the real and.imaginary components of complex scalar field $ = r), the symbol * denotes complex conjugation, Aa = Aa(t,r) is the electromagnetic vector potential, and Fab is the anti-symmetric electromagnetic field strength tensor defined according to Fab = V a A f c - VbAa . In terms of its real and imaginary components (respectively <pi scalar field is written $ = <f>i+i<t>2 • (3.2) 4>i{t,r) and 4>2 = 4>2{t,r)) the (3.3) To make connection with the physical fields, an observer moving with four-velocity na would measure an electric field Ea = Fabnb and a magnetic field Ba £abdFcdnh , where eabcd is the totally antisymmetric tensor of positive orientation. 2 The set of Euler-Lagrange equations for the system described by (3.1) M 5$ dCM 9 ( V a $ ) dCM dAb = 0 , - V „ d$* -' dCM d(yaAb) dC M L 9 ( V a $ * ) o , = 0 yields equations of motion 3 V a V a $ * + 2 i e ( V a $ * ) A a - e2$*AaAa + ie$*VaAa - m ^ * = 0 V ° V a $ - 2 i e ( V a $ ) A a - e1$AaAa - ie$VaAa - m $ 2 $ = 0 , VaFab - i e ( $ * V 6 $ - $ V f c $ * ) - 2e2$$*Ab = 0 . (3.4) (3.5) (3-6) (3.7) (3.8) (3.9) (3.10) I have the freedom to choose an electromagnetic gauge condition. I choose the Lorentz gauge condition VaAa =0. This provides me with an equation of evolution for the temporal component of Aa. Furthermore, since it can be demonstrated that V a J a = 0 , where -Jb EE V a F a f e = ie($*V6<S> - $ V 6 $ * ) + 2e2$$*Ab, I conclude Ja is a conserved current. 4 (3.1.1) (3.12) (3.13) 2eabcd has norm given by e a i 3 C a e a b c d = —24, and eoi23 = 1 in a right-handed orthonormal basis. 3Under decomposition (3.3) it is clear that (3.8) and (3.9) are completely equivalent. 4The choice of sign and normalizing factor on JB are somewhat arbitrary. A different choice of sign would later introduce a factor of —1 in the value of the electric charge density PQ. . CHAPTER 3. CHARGED SCALAR CRITICAL COLLAPSE 31 3.2.2 The Einstein Lagrangian and Field Equations The Einstein Lagrangian is £ g = ^g~R, (3.14) and the total Lagrangian is given by minimally coupling the matter Lagrangian to the geometry via C = £g+aM^M, (3.15) where coupling constant O.M will have to be chosen to give the correct normalization for the stress-energy tensor of the matter field. The Einstein field equations are obtained by variation of the action S[gab,§,§*,Aa} = jCd4x (3.16) with respect to gab. They are Gab = Rab - ^9abR = 8irTab , (3-17) where the stress-energy tensor Tab is given by-Using (3.1) and (3.18) I explicitly obtain Tab = i ( V a $ V 6 $ * + V f e $ V a $ * ) - ^ e [ ( $ V 6 $ * - $ * V 6 $ ) A a + ( $ V 0 $ * - $ * V a $ ) A b ] ,+e2^*AaAb + \FacFbdgcd - ^ a 6 [ ( V c $ - ieAc<f>)(Vc$* + ieAc$*) + l-FcdFcd +m2<S>*$] , (3.19) where CUM = 8TT has been chosen in order that I retrieve the Einstein-Maxwell-Klein-Gordon equations from (3.17). 3.2.3 The Spacetime Metric and Equations While many coordinate systems could be used to study the critical collapse of a massive charged scalar field, one convenient choice is polar-areal. As presented in subsection 2.1.8, the line element of a spherically symmetric spacetime in polar-areal coordinates is ds2 = -a2dt2 +a2dr2 + r2dQ2, (3.20) where a = a(t, r), a = ait, r), dfi2 = d62 + s in 2 6d(j>2, t is the temporal coordinate, and r, 6 and <f> are respectively the radial and angular spatial coordinates. Again, the future-directed unit-norm spatial hypersurface orthogonal vector is na = (1/a, 0,0,0), and the spatial metric is 7ab = diag (a 2 , r2,r2 s in 2 9). (3.21) Clearly, as the divergence of a2 marks the formation of an apparent horizon, this coordinate sys-tem becomes singular and is therefore horizon non-penetrating. W i t h the choice I've made for normalization of the electromagnetic field Lagrangian density, the mass aspect function takes on C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 32 the form 2 V a 2 47rr ™ = o ( l - 3 + 7 ^ ) , . (3-22) where m = m(t, r) is the gravitational mass and q =. q(t, r) is the electric charge of the source. Finally, the Hamiltonian constraint is - + - Airra2p = 0, (3.23) a 2r the momentum constraint is d — 4nraajr = 0, (3-24) and the polar slicing condition is — - - 47rra 2S\ = 0, (3.25) where prime, ', designated partial differentiation with respect to radial coordinate r, overdot, ', designates partial differentiation with respect to temporal coordinate t.5 The explicit forms of p (energy density), jr (momentum density) and Srr (stress tensor) are p = Tabnanb ' . " = \ \Jp) [n*nj + (*r - ie$Ar) + ie$*Ar)) + i (n r) 2 + ^m2$$* , (3.26) jr = -±(Trana) = - ^ Q ^ [(*; + te$*A.)n* + ($ r - ie$4 r )n j ] , (3.27) srr = ± T ; = -(•-2 Va 2 J ' , y ' r ' ." 4 v~~" 2 2 1 L i l * I I J + ( $ r - i e $ A r ) + ze$*A r)] - - (n r) 2 - - m $ 2 $ $ * , (3.28) where n$ = n$(£, r), $ r = $ r ( i , r ) , and n r = n r (£ , r ) are defined below in equations (3.30), (3.31) and (3.38). 3.2.4 Matter Equations in First-Order Form I can rewrite equation of motion (3.9) as V a ( V a $ - ieAa$) - ieAa ( V a $ - ieAa$) - m£$ = 0 . (3.29) Then, computing covariant derivatives of (3.29) for metric (3.20), the equation of motion is a relatively simple expression. I choose to re-express it in first order form. Defining n$(t,r) EE naa ( V a $ - ieAa<&) , ' (3.30) $ r ( t , r ) EE , (3.31) 5However, to eliminate divergences and loss of numerical precision when a and a approach values close to zero, the Hamiltonian constraint and slicing condition are best solved using logarithmic variables A = In a and i 5 h a . C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 33 where dr designates pa r t i a l differentiation w i t h respect to coordinate r, equat ion of mot ion (3.29) becomes the fol lowing t r ip le t 33r3 r 2 - (4V - zeA r $) a — ie a (<3>r - i e A r $ ) Ar - I I $ A t - aam$ 24> , (3.32) a - (ILj>) + ieAt$ 4> = (ILj>) + i e A t $ l , La J (3.33) (3.34) where overdot, ' , again designates pa r t i a l differentiation w i t h respect to t, and I have used the chain rule to rewrite the par t i a l derivative w i t h respect to r as a par t i a l derivative w i t h respect to r 3 , designated 9 r 3 . 6 A s w i t h the scalar field, I choose to wri te and evolve the spat ia l par t of Aa i n first order f o r m . 7 Defining Ilr(t,r) • Atr(t,r) = drAt , n a - (VaAr - VrAa) , a (3.38) (3.39) the Eule r -Lagrange equat ion for the spat ia l component of Aa becomes the following set of three equations n r = - ( j P ) Atr [aa ( I I r ) + Atr] dr(At) , where (3.13) gives Jr = - i e ( $ * $ r - $ $ ; ) - 2e2$$*Ar . (3.40) ,(3.41) (3.42) (3.43) F ina l l y , whi le the Loren tz gauge condi t ion is already i n first order form, I find i t convenient to define .nt(t,r) = naaAa (3.44) TLt(t,r) = Qt 6 By using the chain rule as per numerically unsuitable forms such as at r = 0 can be rewritten more appropriately as dj_ = 5r™ df_ = n r n - i d f _ dr dr dr71 d r n L9! R2 Qr r 2 dr cV 3 (3.45) (3.35) (3.36) (3.37) The first appearance in the literature of numerical relativity for this application of the chain rule was [45]. 7 It can be demonstrated that in spherical symmetry the components Ag and A^, must completely decouple from the system of equations. I am therefore free to set them both to zero for all time. C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 34 and recast the gauge condition as the triplet n t = 39 r 3 r2-{Ar) , (3.46) a fit = n t , (3.47) At = - (nt) , . (3.48) a where the chain rule has again been applied to recast the differential equation in a more appropriate form. 3.2.5 Physical and Conserved Quantities The only non-vanishing electromagnetic field component measured by an observer with four-velocity na can be expressed ER = - a L L , , (3.49), the temporal component of the current four-vector can be written 7 t = i e - ( $ * I I * - , (3.50) a and the conserved electric charge density expressed PQ = -ie-^ ($*n* - $IT£) . (3.51) Outside gravitational sources, where ta = (1,0,0,0) is a timelike Ki l l ing vector field, the quantity P T = \ (5) { U ^ U l + ( ^ - i e ^ A r ) ( ^ ; + ie^Ar)] + ^a(Ur)2+ ^ a m i ^ * (3.52) is a conserved energy density. Then, from (2.104), the total charge, Q= . / pQ (ar2 sm9) drd9d(f) , (3.53) Jcj>=o Je=o Jr=0 will be constant. Similarly, from (2.108), the total energy, /27T P77 r>T2 / / PT (ar2 sin 9) drd0d<j) , (3.54) T,=0 J0=O Jr=ri will also be constant if no gravitational sources reside within r\ <r <r2-3.2.6 Regularity, Boundary Conditions and Dissipation According to the definitions of section 2.4 (cf. [6], [45]), the regularity conditions for functions listed above require that matter variables <5>, FI$, .At, flt, LTt Jt, PQ and pr are, to leading order, even in r about r = 0. Variables $ r , Ar, Ur, Atr, Jr and ER will to leading order be odd. For the leading-order-even variables I enforce regularity at the grid point corresponding to r = 0 by applying a quadratic fix as listed in Appendix C. Meanwhile, the regularity condition of the leading-order-odd variables is enforced by setting the functions equal to zero at the degenerate point. I apply the (approximate) Sommerfeld outgoing radiation condition, to $ at the computational C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 35 domain boundary r = r m a x . As discussed in Chapter 2, this condition adopts the form 3t(r$) + ( ^ ) 5 r ( r$ ) = 0 ' (3.55) for the spherically symmetric metric in polar areal coordinates. Assuming spacetime is essentially flat at r = r m a x , condition (3.55) reduces to <9t(r$) + «9 r(r$) = 0 . (3.56) Observe how (3.55) and (3.56) assume both e and m$ have an insignificant effect on scalar field dynamics at the boundary. It can be shown that, under these assumptions, an approximate Som-merfeld condition can also be written for 11$ as per dtirUg,) + 5 r(rn$) = 0 . (3.57) Assuming r = r m a x is sufficiently large, an approximate Sommerfeld condition can similarly be written for $ r . I implement the these equations (and all others containing partial derivatives in r) using backwards spatial differencing operators. Since the Hamiltonian constraint is solved by integrating outwards, I impose the inner boundary condition o ( i , 0 ) = . l . (3.58) on metric function a(t,r). Function a is set to unity here because this is the flat-space value the metric takes interior to a spherically symmetric matter distribution. Meanwhile, for a{t, r) I impose an outer boundary condition a ( i , r m a x ) = — . (3.59) a[t, 7 * m a x ) The condition is for the outer boundary because I solve the slicing condition by integrating inwards from r = r m a x . Furthermore, the specific choice a _ 1 ( £ , r m a x ) is used because of the analogy with Reissner-Nordstrom spacetime in which the temporal and radial metric functions are reciprocal. Finally, regularity for metric functions a and a requires they be even to leading order in coordinate r about r = 0. I employ Kreiss-Oliger dissipation as discussed in Appendix C. The dissipation is added to all functions whose evolutionary equations explicitly include derivatives with respect to t. The complete set is 4>, $ r , n$, Ar, Ur, flt and a m o m , where a m o m represents the metric function a(t,r) when solved using the momentum constraint (3.24). Since the Kreiss-Oliger dissipation operator has a stencil 8 five points wide, it is only applied at locations two grid points from the bounds of the computational domain (r = 0 and r = r m a x ) . Finally, I find the value e = 0.5 a good choice for the adjustable dissipation parameter in all simulations presented. W i t h the choice of Courant factor A = 0.5, the system of equations evolves stably under the Crank-Nicholson iterative scheme. 3.3 Results 3.3.1 Initial Data Profiles The basic idea is to choose a family of initial data parametrized by a single variable p and let the system evolve. Observing the dynamics, I note whether or not the outcome is formation of a black hole at some future time. I then successively adjust the value of p to the critical value p* which marks the threshold of black hole formation: if p > p* the system forms a. black hole, if p < p* it does not. 8 A stencil is essentially a list of the set of points on which a finite difference operator is dependent. Details and discussion can be found in Appendix C. C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 36 Now, since the scalar field is complex . $ = fa + i(j)2 , (3.60) its conjugate momentum must also be complex n$ = n.i + m2, (3.6i) where LTi = IIi(£, r) and LT2 = I I 2 r ) . From definition (3.30) it is evident the complex components of 11$ are explicitly I i i = J (^ 1 + eAt<t>2) , ' . (3.62) and lJ2 = ^(^2-eAt(pij . • (3.63) In terms of the field variables, electromagnetic coupling parameter, and metric components, the electric charge density (3.51) is explicitly. 2e PQ = ~— (</>in2 - (f>2Tli) = ~ (<Pi<i>2 - fa<i>i ^  eAt(<Pl+$)) . (3.64) So while it may be tempting to ascribe positive charge to, say, scalar field component 4>i and negative charge to 4>2, the dependencies of 111 and II2 on both <p\ and (f>2 indicates that positive and negative charge must simultaneously be attributed to both scalar field components. Families of initial data can therefore possess a variety of charge density profiles, depending on the choice of 4>i, (f>2, I i i , II2 and e, and generic initial data will evolve to a state with both positive and negative charge density distributed within some volume of space.9 For given values of e and m<j>, I choose from one of six different initial data families tabulated below and tune a single free parameter to its critical value. Each of the six families are constructed from initial Gaussian profiles for </>i, (j>2, FIi , II2, At and Ar.w That is, <t>l(t = 0,r) = a i e x p ( - ( ( r -r- i ) /<5i) 2 ) , (3.65) <h(t = 0,r) = a 2 exp(-((r - r 2 ) / < 5 2 ) 2 ) , (3.66) ni( t = 0,r) = a 3 exp ( - ( ( r -r3)/53)2), (3.67) n 2 ( i = 0,r) = 0,4 exp(—((r - r 4 ) / < 5 4 ) 2 ) , (3.68) At(t = 0,r) = d t exp( - ( ( r -n)/St)2), (3.69) 9Furthermore, assume that a system is assembled from two species of massive complex scalar field, both with the same charge coupling parameter e, but one having a large mass parameter m$ and the other a small one. Data could always be constructed where the field with large mass parameter contributes a positive charge density at initial time, while the field with small mass parameter contributes a negative charge density. However, as evident from equation (3.64), generic evolution of the field with initial positive charge density will soon appear to carry a component of negative charge, while the initially negative charge density field will appear to contribute some amount of positive charge. While total charge of the ensemble will be conserved, neither positive nor negative contributions by either field need remain separately constant. This behaviour is borne but by numerical simulation. It is in this sense that massive charged scalar fields are incapable of modelling the generic, behaviour of an ionized gas composed of heavy, always positively charged ions amid a cloud of light, always negatively charged electrons. 1 0From equation (3.43) a nonzero radial electric current requires e, $ and <3>* to be nonzero. In addition, the • conditions for a nonzero current are either Ar be nonzero, <3>r be nonzero, or be nonzero. Clearly, families A, B, C and D of Table 3.2 satisfy the nonzero current condition at initial time if e ^ 0. Charge separation in these four cases can therefore be said to originate at initial time t = 0. "Furthermore, since families E or F do not satisfy the requirements for nonzero radial current at initial time, charge separation can only occur when t > 0. C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 37 and Ar(t = 0,r) = arexp(-((r-rr)/5r)2) . • (3.70) The tuning parameter is always a i , at = ar = 0.0132, and depending on the family, 0 2 , 0 3 and o 4 take on values either 0.0 or 0.0132. For all families r i = r 2 = r3 = r 4 = 10.0 (3.71) and 5± = <52 = 53 = S4 = 2.0 . (3.72) The spatial coordinate range is always 0 < r < 80. Finally, all solutions are computed on a finite difference grid with at least J = 3201 grid points in the radial direction, and using adaptive mesh refinement as outlined in Appendix C. family fc, fc, r i i , n 2 pQ(t = 0,r;e^0) A fc{t = 0, r) = a i exp(-((r - rJ/61)*), fc(t = 0, r) = a2 exp(-((r - r2)/82)2), U1(t = 0,r)=a3eM-((r-r3)/53)2), n 2 ( i = 0, r) = o 4 exp(-((r - r4)/<54)2) ^0 B fc(t = 0,r) = a i exp(-((r - r i ) / « i ) 2 ) , 0 2 ( i = O, r )=O, n i ( t = 0,r) = a 3 exp( - ( ( r -r3)/83)2), n 2 (£ = 0,r) = a 4 e x p ( - ( ( r - 7 - 4 ) / c 5 4 ) 2 ) / O C >i ( t = 0,r) = 01 exp(-((r - r i ) /# i ) 2 ) , 0 2 ( i = 0,r) = o 2 exp(-((r - r2)/<52)2)," n i ( t = 0, r) = a 3 exp(-((r - r3)/53)2), n 2 ( i = 0 , r ) = 0 ^0 D fc{t = 0, r) = a i exp(-((r - n)/<5i)2), <£2(i = 0,r) = 0, n i ( t = 0,r) = a 3 exp(-((r - r 3)/<5 3) 2), n 2 ( i = 0,r) = 0 = 0 E fc(t = 0,r) = 01 exp(-((r - n ) / o i ) 2 ) , & ( * = 0,r) = a 2 exp ( - ( ( r - r 2 ) /<5 2 ) 2 ) , n i ( t = 0,r) =0 , n 2 (t = o,r) = o = 0 F 0i( t = 0, r) = 01 exp(-((r - r-i)/<5i)2), 02(* = O, r )=O, n i ( t = 0,r) = 0 , n 2 ( i = 0 , r ) = 0 = 0 Table 3.2: Crit ical gravitational collapse initial data families A through F . The first column lists the family and the second column specifies the initial data profiles for fc, fc, II1 and $ 2 of that family. The third column lists whether or not the initial data can carry nonzero electric charge density when e ^ 0. C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 38 3.3.2 Resul ts for e = 1.0 w i t h m$ = 0.0 As expected, I recover Choptuik's original discretely self-similar and universal solution [23] in the limit e —> 0 and rn<j> —> 0: For values e = 1.0 with m$ = 0.0 I again observe a discretely self-similar and universal solution, but the near-critical solution now carries electric charge. Discrete Self-Similarity and Universality The solution represents itself as an infinite series of echoes as the scalar field repeatedly attempts to collapse to a black hole, but never quite makes it. Wi th each failed attempt some scalar field is shed, and the succeeding echo occurs on an exponentially smaller spatial scale after an exponentially shorter interval of time. The discrete self-similarity of the critical solution is apparent in successive maxima of the quantity 2m(i, r)/r = (l-a~2)+Q 2/(4irr2) , (3.73) where Q = Q(t, r) is the net charge enclosed by a sphere of radius r at time t. For example, having tuned to jp — p*| ~ 10~ 1 6 , I take the profile of 2m(t,r)/r at the maximum of its third echo for family A and spatially rescale it by a factor exp(A), A « 3.47. I then find the rescaled third echo profile matches that of the preceding (second) echo, consistent with the findings of [66] and [68]. This is displayed below in Figure 3.5. Meanwhile, universality of the solution is manifest in the fact that all critical solutions exhibit like profiles and that the scaling exponents A are essentially the same for all families A through F . Mass and Charge Scaling As the critical solution is approached from above, it would appear the back holes formed become arbitrarily small as p —• p*. This is supported by the results presented in Figure 3.6. Down to machine precision, black hole masses there observe a power-law behavior ln(m B H) = 7 m l n ( ( p - p * ) / p * ) + c i , (3-74) where c\ is a family dependent constant. For family A I find a value 7 M « 0.385, consistent with the results of [66], [68] and analysis of [57]. Not only do black hole masses observe a power-law relationship with respect to the critical parameter, but so do the black hole charges. Supporting numerical results are presented in Figure 3.7. Similarly, this power-law has the functional form 1H(|QBH|) = 1Q In ((p - P*)/P*) + c 2 , (3.75) where c 2 is another family dependent constant. 1 1 However, 7Q > 7 M so the black hole sheds charge more rapidly than mass with each successive echo as the critical solution is approached from above. For family A I find 7 Q . « 0.854, again consistent with [66], [68] and [57], and also their prediction that 7Q > 2 7 M . Explicitly, I show the charge to mass ratio of the black hole solutions in Figure 3.8. Clearly, while'the ratio IQBHI/^BH decreases exponentially as p —> p*, it is possible to form black holes with 0 < |QBH|/TOBH ^ 1 when sufficiently far from the critical solution. n T h e data of Figure 3.7 suggest a small periodic wiggle superimposed on the otherwise straight line fit, consistent with the predictions and results of [57], [66] and [68]. While a similar periodic wiggle is known to exist for the mass scaling relationship of both charged [57], [66], [68] and uncharged [58], [67] massive scalar fields, the behaviour is not apparent in Figure 3.6. The absence of a wiggle can there be attributed to numerical inaccuracy arising with the divergence of polar areal radial metric function a(t, r) just before apparent horizon formation. Of course, the periodic wiggle could be explored further by examining maximum spacetime curvature of subcritical solutions as p —• p* from below. C H A P T E R 3. C H A R G E D S C A L A R C R I T I C A L C O L L A P S E 39 A ^ 3 . 4 7 1 0 0.1 0 .2 0 .3 0 .4 0 .5 0.6 F i g u r e 3.5: Discrete self-similarity for the critical solution of family A with e = 1.0, m$ = 0.0 as demonstrated by spatially rescaling the third echo of 2m(t,r)/r by exp(A) and comparing it to the unsealed second echo. The two echo profiles agree so closely their differences are not apparent on the scale of this figure. C H A P T E R 3. C H A R G E D S C A L A R C R I T I C A L C O L L A P S E •10 F i g u r e 3.6: Power-law scaling of the black hole mass TTIBH as the critical solution is approached from above for family A with e = 1.0 and m * = 0.0. Here, the scaling exponent is 7 M w 0.385. C H A P T E R 3. C H A R G E D S C A L A R C R I T I C A L C O L L A P S E 41 - 2 5 - 2 0 - 1 5 - 1 0 - 5 l n ( ( p - p*)/p*) F i g u r e 3.7: Power-law scaling of the black hole charge QBH as the critical solution is approached from above for family A with e = 1.0 and m$ = 0.0. Here, the scaling exponent is 70 « 0.854. C H A P T E R 3. C H A R G E D S C A L A R C R I T I C A L C O L L A P S E 42 0.8 0.6 X -B 0.4 X CQ Of 0.2 0 I I I I I I I I I I I I I I e=1.0, m # = 0.0 family A i i i i i i i i i i i i i i -25 -20 -15 -10 - 5 ln((p - p-)/P*) F i g u r e 3.8: Charge to mass scaling of the black hole solutions as the critical solution is approached from above for family A with e = 1.0 and m$ = 0.0. The charge of black hole solutions decreases more rapidly than black hole mass as p —• p*, indicating the exactly critical solution wil l be uncharged. Furthermore, solutions with 0 < |QBH|/TOBH < 1 can be obtained when p is sufficiently far from its critical value p*. CHAPTER 3. CHARGED SCALAR CRITICAL COLLAPSE 43 3.3.3 Results for e = 1.0 w i t h m$ = 1.0 When the scalar field mass parameter TO$ is sufficiently larger than zero, the critical solution ceases to be discretely self-similar. Rather, it becomes periodic—resembling an oscillating perturbed boson star with nonzero electric charge. Again, the existence of such solutions is not surprising—not only are charged boson star solutions known to exist [79], [80], [81], but the current understanding of massive scalar critical phenomena would also suggest that critical solutions resembling perturbed charged boson stars should exist for an appropriate set of initial conditions and range of parameter values e and ra$ [31]. However, the emergence and details of such solutions were not previously observed. Figures 3.9, 3.10 and 3.11 show the maximum of 2m(t,r)/r and its location for an e = 1.0, m$ = 1.0 supercritical evolution of the family F when \p — p*\ ~ 10~ 1 6 . Examining Figures 3.10 and 3.11 it is clear the nearly critical solutions are composed of a high frequency component with period « At modulated by a lower frequency of period fa AOt. Assuming the results from [63], [64] and [65] carry over to the situation presented here, the low frequency period « AOt can likely be attributed to excitation of the first harmonic mode of an unstable boson star. However, the occurrence of an additional higher frequency period ( « At or otherwise) has not been noted in previous studies of critical gravitational collapse in massive complex scalar field systems. But note those previous studies involved perturbation of (static) boson-star initial conditions when obtaining critical solutions. It is entirely possible initial conditions of that sort may actually disallow the additional high frequency components as observed in this study. Finally, note the presence of the high frequency mode throughout the entire evolution (not.just over the critical phase 140 < t < 450) here indicates it is not unique to the critical solution, and so must be associated with more generic properties of the scalar field evolution. To investigate further consequences of the additional high frequency mode, the scalar field modulus at r = 0 is plotted as a function of time in Figure 3.12. It is there apparent how the maxima of |$(t, 0)| follow a low frequency period likely equal to that of the first harmonic mode of an unstable boson star. However, the scalar field amplitude also oscillates with a period ss At, dropping to nearly zero with each oscillation. 1 2 Furthermore, careful examination reveals that while the low frequency maxima of |<3>(t, 0)| occur in phase with the low frequency mode maxima of max(2m(i, r)/r), the high frequency extrema of |$ ( i , 0)| occur exactly out of phase with those of max(2m(£, r)/r). This opposing phase correlation is observed throughout evolution, not just during the critical phase 140 < t < 450. Furthermore, the high frequency opposing phase correlation is observed between functions 0)| and max(2m(£, r)/r) for (generic) evolution of this family when p is far from p*. To examine possible correlation of the two dominant modes of oscillation and electric charge, Figure 3.13 displays Q(t,r), the total charge contained within spheres of r, for family F when \p ~ P*\ ~ 10~ 1 6 . The chosen values for r are r = f, r = 2f, r = 3f and r = Af, where f « 2.7 is the average radius of max(2m(£, r ) / r ) over 150 < t < 360. This figure shows how, just as for |$(£,0) | , the low frequency local maxima in Q(t,f) and Q(t,2f) correlate with the low frequency local maxima in max(2m(£, r)/r). Furthermore, a high frequency component is observed in Q(t,f) and Q(t, 2f), as would be expected from the relationship between charge density and scalar field amplitude. Notice how integration of charge density over larger radii effectively averages the dynamics observed in Q(t,r). Finally, it is clear the average value of Q{t,r) tends towards zero from above, indicating small net leakage of positive charge (or inflow of negative charge) from smaller (larger) radii during critical evolution. Finally, to investigate whether the additional high frequency mode stems from electromagnetic coupling, the critical solution was obtained for family F when m$ = 1.0 and e = 0.0. In doing so, it found the high frequency oscillation of period rvAt was again present, as was the fundamental 12Similar behaviour has been noted for the so-called (uncharged, non-critical) phase-shifted boson star solutions documented in Hawley's dissertation [64]. There, numerical evolution suggested the oscillations in scalar field amplitude were likely stable and quasi-periodic. C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 44 F i g u r e 3.9: max(2m(t,r)/r) of the critical periodic state for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields evolved from initial data family F when \p — p*\ ~ 1 0 - 1 6 . After an initial phase of evolution, the solution enters its critical state which lasts over the period 140 < t < 450. The sharp increase of max(2m(i, r)/r) at late time corre-sponds with a(t,r) —> 1, the signal of imminent apparent horizon formation. A n M P E G animation of 2m(t,r)/r over the period 0 < t < 507 leading to black hole formation is available at ht tp: / /$WWW/People/petryk/animations/1 .mpeg where $WWW is currently l a p l a c e . p h y s i c s . u b c . ca . C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 45 I I I I I I I I I I I I I I I I I I L 150 20  250 30  350 t F i g u r e 3.10: A closer view of max(2m(t,r)/r) showing oscillations within the critical periodic state for e = 1.0, m$ .= 1.0 Maxwell-Klein-Gordon fields evolved from family F when |p - p*\ ~ 10~ 1 6 . The average value of max(2ra(£, r ) / r ) over 150 < t < 360 is approximately 0.30. The periodicity of the solution is clearly evident as a low frequency component with period « 40i along with a high frequency component with periodicity « At. i C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 46 2 I L_J I I I I I I I I l _ J I I I I I L 150 200 250 300 350 t Figure 3.11: A view of r(max(2m(£, r)/r)) within the critical periodic state for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields evolved from family "F when \p — p*\ ~ 1 0 - 1 6 . Clearly, the location of max(2m(£, r)/r) oscillates about an average radius of approximately f « 2.7 for 150 < t < 360 during the critical phase of evolution. CHAPTER 3. CHARGED SCALAR CRITICAL COLLAPSE 47 0.15 0.1 +->" ~ 0.05 0 150 200 250 300 350 t F i g u r e 3.12: Modulus of the scalar field amplitude at r = 0 for the critical periodic state of e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields evolved from family F when \p — p*\ ~ 1 0 - 1 6 . Unlike previously studied boson star critical solutions where just the first harmonic mode is excited and 0)| remains :far from zero, the central scalar field amplitude of the highly excited periodic critical solution drops to nearly zero with each high frequency oscillation of period « 4£ . 1 4 Furthermore, while previous perturbed boson star critical solutions displayed coincident peak values of max(2m(£, r)/r) and 0)|, just .the local maxima associated with the low frequency excitation remain coincident for the solution displayed here. In fact, local maxima associated with the high frequency mode of |$(f, 0)| are precisely coincident with the local minima associated with the high frequency mode of max(2m(£ , r ) / r ) i C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 48 0 .005 0 0 .005 0 Q(t.r) . 0 .005 0 0 .005 0 :—1 1 1 1 1 1 1 1 1 1 1 1 1 r—hz 1 , 1 , i i 1 i i , , 1 , 1 2 0 0 i i i i 300 t i i i i i 4 0 0 i i i i i (a) (b) (c) (d) S 0 .25 J i i_ J L 2 0 0 300 t 4 0 0 F i g u r e 3.13: Charge contained within spheres of radius r for the critical periodic state from e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields evolved from initial data family F . A frame displaying max(2m(i, r)/r) is provided for reference. Subfigure (a) dis-plays Q(t,r) for r = f where f « 2.7 is the average radius of max(2m(£ , r ) / r ) over 150 < t < 360. Meanwhile, (b) displays Q(t,r) for r = 2f, (c) displays Q{t,r) for r = 3f, and (d) shows Q(t,r) for r = 4f. From (a) and (b) it is clear that, at small radii, Q(t,r) fluctuates with a low frequency component period s=s 40£ and a high frequency component period of « 4£. While the low frequency mode becomes somewhat irregular at later time, the high frequency component is persistent and very regular throughout evolution. Integration over larger radii effectively averages the fluctuations in Q(t,r). In doing so, it makes clear a tendency for the net charge within r = 4f to slowly decrease in time during the critical phase of evolution. C H A P T E R 3. C H A R G E D S C A L A R C R I T I C A L C O L L A P S E •19 500 400 t- 300 200 100 - 4 0 - 3 0 - 2 0 - 1 0 ln |p - p*| F i g u r e 3.14: Lifetime of the critical periodic state for e = 1.0, m<p = 1.0 Maxwell-Klein-Gordon fields evolved from initial data family F . The exponential scaling of r with |p — p*| is typical of perturbed boson star critical solutions [63], [64], [65], [92], and the scaling exponent is here - y T « 12.3. harmonic mode of period w 40£. The critical solutions for family F with m$ = 1.0 and e = 1.0, and m$ = 1.0 and e = 0.0 are very similar in all but two regards. The first being the obvious lack of charge for case e = 0.0. The second is the lifetime of the critical state for [p — p*j ~ 1 0 - 1 6 . While the critical phase of evolution for the e = 1.0 charged case was 140 < t < 450, the critical phase of the e = 0.0 uncharged case was merely 140 < t < 240, indicating the unstable mode of the charged critical solution appears to grow more slowly than that of the uncharged critical solution when the adjustable parameter of this family is tuned to its critical value to within one part in 10~ 1 6 . This brings discussion to the lifetimes of the charged periodic critical solutions. Figure 3.14 displays the scaling of the critical solution lifetime, T , with In |p — p*|. Clearly, the relationship of lifetime with tuning parameter is r = - 7 r l n j p - p * | + c 3 , (3.76) where 7 r and C3 are solution-dependent constants (cf. [63], [64], [65], [92]).15 The lifetime of the critical solution is measured from the time it takes to form a black hole from the slightly supercritical data. In the present case 7 r ~ 12.3. 1 5This constant is known to be related to the growth factor of unstable modes for uncharged boson stars. The relationship can be verified through perturbation theory analysis about equilibrium boson star solutions. While it is likely that a similar relationship exists for weakly charged boson stars, I am unaware how this relationship could carry over for boson stars with strong electromagnetic coupling. Investigation along these lines are certainly of interest for future calculation. C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 50 Typical of such periodic critical solutions is the presence of a mass gap in the black hole spectrum when tuning to p* from above (i.e.,-they exhibit Type I critical behaviour). This means there is a minimum mass for the black holes formed from supercritical evolution. This behaviour is clearly depicted in Figure 3.15 where the average black hole mass limit is TOBH ~ 0.48. A least-squares fit of the data to TTI-QB. = Ci\n\p — p*\ + c 5 yields values c 4 « —9.2 x 10~ 4 and c 5 « 4.6 x 1 0 - 1 . Also provided is Figure 3.16 displaying how the average charge of these black holes approaches the limit Q B H « 0.0032. A least-squares fit to Q B H — CQIU \p-p* \ +C7 here yields CQ SS 3.6 x 10~ 5 and C7 w 4.1 x 10~ 3 . These results together imply a charge to mass ratio Q B H / ^ B H ~ 1 0 - 2 as p —> p* for this family of solutions. These periodic solutions carrying electric charge represent a new result, because the only critical solutions found prior to this investigation have been uncharged. Finally, the oscillation periods of the critical solution are dependent on e and m$, and the scaling exponent 7 T depends on the initial data family. Of course, the family dependence of perturbed boson star solutions is well known [63], [64], [65], [92] and means the critical solutions are not universal. Table 3.3 lists the properties.of black holes formed from slightly supercritical Type I solutions for each of the families A through F when e = 1.0 and m$ = 1.0. Each family exhibiting Type I critical behaviour has a scaling exponent *yT « 12 and forms a black hole of mass TOBH ~ 0.5. However, the black hole charges vary widely, and the charge to mass ratios of the black holes range from IQBHI/^BH ~ 0.006 for family D to |QBH|/"1BH ~ 0.3 for family B . family PQ(t = 0,r) Q(t = 0,r) 7r TOBH(P -> p*) QBK(P -> P*) A - - - - -B < 0 « -0.46 « 12.4 MO . 48 ' w -0.16 C - - - - -D = 0 = 0 « 11.9 « 0.45 « 0.0026 E = 0 = 0 « 12.3 « 0.47 w 0.0032 F = 0 = 0 « 12.3 « 0.48 « 0.0032 Table 3.3: Black hole properties from gravitational collapse of slightly supercritical Type I solu-tions for families A through F when e = 1.0 and m$ = 1.0. The first column lists the family (cf. Table 3.2), the second column specifies whether the initial charge density is nonzero and the third column lists the total charge carried by the initial data. The fourth column lists the lifetime scaling exponent for the Type I critical solution. The scaling exponents are all close in value. The fifth and sixth columns respectively list the black hole mass and charge from gravitational collapse of slightly supercritical so-lutions. The black hole masses are all close in value, but the charges cover a wide range of values. The charge to mass ratio of the black holes range from | Q B H | / T O B H ~ 0.006 for family D to | Q B H | / T O B H ~ 0.3 for family B . The entries for families A and C are left blank, since those initial data profiles do not exhibit Type I critical solutions for any value of tuning parameter fa. Concluding discussion of the results, I now comment on the final fate of subcritical periodic solutions. In the earlier studies [63], [64] and [65] it was assumed the subcritical periodic solutions were simply described by dispersal of the scalar field to large distances. Since then it was found that slightly subcritical periodic solutions may actually begin dispersal, only to gravitationally recontract to a more compact distribution at later time [92]. After such recontraction to a more compact state, the scalar field distribution may either collapse to form a black hole or again expand to larger radii. If a second dispersal is attempted, the scalar field distribution wil l again recontract to a more compact compact state before either collapsing to a black hole or repeating the process of expansion and contraction. The results I now present displaying this behaviour were found C H A P T E R 3. C H A R G E D S C A L A R C R I T I C A L C O L L A P S E 51 i — i — i — I — i — i — i — i — I — r - m 0 .5 0 _J I I 1 I I I I 1 I i I I I I L - 4 0 - 3 0 - 2 0 - 1 0 ln|p - p*| F i g u r e 3.15: Mass gap for (supercritical) black hole solutions for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields evolved from initial data family F . The average minimum mass of black holes formed from slightly supercritical data remains fairly constant about TTIBH ~ 0.48 as p —* p*. The best fit line is shown for reference. CHAPTER 3. CHARGED SCALAR CRITICAL COLLAPSE 52 0 .04 0 .02 m f) Qf U - • » , • V V " * ' * - 0 . 0 2 h •0.04 J I 1 1 I I I L - 4 0 3 0 - 2 0 ln|p - • p*| 10 F i g u r e 3.16: Minimum charge of (supercritical) black hole solutions for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields evolved from initial data family F . The average charge of black holes formed from slightly supercritical data remains fairly constant about Q B H 5 8 0.0032 as p —* p*. The best fit line is shown for reference. C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 53 independent, and without knowledge, of those eventually published in [92]. Figure 3.17 shows max(2m(£, r)/r) and |4>(£,0)| in time for the slightly subcritical evolution of family F with e = 1.0 and m$ = 1.0. Meanwhile, Figure 3.18 shows max(2m(£, r)/r) and |$(£ ,0) | in time for the slightly subcritical evolution of family F with e = 0.0 and m$ = 1.0. In both cases, the displayed evolution is for a state with \p — p*\ ~ 10~ 1 6 . Observe how the solution with e = 1.0 enters its critical phase of evolution from 140 < t < 450 before attempting dispersal. The solution then attempts recollapse and max(2m(t,r)/r) peaks at t « 660. The charged scalar field attempts a second dispersal, but then undergoes complete gravitational collapse at t « 960. Also observe how the high frequency oscillations with period « 4£ persist throughout the evolution. Furthermore, note the correlation between the high frequency oscillations of |4>(£,0)| and those of max(2m(£,r) /r) , and how the amplitude |$(£,0) | drops to nearly zero with each oscillation period « 4£ from the beginning to the end of evolution. Meanwhile, Figure 3.18 demonstrates how the slightly subcritical evolution of for e = 0.0 enters the critical phase of evolution which lasts from 140 < i < 250 before attempting a seemingly infinite series of attempted dispersals and collapses. The high frequency oscillations of period « it are again observed throughout evolution in both max(2m(£, r)/r) and |4>(£,0)|. While the |4>(£,0)| drops to nearly zero with each oscillation, the cycle averaged value of |4>(t,0)| shows direct correlation with the cycle averaged value of max(2m(£, r)/r). Finally, with each attempted dispersal and contraction on the scale of « 230* (similar to that found in [92]), the v value of max(2m(£, r)/r) asymptotes towards a value « 0.14. Similar behaviour is seen in |4>(£,0)| as its peak values tend to approach ss 0.04. While only presented for family F initial conditions, similar behaviour can be observed for periodic solutions of all families A through F when m<j> = 1.0 and 0.0 < e < 1.0. 3.3.4 Code Testing Finally, the adaptive mesh code I use to obtain the results is demonstrably convergent to second order, and through convergence of conserved quantities can be shown to be accurate to well within a percent for a typical evolution. Also, the suitability of the approximate Sommerfeld boundary condition was briefly investigated and found to contribute an error on the order of a percent in worst case scenarios,. Detailed investigation of these boundary condition effects is certainly of interest in future calculation. In conclusion I provide in Figure 3.19 an explicit demonstration that the code, being convergent, generates lifetime scaling exponents that asymptote towards a specific value. In this figure are plotted the lifetime of the nearly critical periodic solutions for family F when e = 1.0 and m$ = 1.0. The lifetimes were calculated using the adaptive code, but with the mesh refinement effectively disabled by setting a large solution error tolerance. 1 6 Results were thus obtained on fixed finite, difference meshes with J = 1601, J = 801 and J = 401 grid points. Since the critical parameter value p* is resolution dependent, this required recalculation of the critical parameter value for each of the three resolutions. Fit t ing a liner function to the results at each resolution generates values of 7 r w 12.5 for J = 1601, jT « 13.0 for J = 801 and jT « 17.5 for J = 401. Clearly, as resolution is increased, the lifetime scaling exponent approaches the value 7 r « 12.3 obtained using the adaptive code. Of course, this behaviour is entirely expected. See Appendix C for discussion of the adaptive algorithm and refinement by solution error. C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 54 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 t 0 . 2 5 r i i i i i i i i i i i i i i i i i i r 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 t F i g u r e 3.17: max(2m(£ , r ) / r ) and |$(£ ,0) | in coordinate time t for the e = 1.0, m$ = 1.0 fields for slightly subcritical initial data family F , specifically \p — p*\ ~ 10~ 1 6 . After the critical phase of evolution 140 < t < 450, the charged massive scalar field attempts dispersal twice before eventually collapsing to form a black hole at t w 960. The high frequency oscillations of period « At are apparent throughout evolution for both max(2m(t,r)/r) and |<&(t,0)|,'and |$( i ,0) | drops to nearly zero with each of these oscillations. C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 55 0.5 0 i I i r j i i i i i I i i i i I i i i L 0 1000 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 t 0.2 0 .15 0.1 0 .05 0 0 1000 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 t Figure 3.18: max(2m(t,r)/r) and |<3>(t,0)| in coordinate time t for the e = 0.0, m$ = 1.0 fields for slightly subcritical initial data family F , specifically \p — p*\ ~ 1 0 - 1 6 . The mas-sive scalar field exits its critical period of evolution and undergoes an apparently infinite series of dispersals and contractions. The high frequency oscillations of pe-riod w At are apparent throughout evolution for both max(2m(£, r)/r) and |$(£, 0)|, and |$(£,0) | drops to nearly zero for each of these oscillations. It appears that max(2m(£,r)/r) and |$(£, 0)| gradually relax to a less excited state as the evolution proceeds. C H A P T E R 3. C H A R G E D S C A L A R CRITICAL C O L L A P S E 56 Figure 3.19: Lifetime of the critical periodic state for e = 1.0, m«, = 1.0 Maxwell-Klein-Gordon fields evolved from initial data family F on three successive grid resolutions. The lifetime scaling exponent of the nearly critical solution on a grid of J = 401 points is 7 R ss 17.5, on a grid of J = 801 points is 7 R as 13.0, and on a grid of J = 1601 points is 7 R ss 12.5. Wi th increase in resolution, the scaling exponent thus approaches a value near 7 R ss 12.3 as found with the adaptive code. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 57 C H A P T E R 4 C H A R G E D S C A L A R A C C R E T I O N To summarize, Chapter 1 introduced and motivated the research and results to be presented in this dissertation, Chapter 2 covered the physical formalism to be used in the investigations of following chapters, and Chapter 3 provided discussion and results for the critical gravitational collapse of spherically symmetric charged massive scalar fields. This chapter wil l cover the scattering and accretion of axially symmetric charged massive scalar fields in Kerr spacetime. The length of this chapter warrants outline of the physics to be investigated and results obtained through numerical solution. Table 4.1 has also been provided for reference and summary. In this chapter, I first provide discussion of theory and previous work on the subject of s = 0 (scalar) and s = 1 (electromagnetic) perturbations of black hole spacetimes. Specifically, §4.1.1 outlines the effective potential formulation for Schwarzschild spacetime, the trapping of massive scalar perturbations in the effective potential formulation, and the scattering of both scalar and electromagnetic perturbations from the effective potential perspective. §4.1.2 outlines the physics of glories and orbiting resonances in Schwarzschild spacetime from the perspective of the effective potential formulation. §4.1.3 outlines the effective potential formulation of s = 0 and s = 1 perturbations in Kerr spacetime, and §4.1.4 covers the physics of superradiant scattering and some related topics. §4.1.5 and §4.1.6 close discussion of the underlying theory by respectively covering the topic of black hole perturbation decay at late time and reviewing related work in the area of charged scalar perturbations. Table 4.1 references these subsections in the context of physics investigated through numerical simulation as presented later in the chapter. Following the theory is a section which details the system of equations to be used in numerical simulation. This includes discussion of the Kerr spacetime (§4.2.1), the matter Lagrangian and equations of motion (§4.2.2 and §4.2.3), physical fields and conserved quantities (§4.2.4). The section ends by discussing some aspects of the numerical solution method, including regularity, boundary conditions and dissipation (§4.2.5), and then strategies for adaptive mesh refinement and parallel computation (§4.2.6). The remainder of this chapter presents numerical solution of the equations presented in §4.2 to investigate the theory of §4.1. Specifically, §4.3.1 outlines the prescription of initial data, §4.3.2 presents results for orbiting resonances and mode trapping of massive s = 0 perturbations and §4.3.3 presents results displaying orbiting resonance and superradiant scattering of $ = 1 pertur-bations. §4.3.4, §4.3.5 and §4.3.6 then present solutions for electromagnetically coupled s = 0 and s = 1 perturbations which display orbiting resonance, charge separation, dynamo-like behaviour and collimation along the axis of black hole rotation. Again, Table 4.1 provides quick and con-venient summary of the observed physics in relation to predictions from the theory of black hole perturbations. The last few subsections of the chapter provide validation of the numerical solutions through convergence testing (§4.3.7) and independent residual testing (§4.3.8) the numerical imple-mentation, and by testing the approximate outgoing radiation boundary condition used throughout (§4.3.9). C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 58 s phenomenon, subsection(s) e, a/M . detail(s) 0 orbiting resonance, §4.1.1, §4.1.2, §4.1.3, §4.3.2 0.0, 0.0, > 0.0 orbital period ss 70t, orbital attenuation ~ 10 - 1 , log(T) ~ -kt, k ss 1 x. 10~2 mode trapping, §4.1.1, §4.1.3, §4.3.2 0.3, 0.0, > 0.0 log(T) kt, k ss 5 x 10" 3 rapid infall, §4.1.1, §4.1.3, §4.3.2 2.4, 0.0, > 0.0 log(T) ~ -kt, k ss 7 x 10~2 energy collimation, §4.3.2 0.3, 0.0, > 0.0 / pT(t,r,0)dtdr ~ / PT{t,r,ir)dtdr » / pT(t,r, w/2)dtdr 1 orbiting resonance, §4.1.1, §4.1.2, §4.1.3, §4.3.3 0.0, 0.0,. > 0.0 orbital period w 70t, orbital attenuation ~ 10 - 1 , log(T) ~ -kt, k w 1 x 10" 2 superradiant scattering, 0.0, 0.0, - 0.8 amplification ss 0.05% §4.1.3, §4.1.4, §4.3.3 0.0, 0.0, 0.98 amplification ss 1.3% 0.0, 0.0, > 0.998 amplification ss 1.6% 0, 1 orbiting resonance, §4.1.1, §4.1.2, §4.1.3, §4.3.4, §4.3.5 0.0, < 0.6, > 0.0 orbital period ss 70t, orbital attenuation ~ 10 _ 1 , log(T) kt, fc w 1 x 10~2 orbiting resonance, §4.1.1, §4.1.2, §4.1.3, §4.3.6 0.6, 0.6, > 0.0 orbital period ss 70£, orbital attenuation ~ 10 _ 1 , log(T) kt, fc ss 6 x 1 0 - 3 charge separation, §4.3.4, §4.3.5, §4.3.6 > o.o, > 0.0, > 0.0 dtpQ(t,r,9) / 0 for t > 0 dynamo-like behaviour, §4.3.4, §4.3.5, §4.3.6 > o.o, > 0.0, > 0.0 dtJ<t,{t,r,6) ^ 0 for t > 0 energy collimation, §4.3.6 0.3, 0.6, > 0.0 / pT(t,r,0)dtdr •ss Jpx(t,r, ir)dtdr » /pT(t,r,n/2)dtdr, J PT{<s>)(t,r, 0)dtdr » /PT(A)(t,r,0)dtdr » /PT(*A){t,r,0)dtdr Table 4.1: Predicted and observed physics of the s = 0 (scalar) and s = 1 (electromagnetic) perturbations in Kerr spacetime. The second column lists phenomena which arise and chapter subsections covering the relevant theory and numerical solutions. The third column specifies the scalar field mass parameter m$, the fourth lists the value of electromagnetic coupling parameter e, and the fifth column lists a/M where M is the black hole mass and a is the black hole angular momentum per unit mass. The sixth column summarizes some of the most significant details from numerical simulation. This table is to be treated as an index, and the reader is referred back to the listed chapter subsections for notation, detail and complete discussion. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 59 4.1 Scattering and the Physics of Black Hole Perturbations 4.1.1 The Effective Potential Formulation for Schwarzschild Spacetime The programme of quasi-normal mode analysis has provided insight that, in part, helps explain the early and intermediate time behaviour in the scattering phenomena I study. Consider a pertur-bation * = ty(t,r,9,4>) in the form of a massless, uncharged Klein-Gordon field in Schwarzschild spacetime. Recall this spacetime is described by the line element tf^^jtf+^j'V^ (4.1) in the usual (t, r, 0,4>) Schwarzschild coordinates, where M determines the black hole mass and dQ.2 = d92 + s in 2 6d(f)2 . (4.2) B y instead choosing to use the (t,r*, #,</>) radial tortoise coordinate system, denned in relation to the (t, r, 9, <j>) coordinates by _d_ dr* and expanding the perturbing field in spherical harmonics Yim(9, <p). as per *(*, r, M ) = £ ^^Yim(9,<t>) , (4.4) l,m the radial equation of motion for the perturbing field is simply 82 82 dr2 dt2 V , ( r ) j ^ ( t , r ) = 0 , (4.5) where V](r) is the effective or Regge- Wheeler potential (cf. [132]) « M - ( i - ^ ) ( ^ + £ ) - . . • ' . • " ( « ) Furthermore, if a harmonic time decomposition i>t(t,r) = J ^ ( w . r j e - ^ d w (4.7) is assumed, then equation (4.5) becomes the ordinary differential equation (J^+"2-Vi(r))Mu,r)=0, (4.8) where VJ is again given by (4.6). More generally, for axial perturbations (i.e., those which transform as (—1) ' + 1 under a change of parity) where s = 0,1,2 is the spin of the perturbing field. For massless Klein-Gordon fields s = 0, for electromagnetic radiation perturbations s = 1, and for gravitational perturbations s = 2. The term 2M/r is interpreted as the Newtonian contribution, 1(1 + l)/r2 is the so-called centrifugal barrier, CHAPTER 4. CHARGED SCALAR ACCRETION 60 F i g u r e 4.1: The effective (Regge-Wheeler) potential for massless scalar perturbations in M = 0 Minkowski spacetime. Displayed is the potential for modes I = 0 through I = 10. / = 0,1 = 5 and I = 10 are highlighted for reference. Since M = 0, only the centrifugal barrier contributes to scattering of scalar perturbations. This indicates scalar modes cannot be trapped in the Minkowski spacetime. and 2(1 — s 2 ) M / r 3 is the additional term from relativistic effects. Notice how the relativistic term is dominant for sufficiently small r. Purely radial modes are those with I = 0. Equations (4.6) and (4.9) show the effective potential has a peak near r = 3 M (corresponding with the unstable circular photon orbit from the null geodesic equations). Figures 4.1, 4.2 and 4.3 show the effective potential for scalar modes I = 0 through I = 10 for M = 0, M = 1 and M = 5 spacetimes, respectively. Except for the case M = 0, it is clear that a change in black hole mass M rescales the Vj, but leaves them otherwise unchanged. In the exceptional M = 0 case, only the centrifugal barrier contributes. Figure 4.4 displays the effective potential for electromagnetic modes I = 0 through I = 10. Besides the vanishing I = 0 mode, the electromagnetic effective potential is similar to that for scalar perturbations. Because the effective potential has at most a single extremum (in this case a maximum) massless scalar and electromagnetic perturbations are expected to scatter without becoming trapped. Now, allowing for massive scalar perturbations in Schwarzschild spacetime, the effective poten-tial becomes m B (, - m f g ± l > + J & z j f H + 0 ~ » K » - 4 4 ) , ( 4 . 1 0 ) being the mass parameter of the perturbing scalar field. Figures 4.5 and 4.6 show the effective potential for scalar modes I = 0 through I = 10 in a spacetime with M = S, respectively for C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 61 i l I | I I l I r F i g u r e 4.2: The effective (Regge-Wheeler) potential for massless scalar perturbations in Schwarzschild spacetime with M = 1. Displayed is the potential for modes I = 0 through / = 10. I = 10. I = 0, I = 5 and I = 10 are highlighted for reference. Scalar perturbations can scatter off the centrifugal barrier and also off the peak near r = 3 M , but cannot be trapped in the spacetime. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 62 F i g u r e 4.3: The effective potential for massless scalar perturbations in Schwarzschild spacetime with M = 5. Displayed is the potential for modes I = 0 through Z = 10. I = 10. / = 0,1 = 5 and I = 10 are highlighted for reference. Again, while scalar perturbations scatter off both the centrifugal barrier and potential peak near r = 3 M , they do not experience trapping in the spacetime. The change in black hole mass simply shifts the location of the potential maximum to larger radii. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N F i g u r e 4.4: The effective potential for electromagnetic perturbations in Schwarzschild spacetime with M = 5. Displayed is the potential for modes I = 0 through I = 101 { = 10. I = 0, I = 5 and / = 10 are highlighted for reference. As in the case of scalar perturbations, scattering off the centrifugal barrier and potential peak near r = 3 M does not allow the possibility of mode trapping. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 61 Figure 4.5: The effective potential for = 0.3 massive scalar perturbations in Schwarzschild spacetime with M = 5. Displayed is the potential for modes I = 0 through Z = 10. I = 10. I = 8,1 == 5 and Z = 10 are highlighted for reference. The nonzero scalar perturbation mass introduces a local minimum of potential about which perturbations can be trapped. = 0.3 and « # = 0.6. This shows the scalar field mass parameter can modify the effective potential by either removing the maximum, or shifting its location to larger radii and possibly introducing a local minimum. The details depend on the black hole mass, the mode number, and the scalar field mass parameter. Figures 4.7 and 4.8 show the effective potential for the scalar modes with local minima for = 0.3 and = 0.6 respectively. For the case = 0.3 the minimum occurs near r = 45 for mode I = 5, and for m$ = 0.6 it occurs near r = 35 for I = 10. Since the effective potential possess a local minimum, massive scalar perturbations of an appropriate mode and energy can become trapped and persist indefinitely in a region of spacetime not far from the black hole. The effect is like that of mirrors reflecting the modes back and forth between two radii straddling the potential minimum. Also note that trapping does not occur in the M = 0 Minkowski limit, since then the only contributions to the potential are a positive term 1(1 + l ) / r 2 , and a constant m\. Deeper analysis of the effective potential reveals interesting behaviour. The effort is simplified if one considers high or low frequency limits for scattered perturbations. In these cases closed form approximations can be obtained. The results for massless scalar and electromagnetic perturbations clearly indicate that, in the high frequency limit wM —> oo, perturbations are hardly affected by the effective potential barrier. This means that such incident perturbations are almost completely absorbed by the black hole (it has a large transmission amplitude, Aram and small reflection amplitude, A r e f ) . The opposite is true in the low frequency limit uM —> 0. In that case, almost all C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 0.3 0 .2 0.1 0 1 1 1 1 1 1 1 1 1 I I , - ll/ - 1 = 1 0 ~ m - 1=5 1 - 1=0 II i i l i i II 1 i i i i 1 i i i i 1 i i i i 0 10 2 0 3 0 4 0 50 r F i g u r e 4.6: The effective potential for m* = 0.6 massive scalar perturbations in Schwarzschild spacetime with M = 5. Displayed is the potential for modes I = 0 through I = 10. I = 10. I = 0, I = 5 and I = 10 are highlighted for reference. Again, the scalar perturbation mass gives rise to a local minimum of potential where perturbations can be trapped. Furthermore, the location of this minimum changes with perturbation mass. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 66 Figure 4 .7 : The I = 5 effective potential for = 0.3 massive scalar perturbations in Schwarzschild spacetime with M = 5. This figure details the potential for I = 5 scalar modes. These modes exhibit the deepest local minimum of potential and should therefore experience the greatest trapping in the M = 5 spacetime. The local minimum is in this case located near r = 45. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 67 Figure 4 . 8 : The I = 10 effective potential for my = 0.6 massive scalar perturbations in Schwarzschild spacetime with M = 5. The I = 10 modes experience the deepest local minimum for the M = 5 spacetime. The minimum is located near r = 35 for this choice of perturbation mass. This designates the location about which = 0.6 perturbations are likely to be trapped. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 68 of the incident perturbation is scattered off the potential barrier (AtTim is low, and Aref is high). The analysis is well documented, and I refer the reader to [49], [51] and references contained therein. Finally, it should be clear from this discussion that the radial profile of an ingoing perturbation will have an effect on the transmission and reflection amplitudes.- Specifically, steep gradients in the radial profile imply rapid changes in amplitude as the perturbation moves past a fixed radius. This means the high frequency content of such a perturbation will be relatively large. The converse is true with perturbations possessing a relatively flat radial profile. This effectively means sharply peaked distributions will have smaller A t r a n (and larger Are{) that those with a natter profile. For demonstrative purposes, I have computed the scattering of initially ingoing massless scalar perturbations from a Schwarzschild black hole in spherical symmetry. The radial profile of the scalar perturbation is specified at initial time with a Gaussian (4.11) where ay is the Gaussian amplitude, ry determines the center of the Gaussian profile, and 5q? sets its width. As expected, the transmission increases (and reflection decreases) with decreasing <5^, as such profiles have steeper radial gradients. Denoting the energy of the set of ingoing modes Tm and the energy of that set upon reflection T o u t , the fractional transmitted energy, Z + 1, is Z + l = ^ ; (4.12) in Figure 4.9 shows this fractional transmitted energy as a function of 0 < 5y/M < 25. 4.1.2 The Glory Effect and Orbiting Resonances The phenomena of glories and orbiting resonances can also be obtained as limiting cases in the analysis of (4.10). The details are again summarized in [49] and [51], while a more recent and rather thorough investigation of scalar glories for both Schwarzschild and Kerr spacetime has been presented in [54]. The glory is a diffraction phenomenon that happens when a differential scattering cross-section diverges in either the forwards or backwards directions. The phenomenon is known to occur in optics, quantum mechanics, and also in the scattering of perturbations off a black hole, among others. In the case of black hole scattering, glories arise as a consequence of strong gravitational interaction of high frequency perturbations through angles near 9 = kn, k being an integer. The generic relationship between total scattering angle 0 and impact parameter 6 = 6(9) for massless perturbations of this sort is [38], [39], [102], 6«3v / 3M + 3.48Mexp(-9) , (4.13) providing b ~ 2M. The impact parameter for the first backwards glory 9 = 7r is thus b = 6giory w 5.35M. Figure 4.10 depicts scattering trajectories for angles near 9 = n. The differential scattering cross-section for massless perturbations of arbitrary spin near angles 0 = krr is given by [43], [155] Jl(cvb sin 9) , (4.14) where b = 6(9) is provided by (4.13), s is the perturbation spin, and J2s (w6sin9) is the standard Bessel function with argument w6sin9. From (4.14), it is apparent that glories constructively interfere for massless (s = 0) scalar fields, but destructively interfere for (s = 1) electromagnetic and (s = 2) gravitational perturbations. The phenomenon known as orbiting resonance occurs when a perturbation scatters through an angle 9 > 2n. The perturbations can be deflected through several multiples of 2-K before escaping to <£(£ = 0,r) = ay exp ry da dll db « 2TTO;62 dQ C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 69 Figure 4.9: F rac t iona l t ransmi t ted energy, Z + 1, for spherical ly symmetr ic massless scalar per-turbat ions scat ter ing off a Schwarzschi ld black hole. T h e profile is an i n i t i a l l y ingoing Gauss ian profile of w i d t h 8%. T h e fractional t ransmi t ted energy is p lo t ted as a func-t ion of /M, where M is the black hole mass. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 70 F i g u r e 4.10: In the glory effect, perturbations with impact parameters b = 6gi0ry deflect through an angle of 0 = w. Impact parameters larger that 6 g i o r y scatter through smaller angles, while smaller impact parameters scatter through angles larger than 0 = IT. This figure depicts scattering trajectories for angles near 0 = n. C H A P T E R 4 . C H A R G E D S C A L A R A C C R E T I O N 71 Figure 4.11: Orbiting resonances require impact parameters, 6, smaller than those for a glory, ''glory In an orbiting resonance, the total scattering angle is 0 = 2-rr or larger. This figure depicts the trajectory for a resonance which orbits the black hole several times before escaping to large distances. distances much larger than r ~ 2M. However, the cross-section observes approximately exponential decay in O. For example, equation (4.14) shows that a second backwards glory (scattered through angle O = 3ir) is suppressed by at least exp (—2ir) relative to the first backwards glory (with scattering angle O = ir). An resonance of several orbits is depicted in Figure 4.11. 4.1.3 The Effective Potential Formulation for Kerr Spacetime The calculations become further complicated when considering rotating black hole spacetimes. In the simplest case of Kerr spacetime1, the perturbation equations can be separated into radial and angular parts if, rather than spherical harmonics, the expansion of the perturbing field is performed in terms of the spin-weighted spheroidal harmonics Sim(9) as per nt,r,9,d>) = V ^ # ^ 5 i m ( e ) e - ™ * . (4.20) Then, assuming a harmonic time decomposition lfcm(t,r)= / $bn(u,r)e-**du , (4.21) where 1 Recall the line element for Kerr spacetime in (t, r, 0,0) Boyer-Lindquist coordinates is , , / 2Mr\ l 2 4Mrasm26 , , E , , A s i n 2 0 . <fe2 = — ( 1 J dt dtd<t> + —dr2 + Ed0 2 + d<t>2 , (4.15) E = r 2 + a 2 cos2 9 , (4.16) A = r2 - 2Mr + a2 , (4.17) A EE ( r 2 + a 2 ) 2 - a A s i n 2 0 , (4.18) and (4.19) is the black hole angular momentum per unit mass, J designating the black hole angular momentum. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 72 and using the (£, r*, #, </>) tortoise coordinate system related to the standard (t,r,6,< Lindquist coordinates through d _ A d dr* r2 + a2 dr ' the radial part of the perturbation equation becomes X-r2(l-s)(2-s)m%/2) Boyer-(4.22) d2 K2 - 2is(r - M)K + A (Airujs • dr} + (r2 + a2)2 G 2 _ d G dr* where K A G Xlm E being the separation constant E = 1(1 + r" + a )u) — am , E — s(s + 1) + a2u)2 — 2amw , s(r - M) + (r2 + a2) 2\2 ( r 2 + a 2 ) 1 / 2 A s / 2 ^ ( Im i) 2maus2 1(1 + 1) + O ( a2ml 11 -Xim(u,r) = 0 , (4.23) (4.24) (4.25) (4.26) (4.27) (4.28) A = r 2 — 2Mr + a2, M is the black hole mass, a is the black hole angular momentum per unit mass, s is again the spin of the perturbing field, and my is the mass parameter of the scalar (s = 0) perturbations. It is clear from (4.23) the effective potential is now dependent on the perturbation frequency, and that it reduces to the expected Schwarzschild form (4.10) in the limit a —» 0. A l l the scattering phenomena previously presented will therefore play a role in Kerr spacetime. 4.1.4 Polarization, Superradiance, and the Black Hole Bomb Two new scattering effects are present for perturbations on Kerr spacetime which are absent from the Schwarzschild black hole scenario discussed in subsection 4.1. They are polarization and su-perradiant scattering of the incident perturbation. Both electromagnetic and gravitational per-turbations can be partially polarized by scattering off a rotating black hole. [98] and [99] present the first calculations demonstrating this polarization effect for electromagnetic perturbations in Kerr spacetime. The calculations are involved, but essentially different polarization states scatter differently in the rotating black hole spacetime. It should, in principle, be possible to observe these effects in the numerical evolutions I present. The superradiant scattering effect should also play a role in my computations. In superradiant scattering an incident perturbation is amplified as it scatters off the black hole, the additional energy being drawn from the rotation of the black hole. Some of the earliest studies of superra-diant scattering were [151], [152], [153], [5], [101], [124], [137], [138] and [141]. The appearance of superradiant scattering is intimately connected with the change of character of Ki l l ing vector ta = (1,0,0,0) in different regions of the Kerr spacetime. For radii r > rs = M + \JM2 — a? cos 2 8, ta is timelike. However, ta becomes spacelike for r < rs. This implies energy densities pr can be negative within the ergoregion, defined by r < rs. Under these circumstances an incident perturbation can carry negative energy into the black hole, leaving a reflected component with greater amplitude and energy than that which was incident. From analysis of (4.23) the incident CHAPTER 4. CHARGED SCALAR ACCRETION 73 perturbation will carry negative energy when 0 < LO < m f i H , (4.29)' Qn being the horizon angular velocity Q H = , (4-30) where r+ = M + \ / A f 2 — a 2 marks the location of the event horizon. Equation is known as the superradiance condition, since it can be shown that superradiance must occur the condition is met. Clearly, this requires the perturbations be prograde (LO > 0), and also that mode numbers satisfy m > 0. [141] found this superradiant amplification is maximal for a = M, and drops sharply with decreasing a, or with increasing mode numbers I and m. Denoting the energy of an ingoing mode T i n and the energy of that mode upon reflection Tout, [141] numerically determined the maximum of fractional energy change, Z, given by Z = ^ - l , .(4.31) in for a single mode is 1.38 for (s = 2) gravitational perturbations, and 1.044 for (s = 1) electromag-netic perturbations. The maximal value of Z was calculated at 1.003 for (s = 0) massless scalar perturbations in [124]. More recent investigations have shown that the maximal value of Z can be increased to near 1.07 for scalar perturbations confined to a toroidal wedge in the equatorial plane of the Kerr black hole [142], [1]. For the same geometry, Z decreases to approximately 1.003 in the case of confined electromagnetic perturbations [1]. Confinement of plasma in a toroidal ring by magnetospheric dynamics was the motivation for these two investigations. Because massive scalar perturbations are dispersive, it is difficult to define notions of purely ingoing and outgoing modes when > 0. None the less, modes can be reflected superradiantly. The superradiant condition is again (4.29). It was pointed out to me by [48] that such massive perturbation modes possess growing mode instabilities when they become trapped by the effective potential of the black hole spacetime. . If modes are trapped, then superradiant scattering can amplify the trapped modes over and over, thereby leading to a growing mode instability. The effect is analogous to the so-called black hole bomb first discussed by Press and Teukolsky [124]. The first work to note this instability for massive scalar perturbations was [37]. Later studies include [156], [41], [50] and [140]. One implication is that through this mechanism, a black hole can, in principle lose all of its angular momentum. However, these unstable massive scalar modes are slowly growing, with a growth times similar to the age of the universe for solar mass black holes. These instabilities are thus not expected to play a role in my calculations (which cover significantly shorter timescales). Recalling that scalar field models can represent the pions, note how this time scale is also quite large relative to TT°,'TT+ and n~ lifetimes of 8.4 x 1 0 _ 1 7 s and 2.6 x 1 0 _ 8 s respectively. Furthermore, for typical IT0, TT+ and TT~ masses of ~ 2.4 — 2.5 x 1 0 _ 2 8 k g , the growth rate is found to be maximal for black hole masses M <C 10 1 5 kg. Such black holes are far less massive than solar mass stars, where M = MQ ~ 2.0 x 10 3 0 kg. 4.1.5 Late Time Decay, and the Coupling of Electromagnetic and Gravitational Perturbations Perturbations are known to exhibit exponential amplitude decay at sufficiently late time. Like the intermediate phase of evolution, these asymptotic late time dynamics also originate from the spacetime curvature. From analysis of (4.10), it has been shown that decay rates of massless (s = 0) scalar, (s = 1) electromagnetic, and (s = 2) gravitational perturbation amplitudes in Schwarzschild C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 74 spacetime observe the late-time relationship | * ( £ , r ) | . ~ £ - ( 2 ' + p + 1 ) , (4.32) where P = 1 if the field is initially static, and P = 2 otherwise [130], [131]. Meanwhile, it was found by [89] that, for massive Klein-Gordon perturbations, the late-time decay rate is independent of the mode number and adopts the form | * ( £ , r ) | ~ i " 5 / 6 . (4.33) While the authors of [89] have studied the massive scalar problem in detail in [88] and [90], the first to begin investigation of this system were Novikov and Starobinski [139], and Burko [14]. While, it was shown in [15] that (4.33) also holds for sufficiently late times in Kerr spacetime, regardless of the black hole angular momentum', the power-law decay of massless scalar, electromag-netic, and gravitational perturbations differ from (4.32) when the black hole rotates. Specifically, [74] [75] and [76] studied the r\u>\ -C 1 and a\u\ -C 1 limits of massless perturbations. It was found the decay of massless (s = 0) scalar perturbations at timelike infinity obeys' | * ( £ , r ) | - £ " ( 2 i + 3 ) , (4.34) when |m| < I < \m\ + 1, and |*( t , r ) | ~ £ - ( '+ l " l !+ p + 1 ) , (4.35) when |m| + 2 < I, where P = 0 if / — |m| is even and P = 1 otherwise. Meanwhile, electromagnetic and gravitational perturbations were found to obey | * ( £ , r ) | ~ £ - ( 2 m a x ( l m | ' s ) + 3 ) , (4.36) when max(|m|, s) < I < max(|ra|, s) + 1, and | * ( £ , r ) | ~ £ - ( ' + m a x ( l m l ' s ) + 1 ) , (4.37) when max(|m|,s) + 2 < I. While results (4.34) through'(4.37) hold for perturbations consisting of a single mode /, [74] demonstrated that when a perturbation is initially a superposition of many modes, the decay law becomes | * ( £ , r ) | ~ £ - ( 2 m a x ( | m | > s ) + 3 ) , . ' (4.38) for all s. These results contradict the previously widely held belief that late time decay of perturba-tions in gravitational collapse is universal (i.e., independent of s), and are presumably a consequence of the coupling of modes with different I for rotating black hole spacetimes. Furthermore, these results demonstrate the possibility of slower decay rates than in the non-rotating case. In a more general pair of studies [3] and [55] found the decay rate of massless scalar perturbations reduces to |*(*,r) | -t-1 , (4.39) for a maximally rotating black hole (i.e., a = M). This slower decay rate is apparently the result of superradiant scattering. In principle, all these late time relations for scalar and electromagnetic perturbations should be observable for the system I study in this dissertation. However, since my numerical evolutions are relatively short, I do not expect to get good estimates of any power law decay. Furthermore, since the chosen computational domain only covers finite radii, the results obtained for asymptotically large distances will not be attainable. Spatial compactification would be one way to correct this second restriction. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 75 4.1.6 Charged Scalar Perturbations in Static and Stationary Spacetimes Finally, the effective potential formulation of black hole perturbations has been studied for charged scalar fields in static and stationary spacetime. The analysis was performed in the context of weakly charged perturbations about a Reissner-Norstrom black hole in [69], [70], [86], [87]. In [37] and [50] analysis was also carried out for weakly charged perturbations, but their spacetime was Kerr-Newman (i.e., the charged analogue of Kerr). In all these studies it was assumed the scalar perturbations carry test charge in the sense that while they could be electromagnetically affected by the charge of the black hole, they themselves did not contribute to electric or magnetic fields. These charged perturbations did not electromagnetically self-interact. In the limit where the black hole charge vanishes, this system is thus exactly that of uncharged scalar perturbations in Schwarzschild spacetime. It is through this approximation the equations of a charged scalar perturbation become linear and allow an effective potential formulation. 2 The system I study—described in the next section—assumes a fundamentally different ap-proximation. The scalar perturbations are massive and electromagnetically self-interacting. But while self-interacting, this system assumes the charge carried by these perturbations doesn't in-fluence the curvature of the spacetime in which they evolve. However, this does not imply weak electromagnetic interaction, or even limitation to a regime in which.electromagnetic interaction is subordinate to all gravitational effects—consider a common crystalline substance where electromag-netic binding energy is orders of magnitude stronger than gravitational attraction. Furthermore, the system assumes the field mass is likewise too small to affect the curvature of spacetime. Since the influence of charge and mass on spacetime must be through the stress-energy tensor, the as-sumptions are essentially a statement that the local energy density of the field must be small. The notion of small can be roughly quantified through comparison with the Schwarzschild solu-tion. Since the Schwarzschild radius is Rs = 2 M , the average energy density of the black hole is PTS ~ M/(^irRs) = 3/(327rM 2). Therefore, the assumption that'the perturbation energy density PT is small means PT/PTS ~ 3pr/(327rM 2) <C 1. For the results presented in section 4.3, where M = 5, the maximum energy densities are PT « 1-10, so the assumption PT/PTS -C 1 holds. The system of equations for these perturbations are nonlinear and so cannot be cast in the form of an effective potential problem. Being nonlinear, the system must be solved numerically. To my knowledge, the only other work that has analysed the nonlinear problem is [71], where the authors investigated the late-time decay of massless charged scalar perturbations in a dynamic spherically symmetric spacetime. Investigation of the nonlinear charged model in rotating spacetime has until now been lacking—the results I present are the first of their kind. 2The effective potential formulation for charged spacetime backgrounds is known to couple electromagnetic and gravitational perturbations. A purely electromagnetic incident perturbation can by this mechanism scatter as a purely gravitational perturbation. This could lead to interesting consequences for both superradiant scattering and eventually gravitational wave detection. The effect is apparently dependent on the existence of a background electric field, which is not the case for the system I investigate. However, some of the earliest works on this topic are [18], [154], [104], [53], [105], [19] and [118]. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 76 4.2 System of Equations for Charged Scalar Accretion 4.2.1 The Spacetime Metric The rotating black hole spacetime is described by the. Kerr metric. In (i, r, 9, <f>) Kerr-Schild coor-dinates the line element takes the form , •> ( 2Mr\ l 9 - / 2Mr\ , , „ ,n2 . , „ ( 2 , 2 M r a 2 s in 2 9\ ,,2 ds2 = - { l - ^ ^ \ d t 2 + i l + ^^)dr2+Yd92+sm2e{r2 + a2 + j # 2 / 4 M r \ , , / 4 M r a s i n 2 6 > \ , , , „ . 2 „ / 2Mr\ , „ + f j diaY - f J dtd<j> - 2a sin 2 <9 f 1 + j drd<p , (4.40) where S E E r 2 + a 2 cos 2 9 , (4.41) a = ^ , (4.42) t is the temporal coordinate, and r, 0 and (j> are respectively the radial and angular spatial coordi-nates, M represents the total mass of the black hole spacetime and J its total angular momentum. The metric (4.40) diverges at E = 0 and the scalar invariant RabcdRabcd indicates the root r = a cos 9 is a true curvature singularity. The singularity is a ring of radius a. Because the form (4.40) is horizon penetrating, it doesn't immediately reveal the existence of other special surfaces in the spacetime such as A = 0; where A = r2 + a2 - 2Mr . (4.43) When a2 < M2 the root r = r+ = M + \/M2 — a? marks the location of the event horizon. It is geometrically a sphere in the given coordinates. The second root r = r_ = M — VM 2 — a2 is a Cauchy horizon. 3 When a 2 > M2 there is no event horizon and the ring singularity is naked. Since in this instance there is no event horizon, the metric does not describe a black hole spacetime. Such a spacetime would be in violation of the Cosmic Censor Conjecture as put forth by [123] which states no such singularities should be visible to any observer. The spacetime (4.40) possesses two Ki l l ing vector fields ta = (1,0,0,0) and sa = (0,0,0,1). ta is timelike and sa spacelike in the portion of spacetime where r > rs = M + VM2 — a2 cos 2 9. However, when r+ < r < rs the Ki l l ing vector field ta becomes spacelike. The region r+ < r < rs is known as the ergosphere. From the perspective'of a stationary observer at infinity, a timelike orbit cannot remain stationary within the ergosphere, but must rotate in the direction of rotation of the black hole, r = r s is thus called a stationary limit surface in this strong example of the frame-dragging effect. Figure 4.12 depicts the rotating Kerr black hole spacetime, and Figure 4.13 displays its causal structure. Finally, the future-directed unit-norm spatial hypersurface orthogonal vector is (4.44) 3 Based upon arguments that an observer crossing r = r_ will see entire histories evolved from a Cauchy surface of the Kerr spacetime in a finite period of time, the Cauchy horizon at r = r_ is thought to be unstable, as- first put forth by Penrose in [122]. However, definitive results, numerical or otherwise, detailing the formation and evolution of the Kerr Cauchy horizon are apparently lacking. But if the results known for horizons of charged spherically symmetric black holes carry over, then perturbations of the Kerr spacetime could lead to the formation of null singularities which then merge with the Cauchy horizon. However, as in the charged spherically symmetric case, this null singularity may still be traversable by observers willing to suffer finite tidal forces. While the Kerr metric may not accurately describe the interior of a black hole formed by the collapse of matter with angular momentum, it is assumed to accurately depict the exterior spacetime after a stationary state has been achieved. Since my interest is with the dynamics of fields outside the event horizon r = r+, I can excise the portion of spacetime r < r+ without additional concern.. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 77 I F i g u r e 4.12: Horizons of Kerr spacetime: side view (a) and top view (b). r = r+ marks the event horizon, r = rs is the stationary limit surface, and Qa is the horizon angular velocity. The ergosphere is the region r+ < r < rs. The ring singularity is shrouded by the event horizon. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 78 F i g u r e 4.13: Penrose diagram showing causal structure of Kerr spacetime (cf. [62]). I and F depict asymptotically flat regions of the spacetime outside the black hole. II and II' are interior to the event horizon at r = r + , while interior regions III and III' are separated from II and II' by the Cauchy horizons at r = r_ . 3+ represents future null infinity and 3~ past null infinity, while i~ represents past timelike infinity, i+ future timelike infinity and i° spacelike infinity. The singularity at r = 0 is timelike, and can thus be avoided by a timelike trajectory originating on the Cauchy surface (red). C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 79 where (TT). ("») * ,= (. + ? f ) , (4-46) and the spatial metric is encapsulated by the hypersurface line element 2Mra? s in 2 6 da2 = + o r 2 + E o t f 2 + s i n 2 2 i 2 2a s in 2 0 ( 1 + ) drd0 . (4.47) 4.2.2 The Matter Lagrangian and Equations of Motion The Lagrangian for a massive electromagnetically coupled complex scalar field is again CM = ^ [ - ( V a * - ; eA>$) (V a $* + j e A a $ * ) - - F a b F a 6 - m $ 2 $*$] V (4.48) where m$ is the scalar field mass parameter, e sets the strength of electromagnetic coupling between the real and imaginary components of complex scalar field 4? = $>(t,r,d) (respectively 4>i = 4>i(t>r>9) a n d 4>2 = 4>2(t,f,9)), the symbol * denotes complex conjugation, Aa = Aa(t,r,9) is the electromagnetic vector potential, and Fab is the antisymmetric electromagnetic field strength tensor. The set of Euler-Lagrange equations for the system described by (4.48) yields equations of motion V a V a $ * + 2 i e ( V a $ * ) A a - e2$*AaAa + i e $ * V a A a - mg<S>* = 0 , (4.49) V a V a $ - 2ie(Va$)Aa - e2$AaAa - z e $ V a A a - m $ 2 $ = 0 , (4.50) VaFab - «e($*V f c $ - $ V f c $ * ) - 2e2$$*Ab = 0 . (4.51) where (4.49) and (4.50) are completely equivalent. I also choose the Lorentz gauge condition V a A a = 0 , (4.52) the conserved current is - Jb = VaFab = ie(<E>*Vb$ - $ V b $ * ) + 2e 2 $$*A b , (4.53). and the stress-energy tensor is once again Tab = i ( V a $ V b $ * + V 6 $ V a $ * ) - i z e [ ( $ V f c $ * - $ * V 6 $ ) A a - ( - ( $ V a $ * - $ * V a $ ) A b ] +e2$$*AaAb + \FacFbdgcd - ^ a 6 [ ( V c $ - ieAc*)(Vc$* + ieAc$*) + -JcdFcd +m£$*$] . (4.54) 4.2.3 Matter Equations in First-Order Form Equation of motion (4.50) can be rewritten as V a ( V a $ - ieAa$) - ieAa ( V a $ - ieAa$) - = 0 . (4.55) CHAPTER 4. CHARGED SCALAR ACCRETION 80 Computing covariant derivatives with respect to metric (4.40), the equation of motion adopts a rather complicated form. I choose to rewrite (4.55) in terms of first-order temporal and spatial derivatives. To recast it in a more tractable form I first define field variables II$(£,r , 9) = nay/=g(Va$-ieAa$) , (4.56) &r(t,r,9) = 9 r $ , (4.57) $e(t,r, 9) = 9e$ , (4.58) Atr(t,r, 9) = drAt , (4.59) Ate(t,r, 9) = deAt , (4.60) Arr(t,r, 9) = (4.61) Ar0(t,r, 9) = doAT,, (4.62) Aer(t,r,6) = drAe , (4.63) Aee{t,r,6) = deAe , (4.64) A ^ r ( £ , T - , 0 ) = drA(f) , (4.65) A<j>e{t,r, 9) EE deA^ , (4.66) and auxiliary metric functions Mt,r, 9) U<i>(t,r, 9) et4>(t,r, 9) 9t<t> sin# 9<t>4> sin# _9t4_ s in 2 6 sin where 9t4> 2Mra s in 2 9 = sin r2 + a2 + 2Mra2 s in 2 9 and ( 2 2 > \9ir94>4> ~ 9U9rrj 9tr s i n 2 9 (4.67) (4.68) (4.69) (4.70) (4.71) (4.72) (4.73) C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 81 Under these transformations equation of motion (4.55) becomes the set 1 n. \ 9rr -.Ai $ + ie (drgrr) f<j><j> 3/2 9rr 9tr^9rr \J 9rr SmU^/grr cos (9 sin 69 {drgtr) ft<t> , . (drft<t>) A.9 ^ = A 0 6 I 2—A<* + ( ? r r ~ V^r-r y<?rr \J9rr9tr 5/2 yrr A _ J<P<P A r 3/2 r r / 9rr9tr 5t r • m 9rr9tr 2 Z' 9t(j>y/9rrft4> . /<£4> 3> (dr9rr) f<t>4> (9rf</><!>) 5/2 9rr 3/2 4?r = 9 r $0 = de ie \ A t - ^ A r ) $ + ^ r + ie U - ^ A r U + ^ $ 5?r 53/2 sfgTr y2 ff2r • Srr / 9rr (4.74) (4.75) (4.76) (4.77) / Qrr (9trf<t«t> ~ 9t4>ft<t>9rr) 9tr /Qrr IL, 9t $ = ie [At - ^ A r $ + ^ $ 9tr , 9tr Qrr \JQrr (9trU<t> - 9t<t>ft<p9rr) where the overdot,', again designates differentiation with respect to coordinate t. I similarly choose to evolve the four-vector potential equations of motion (4.51) in first-order form. For this purpose I define auxiliary variables Ur(t,r,0) = na^~9 (VrAa - V 0 A r ) , He(t,r,0) = na^{VeAa-VaA6) , n+(t,r,0) = n a \ / ^ ( V ^ a - V a A ^ ) , (4.78) (4.79) (4.80) CHAPTER 4. CHARGED SCALAR ACCRETION 82 under which (4 .51) become 1 (9tr (9e9rr) sin 9 9ire<t>4> Ur = -ie^cosfl . (dBgtr)f?4,^(frr~sm6 [ftr^H fi<f>9rr^ 3/2 9rr + 9tr fix (degrr) sin 9 \ _ * - + fi^VgVrCOS 9)Arg + -f= (deAr0) + fi<t>9rrj [ftr^H fid> (degw) sin 9 \fg~rr ft<f> (drgrr) smt / 9rr ^ J \J~9~rr P2 grr sin a 1 / 5 2 r ( 9 e 5 r r ) e ^ s i n 6 i g^e^ cos 9 (de9tr) / ^ y ^ s i n g 3/2 9rr 9tr + fi^^gVr cos9 - (degt4>) et4>^/g^sm9 A6r - -^^= (deAdr) sin t fg~rr 3/2 9trgrr A^r + _ ft<t— (drA4 9tr\l9rr (dcose9tr) 9t<t>) A^e -{?tre<t>4> ~ fi^rr) ( 1 9tr (9t9 (9tr (9 C , (9trH4> ~ fi^rr) 2 9lr 3 fftr/t^ (drgrr) 2 5 r r - f t r / t * (3r/W) + 2 (dr<?(r) ) n r + ^ ( 9 r n r ) + — - — ^ fi,rr 9rr 9tre<t>4> (degtr) 9rr)^6 - 9tr 9rr {9tre4>4> fi<ji9rr^ (ptr^U ~ fi<t,9rr) (gtret<p (dTgrr) + 9tr (dret^) grr (gtr (degrr) o / c i \ \ T T , (9trU<t> - 9t<t>ft<t>9rr) -I (Orgtr) Zt4>9rr) l-i-j, H ^ ; = = Jr , 9trV9r (4 .81 ) CHAPTER 4. CHARGED SCALAR ACCRETION 83 Tie - % (d8Arr) + Qrr 9tr (drgrr) f4><f>e-<t><t> , 0 (drgtr) gt<j>ft4>e<t>4> [ftr^H - ft<t>9rr) 5/2 iJrr + 2-gtrJg rr r.gt^fttpidrgrr)^ gt<t>ft<i> (dre^) 2 (drgt<t>) ft<t>e<t><t>\ A ~T7* ' 1 J= J= A r 6 V yrr V 9rr I 3/2 grr g2r(drgrr) f<t,<j>&4>4> . 0 (drfltr) 9t<j>ft4>^ ^gt4>ft<f> (dr9rr) ^ 5/2 yrr + 2 gtrJg rr 3/2 Qrr Uet4>(drgH) _ 2(dr9^)U^A A e r + M { d r A e r ) + _U±_ { d e A r j > r y Qrr / / Qrr l 3/2 grr gtry/grr. gtret<t> {drf<j,4,) gtr (drft<f>) &<f<t> , (drfftr) ft^,y/9rT (9treU + farr) V V9rT (drgtr) ft<t>e^><t> ft<t> (drgrr) (drft<t>) V ^ - N 9tr /Qrr I rr gtr ( 1 gfr (drgrr {9tre4>4> - ft4,9rr) \ 2 gl {Or 9tr (°rgtr birr Qrr 3 Qtrft<f> (drgrr) 2 Qrr -2gtrU (drft,) + 3 (dr9tr)fa ) Tie + — (drKe) + { 9 l U \ ^ ^"he , (4.82) 9rr 9tr\9rr C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 84 - (degtr) 9t<Pf<p4>') Are + (9tr9t<j> (d$9rr) f<p<p 9rr 1 9trft<t> (deg<f,4>) + gtr (degt<p) U<t> 'gtrgt4> (degrr) f<t><f> +gtr (degt^fu - (degtr) gt<f>U<t>- J Ag. 1 (drgrr) ftptp 9trft<j> (deg4,$) rr I J (pep A , J<P<P fa A \ 5/2 + ~~3/2 \°rA<t>r) 9rr ' 9rr -ft4>\fg~rr~COS6 + glrH4> c o s & + gtr (9cos0. 2 Q , f?4, (de9rr) sin 9 (degtr) ftcpVgrTsin 6> (de9t<p) e ^ v ^ s i n f l A^g + - — ( c ^ A ^ ) + 9rr (gtre<p<p ~~ ft<p9rr^J 9tr9t<p 9rr \9t ItrhUrr (et<t> (drg<j><p) ~ (drgt<j>) e ^ ) IL-(gtr (degrr) - (degtr) grr) Tie i gl (drgr gtr (dre^ gl (^rfftr (-9tr«^ + ft<p9rr) ^ 2 1 9trftd, (drgrr 9rr gtr (drgt<p) + (drgtr) ft<p 1 LT^ + — (d rII^) / Qrr 2, grr (9trf<t>4> ~ 9t<f>ft<f>9rr) 9tr V9rr (4.83) A8r = dr dr dr gl /9rr (9trU<P ~ gt<j>ft<l>grr) TLr Ate - ^ Are+ ^ Aer gtr 9rr gl / Qrr (gtrU<t> ~ gt<t>ft<t>grr) Tie gtr_ lrr gl Q  \JQrr (glf<t><p - 9t<pft<p9rr) TI. (4.84) (4.85) (4.86) Are Aee gl yjQrr (gtrf<t><t> " 9t4,ft<p9rr) A gtr A , gtr & At$ Are H Aer TLr gir = de 9tr 19 iQrr {QtrU<f> - Qt<pft<pQrr) / Qrr (9trf4"t> - 9t<pft<t>grr) Lb (4.87) (4.88) (4.89) C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 85 Ar = Atr+ „ 2 / t r _ t _ ,nv , (4.90) /Qrr {9trf<j"t> ~ 9t<j>ft<t>9rr) Ae = Atg - 9^Ar6 + ^ A 6 r + „ 2 , 9* j , (4.91) 9rr grr sj9rr \9trJ<t>4> ~ 9t<t>H<t>9rr) 2 A* = g t r - n ^ , (4.92) 9rr \/9rr \9trJH ~ 9t<j>Jt4,9rr) where from (4.53) Jr = - i e ( $ * $ T . - $ $ ; ) - 2 e 2 $ $ * A r , (4.93) Je = - i e ( $ * $ e - <f>$*6) - 2e2$<f>*Ae , (4.94) ' Jj, = - 2 e 2 $ $ * A , (4.95) and I have used the chain rule to rewrite some partial derivatives with respect to 9.4 Finally, the Lorentz gauge condition (which evolves component At) is already in the first order form At = l- (9_Udre^ + gl (dr9tr) ^ _ 9tr& (dr9rr) _ ^ ^ ^ „ s F2 A A ,09tr . , 1 ' (9tr (^rffrr) eU gAr{dre^) + {drgtr)it4>- ) At + 2—Atr + j - r I - 3 — ~ 2 1 9 r T \92r^U - ft<t>9rr) \ 9"~ 9 ^ 29tr {dr9tr)e4,4, + 2g2r (drgt4>) et4, _ g2tr (drgrr) eu g2tr ( d r e ^ ) \ A 9rr 9rr 9^r 9rr ) 9 t T (9tre<i"i> - ft<j>9rr ~ H4>) Arr ~ . ^^ °°S 9 ^ ^—^AQ 9lr \9treH ~ ft<j,9rr) 9rr [9tre4"t> ~ Jt^rr) q2 1 t r -Aee ; r- (gtr (dret^) - (drgtr) et<j,) A^ ( 2 \ ( 2 ' 2 \9rr \ 9tre<t>4> ~^ ft(j>9rr J 9rr ( 9tre4»t> ~ ft<j>9rr J /i • 1 •• ^ n r . (4.98) 9rr (g?re<t>4> ~ ft<j>9rr^ . 9^ {9trf<p4> ~ 9t<j>ft<j>9rr) 4.2.4 Phys ica l and Conserved Quantit ies In axial symmetry, none of the components Ea or Ba generically vanish. 5 This differs from the spherically symmetric case presented in Chapter 3 where Er was the only nonzero component. Explicitly, using definitions Ea = Fabnb ' (4.99) 4Applying the chain rule 89 89 8cos6 8cos6 transforms numerically unsuitable forms at 9 = 0 or 9 = n such as . " 1 9 / df df_8cose df __sine_oi_ (4.96) (4.97) sin(? 89 8 cos 9 5Regularity will, however, require Eg, Ex., B$ and to vanish on the axis of symmetry (i.e., along 9 = 0 and C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 86 and for an observer with four-velocity na, Ba = -l-tabcdFcdnb (4.100) Er = - , o' 9 1 t _. , (4.101) Ee = -rs-i —7 ;TLe , (4.102) (9trU<l> ~ 9t4>ft<f>9rr) 9tr ~ 9t<t>ftij>9rr) 2 9tr (9trU</> ~ 9t<pft(/>9rr) E<t> = ~<„2t t ^ n » » (4-103) Br = 9 t r ^ ^ {gtridcoseAj,) +Jt4>Are - ft<t,A6r) , (4.104) and Be = . . - V , (4.105) ^ / 5 r r sin 0 -Btf. = p — r- {gtrf<t><t>Are ~ gtrf<t>'<t>Aer — ft<t>grrA<f,e) • (4.106) l9rr (j3tretj>(p ft(j>9rr^j Meanwhile, the temporal component of the current four-vector is 2 Jt = -ie^ ($*4V - $$;) - ie 2 t 9 t r _ t _ , ($*II* - $nj ) - 2e2^<M>*Ar , (9trU<f> - 9t<t>ft<t>9rr) (4.107) and the conserved electric charge density can be expressed PQ = i e . 2 , g « r , r (4>*IT$ - $n|) . (4.108) {9trJ<t><t> - 9t<t>Jt4>9rr) Contracting the stress-energy tensor Tab of (4.54) with na from (4.44) and ta = (1,0,0,0) . obtains a conserved energy density, PT = PT($) + PT(A) + PT{$A) , (4.109) where PT($) is the contribution from the scalar field terms of Tab, PT(A) LS the contribution from electromagnetic field variable terms, and PT{<S>A) is the contribution from terms involving the in-C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 87 teraction of $ and Aa through coupling constant e.6 In detail, -$*$ r ) + 9tr 9rr & gr'r I z 9tr -At 9tr {9trf<i><t> — 9t<fift<p9rr) 9rr{9tr f<t>4> ~ 9t<j>ft<l>9rr) 9treU 2 JgT 2 9%2(9\r^4> - ft4,9rr) V <?rr / Ar) ($114 - $*n*) 5?, ^ V9rr\9tre4><t> Jtd>9rr) 9tr ^9rr{.9trfH ~ 9t<j>ft<ji9rr) ( $ r n | + $*n*) + - - n $ n | , , 6Before substituting auxiliary variables, PT(&) n a s * n e form P T ( * ) 1 1 2 (9t2r900 " 9%9rr) \ g%2 g%2 s/9~-2 yf9r~r~{9trf<t>4> ~ 9t4>fttj>9rr) (4.111) 2 (9tr9d>4> - sLflrr) 92r94>4> _ 9$±_ _ 9t<t> \ . 1 g2trsm29 2 ^9rT(girg^ - 5203rr) • (4.110) C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 88 PT(A) — 9trH<t> -A. 4 9tre<t>4> 4 P r r 2 ( 5 t 2 r e ^ - f^grr)2 * 2 9 ^ ( ^ r ^ ~ fi^rr)' _ i 9tret<f, A A ^ , 1 5tV ' ' 2 ^9rTi.9tre<t><t> ~ ft(j>9rr)' -ArgAer + 4 9rr {.92Te4"t> ft<t>9rr)2 + 9tret<t> 2 \/9rr(9tre4>4> ft<f>9rr)' 4 JgTr{.92r&W - ft<t>9rr)' ;AgrA4 1 5?r (9 c o seA) 49r(-2(g2r9^-92t^9rr) 2 , 1 - A S i r 0 / 2 f2 \2 (^coseA-) He * 9rr(9tre<t>4> Jt<j>9rr) 1 9tr ^ 9rr{g2rf<i><t> ~ 9t4>ft<t>9rr){9tre<t><t> ~ ft<j>9rr) ^ $ 9tret<t> + 2 9rr{gtrf4"t> ~ 9t<t>ft<t>9rr)(gtre4>4> ~ ft<t>9rr) 1 fltr 2 grr{g2rg<t><t> - g\^grr){.glrS^ - g^ft^grr) 1 S t r ^ t f 4 ffrr2(5tr./W ~ 9t4>ft4>9rr)2{9tr^U ~ ff^rr) -Hi g\r&fi> 2 T/g~r7(gtrf<txt> — 9t4>ft4>9rr)2{gtre<t"t> ~ ft<f>9rr) + + :ITZ 4 ^/9^{g2rfu ~ gt</>ft<t,grr)2(g2re<t><t> - ffyrr) 1 " TT2 4 y^gr7(g2r9<i>4> ~ 5^0rr)(fl&./<M> - 9t<t>ft<l,9rr)' n i , (4 .112) PT($A) 1 9tr ( e<t><t> 2 g % 2 I ( g l e ^ -ft^rr) - 1 A - gtr&t4> /grr(gtre<t"t> ft<j>9rr) A-rA-rf-+ S i r 9tr 2 vKfrr~{g2re<i><j> - ft^rr) . 2 y/g^[9tr9^ ~ 9t4>9rr) * 2 A2, - - /TTA2 9tr y/9rr : A A ie ~~2 9tv + ^=At g%2 \{gtr^4> - ffarr) 1 A - gtr&t4> I'9rr{9tre-<t»t> ft<j>9rr) 9tr [gtrfu - 9t<j>ft<j>9rr) 9rr(9tre4>4> ft<j>9rr) A t ($n;-$*n$)l . (4 .113) Similarly, contracting Tab with na and sa = ( 0 , 0 , 0 , 1 ) yields the conserved angular momentum density PS = PS(A) + PS(S>A) , ( 4 .114 ) C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 89 where _ 1 9trf<t><t> A JJ S { A ) ^ 9rr{9trU<p ~ gt^ft^rr)2 ^ 1 9trft<p A ^ A^gUg (4.115) 2 (9trU<p - 9t<pftcp9rr)2 1 gfr sin 9 2 (g?rU<p - 9t<pft^9rr)2 is the electromagnetic field contribution, and PS(*A) = %4f„2 f 9 \ f „ ^ ( $ n | - ^ I l g ) (4.116) 2 {gtrJ4>4> - 9t4>Jt<p9rr) is from the coupling of scalar and electromagnetic fields. Then, from (2.104), the total charge Q = r r r PQ ^  &f«- ^, - ^ J<p=0 Jg=0 Jr=0 gtr from (2.108), the total energy 27T /*7T rOO T = I / / PT v a " x a t r J v \ drdOdfi (4.118) =0 Je=0 Jr=0 9tr / 9rr (gtrU<t> - 9t<j>ft<j>9rr) and, from (2.109), the total angular momentum S= / / / p s v a " v a t r ' ^ 2 ^VJ*va"'drd9dct> (4.119) 7rf>=o 7e=o i r = o ^ f* r°° f9r~r {glfw - 9t4>ft4>9rr) =0' 9tr will all be constant, providing ta is timelike everywhere the densities PQ, PT and ps are non-zero. 4.2.5 Regularity, Boundary Conditions and Dissipation From the definitions qf section 2.4 (cf. [6], [45]), regularity for the functions listed above requires $, 4>r, Ar, A r r , Jr, Agg, ILj, Af, Aj,r, J<i>, At, Atr, Er, E^, Er, BQ, Jt, PQ, PT and ps be, to leading order, even in 9 on axis (i.e., at 9 = 0 and 9 = 7r). In addition to being even, UQ, A^, A$r, J<j,, Etf, and B^ must also vanish along the axis of symmetry (i.e., they are at least quadratic in 9 on axis). Meanwhile, remaining variables 11$, Arg, Ur, Ag, A$r, Jg, A^g, LL^, Atg, Eg and Bg are, to leading order, odd. For the leading-order-even variables I enforce regularity on grid points corresponding to 9 = 0 and 9 = TT by applying a quadratic fix as listed in Appendix C. I also apply interpolation in the direction of 9 on the next-to-axis points for these variables using a 3rd order Lagrange interpolant as discussed in Appendix C. Numerical testing shows the additional interpolation helps ensure smooth and stable evolution along the axis of symmetry. Variables Ug, Atp, A<pr, J$, Ej, and B$ are exceptional cases. I enforce regularity by setting them equal to zero on the axis and imposing a quadratic fix on the next-to-axis points. Meanwhile, regularity of the leading-order-odd variables is enforced by setting the functions equal to zero at the degenerate point. As with the leading-order-even variables, interpolation is applied next-to-axis. Because their counterparts in the massless electromagnetically uncoupled (i.e., e = 0) theory can be expressed as wave equations, I apply the (approximate) Sommerfeld outgoing radiation CHAPTER 4. CHARGED SCALAR ACCRETION 90 condition to <3>, Ar, Ag, A+ and At. As discussed in Chapter 2, the condition can here be written • ' ™^> + ( T ^ ^ ) ^ ™ ) = °» ( 4 - 1 2 0 ) where u = u(t, r, 9) are one of $, Ar, Ag, A^ or At. The condition is applied on the outermost grid points (i.e., at the points corresponding to the largest value of radial coordinate r = r m a x ) with the exception of those lying on the axis of symmetry (9 = 0 or 9 = ir) and those next-to-axis (i.e., at 9 = A9 or 9 = n — A9). Therefore, a backwards spatial differencing operator is used to implement dr. The Sommerfeld condition for $ provides an algebraic equation for boundary values of 11$ as well. Fields Er, Eg and E^ can similarly be rewritten as a set of wave equations. Sommerfeld conditions for the electric field components provide outer boundary conditions for I I r , n<j and 11^, which are again applied at all points r = r m a x except those on axis or next-to-axis.' Backwards difference operators are similarly used at r = r m a x for $ r , Arr, Agr, A^r and Atr, while $g, Arg, Jr, Agg, Jg, A^g, J<t>, Atg, Er, Eg, Br, Bg, Bj,, Jt, pQ, pT and ps require no special treatment. I avoid both the Cauchy horizon at r = r_ and ring singularity at r = a cos 9 by excising grid points within a surface of constant r = r e x , r_ < rex < r+, from the computational domain. I have found that numerical evolution is stable and smooth if I apply a mixture of extrapolation (in coordinate r) and interpolation (in 9) to the time-dependent grid functions u{t,r,6). Formally, I use the linear combination "< ( + A T'-'S» - A^A« <*r>+s£s? ( p » ' ) • ( 4 - m ) where P 2 *+ m = 3w(i + At, r + Ar, 9) - 3u{t + At, r + 2Ar , 9) + u{t + At, r + 3Ar , 9) (4.122) is the 2nd order extrapolating polynomial in r, P 3 '+ A* = -^u(t + At,r,9-2A0) + ^u(t+At,r,9-A9) + ^u(t+At,r,6+A9)-^u(t+At,r,9+2A9) 0 o o D (4.123) is the 3rd order Lagrange interpolating polynomial in 9, At designating the interval in coordinate t, Ar the interval in r and A the interval in 9. Factors A9 Ar + A6 ' and Ar (4.124) (4.125) A r + A9 provide weights adjusted to differences in intervals A9 and A r . This weighted combination of interpolation and extrapolation is applied at all points r = r e x which are neither on axis nor next-to-axis. I employ Kreiss-Oliger dissipation as outlined and summarized in Appendix C. Dissipation is added to all equations of motion explicitly involving partial derivatives with respect to t. This set of functions is <3>, $ r , §g, 11$, Ar, Arr, Arg, TLr, Ag, Agr, Agg, Tig, A$, A^r, A^g, n ,^ and At. The Kreiss-Oliger dissipation operators have stencils five points wide. For coordinate r they can then be applied only at locations two grid points from the boundaries r = r e x and r = r m a x - Similarly, dissipation in coordinate 9 can be applied only when, at minimum, two grid points away from boundaries 9 = 0 and 9 = ir. Therefore, dissipation in both r and 9 is applied only in the bulk of the computational domain—where r e x + 2 A r < r < r m a x — 2 A r and 2A9 < 9 < ir — 2A9—while dissipation in 9 alone is applied at r = r e x + A r and r > r m a x — A r . Formally, if u = u(t,r,9) C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 91 is one of the functions whose equation contains an explicit derivative in time, then, using the Crank-Nicholson scheme without dissipation, its expression wil l be u{t + At, r, 9) — u(t, r, 9) 1 . _, ' . . . , , , „ „ v - i = - [R(t + At, r, 6) + R(t, r, 9)} , . (4.126) where the left hand side is—to O ^ ( A i ) 2 ^ accuracy—the time derivative of u centered between times t and t + At, and the right hand side represents—to O (i[Ar)2^j and O ^ (A0 ) 2 j—al l other terms in the equation time averaged between steps t and t + At. Then, Kreiss-Oliger dissipation is implemented by adding specific terms on the right hand side of equation (4.126). In this case, u(t + At, r, 9) - „(*, r, 9) _ 1 ^ + ^ ^ & ) + R ^ ^ Q ) ] At 2 1 , [Dr(t + At,r,9) + Dr(t,r,9)} + i [De(t + At,r, 9) + De(t,r,9)] (4.127) when r e x + 2 A r < r < r m a x - 2 A r and 2A6> < 9 < TT - 2A9, and u(t + At,r,9) -u(t,r,9) 1 At 2 [R(t + At,r, 9)+R{t,r, 9)} + i [De{t + At,r, 9) + De{t,r, 9)} (4.128) when 2A# < 9 < TT — 2A9 and r = r e x + A r or r > r m a x — A r , where Dr(t,r,9) = -'^^[u(t,r+ 2Ar,9) - 4u(t,r + Ar,9)+ 6u(t,r,9) -4u (£ , r - A r , 9) + u(t, r - 2Ar, 9)) , (4.129) De{t,r,9) = - ^ t [ u ( t , r , 9 + 2A9)-4u(t,r,9 + A9)+6u(t,r,9) -iu(t,r,9 - A9) +u(t,r, 9 - 2A8)} , (4.130) and 0 < e r < 1 and 0 < eg < 1 are adjustable parameters. I use values er = 0.5 and eg = 0.5 for the adjustable dissipation parameters in directions r and 9, respectively. Using a Courant factor A = A t ~ 0.25 , (4.131) V/ i ( ( A r ) 2 + (A0) 2 ) this system of equations evolves stably under the Crank-Nicholson iterative solution scheme. 4.2.6 Strategies for A M R and Parallel Computation I have three functional numerical implementations that solve the dynamics of the Maxwell-Klein-Gordon system: (1) a uniprocessor unigrid code; (2) a multiprocessor unigrid code; and (3) a uniprocessor adaptive mesh code. 7 The uniprocessor unigrid code was written with the aid of R N P L 7While I attempted to incorporate a multiprocessor adaptive code, I was unable to find conditions at once compatible with the regularity conditions and processor distribution process. Specifically, all tested implementations were found to introduce high frequency errors during the adaptive regridding process. It is suspected this problem could be easily resolved by switching to another coordinate system where regularity is not at issue. CHAPTER 4. CHARGED SCALAR ACCRETION 92 ([94], [95], [96], [97]), and was used exclusively in obtaining results for the uncharged Maxwell-Klein-Gordon equations. M y multiprocessor unigrid code and uniprocessor adaptive mesh codes make use of the P A M R ([128], [129]) library. While I use the multiprocessor unigrid code extensively in solving the electromagnetically coupled dynamics, I have not found mesh refinement very beneficial. This is discussed below. In the parallel implementation it is necessary to ensure the computational domain is properly distributed among the compute nodes. While P A M R takes care of the processor distribution and information passing processes, I must manually specify the ghost region 8 widths. For the example given in Appendix C, a ghost width of a single point was appropriate. However, in the present case, the chosen difference equations involve not only Kreiss-Oliger dissipation operators, but also interpolation and extrapolation operators. Simple counting reveals these operator stencils can extend as many as 3 points in each grid direction from their centers. This means ghost regions at least 3 points wide are required for complete information communication. However, more than 3 ghost points are not required. Using a wider ghost region is therefore unnecessary and will only decrease computational efficiency. I exclusively use ghost regions with exactly 3 points in each grid direction for this reason. Furthermore, it is important to use no more processors than necessary. Specifically, since there is a cost associated with inter-node communication from the transfer of data on ghost regions, choosing too many compute nodes could result in a total ghost point count comparable to the number of points on the global domain itself. I have found that 64 processors is typically a good choice for global domains having a total of 3.2 x 10 5 grid points or more. The uniprocessor adaptive mesh code uses the modified Berger and Oliger time stepping algo-rithm as discussed in Appendix C, and implementation again uses the P A M R libraries. Central to this algorithm is the relative solution error that compares results on successive levels of resolution. It is important to consider appropriate grid functions in this measure of error. A good choice for the system includes the functions representing physical fields Ea, Ba and J Q , and conserved densities pr, ps and PQ. Auxiliary fields and vector potential components could always be in-cluded in the measure of error, but the physical fields and conserved densities are ultimately the quantities of interest to physics, so ensuring they are properly resolved through the measure should be sufficient. Forming the measure and from these quantities and using a number of different er-ror tolerances, I have tested the adaptive implementation by evolving massive electromagnetically coupled Maxwell-Klein-Gordon perturbations on an a = M = 5 Kerr spacetime. While evolution was smooth and otherwise well behaved, I have found the regions of adaptive refinement quickly grow and cover the greatest proportion of the computational domain, indicating the resolutional requirements become nearly uniform across most of the domain. This behaviour is depicted in Figure 4.14 for a maximum of four allowed levels of resolution. Considering the computational overhead of the adaptive refinement procedure, it is not clear that long term advantages can be gained from this implementation. The efficiency and cost is likely comparable to that of the parallel implementation where I simply evolve on a grid of sufficient uniform resolution. While the adaptive measure of solution error could probably be optimized, the improvement wil l probably not be great for the choice of spatial domain, typical initial conditions and parameter values e and m$. I have therefore not investigated mesh refinement in any more detail. 9 8 Effectively, the ghost region of a parallel finite difference code is that set of grid points whose values must be communicated between processors. The size and form of a ghost region is determined by the computational domain of dependence, which is itself established through the choice of specific finite difference approximations and thus stencils. Details and discussion can be found in Appendix C. 9 Of course, if the evolutionary details were known ahead of time, a non-uniform grid could be used in lieu of adaptive mesh refinement. Clearly, these non-uniform grids would have to be chosen on a case by case basis to accommodate resolutional requirements of the unfolding dynamics. So while much simpler to implement than adaptive refinement, the case-specific nature of non-uniform grid methods pose a serious limitation for the technique. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 93 F i g u r e 4.14: Adaptive mesh refinement (to a maximum of four levels) of the energy density for massive electromagnetically coupled perturbations in for a = M = 5 Kerr spacetime. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 6 < ir, and the corners of the domain are labelled with their (r, 6) ordinate values. The amplitude is scaled linearly and the colour gradient is scaled logarithmically. Refined regions quickly grow to cover a large proportion of the computational domain. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 94 4.3 Results 4.3.1 Initial Data The initial value problem for the Maxwell-Klein-Gordon system in Kerr spacetime amounts to specification of all the variables in a manner consistent with regularity and gauge conditions, and satisfying the electromagnetic constraint equation DaEa = pQ , (4.132) where DA is the spatial covariant derivative operator, and P Q , is, according to equation (4.108), PQ = I E F 2 t 9 * t T ( * * n * " $ n * ) • ( 4 - 1 3 3 ) Following the procedure of section 2.2.1, I decompose Ea into its longitudinal and transverse parts, respectively La and Ta, to obtain Ea=La + Ta. (4.134) Then introducing the scalar potential, U = U(r,6), as per DaU = La, (4.135) (4.132) becomes DaDaU = DaEa = PQ , (4.136) where the left hand side is explicitly 1 92 DaDaU = r - [2grr(dre^)(drU) - {drgrr)e^(drU) + 2grre4>4,(drrU) grr \9tre4>4> ~ ft^rr) +grr{degrr){dgU) -2g2rrcos9{dcoseU) + 2g2r(deeU)} . (4.137) After solving (4.136) using the methods of Appendix C, I reconstruct the longitudinal electric field components, which are, from (4.135), Lr = drU , . • (4.138) Lg = 3gU , (4:139) = 0 . (4.140) I can then specify any divergenceless set of components for the transverse part Ta. For simplicity, I choose Tr . = 0 , (4.141) Tg = 0 , (4.142) T 0 = 0 . (4.143) C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 95 The completely reconstructed electric field at initial time is then Er Ed, = drll , = d9U , = • 0 . (4.144) (4.145) (4.146) Having solved for Ea, the final step is to then initialize n r , Tig and n^, in accordance with (4.101), (4.102) and (4.103), meaning n r = -ne = -IL* = -9trU<P ~ 9t<pft(p9rT 9tr 9trU<t> ~ 9tcj>ftcj>9rr 9tr 9trfu ~ 9t(pft<p9rr 9tr Er Ee , E, (4.147) (4.148) (4.149) Having satisfied the initial data constraint, I can freely specify the remaining variables in accor-dance with their definitions and regularity conditions. For simplicity I choose Gaussian profiles for the complex components of 4>(i,r, 9) = 4>i(t,r,9) + i<f>2(t,r,9), Tl$(t,r,9) = LTi(t,r, 9) + iU2(t,r,9), and the vector potential components Aa(t,r, 9).10 That is, M0,r,6) = aiexp :-((*• - r - i ) /5n) 2 ) exp (--((61- 9i)/Seif) , (4.150) h(0,r,9) = 02exp - r2)/5r2)2) exp (--((9- 0 2 ) / M 2 ) , (4.151) IIi(O,r,0) = a3exp - r3)/8r3)2) exp (--((9- 9s)/6e3)2) , (4.152) TL2(0,r,9) = a4 exp - r 4)/<5 r 4) 2) exp (- 94)/Se4)2) , (4.153) At(0,r,9) = arexp :-((»• - rt)/Srt)2) exp (-•w-9t)/6et)2) , (4.154) Ar(0,r,9) = arexp - rr)/5rr)2) exp (- 9r)/Ser)2) , (4.155) A0(P,r,B) = aeexp [-((.r - re)/5Te)2) exp (- 9e)/5ee)2) , (4.156) ^ ( 0 , r , 9) = exp - r^/Srj,)2) exp ( -((6>--0*)/<W)2) , . (4.157) where a i , a2, a3, a 4 , at, a r , ag and are constants setting the initial amplitudes, r\, r2, r%, r 4 , i"t, Tri r6 and determine the Gaussian peaks in the radial direction, 8\, 92, # 3 , # 4 , 9t, 9r, 9$ and 94, determine the Gaussian peaks in the 9 coordinate direction, Sri, Sr2, 5rs, Sr4, Srt, Srr, 5re and 5r<j) determine the Gaussian widths in coordinate r, and 5g\, 8g2, $63, 5g4, 5gt, Sgr, 5gg and 5$$ determine widths in 9. A l l remaining variables are initialized according to definitions in sections 1 0 F r o m equations (4.93) and (4.94), nonzero electric current components Jr and Jg require e, $ and «J>* to all be nonzero. The condition for nonzero Jr additionally requires that either AT, <J>r or <3>* be nonzero. Similarly, the condition for nonzero Jg additionally requires that either Ag, <bg or be nonzero. Clearly, the chosen initial data (4.150)-(4.162) satisfies the nonzero conditions for both Jr and Jg when e / 0. Therefore, it can be said charge separation begins at t = 0 for this specification of the Maxwell-Klein-Gordon system. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 96 4.2.3 and 4.2.4. Unless otherwise specified, I use values a-i = c-2 = 0-3 = <i4 = at = ar = a-e = = 1.0 , (4.158) n =r2= r3 = r4=rt = rr = re=r<f,= 25.0 , (4.159) ' 0i = 02 = 03 = 9A = Bt = 0r = Be = = TT/2 , (4.160) 5ri = 8r2 = 5r3 = 5rA = $rt = 5rr = $r0 = <5rc£ = 5.0 , (4.161) Sei = $$2 = $03 = $04 = Set = S0r = S00 = $04, = 0.3 . ' (4.162) The chosen Kerr spacetime is exclusively mass M = 5 with variable angular momentum per unit mass, a. Finally, the computational domain always covers the coordinate range 5 < r < 50 and 0 < 9 < Tr. Finally, while the form of metric (4.40) suggests symmetry about the equatorial plane (9 = 7r/2), this symmetry is for the spacetime metric and not for matter evolving in the background spacetime. Since the Maxwell-Klein-Gordon fields are here treated as perturbations not affecting the spacetime, they are free to evolve in a manner which is not symmetric about the equatorial plane. Observe how the equations of motion and constraint have been written in a general form consistent with the assumption of axial symmetry and therefore allow a more generic evolution—one that need not respect any symmetry about the equatorial plane. 1 1 As shown in section 4.3.3, the choice of initial conditions (4.132), (4.147)-(4.149), and (4.154)-(4.157) do not generate electromagnetic fields with covariant components all symmetric about 9 = TT/2. Furthermore, since the difference approximations used are accurate to only O (^(A#)2^, slight asymmetries about the equatorial plane will be expected throughout evolution regardless of how initial conditions are chosen. 4.3.2 K l e i n - G o r d o n Dynamics : Orb i t i ng Resonances and Mass ive M o d e Trapping for s = 0 Per turbat ions W i t h initial data specified as discussed above, I evolve massive s = 0 perturbations on,a Kerr spacetime. The black hole mass is set to M = 5, and the angular momentum per unit mass is chosen to be either a = 0 or a = M. I vary the Klein-Gordon mass parameter m$ = my between m$ = 0.0 and m$ = 2.4. The computational domain covers radial coordinate range 5 < r < 50 and angular coordinates 0 < 9 < IT. The computational domain has J = 401 points radially and K = 201 points in the 9 angular direction. The numerical evolutions span the range of coordinate time 0 < t < 200. For a = M = 5, m$ =0.0 evolution, perturbations clearly display the glory effect and orbiting resonances. Figure 4.15 displays the normalized energy density within 5 < r < 5 0 , O < 0 < 7 r and 0 < <f> < 2ir by a sequence of steps from the evolution beginning at t = 0 and leading to a final step at t = 77.11 shown in Figure 4.16. The orbit resonance period is « 70i and each successive orbit is attenuated by a factor of ~ 1 0 - 1 . Note how the orbital frequency w l/(70£) is more than twice that of the unstable circular photon orbits occurring at r = 3 M . 1 2 While strong gravitational 1 1 While an equatorially symmetric code could be written for the Maxwell-Klein-Gordon system, it would not allow study of any physics unique to the non-equatorially symmetric system. 1 2 The innermost stable circular orbit of the Schwarzschild spacetime described by (4.1) occurs at radius r = 6M, while the innermost unstable circular orbit occurs at r = 3M. Prom the study of timelike geodesies in Schwarzschild spacetime it can be shown that circular orbits in the equatorial plane are given by or r^ where r is proper time and L is the angular momentum per unit rest mass as measured by a static observer at . infinity. The proper time for the innermost stable circular orbit of an M = 5 Schwarzschild spacetime is therefore 1 8 ° 0 , r , while the proper time for the innermost unstable circular orbit in the same spacetime is . More cannot be said about the orbital period without specifying a value for L. Instead consider the null geodesies of Schwarzschild C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 97 interaction is observable in the form of orbiting resonances, neither mode trapping nor superradiant scattering are apparent from the calculations. While mode trapping is expected only for massive scalar perturbations, superradiant scattering could, in fact, be present for any scalar perturbation mass. However, the effect is expected to be only on the order of a fraction of a percent [124]. Figure 4.17 displays the integrated normalized energy density within the computational domain for 0 < t < 200 for both a = 0 and a = M. Clearly, the addition of black hole angular momentum has but a small effect on the scalar perturbation evolution. 1 3 Predominantly, addition of black hole angular momentum decreases the perturbation energy flux through surfaces r = 5 and r = 50. While not exact for either a = 0 or a = M, the late time decay of integrated normalized energy, T, is approximately a power law of form with constant of proportionality fcwlx 10~ 2 . A local maximum in this integrated energy density would be sufficient for identification of superradiant scattering. But since scalar perturbations are flowing out of the computational domain at both the r = 5 excision surface and the r = 50 outer boundary, a better test would also incorporate the energy flux through the surface r = 50. A local maximum in the integrated energy within the computational domain plus that of the outflowing flux would constitute a necessary condition for superradiant scattering. So while not apparent in the data shown, superradiant scattering may, in fact, be occurring in the evolution of the massless scalar perturbations. Finally, Figure 4.18 displays the normalized energy density at t = 53.98. Results presented in section 4.3.3 show a superradiant scattering peak for s = 1 electromagnetic perturbations at this time (cf. Figures 4.29 and 4.30). spacetime. It is known there are no stable circular orbits and just a single unstable circular orbit at r = SM whose equation in the equatorial plane is The period in coordinate time of a unstable circular photon orbit in M = 5 Schwarzschild spacetime is therefore y/330n ~ 163.24. 1 3In fact, a comparison of the normalized energy densities reveals that a = 0 and a = M evolutions are morpho-logically nearly identical. log(T) kt (4.165) (4.164) C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 98 1.00e-07 1.00e-05 1.00e-07 1.00e-05 1.00e-07 1.00e-05 F i g u r e 4.15: The glory effect and orbiting resonances for a = M = 5, m$ =0.0 scalar perturba-tions. The figure displays the normalized energy density in a sequence of steps. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < ir, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude is scaled linearly, while the scale of the colour gradient is logarithmic. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 99 F i g u r e 4.16: The glory effect and orbiting resonances for a = M = 5, m$ = 0.0 scalar per-turbations at late time. The figure displays the normalized energy density at a final step of the evolution. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown, the computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude is again scaled linearly and the colour gradient logarithmically. A n M P E G animation showing orbiting res-onance through normalized energy density for the period 0 < t < 108 is avail-able at http://$WWW/People/petryk/animations/2.mpeg where $WWW is currently laplace.physics.ubc.ca. As discussed in section 4.1, setting the scalar perturbation mass to nonzero values may permit mode trapping in the minimum of the effective gravitational potential. The details depend on the specific mode, and the magnitude of the perturbation mass, m$, relative to the mass of the black hole, M. Specifically, Figures 4.5 and 4.7 demonstrate that I = 5 can be the only trapped mode of perturbation mass m$ = 0.3 in a spacetime with a = 0 and M = 5. From equation (4.10) it can be deduced that, for r < 50, all modes with I < 5 experience only a potential well centered within the event horizon. The only well for modes I > 5 is again within the event horizon, but now additional scattering is observed because of a local maximum in the potential near r = 3 M . Comparing with Figures 4.3, 4.6 and 4.8 it is clear that, for a given spacetime, trapping of low I modes occurs for smaller values of m$, and trapping of higher I modes takes place for larger m<j>. Furthermore, if m$ is sufficiently large, all lower I modes are expected to experience only a deep potential well and therefore fall into the black hole. Figure 4.19 displays the integrated normalized energy density for m<j> = 0.0,0.3,0.6,1.2, and 2.4 massive scalar perturbations in an a = M = 5 Kerr spacetime. m$ = 0.3 perturbations experience the greatest trapping and persist longest on the computational domain, while the m$ = 2.4 perturbation total energy decays most rapidly. A t late time, t ss 200, the total energy of m<j> = 0.6 perturbations most closely matches that of the m$ = 0.0 perturbations. Superradiant scattering peaks are not observed for any of these cases, C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 100 0 -0 .5 - 1 -1 .5 - 2 - 2 .5 1 I 1 1 1 \ V i i i i i i i i i i i i i \--.. \ ••• --\ \ \ \ \ —«v —• N \ \ \ \ , , , , i i i i \ --. \ \ \ -\ " V , 1 , , , ' , 1 , , , , 1 0 50 100 150 20C t F i g u r e 4.17: Integrated normalized energy density for massless scalar perturbations. Plotted is log(T), the logarithm of the total energy contained within 5 < r < 5 0 , 0 < # < 7 r and 0 < <j> < 2n, as a function of coordinate time t. The dashed line represents data for a/M = 0/5 = 0 and the dotted line data for a/M = 5/5 = 1. The decay rate of integrated energy is approximately a power law at late time. A local maximum signalling superradiant scattering is not observable for either evolution. Black hole angular momentum effectively decreases the overall decay rate of these massless scalar perturbations. Note the decrease in time of this quantity results from the energy flux across surfaces r = 5 and r = 50, and thus out of the computational domain. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 101 F i g u r e 4.18: Normalized energy density at fc = 53.98 for a = M = 5, m$ = 0.0 scalar pertur-bations. It is at this time that superradiant scattering peaks are seen for electro-magnetic perturbations (cf. Figures 4.29 and 4.30), but here there is no evidence of superradiance for the massless scalar perturbations. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown, the computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the (r, 6) ordinate values at the domain corners are provided. The amplitude is again scaled linearly and the colour gradient logarithmically. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 102 but this test is again sufficient but not necessary for superradiance. The detailed structure of massive scalar perturbation evolution k r a = M = 5 Kerr spacetime is displayed for m$ = 0.3 in Figures 4.20 and. 4.21, and for m$ = 2.4 in Figure 4.22. Figures 4.20 and 4.21 reveal a rather detailed (normalized) energy density distribution as the m$ = 0.3 perturbations evolve on the computational domain. While the evolution markedly different from that of m$ =0.0, there is still some evidence of orbiting resonances very near the excision surface at r = 5, but it is not possible to provide a reasonable estimate for either the resonance orbit period or their attenuation. However, the perturbations now persist for a extended period of time in the vicinity of the black hole. Furthermore, there is some evidence of additional energy density concentration along the axis of black hole rotation. Meanwhile, Figure 4.22 shows how the m$ = 2.4 scalar perturbations remain localized throughout evolution as they become tidally distorted and consumed by the black hole within a relatively short period of time. The evolution of m$ = 1.2 perturbations (not displayed) are very much similar in these respects. Finally, as expected, the effect of black hole angular momentum are small for (s = 0) scalar perturbations. Morphologically, the a = 0 and a = M evolutions are nearly identical for m<j> = 0.3 and also for m$ = 2.4. Figures 4.23 and 4.24 compare a = 0 and a = M integrated normalized energy densities for the respective m$ = 0.3 and m$ = 2.4 evolutions. The results again show little difference in the total energy within 5 < r < 5 0 , O < 0 < 7 r and 0 < </> < 2TT as a function of time between a = 0 and a = M spacetimes. The approximation of total integrated energy, T, to a power law of form log(T) ~ -kt (4.166) is worse than for the massless perturbations. However, if a power law is assumed at late time, the decay constant would be k s=s 5 x 10~ 3 for m$ = 0.3 perturbations, and fc«7x 10~ 2 for m$ = 2.4 perturbations. The lower decay rate for m$ perturbations results from mode trapping about a local potential minimum, and the increased decay rate for m$ = 2.4 perturbations derives from the rapid infall of perturbations in a steeper potential well. These results differ from the case of (s = 1) electromagnetic perturbations presented below. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 103 0 i i i i | i i i i | i i i i j » i. i ,i j ' •** ! ^:^~-_ \ ~~ ~~ ~ \ \ \ - 5 \ -\ \ o \ \ \ \ \ \ -10 \ \ X \ \ \ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 50 100 150 20C F i g u r e 4.19: Integrated normalized energy density for massive scalar perturbations. Plotted is log(T), the logarithm of the total energy contained within 5 < r < 50,0 < 9 < n and 0 < <f> < 2TT, as a function of coordinate time t. A l l data is for the a/M = 5/5 = 1 Kerr spacetime. The dashed red line is for m$ = 2.4, the dotted red line is for m<j> = 1.2, the dashed blue line is m$ =0.6, the dotted blue line is m<j> = 0.0, and the dashed black line is m$ = 0.3. Clearly, m$ = 0.3 perturbations experience the greatest trapping and persist longest on the computational domain. In none of the cases is a local maximum signalling superradiant scattering apparent. The decrease in time of this quantity results from the energy flux across surfaces r = 5 and r = 50, and thus out of the computational domain. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 104 1.00e-07 1.00e-05 I 1.00e-07 1.00e-05 t=77.11 I t=96.39 F i g u r e 4.20: Mode trapping and orbiting resonances for a = M = 5, m$ = 0.3 scalar per-turbations. The figure displays the normalized energy density evolution in a sequence of steps. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown, the computational domain covers the range 5 < r < 50 and 0 < 0 < w, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude is again scaled linearly and the colour gradient logarithmi-cally. A n M P E G animation of the evolution over the period 0 < t < 108 is avail-able at ht tp:/ /$WWW/People/petryk/animations/3.mpeg where $WWW is currently l a p l a c e . p h y s i c s . u b c . c a . C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 105 t-134.94 1.00e-07 1.00e-05 F i g u r e 4.21: Mode trapping and orbiting resonances for a = M = 5, m$ = 0.3 scalar pertur-bations at late time. The figure displays the normalized energy density evolution in a sequence of steps. The directions of coordinate vectors ra ~ (0,1,0,0) and 6a ~ (0,0,1,0) are again shown, the computational domain again covers the range 5 < r < 50 and 0 < 8 < TT, and the corners of the domain are labelled with their (r, 6) ordinate values. The amplitude is again scaled linearly and the colour gradient logarithmically. CHAPTER 4. CHARGED SCALAR ACCRETION 106 1=0.00 I 1=19.28 1.00e-07 1.00e-05 1 1.00e-07 1.00e-05 t=77.11 I 1=96.39 1.00e-07 1.00e-05 I 1.00e-07 1.00e-05 F i g u r e 4.22: Localization, tidal distortion, and rapid decay of m$ = 2.4 scalar perturbations in a = M = 5 Kerr spacetime. The figure displays the normalized energy density evolution in a sequence of steps. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown, the computational domain covers the range 5 < r < 50 and 0 < 0 < sr, and the corners of the domain are labelled with their (r,9) ordinate values. The amplitude is again scaled linearly and the colour gradient logarithmically. A n M P E G animation of the evolution over the period 0 < t < 108 is available at ht tp: / /$WVW/People/petryk/animations/4.mpeg where $WWW is currently l a p l a c e .phys ics .ubc. ca . C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 107 0 -0 .5 - 1 1 1 1 1 * \ i i i i i i i i i i i > i i i "\ -\ V \ ' - - . V--. N \ N -v — . \ -i i i i ~\ , , , , l , , , , 1 , , i , 0 50 100 150 20C F i g u r e 4.23: Integrated normalized energy density for TO$ = 0.3 scalar perturbations. Plotted is log(T), the logarithm of the total energy contained within 5 < r < 50, 0 < 9 < ir and 0 < <f> < 27T, as a function of coordinate time t. The dashed line represents data for a/M = 0/5 = 0 and the dotted line data for a/M = 5/5 = 1. Black hole angular momentum decreases the overall decay rate for these massive scalar perturbations. The decrease in time of this quantity results from the energy flux across surfaces r = 5 and r = 50, and thus out of the computational domain. . J " 4.3.3 Maxwell Equation Dynamics: Orbiting Resonances and Superradiance for s = 1 Perturbations With initial data again specified according to section 4.3.1,1 evolve s = 1 perturbations on a Kerr spacetime. The black hole mass is again set to M = 5, and the angular momentum per unit mass is chosen somewhere in the range a = 0 through a = M. The computational domain again covers radial coordinate range 5 < r < 50 and angular coordinates 0 < 9 < IT. The computational domain has J = 401 points radially and K = 201 points in the 9 angular direction and numerical evolutions span the range of coordinate time 0 < t < 200. In terms of physical fields, the chosen initial data amounts to an everywhere vanishing electric field and a non-vanishing nontrivial magnetic field at t = 0. For example, in the case a = M, the initial magnetic field components are distributed as shown in Figure 4.25. The resulting energy density distribution is shown in the first frame of Figure 4.28. While the electric field components are initially vanishing, their time derivatives—and also those of the magnetic field components—are not, implying nontrivial evolution. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 108 O -10 n 1 1 1 1 1 1 1 1 1 1 1 1 1 r \ \ \ \ \ \ \ \ \ \ \ V _ l I I I I I I I I I I L 50 100 t 150 20C Figure 4.24: Integrated normalized energy density for m$ = 2.4 scalar perturbations. Plotted is log(T), the logarithm of the total energy contained within 5 < r < 50, 0 < 9 < IT and 0 < ((> < 2n, as a function of coordinate time t. The dashed line represents data for a/M = 0/5 = 0 and the dotted line data for a/M = 5/5 = 1. Black hole angular momentum has little effect on the evolution of these massive scalar perturbations. Again, decreases in time of this quantity result from the energy flux across surfaces r = 5 and r = 50, and thus out of the computational domain. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 109 (a) (b) (c) F i g u r e 4.25: Initial magnetic field distribution for a = M = 5, electromagnetic perturbations. (a) displays Br, (b) displays Bo, and (c) shows B$. The directions of coordinate vectors ra ~ (0,1,0,0) and 8a ~ (0,0,1,0) are shown in (a). The computational domain covers the range 5 < r < 50 and 0 < 6 < ir, and the corners of the domain are labelled with their (r, 8) ordinate values. The amplitude and colour gradient are both scaled linearly. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 110 For evolutions a = 0 through a = M, electromagnetic perturbations clearly display the glory ef-fect and orbiting resonances. Figure 4.26 displays the normalized energy density within 5 < r < 50, 0 < 9 < 7T and 0 < <f> < 2TT by a sequence of steps from the evolution beginning at t = 0 and lead-ing to a final step at t = 77.11 shown in Figure 4.27. Strong gravitational interaction is clearly manifest in the form of orbiting resonances. Similar to the case of massless scalar perturbations, the resonance orbit period is « 70i, corresponding to an orbital frequency more than twice that the unstable circular photon orbit at r = 3 M (cf. equation (4.164) and related discussion). Fur-thermore, the attenuation between successive orbits is again ~ 1 0 - 1 . While an effect akin to mode trapping is not expected for electromagnetic perturbations, superradiant scattering could be present, depending on the initial data and value of angular momentum parameter a. The maximum superradiant amplification is expected to be on the order of four percent in an a = M spacetime. Figure 4.28 displays the integrated normalized energy density within the computational domain for 0 < t < 200 for a = 0 ,0 .8M and a = M. Clearly, the addition of black hole angular momentum has a measurable effect on the evolution of electromagnetic perturbations. A t late time the total integrated energy, T , can be roughly approximated by the power law log(T) ~ -kt , (4.167) where the decay constant is fc « 1 x 10~ 2 . Generally, the decay constant decreases with in-creasing black hole angular momentum. However, the total integrated energies are nearly iden-tical for all a > 0.998. Figure 4.29 displays the integrated normalized energy densities for a — 0,0 .8M, 0.98M, 0.998M and a = M for 47.5 < t < 57.5. The local maxima of the integrated energy densities for a = 0 .8M, 0.98M, 0.998M and a = M at t = 53.98 are signals of superradiant scat-tering. From the data I deduce a lower estimate on the superradiant amplification in the amount of w 0.05% for a = 0 .8M, « 1.3% for a = 0 .98M and w 1.6% for a > 0.998M. Observe how the largest value measured here is ss 1/3 the maximal value of 4.4% calculated in the study of s = 1 perturbations by Teukolsky and Press [141]. Again, since perturbations are flowing out of the computational domain at the r = 50 outer boundary, a better estimate for the total superra-diant amplification would also incorporate the energy flux through that surface. The superradiant amplification could thus be a couple percent larger than given by these lower estimates. For ref-erence, Figure 4.30 displays the normalized energy density at the maximum of the superradiant scattering peak for the a = M spacetime perturbations. The concentration of energy density in the plane 9 = TT/2 within the ergosphere near the excision surface is markedly different from the case of scalar perturbations at t = 53.98 as depicted in Figure 4.18. Apart from the resonance and superradiant scattering amplitudes, the energy densities are quite similar for electromagnetic perturbation evolutions in a = 0 ,0 .8M, 0.98M and 0.998M <a<M spacetimes. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 111 F i g u r e 4.26: The glory effect and orbiting resonances for a = M = 5, electromagnetic perturba-tions. The figure displays the normalized energy density in a sequence of steps. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude is scaled linearly, while the scale of the colour gradient is logarithmic. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 112 F i g u r e 4.27: The glory effect and orbiting resonances for a = M = 5, electromagnetic pertur-bations at late time. The figure displays the normalized energy density at a final step of the evolution. The directions of coordinate vectors ra ~ (0,1,0,0) and 6a ~ (0,0,1,0) are shown, the computational domain covers the range 5 < r < 50 and 0 < 9 < TT, and the corners of the domain are labelled with their (r, 6) ordi-nate values. The amplitude is again scaled linearly and the colour gradient log-arithmically. A n M P E G animation showing orbiting resonance through normal-ized energy density for the s = 1 perturbations over period 0 < t < 108 is avail-able at ht tp:/ /$WWW/People/petryk/animations/5.mpeg where $WWW is currently l a p l a c e . p h y s i c s . u b c . c a . For an understanding of the magnetic and electric field dynamics, I compose the products BaBa and EaEa for the a = M = 5 spacetime evolution. Figure 4.31 displays the square of the magnetic field at times t = 0, 19.28, 38.55, 53.98, 57.83 and t = 77.11. Similarly, Figure 4.32 displays the square of the electric field at t = 0, 19.28, 38.55, 53.98, 57.83 and t = 77.11. Observe that while the BaBa is essentially zero in the equatorial plane 6 = n/2 outside the ergoregion, the quantity EaEa is not. This indicates that the energy density outside the ergosphere about the equatorial disk is carried predominantly by the electric field. Furthermore, the magnetic field amplitude is substantially smaller than the electric field amplitude along the axis of symmetry where 6 = 0 or 9 = TT. However, note that, in curved spacetime, EaEa and BaBa are merely proportional to the contributions from electric and magnetic fields. The relationship between these squares and the energy density is only made an equality through inclusion of multiplicative factors that include metric components. Therefore, the sum of squares 1 (EaEa + BaBa) is not generally a conserved quantity. The identification between this specific function and a conserved density can only be made in the exceptional case of flat spacetime. Furthermore, Figure 4.33 plots the integrated squares over the computational domain, f9r~r (gtrf<p<P ~ 9t<pft<p9rr) B2= / / BaBa v V J t r ' w 2 = g I drd9dcj> , (4.168) 7,4=0 J6=0 Jr=5 9tr C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 113 - 0 . 5 -2 1 1 1 1 j 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I ; x -1 1 1 1 1 1 1 1 \ \ i V i i i i i i i i i i i i 0 50 100 150 20C F i g u r e 4.28: Integrated normalized energy density for electromagnetic perturbations in M = 5 Kerr spacetime. Plotted is log(T), the logarithm of the total energy contained within 5 < r < 50, 0 < 9 < IT and 0 < <j> < 2ir, as a function of coordinate time t. The dashed red line represents data for a/M = 0, the dotted red line data for a/M = 0.8, and the dashed black line data for a/M = 1. Clearly, black hole angular momentum decreases the overall decay rate of electromagnetic perturbations. Any decrease in time of the energy results from the flux across surfaces r = 5 and r = 50, and thus out of the computational domain. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 114 T 1 1 1 1 1 r -0 .14 W -0 .16 \ \ m a -0 .18 \ - \ \ \ \ \ \ \ \ •8 50 52 54 56 F i g u r e 4.29: Integrated normalized energy density for electromagnetic perturbations in M = 5 Kerr spacetime. Plotted is log(T), the logarithm of the total energy contained within 5 < r < 50, 0 < 9 < ir and 0 < <j> < 1-K, as a function of coordinate time t. The dashed red line represents data for a/M = 0, the dotted red line data for a/M = 0.8, the dashed blue line data for a/M = 0.98, the dotted blue line (nearly coincident with the dashed black line) data for a/M = 0.998, and the dashed black line data for a/M = 1. The peaks in integrated energy densities at t = 53.98 for a > 0 . 8 M are a mark of superradiant scattering. t C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 115 F i g u r e 4.30: Normalized energy density at the superradiant scattering peak time t = 53.98 for electromagnetic perturbations in an a = M = 5 Kerr spacetime. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown, the computational domain covers the range 5 < r < 50 and 0 < 6 < IT, and the (r, 9) ordinate values at the domain corners are provided. The amplitude is scaled linearly and the colour gradient logarithmically. A concentration of energy density in the plane 9 = TT/2 within the ergosphere near the excision surface is the manifestation of superradiant scattering. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 116 E 2 = r r r EAE* ^ ^ R U \ . ( 4 , i 6 9 ) J<t>=0 Je=0 Jr=5 9tr After an initial period of oscillation where magnetic fields convert to electric fields, and vice-versa, the integrated amplitudes become closely matched. At late time, the decay rates can be roughly approximated by the power laws log(S 2 ) ~ -kt , (4.170) and log(.E 2) ~ -kt , (4-171) where k « 1 x 1 0 - 2 in both (4.170) and (4.171). Finally, except for the lack of superradiant amplification and even smaller B A B A contribution along the axis of symmetry, evolution of B A B A and E A E A are very similar in the a = 0 and a = M spacetimes. The integrated densities B 2 and E2 are also similar in the cases a = 0 and a = M. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 117 1.00e-05 1.00e-03 • 1.00e-05 1.00e-03 1.00e-05 1.00e-03 I 1.00e-05 1.00e-03 F i g u r e 4.31: BaBa for a = M = 5, electromagnetic perturbations. The figure displays the square of the magnetic field in a sequence of steps including the superradiant scattering peak at t = 53.98. The computational domain covers the range 5 < r < 50 and 0 < 0 < sr, and the corners of the domain are labelled with their (r, 6) ordinate values. The amplitude is scaled linearly, while the scale of the colour gradient is logarithmic. The magnetic field amplitude is negligible outside the ergoregion in the plane 9 = n/2, and much smaller than the electric field along the axis of symmetry where 9 = 0 and 9 = IT (cf. Figure 4.32). C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 118 l.OOc-05 1.00e-03 I 1.00c-05 1.00e-03 1=57.83 I 1=77.11 1.00c-05 1.00e-03 I 1.00c-05 1.00e-03 F i g u r e 4.32: EaEa for a = M = 5, electromagnetic perturbations. The figure displays the square of the electric field in a sequence of steps including the superradiant scattering peak at t = 53.98. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude is scaled linearly, while the scale of the colour gradient is logarithmic. The electric field amplitude is substantially larger than the magnetic field amplitude along the axis of black hole rotation (cf. Figure 4.31). CHAPTER 4. CHARGED SCALAR ACCRETION 119 20C F i g u r e 4.33: Integrated squares of electric and magnetic fields in a = M = 5 Kerr spacetime. Plotted are l'og(732) and log(iJ 2 ) , the logarithm of the integrated squares contained within 5 < r < 50, 0 < 9 < ir and 0 < <f> < 2ir, as a function of coordinate time t. The dashed black line represents log(S 2 ) , and the dotted black line log(.E 2). Initially, the field amplitudes undergo substantial oscillation, but the decay of in-tegrated electric and magnetic amplitudes become closely matched at intermediate and late time. The effects of black hole rotation are more strongly manifest in the evolution of electromagnetic angular momentum. The angular momentum density for electromagnetic perturbations is given by equation (4.115). Figure 4.34 displays the evolution for a = M in a sequence of steps, beginning at t = 0, and leading to the final displayed step at t = 77.11 in Figure 4.35. 1 4 Meanwhile, Figure 4.37 shows the evolution sequence for o = 0, leading to the final displayed frame in Figure 4.38. Observe how the a = M density has much larger amplitude and steeper gradients than the a = 0 data. Furthermore, Figures 4.36 and 4.39 respectively display the angular momentum densities at the superradiant scattering peak time t = 53.98 for a = M and o = 0. It is there evident how strongly the electromagnetic perturbation angular momentum is affected by the black hole rotation. Finally, Figure 4.40 shows the angular momentum density integrated over the computational domain as a function of time for the M = 5 spacetimes with a = 0, a = 0 . 8 M and a = M. Also, Figure 4.41 shows the integrated angular momentum for a = 0 ,0 .8M, 0.98M, 0.998M and a = M for 47.5 < t < 57.5. Notice that, while highly variable for a > 0 . 8 M spacetimes, the total angular 1 4This nontrivial evolution of electromagnetic perturbation angular momentum introduces an interesting possibility for black hole formation. Namely, not only can black holes be formed from pure electromagnetic radiation, but such black holes can also carry angular momentum. This makes it possible to generically study black -.hole angular momentum scaling in critical phenomena at the threshold of black hole formation in both axial symmetry and in four dimensional spacetimes without symmetries. 5 C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 120 momentum, S, tends towards 5 = 0. In the standard picture of superradiant scattering, amplification of incident perturbations oc-curs at the expense of black hole rotational energy, and therefore at the expense of black hole angular momentum [151], [152], [153], [5], [101], [124], [137], [138] and [141]. A net flux of angular momentum should thus be observed from a black hole during superradiant scattering events. Fig-ure 4.36 shows there is very large angular momentum density in the region of the event horizon and ergosphere, and negligible angular momentum density at the r = 50 outer boundary, when t = 53.98 in the a = M = 5 Kerr spacetime. Meanwhile, Figure 4.39 similarly shows negligible angular momentum density at the r = 50 domain boundary, and a small (but not insignificant) density near the event horizon at t = 53.98 in the a = 0, M. = 5 spacetime. Any substantial changes of total angular momentum in these spacetimes around t = 53.98 wil l then be due to flux to or from the black hole itself. Figures 4.40 and 4.41 show marked increases in total angular momentum for the 0.8 < a/M < 1, M = 5 spacetimes around t = 53.98, but insignificant change in angular momentum for the a = 0, M = 5 spacetime around the same period. But t = 53.98 is precisely the time when total electromagnetic energy within r = 50 in the 0.8 < a/M < 1, M = 5 spacetimes show their largest increase in value (cf. Figure 4.29). Then, according to the standard picture, the marked increases of electromagnetic angular momentum around t = 53.98 in these spacetimes are the direct result of superradiant amplification drawn from the rotation of the black hole itself. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 121 -1.00e-04 1.00e-04 1 -1.00e-04 1.00e-04 1=38.55 1 1=48.19 -1.00e-04 1.00e-04 I -1.00e-04 1.00e-04 F i g u r e 4.34: Angular momentum density for electromagnetic perturbations in a = M = 5 Kerr spacetime. The figure displays angular momentum density density in a sequence of steps. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 6 < sr, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude and colour gradient are both scaled linearly. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 122 t=77.11 F i g u r e 4.35: Angular momentum density for electromagnetic perturbations at late time in a = M = 5 Kerr spacetime. The figure displays angular momentum density density at a final step of the evolution. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown, the computational domain covers the range 5 < r < 50 and 0 < 9 < TT, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude and colour gradient are again scaled linearly. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 123 F i g u r e 4.36: Angular momentum density for electromagnetic perturbations at superradiant scat-tering peak time t = 53.98 in o = M = 5 Kerr spacetime. The directions of co-ordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown, the computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude and colour gradient are again scaled linearly. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 124 -l.OOc-04 l.OOc-04 1 -l.OOc-04 1.00e-04 t=38.55 i t=4S.19 -1.00e-04 l.OOc-04 I -l.OOc-04 1.00e-04 F i g u r e 4.37: Angular momentum density for electromagnetic perturbations in a = 0, M = 5 Kerr spacetime. The figure displays angular momentum density density in a sequence of steps. Notice the general decay in electromagnetic angular momentum relative to that of the case a = M. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < 7r, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude and colour gradient are both scaled linearly. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 125 t=77.11 -1.00e-04 1.00e-04 F i g u r e 4.38: Angular momentum density for electromagnetic perturbations at late time in a = 0, M = 5 Kerr spacetime. The figure displays angular momentum density density at a final step of the evolution. Notice how the angular momentum density distribution here differs from that of the case a = M. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown, the computational domain covers the range 5 < r < 50 and 0 < 9 < ar, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude and colour gradient are again scaled linearly. 4.3.4 Massless M a x w e l l - K l e i n - G o r d o n Equa t ion Dynamics I: Orb i t i ng Resonances, Charge Separation and Dynamo-Like Behaviour for e = 0.6 Electromagnet ical ly Coupled s = 0 and s = 1 Per turbat ions I again specify initial data according to section 4.3.1, but here I evolve m<t> = 0.0 electromagnetically coupled s = 0 and s = 1 perturbations on a Kerr spacetime by specifying nonzero charge coupling constant e. The black hole mass is set to M = 5, and the angular momentum per unit mass is again chosen somewhere in the range a = 0 through a = M. As before, the computational domain covers radial coordinate range 5 < r < 50 and angular coordinates 0 < 9 < w, but now resolutional requirements demand a computational domain with J = 801 radial points and K = 401 points in the 9 angular direction. Because of the computational demands, these calculations were performed in parallel across 64 processors with processor overlap regions being 3 grid points wide. The numerical evolutions again span the range of coordinate time 0 < t < 200. Since I choose the charge coupling parameter e = 0.6 in a following analysis of charged massive scalar fields, I focus discussion mostly on the e = 0.6 case but later compare these results to the evolution of both e < 0.6 and e > 0.6 coupling parameter data. In terms of physical electromagnetic fields, the chosen initial data again amounts to an everywhere vanishing electric field and a non-vanishing nontrivial magnetic field at t = 0. For the case a = M, the initial magnetic field components are distributed as shown in previous Figure 4.25. However, the coupling between scalar and electromagnetic fields now gives rise to a nontrivial four-current distribution at initial time. The nonlinear dynamics are C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 126 t=53.98 -1.00e-04 1.00e-04 Figure 4.39: Angular momentum density for electromagnetic perturbations at t = 53.98 in a = 0, M = 5 Kerr spacetime. The density distribution has significantly smaller gradients than in the case a = M The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown, the computational domain covers the range 5 < r < 50 and 0 < 6 < TT, and the corners of the domain are labelled with their (r, 8) ordinate values. The amplitude and colour gradient are again scaled linearly. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 127 Figure 4.40: Integrated angular momentum density for electromagnetic perturbations in M = 5 Kerr spacetime. Plotted is 5, the total angular momentum contained within 5 < t < 50, 0 < 9 < TT and 0 < <j> < 2ir, as a function of coordinate time t. The dashed red line represents data for a/M = 0, the dotted red line data for a/M = 0.8, and the dashed black line data for a/M = 1. Clearly, perturbation angular momentum is highly variable in rotating black hole spacetimes, but tends towards S = 0 at late time. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 128 Figure 4.41: Integrated angular momentum density for electromagnetic perturbations in M = 5 Kerr spacetime. Plotted is S, the total angular momentum contained within 5 < r < 50, 0 < 6 < 7r and 0 < 0 < 27T, as a function of coordinate time t. The dashed red line represents data for a/M — 0, the dotted red line data for a/M = 0.8, the dashed blue line data for a/M = 0.98, the dotted blue line (nearly coincident with the dashed black line) data for a/M = 0.998, and the dashed black line data for a/M — 1. The differences between a = 0 and a > 0 data are significant. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 129 rather detailed as discussed below. In addition to orbiting resonances, I find the electromagnetically coupled fields display both charge separation and dynamo-like behaviour. Starting with e = 0.6, a = M = 5 initial data, the total normalized energy within the compu-tational domain is displayed as a sequence of steps starting at t = 0 in Figure 4.42 with a final displayed time at t = 77.11 as shown in Figure 4.43. Gravitational interaction, in the form of orbiting resonances are again observed, and the resonance period is again w 70£, corresponding to an orbital frequency more than twice that the unstable circular photon orbit at r = 3M (cf. equa-tion (4.164) and related discussion). Furthermore, the attenuation between the first two successive orbits is again ~ 10_1. Mode trapping is not apparent and neither is superradiant scattering15. Figure 4.44 shows the total normalized energy within the computational domain for 0 < t < 150 for a = 0,0.8M and a = M spacetimes.16 It is there apparent how black hole angular momentum decreases the decay rate of electromagnetically coupled perturbations. Furthermore, at late time, the decay of perturbations in the a = M spacetime can be approximated as the power law log(T) ~ -kt , (4.172) with decay constant k w 1 x 10-2, similar to the cases of s = 1 and massless s = 0 perturbations. • The evolution of energy densities are otherwise similar in the a = 0,0.8M and a = M spacetimes. 1 5Again, while this test does not compensate for flux lost through the outer boundary r = 50, some superradiant amplification could still be present 1 6While the evolution was extended to t 200, a very slowly growing instability is manifest some time after t = 175 at r = 50 for cases a = 0.8M and a = M. I therefore limit examination to the shorter time interval. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 1 3 0 1.00e-07 1.00e-05 I 1.00e-07 1.0()e-05 1.00e-07 1.00e-05 i 1.00e-07 1.00e-05 Figure 4.42: T h e glory effect and orb i t ing resonances for a = M = 5, m$ = 0.0, e = 0.6 coupled per turbat ions. T h e figure displays the to ta l normal ized energy density i n a sequence of steps. T h e directions of coordinate vectors ra ~ (0 ,1 ,0 ,0 ) and 9a ~ (0,0,1,0) are shown i n the first frame. T h e computa t iona l doma in covers the range 5 < r < 50 and 0 < 0 < TV, and the corners of the domain are label led w i t h their (r, 6) ordinate values. T h e ampl i tude is scaled l inearly, whi le the scale of the colour gradient is logar i thmic . C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 131 Figure 4.43: The glory effect and orbiting resonances for a = M = 5, m$ = 0.0, e = 0.6 coupled perturbations at late time. The figure displays the total normalized energy density at a final step of the evolution. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown, the computational domain covers the range 5 < r < 50 and 0 < 9 < IT, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude is again scaled linearly and the colour gradient logarithmically. According to equations (4.109) through (4.113), the total conserved energy density can be decomposed into two terms independent of couping parameter e plus one dependent on e. The e-independent terms are the energy density of purely scalar contributions, pr(*) > a n d the energy density of purely electromagnetic field contributions, PT(A) • The e-dependent term, pr(*A) > couples the scalar and electromagnetic fields. While the total PT = PT(*) + PT(A) + PT($A) (4.173) is a conserved quantity, the individual components are not. In fact, the coupling term PT($A) isn't even strictly positive in flat spacetime. Figures 4.45. 4.46, and 4.47 respectively depict sequences of evolution of PT($) > PT(A) and PT($A) • Observe how the purely scalar contribution is the dominant contributor for the displayed phase of evolution. Also note how purely electromagnetic contributions increase rapidly at very early time and then remain fairly large in amplitude for the remaining evolution. Furthermore, the coupling term PT($A) is small in amplitude throughout evolution, except along 9 = 0 and 9 = TT at the time when perturbations first scatter off the steep gravitational potential near the black hole event horizon. Figure 4.48 displays the total normalized energy contained in the computational domain, T , and also the three components T ($ ) , T ( A ) , and T ( $ A ) contributing to the total. It is there apparent that T(4>) is the dominant contributor until t ss 90, after which T(A) dominates. It is also clear that T ( $ A ) is sometimes positive, and at other times negative. Finally, there appears to be a strong correlation between T($) and T(4>A) during the period 20 < t < 40, corresponding to the first perturbation scattering from the steep CHAPTER 4. CHARGED SCALAR ACCRETION 132 — i 1 1 1 j 1 1 1 1 p • I , -0.5 to o - 1 h -1 .5 ~i 1 r V v x 50 100 \ 15C F i g u r e 4.44: Integrated normalized energy density for e = 0.6 coupled massless perturbations in M = 5 Kerr spacetime. Plotted is log(T), the logarithm of the total energy contained within 5 < r < 50, 0 < 6 < ir and 0 < <p fs 27r, as a function of coordinate time t. The dashed red line represents data for a/M = 0, the dotted red line data for a/M = 0.8, and the dashed black line data for a/M = 1. Clearly, black hole angular momentum decreases the overall decay rate of electromagnetic perturbations. Any decrease in time of the energy results from the flux across surfaces r = 5 and r = 50, and thus out of the computational domain. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 133 gravitational potential. The correlation between the electromagnetic component T(A) and T($A) appears to be rather weak during this event. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 134 Figure 4.45: Evolution of the scalar contribution to the total energy for a = M = 5, m<j> = 0.0, e = 0.6 coupled perturbations. The figure displays the normalized scalar contri-bution to the energy density in a sequence of steps. The directions of coordinate vectors ra ~ (0,1,0,0) and 6a ~ (0,0,1,0) are shown in the first frame. The com-putational domain covers the range 5 < r < 50 and 0 < 8 < n, and the corners of the domain are labelled with their (r, 6) ordinate values. The amplitude is scaled linearly, while the scale of the colour gradient is logarithmic. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 135 1=38.55 t=57.83 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ B l.()0e-07 1.00e-05 1.00e-07 1.0()e-()5 F i g u r e 4.46: Evolution of the electromagnetic contribution to the total energy for a = M = 5, m<j> = 0.0, e = 0.6 coupled perturbations. The figure displays the normalized electromagnetic contribution to the energy density in a sequence of steps. The directions of coordinate vectors ra ~ (0,1,0,0) and 6a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 8 < ar, and the corners of the domain are labelled with their (r, 6) ordinate values. The amplitude is scaled linearly, while the scale of the colour gradient is logarithmic. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 136 F i g u r e 4.47: Evolution of the coupling terms contributing to the total energy for a = M = 5, m$ = 0.0, e = 0.6 perturbations. The figure displays the normalized coupling contribution to the energy in a sequence of steps. The directions of coordinate vectors r ° ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < TT, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude is scaled linearly, while the scale of the colour gradient is logarithmic. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 137 : r < S 0.5 o k V - 0 . 5 ~i 1 r ~ i 1 1 r ' W \ \ \ — \ I V \ J v _i i i i_ 50 100 ISC F i g u r e 4.48: Decomposition of integrated normalized energy density for e = 0.6 coupled massless perturbations in a = M = 5 Kerr spacetime. Plotted are the total energy and contributions to the total contained within 5 < r < 5 0 , 0 < # < 7 T and 0 < <j> < 2n, as a function of coordinate time t. The dotted black line represents the total normalized energy, the dashed black line represents the normalized scalar field contribution, the dotted red line is the normalized electromagnetic contribution and the dashed red line the normalized contribution from electromagnetic coupling terms. I again analyze the magnetic and electric field dynamics through analysis of BaBa and EaEa. Figure 4.49 displays BaBa, while Figure 4.50 shows EaEa. As in the case of pure electromagnetic perturbations, the magnetic field norm is substantially smaller than that of the electric field in the equatorial plane (8 = TT/2) and along the axis of symmetry (8 = 0 and 8 = IT). The integrals of these norms are displayed in Figure 4.51. After undergoing substantial change at early time, these integrated amplitudes become comparable and very roughly decay according to l o g ( £ 2 ) ~ -kt , (4.174) and l o g ( £ 2 ) ~ -kt , (4.175) where k w 1 x 10~ 2 similar to the case of pure electromagnetic perturbations. Finally, the evolution of BaBa and EaEa are qualitatively very similar in both a = 0 and a = M spacetimes. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 138 F i g u r e 4.49: B A B A for a = M = 5, e = 0.6 coupled perturbations. The figure displays the square of the magnetic field in a sequence of steps. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 0 < IT, and the corners of the domain are labelled with their (r, 6) ordinate values. The amplitude is scaled linearly, while the scale of the colour gradient is logarithmic. The magnetic field amplitude is negligible outside the ergoregion in the plane 9 = TT/2. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 139 1.00e-05 1.00e-03 1 1.00e-05 1.00e-03 1=77.11 l.()0e-05 1.00e-03 F i g u r e 4.50: EaEa for a = M = 5, e = 0.6 coupled perturbations. The figure displays the square of the electric field in a sequence of steps. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude is scaled linearly, while the scale of the colour gradient is logarithmic. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 140 i 1 1 1 1 rr 1 1 1 1 1 r 3.5 O m 2.5 1.5 V - . I i i i i I i i i i_ 50 100 15C F i g u r e 4.51: Integrated squares of electric and magnetic fields in a = M = 5 Kerr spacetime for e = 0.6 coupled perturbations. Plotted are log(J32) and log(E2), the logarithm of the integrated squares contained within 5 < r < 50, 0 < 9 < IT and 0 < <j> < 2ir, as a function of coordinate time t. The dashed black line represents log(B 2 ) , and the dotted black line log(.E 2). Initially, the field amplitudes undergo substantial change, but the decay of integrated electric and magnetic amplitudes become comparable at later time. Before analyzing the physics of charge separation and dynamo-like behaviour unique to electro-magnetically coupled s = 0 and s = 1 perturbations I now include discussion of angular momentum density evolution. I display the conserved angular momentum density in the sequence of steps shown in Figure 4.52. Similar to the case of purely electromagnetic perturbations, the chosen initial data is for zero net the angular momentum at all point in the computational domain. However, the' distribution becomes nontrivial in short order as shown. Furthermore, as seen in the case of un-coupled electromagnetic perturbations, the distribution becomes sharply peaked in the ergoregion at intermediate time before evolving to a somewhat less dynamic scenario. Also observe that the local angular momentum density amplitudes are enhanced by an order of magnitude over those of the uncoupled electromagnetic perturbations. This effect is clearly due to the addition of charge coupling. Furthermore, in Figure 4.53, I display the same evolution sequence in a = 0 spacetime. There it is clear that the black hole rotation amplifies the angular momentum density amplitude. This is again similar to the situation observed for uncoupled electromagnetic perturbations. To conclude, I display the integrated angular momentum density for a = 0 , 0 . 8 M and a = M space-times in Figure 4.54. It is there evident that the total angular momentum is especially variable for a > 0.8M spacetimes during the evolutionary period 40 < t < 70. C H A P T E R 4 . C H A R G E D S C A L A R A C C R E T I O N 141 -1.00e-03 1.00e-03 F i g u r e 4.52: Angular momentum density for a = M = 5, e = 0.6 coupled perturbations in a sequence of steps. The directions of coordinate vectors r ° ~ (0,1,0,0) and 6a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 6 < r, and the corners of the domain are labelled with their (r, 8) ordinate values. The amplitude and colour gradient are both scaled linearly. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 142 -1.0()e-()3 1.00e-03 I -1.00e-03 1.00e-03 t=77.11 -1.00e-03 1.00e-03 Figure 4 .53 : Angular momentum density for a = 0, e = 0.6 coupled perturbations in a sequence of steps. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < TT, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude and colour gradient are both scaled linearly. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 143 ' 1 1 1 1" 200 c o 100 I r X A ft 5 0 100 15C Figure 4 .54: Integrated angular momentum density for e = 0.6 coupled massless perturbations in M = 5 Kerr spacetime. Plotted is 5, the total angular momentum contained within 5 < r < 5 O , O < 0 < 7 r and 0 < <j> < 2n, as a function of coordinate time t. The dashed red line represents data for a/M = 0, the dotted red line data for a/M = 0.8, and the dashed black line data for a/M = 1. The total angular momentum is especially variable for o > 0.8M spacetimes during 40 < t < 70. In this subsection I have so far discussed the dynamics of energy and angular momentum for the coupled s = 0 and s = 1 perturbations on Kerr spacetime. It is fair to say that, while quantitatively different, the energy and angular momentum dynamics are in many ways similar to those of un-coupled perturbations previously discussed. Because the electromagnetically coupled perturbations carry charge, two physically unique features appear in the field dynamics—the separation of electric charge and dynamo-like behaviour of electric current. The evolution of electric charge density in a = M = 5 spacetime is displayed as a sequence in Figure 4.55. There it is seen how an everywhere zero net charge density quickly evolves into regions of separated net positive and negative charge. Note that this process is not particular to rotating black hole spacetimes. For comparison, I display the analogous sequence for a = 0, M = 5 spacetime in Figure 4.56. The early dynamics of PQ are nearly identical for a = M and a = 0, and only significantly deviate in magnitude after the initial scattering from the steep gravitational potential. For further comparison I plot the total charge contained in the computational domain, Q, for a = 0,0.8M and a = M spacetimes in Figure 4.57, where it is observed how closely the total charges agree at early time for all black hole angular momenta. In addition, observe how the choice of initial data prefers to transfer positive charge through the event horizon and outer computational boundary for 40 < t < 50 and transfer negative charge for 70 < t < 100. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 144 F i g u r e 4.55: Charge density showing separation in the a = M = 5 spacetime for e = 0.6 pertur-bations. Both amplitude and colour gradient are scaled linearly. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < IT, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 145 t=19.28 -1.00e-03 1.00e-03 F i g u r e 4.56: Charge density showing separation in the a = 0, M = 5 spacetime for e = 0.6 perturbations. The amplitude and colour gradient are again scaled linearly. The directions of coordinate vectors ra ~ (0,1,0,0) and 6a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < TT, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 146 100 h - 1 0 0 h -200 15C Figure 4.57: Integrated charge density for e = 0.6 coupled massless perturbations in M = 5 Kerr spacetime. Plotted is Q, the total charge contained within 5 < r < 5 0 , 0 < # < 7 r and 0 < (f> < 2TT, as a function of coordinate time t. The dashed red line represents data for a/M = 0, the dotted red line data for a/M = 0.8, and the dashed black line data for a/M = 1. The choice of initial data preferentially transfers positive charge through the event horizon and outer computational boundary for 40 < t < 50 and negative charge for 70 < t < 100. As the coupled perturbations carrying charge and momentum display interesting dynamics, it is also surmised the electric current should reveal behaviour worth studying. While the evolution of pQ encapsulates information regarding current components Jr and Jg, it doesn't completely reveal the behaviour of the axial current component J^. For this reason I plot an evolutionary sequence for J<f, in a = M = 5 Kerr spacetime in Figure 4.58. Most generally, I consider the dynamo-like behaviour to be the dynamic interplay of electric current and electromagnetic field, converting between kinetic and non-kinetic contributions of the system's total energy. As there is no outside mechanism to constrain the axial current of the coupled perturbations, they must necessarily display dynamo-like behaviour. However, observe how undergoes directional reversal throughout evolution. The choice of initial conditions clearly disallows a single persistent current loop. 1 7 Instead, the loop oscillates in a nontrivial manner lacking a regular period of oscillation. 1 7 T h i s behaviour can be simply explained as a manifestation of Faraday's Law. Recall the differential form of Faraday's law in flat spacetime can be expressed dB v x £ + — = 0 , (4.176) at which shows that a time dependent magnetic field induces an electric field. The electric field, then, will induce a current (or change in current) in a direction to counteract, the change in magnetic field. In the case examined here, C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 147 The same is true for evolution in a = 0, M = 5 spacetime as shown in Figure 4.59. This is substantiated in part by Figures 4.60 and 4.61 which respectively display the global maxima and minima of for a = 0 and a = M spacetimes. While a = 0 perturbations have larger global maxima at early time, these maxima quickly decay and the a = M becomes dominant. Especially note how the maximum, max(J^), peaks during the period 20 < t < 50. Meanwhile, the global minima, min( J^), evolve with comparable amplitude for early time until a = M perturbations begin dominating when 40 < t < 80. The global maxima and minima were also analyzed for a = 0 . 8 M (not displayed) and were found to be qualitatively similar to those of the a = M spacetime. It can therefore be concluded that while black hole angular momentum initially suppresses the evolution of Jcp, it later acts to amplify this axial current density. there is an axial current loop at initial time as shown in Figure 4.58. There is also a magnetic field at initial time, whose square is shown in Figure 4.49. Note how this initial magnetic field distribution—the same as that studied in section 4.3.3—has a nonzero time derivative. As this magnetic field changes in time it induces an electric field as supported by results shown in Figures 4.50 and 4.51. Of course, the resulting induced change in current by the electric field must be in the direction opposing the change in magnetic flux. Now consider the scenario where a stationary charged disk generates a current loop around a black hole and persists indefinitely unless acted on by an external agent. If an electromagnetic wave passes through part of the loop, a current fluctuation will then be induced in order to oppose the change in magnetic flux enclosed by the loop. And so it is by this mechanism that one could imagine electromagnetic radiation destabilizing a stationary charged accretion flow in the spacetime of a black hole. Note there was no appeal to a property unique to black holes in this destabilization mechanism, and to the argument should also hold for other gravitating celestial objects like stars. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 148 F i g u r e 4.58: Displayed is the axial current, J ^ , oscillation in a = M = 5 Kerr spacetime for e = 0.6 coupled perturbations. Both amplitude and colour gradient are scaled linearly. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < ir, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 149 Figure 4.59: Displayed is the axial current, J ^ , oscillating in a = 0, M = 5 Kerr spacetime for e = 0.6 coupled perturbations. Both amplitude and colour gradient are scaled linearly. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < w, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 150 0.3 \i x 15C F i g u r e 4.60: Global maxima of in the computational domain, max( J^), as a function of t for e = 0.6 coupled massless perturbations in M = 5 Kerr spacetime. The dotted black line represents data for a/M — 0 and the dashed black line data for a/M = 1. To gain additional insight on the nature of charge and current in this model, consider the ratios pc)/e and Ja/e. From equation (4.108), the first of the ratios is — = v 2 f g ' r t \ (**n* - * n j ) , (4.177) e \9trJ4><t> - 9tcj>Jt<t>9rr) and may be interpreted as a conserved scalar field density. However, recall that charge density can be both positive and negative. The proper interpretation of PQ/C is therefore the density of the scalar field carrying charge with sign e/|e|. Negative values of pcj/e simply indicate the scalar field is carrying charge of opposite sign at those locations. Furthermore, comparing PQ of (4.108) to Ps($A) from (4.116) obtains 2 PQ = —7-PS(.4>A) , (4.178) revealing the connection between charge density and the angular momentum contribution from the coupling of scalar and electromagnetic fields. Meanwhile, (4.93) and (4.94) obtain the quantities — = -i(<E>*$r - $ $ ! ) -2e$$*Ar (4.179) e and f — = -i ($*$e - - 2e$$*Ae , (4.180) e , C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 151 -0 .5 c 6 F i g u r e 4.61: Global minima of in the computational domain, min(J^,), as a function of t for e = 0.6 coupled massless perturbations in M = 5 Kerr spacetime. The dotted black line represents data for a/M = 0 and the dashed black line data for a/M = 1. CHAPTER 4. CHARGED SCALAR ACCRETION 152 which represent scalar field velocities in the r and 8 coordinate directions. Now, from (4.95), ^ = -2e$$*A^, , (4.181) e which represents a scalar field velocity in the axial direction—specifically, the axial velocity of scalar field carrying charge of sign e/|e|. From (4.56), the quantity LT$ appearing in (4.177) is n$ = n a v / = < ? ( V a $ - i e A a $ ) , (4.182) where, from (4.73), /— _ 9trU<t> ~ 9t<pft4,g rr /, 1 0 0 \ v-g = 2 • (4.183; 9tr Substituting (4.182) and (4.183) into (4.177) and taking the limit e —> 0 obtains l im ^ = -nai ( $ * V a $ - $ V a $ * ) . (4.184) e—>o e Similarly, taking the e -> 0 limits of (4.179)-(4.181) yields l im — = - i ( $ * $ r -<M>!) , (4.185) e—»o e • l im — = -i ( $ * $ e - $ $ ; ) , . (4.186) e—>0 e and l i m ^ = 0 . (4.187) e^O e Observe how, consistent with, the assumptions of axial symmetry, the scalar field velocity in the axial direction vanishes in-the limit e —> 0. Furthermore, the relationship between the Noether current of a (massless and uncharged) complex scalar field, J a = - i ( $ * V a $ - $ V a $ * ) , ' (4.188) and the quantities appearing in (4.184)-(4.187) is clear. Of course, PQ, Jr, Je and are observer dependent quantities. As far as that goes, so are E A , Ba, PT, and ps- But it is well known from both special and general relativity that densities and electromagnetic quantities are dependent on the frame in which they are measured. This discussion concludes subsection 4.3.4. 4.3.5 Massless M a x w e l l - K l e i n - G o r d o n Equa t ion Dynamics II : Compar ing Electromagnet ical ly Coup led s = 0 and s — 1 Per turbat ions W i t h Different Values of Parameter e Having examined the effects of charge coupling e = 0.6 in the massless Maxwell-Klein-Gordon system for different values of the black hole angular momentum parameter a, I now investigate how changing e alters the observed dynamics in a = M = 5 Kerr spacetime. Specifically, I investigate e = 0.0,0.3,0.6,1.2 and e = 2.4. The computational domain again spans 5 < r < 50 and 0 < 8 < n, and the initial data is specified as in the preceding subsection. For e = 0.0,0.3 and e = 0.6 I use a finite difference grid with resolution J = 801 radial points and K = 401 angular points. This provides an accuracy to within « 1% before the perturbations first cross the ergosphere and then the excision surface at r = 5. For e = 1.2 a resolution of J = 1601 radial C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 153 points and K = 801 angular points is used to provide the same level of accuracy. For e = 2.4, a resolution of at least J = 3201 radial and K = 1601 angular points is required before this order of accuracy is achieved. 1 8 The computational requirements are thus very large for e = 1.2 and e = 2.4, and so I compute the evolution just past the first excision surface crossing for these two coupling parameter values. 1 9 Finally, the calculations for 0.0 < e < 1.2 are performed in parallel across 64 processors with processor overlap regions 3 grid points wide, and the calculations for e = 2.4 are performed across 256 processors with overlap regions 3 points wide. While computational demands have restricted the length of high e evolutions, I have complete data for e = 0.0,0.3, and e = 0.6 over 0 < t < 150 for the a = M = 5 Kerr spacetime. The data for e = 0.0 are essentially the sum of results from sections 4.3.2 and 4.3.3. Furthermore, the case e = 0.6 has already been discussed in section 4.3.4. While the results for e = 0.3 should display behaviour between those of e = 0.0 and e = 0.6,1 quantify the similarities and differences through the total normalized energy, total angular momentum and charge contained in the computational domain. Figure 4.62 displays the energy, Figure 4.63 the angular momentum, and Figure 4.64 the charge. While proportionately greater energy persists over time for for e = 0.3 and e = 0.6 than for e = 0.0, there is reasonably close agreement between the cases e = 0.3 and e = 0.6. While the differences are small, it can still be argued that addition of modest charge coupling 1 8 The need for higher accuracy when e is large can be inferred from equation of motion (4.50), V a V a * - 2ie(V a*)A a - e 2 * A a A a - i e * V a A a - m $ 2 * = 0 . (4.189) For simplicity, consider solutions of this equation in a flat spacetime with one spatial dimension and one temporal dimension. Points in this spacetime are designated by the value of spatial coordinate x and the value of temporal coordinate t. Furthermore, consider solutions of equation (4.189) of the form 5> = a$ expi(fcir — LJt) , (4.190) where a$ is the amplitude, fc is the wavenumber and LJ is the solution frequency. When e = m$ = 0, equation (4.189) becomes V ° V „ $ = 0. (4.191) The dispersion relation obtained by substituting (4.190) into (4.191) is simply u? = k2 . (4.192) When e = 0 and ^ 0, equation (4.189) becomes V a V a * - m $ 2 * = 0 . (4.193) Assuming (4.193) has solutions of the form (4.190), the dispersion relation becomes u2 = k2 +m% . (4.194) This shows large values for m$ result in correspondingly large values of LO for a given wavenumber k. Numerically, this implies a need for higher resolution to obtain a solution with the same accuracy as in the case m$ = 0. Similarly, the need for high resolution can be inferred when m$ = 0 and e ^ 0 is large. Using the Lorentz gauge condition V a A ° = 0 and assuming m$ = 0, equation (4.189) becomes V a V a * - 2ie(V a*)A a - e 2 A a A a $ = 0 . (4.195) Solutions (4.190) then obtain a dispersion relation LO2 + 2eAtw = k2 + 2eArk + e2AaAa . (4.196) Comparing with (4.193) and (4.194), the quantity ( e 2 A a A a ) 1 ^ 2 plays a role similar to that of a t and r dependent scalar field mass. Furthermore, it is clear from (4.196) that increasing e correspondingly increases the solution frequency LJ for a given wavenumber fc. This, in turn, suggests the need for higher resolution to numerically obtain the same solution accuracy as in the case e = 0. 19Finally, while adaptive mesh refinement is not clearly beneficial for the evolution of e < 0.6 perturbations from the specified initial conditions on the given computational domain (as discussed in preceding section 4.2.6), cases with sufficiently large coupling may benefit from the technique. Specifically, my results demonstrate how (at least for the early phase of evolution) a relatively high resolution is required in a proportionately small region of the computational domain for cases such as e = 2.4. The utility of adaptive mesh refinement was, however, not investigated in any more detail. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 154 0 , W . -0.5 \ \ o - 1 -1.5 0 50 100 150 t F i g u r e 4.62: Total normalized energy for the massless electromagnetically coupled perturbations in a = M = 5 Kerr spacetime. The dashed red line represents data for e = 0.0, the dotted red line data for e = 0.3 and the dashed black line data for e = 0.6. The decay rates of charged perturbations are smaller than for the case e = 0.0. Any decrease in time of this quantity results from the energy flux across surfaces r = 5 and r = 50, and thus out of the computational domain. decreases the overall decay rate of total perturbation energy. 2 0 Meanwhile, note that only the s = 1 component of perturbations contributes to the total angular momentum of the e = 0.0 data. However, an additional coupling term contributes with the introduction of nonzero e. It is clear from Figure 4.63 how the addition of electromagnetic coupling increases the amplitude of angular momentum remaining in the computational domain as evolution progresses. Furthermore, the maximum amplitude of this total angular momentum increases with parameter e. Finally, while the case e = 0.0 has zero charge density, I still display the e = 0.0 data along with those of e = 0.3 and e = 0.6 in Figure 4.64. The charge remaining in the computational domain varies in time, and the difference between the cases is more significantly affected by parameter e than by black hole angular momentum a as discussed in section 4.3.4. Also note that it cannot be determined in advance whether net positive of negative charge will preferentially leave the domain. Rather, the details must be determined through numerical evolution. Overall, the effect of changing e is notable over the period of evolution. 2 0It is not possible to definitively conclude what trends in decay rate will arise with increased electromagnetic coupling parameter without further computation. From the data of Figure 4.62 it is clear that energy decay rates are not ordered according to the value of e. A similar trend was observed in the massive uncharged perturbation decay rates of Figure 4.19, where the value m<j, = 0.3 gave rise to mode trapping, and m$ > 1.2 resulted in rapid energy decay by infall onto the black hole. It is possible that the behaviour displayed in Figure 4.62 is the beginning of a similar trend with the electromagnetic coupling parameter. CHAPTER 4. CHARGED SCALAR ACCRETION 155 200 h 100 ji J f i l l / ! 11 / ! \l A / H / A srf\ .\S /\ * \ - V d*-50 100 150 F i g u r e 4.63: Total angular momentum for the massless electromagnetically coupled perturbations in a = M = 5 Kerr spacetime. The dashed red line represents data for e = 0.0, the dotted red line data for e = 0.3 and the dashed black line data for e = 0.6. The total angular momenta are more dynamic for charged perturbations than for e = 0.0. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 156 200 100 h a? - 1 0 0 h - 2 0 0 1 'l V I i i A. l / \ / _ l I I I_ _1 I I 1_ 50 100 150 F i g u r e 4.64: Total charge for the massless electromagnetically coupled perturbations in a = M = 5 Kerr spacetime. The dashed red line represents data for e = 0.0, the dotted red line data for e = 0.3 and the dashed black line data for e = 0.6. CHAPTER 4. CHARGED SCALAR ACCRETION 157 Investigating the evolution of high e perturbations it is further apparent that dynamics of large e fields are much richer than those with small e. Figure 4.65 displays the total normalized energy density of e = 2.4 in a sequence of steps from t = 0 through t = 16.06. 2 1 To emphasize, I provide an enlarged view of time step t = 16.06 in Figure 4.66. Comparing with the results of sections 4.3.2, 4.3.3 and 4.3.4 it is clear how the morphology changes and the density gradient increases with charge coupling. There is also a tendency is for density amplitudes to grow with increasing e. Furthermore, notice how early evolution of the e = 2.4 data would suggest perturbations with large coupling are likely to persist in the computational.domain for an extended period of time. That is, the rate at which large e perturbations are likely to decay—either by passing through the r = 5 excision surface or escaping through the r = 50 outer boundary—is probably smaller than for small or vanishing e. This, in turn, suggests the possibility of an effect similar to mode trapping for massless electromagnetically'coupled perturbations. However, the origins of such a phenomenon would have to be investigated by evolving the fields in spacetimes with differing black hole mass and angular momentum. The rich dynamics are also manifest in quantities like the angular momentum density of Figures 4.67 and 4.68, and the charge density of Figures 4.69 and 4.70. Again, comparing with previous results it is clear how the dynamics are enriched as both the density gradients and amplitudes increase with e. 2 1 The evolutionary dynamics for e = 0.3 lie between those of e = 0.0 and e = 0.6, and the dynamics of e = 1.2 lie between those of e = 0.6 and e = 2.4. Neither e = 0.3 nor e = 1.2 are displayed. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 158 1.00e-07 1.00e-05 II 1.00e-07 l.()0e-05 t=12.85 1 t=16.06 1.00e-07 1.00e-05 I 1.00e-07 1.00e-05 F i g u r e 4.65: Displayed is the total normalized energy density in a = M = 5 Kerr spacetime for e = 2.4 massless perturbations for the early phase of evolution. Amplitude is scaled linearly and the colour gradient logarithmically. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 6 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 159 F i g u r e 4.66: Displayed is a closer view of the total normalized energy density at t = 16.06 in a = M = 5 Kerr spacetime for e = 2.4 massless perturbations. Amplitude is again scaled linearly and the colour gradient logarithmically. The computational domain and coordinate vectors would be the same as those shown in Figure 4.65. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 160 -l.()0e-03 1.00e-03 I -1.00e-03 1.00e-03 t= 12.85 I t= 16.06 -1.00e-03 1.00e-03 J -1.00e-03 1.00e-03 F i g u r e 4.67: Displayed is the total angular momentum density in a = M = 5 Kerr spacetime for e = 2.4 massless perturbations for the early phase of evolution. Both amplitude and colour gradient are scaled linearly. The directions of coordinate vectors ra ~ (0,1,0,0) and 9 a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < ar, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 161 Figure 4.68: Displayed is a closer view of the total angular momentum density at t = 16.06 in a = M = 5 Kerr spacetime for e = 2.4 massless perturbations. Amplitude and colour gradient are both scaled linearly. The computational domain and coordinate vectors would be the same as those shown in Figure 4.67. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 162 -1.00e-03 1.00e-03 1 -1.00e-03 1.00e-03 F i g u r e 4.69: Displayed is the charge density in a = M = 5 Kerr spacetime for e = 2.4 massless perturbations for the early phase of evolution. Both amplitude and colour gradient are scaled linearly. The directions of coordinate vectors r° ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < ar, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 163 F i g u r e 4.70: Displayed is a closer view of the charge density at t = 16.06 in a = M = 5 Kerr spacetime for e = 2.4 massless perturbations. Amplitude and colour gradient are again scaled linearly, and the computational domain and coordinate vectors would be the same as those shown in Figure 4.69. 4.3.6 Massive Charged Maxwell-Klein-Gordon Equation Dynamics: Collimation, Orbiting Resonances and Decay of m<j> = 0.3, e = 0.6 Electromagnetically Coupled s = 0 and s = 1 Perturbations Now that I have examined massless coupled and uncoupled s = 0 and s = 1 perturbations in sufficient detail, I am in a position to examine the physics of massive coupled perturbations in M = 5 Kerr spacetime. To summarize preceding results, massless s = 0 perturbations display the effect of orbiting resonance, while massive s = 0 perturbations display mode trapping. Furthermore, for the value m$ = 0.3 which displays the strongest trapping, the s = 0 perturbations appear to display preferential scattering along the axis of black hole rotation. While s = 1 perturbations also exhibit orbiting resonance, they do not display either trapping or preferential scattering in the spacetime. However, there is evidence in the form of total energy increase and angular momentum flux from the black hole to support superradiant scattering of the s = 1 perturbations. Finally, while massless electromagnetically coupled s = 0 and s = 1 perturbations display orbiting resonance, charge separation, and dynamo-like behaviour, neither trapping, preferential scattering, nor superradiant amplification are in evidence from analysis of the choice e = 0.6. I now demonstrate how a choice of nonzero mass and charge coupling parameter together lead to new dynamics. The natural choice for mass parameter is m$ = 0.3—since that is the value at which mode trapping is greatest (for the given initial conditions), it is hypothesized the choice wil l provide a substantial period over which the perturbations will persist and interact electromagnetically in the black hole spacetime. The choice for charge coupling parameter is not as obvious. While values as large as e = 2.4 may be desired, the resulting intricate dynamics would require very high resolution and correspondingly large computational resources. However, the intermediate choice e = 0.6 C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 164 has demonstrated sufficient coupling for interesting electromagnetic effects while requiring modest finite difference grid resolution. I therefore compute the evolution for m$ = 0 . 3 perturbations with e = 0.6 on M = 5 Kerr spacetimes of rotation a = 0 ,0 .8M and o = M. The computational domain spans the coordinate range 5 < r < 50, 0 < 9 < IT, and the period 0 < t < 200. The data is computed in parallel across 64 processors with overlap regions 3 grid points wide. The finite difference grid itself has J = 801 radial and K = 401 angular points, providing sufficient resolution to maintain an accuracy of ss 1% until perturbations first cross the ergosphere. Examining first the total normalized energy density of Figures 4.71 and 4.72, it is apparent that evolution in the a = M = 5 spacetime displays behaviour different from that seen in the other choices for m$ and e. Except for the case m<j> = 0.3, there was previously no evidence for preferential scattering along the axis of black hole rotation. In the present case, addition of charge coupling results in somewhat different focusing of the .energy density and greater collimation. To understand contributions to the observed dynamics, I also display the normalized scalar energy density, pr($) (Figures 4.73 and 4.74), and the electromagnetic contribution, PT(A) (Figures 4.75 and 4.76). For completeness, I also display the corresponding evolutionary sequence for PT($A) (see Figures 4.77 and 4.78). From PT(*) and PT{A) it is apparent how the scalar component accounts for the greatest contribution to the collimated energy density. I then conclude that while electromagnetic coupling has an effect on the collimation of energy, it does not seem to be more significant in this regard than the effect of massive mode scattering from the black hole potential. One case where electromagnetism could have a profound influence is in the collimation of large m$ scalar fields that would otherwise not reflect off the spacetime potential barrier. This hypothesis wil l have be be tested with future computation. Finally, in Figure 4.79 I display the total normalized energy in the computational domain, T , along with individual contributions T ($ ) , T(A) and T(&A). It is there obvious that T($ ) is the dominant component throughout evolution. T(A) is initially very small, but quickly increases by orders of magnitude at early time. During this phase both T($ ) and T($A) decrease sharply. Afterwards, T(A) remains large, and T ( $ A ) increases at the expense of T($ ) . After t = 20 the changes in each of the components are less dramatic. Again, T ( $ A ) takes on both positive and negative values during evolution. Before proceeding, note also the presence of orbiting resonances as displayed in the data, and notice how PT(A) is the principal contributor. As in the case of massless e = 0.6 perturbations, the period between the first two orbits is ss 70i and the attenuation is ~ 1 0 _ 1 . Again, the orbital period here corresponds to a frequency more than twice that of unstable circular photon orbits at r = 3 M (cf. equation (4.164) and related discussion). C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 165 1.00e-07 1.00e-05 i 1.00e-07 1.00e-05 t=38.55 f t=48.19 1.00e-07 1.00e-05 I 1.00e-07 1.00e-05 F i g u r e 4.71: Displayed is the total normalized energy density in a = M = 5 Kerr spacetime for e = 0.6, m$ = 0.3 perturbations. The amplitude is scaled linearly and the colour gradient is scaled logarithmically. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 166 t=77.11 t= 154.22 1.00e-07 1.00e-05 F i g u r e 4.72: Displayed is the total normalized energy density at t = 77.11 and t = 154.22 in a = M = 5 Kerr spacetime for e = 0.6, m$ = 0.3 perturbations. The ampli-tude is scaled linearly and the colour gradient is scaled logarithmically. The di-rections of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < 7r, and the corners of the domain are labelled with their (r, 9) ordi-nate values. A n M P E G animation showing evolution of normalized energy den-sity for e = 0.6, m<j> = 0.3 perturbations over the period 0 < t < 154 is avail-able at http://$WWW/People/petryk/animations/6.mpeg where $WWW is currently l a p l a c e . p h y s i c s . u b c . c a . C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N t=0.00 1.00e-07 1.00e-05 I 1.00e-07 1.00e-05 t=38.55 i t=48.19 1.00e-07 1.00e-05 1 1.00e-07 1.00e-05 F i g u r e 4.73: Displayed is the normalized scalar energy density in a = M = 5 Kerr spacetime for e = 0.6, m$ = 0.3 perturbations. The amplitude is scaled linearly and the colour gradient is scaled logarithmically. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 168 t=77.11 t= 154.22 1.00e-07 1.00e-05 Figure 4.74: Displayed is the normalized scalar energy density at t = 77.11 and t = 154.22 in a = M = 5 Kerr spacetime for e = 0.6, = 0.3 perturbations. The amplitude is scaled linearly and the colour gradient is scaled logarithmically. The directions of coordinate vectors ra ~ (0,1,0,0) and 9A ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 6 < it, and the corners of the domain are labelled with their (r, 6) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 169 F i g u r e 4.75: Displayed is the normalized electromagnetic energy density in a = M = 5 Kerr spacetime for e = 0.6, m$ =0.3 perturbations. The amplitude is scaled linearly and the colour gradient is scaled logai ithmically. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 170 t=77.11 t=154.22 1.00e-07 1.00e-05 F i g u r e 4.76: Displayed is the normalized electromagnetic energy density at t = 77.11 and t = 154.22 in a = M = 5 Kerr spacetime for e = 0.6, m$ = 0.3 perturbations. The amplitude is scaled linearly and the colour gradient is scaled logarithmically. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 0 < 7r, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 171 -1.00e-05 l.OOc-05 I -1.00e-05 1.00e-05 1=38.55 § i . 4S.19 -1.00e-05 1.00c-05 1 -1.00e-05 1.00c-05 Figure 4.77: Displayed is the normalized coupling energy density in a = M = 5 Kerr spacetime for e = 0.6, = 0.3 perturbations. The amplitude is scaled linearly and the colour gradient is scaled logarithmically. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 6 < w, and the corners of the domain are labelled with their (r, 0) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 172 To determine how much effect black hole rotation has on collimation I also evolve the m$ =0.3, e = 0.6 perturbations in M = 5 Kerr spacetimes with a = 0 and a = 0 .8M. The results are nearly identical in all respects to the case a = M. I simply display the total normalized energy density at t = 77.11 for a = 0 in Figure 4.80. To further quantify similarities I plot the total normalized energy, T , contained in the computational domain for 0 < t < 200 for a = 0 , 0 . 8 M and a = M in Figure 4.81. It is there seen that black hole angular momentum has a rather small effect in the evolution of the coupled massive perturbations. However, while the differences are small, it can still be seen that black hole rotation decreases the perturbation energy decay rate at intermediate and, late time. At late time, this decay rate can be approximated with the power law log(T) ~ -kt . (4.197) with the decay constant k « 6 x 10~ 3 . The proximity of this value to w 5 x 10~ 3 measured for uncoupled m$ = 0.3 perturbations in a = M = 5 Kerr spacetime further supports the conclusion that massive scalar effects have a dominant influence in the dynamics. Detailed analysis of the remaining conserved quantities (angular momentum and charge) and axial current component are not especially illuminating since their behaviour can be largely deduced from the results of preceding sections. It is almost sufficient to state they evolve in a nontrivial and morphologically unique manner for e = 0.6, m$ = 0.3 perturbations. For completeness, I provide just a few steps of each evolution. Figure 4.82 depicts the angular momentum density at t = 0,77.11 and t = 154.22 for black hole parameter values a =. M — 5. As in the case of massless electromagnetically coupled perturbations, the initial angular momentum density is zero everywhere, but quickly evolves in a rather detailed manner. The complexity of its dynamics lies somewhere between those of massless perturbations with e = 0.6 and those with e = 2.4 (cf. Figures 4.52, 4.67 and 4.68). At both intermediate and late time there is noticeable correlation between the angular momentum and orbiting resonances seen in the energy density distributions of Figure 4.72. To analyze the effects of black hole rotation I have also evolved perturbations on the a = 0 and a = 0 .8M spacetimes. As in the case of massless e = 0.6 perturbations, comparison reveals the black hole rotation has a noticeable effect on the evolution. Figure 4.83 shows the angular momentum density at t = 77.11 in a = 0, M = 5 spacetime. Comparing with Figure 4.82, it is evident that black hole rotation dramatically amplifies the density and its gradients. This is especially true in the ergoregion and surrounding spacetime at intermediate times near t = 77.11. For further analysis I display the total integrated angular momentum in Figure 4.84 for the period 0 < t < 200 for a = 0 ,0 .8M and a = M Kerr spacetimes. Also similar to the massless e = 0.6 solutions is the high variability of total angular momentum over an intermediate phase of the evolution (cf. Figure 4.54). In the present case, this variability peaks during 50 < t < 80 when the electromagnetic component is largely concentrated in the ergoregion (cf. Figure 4.76). Thus, the total angular momentum is strongly affected by black hole angular momentum, predominantly associated with the electromagnetic perturbations, and not dramatically altered by simply increasing m$ from zero to m$ = 0.3. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 173 -1.00e-05 1.00e-05 •1.00e-05 1.00e-05 F i g u r e 4.78: Displayed is the normalized coupling energy density a t £ = 77.11ina = M = 5 Kerr spacetime for e = 0.6, m$ = 0.3 perturbations. The amplitude is scaled linearly and the colour gradient is scaled logarithmically. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 174 ~i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r < 0.5 \ \ A \ _l I I I L 5 0 100 t 150 200 Figure 4.79: Decomposition of integrated normalized energy density for m$ = 0.3, e = 0.6 per-turbations in a = M = 5 Kerr spacetime. Plotted are the total energy and contri-butions to the total contained within 5 < r < 5 O , O < 0 < 7 r and 0 < ^ < 27T, as a function of coordinate time t. The dotted black line represents the total normalized energy, the dashed black line represents the normalized scalar field contribution, the dotted red line is the normalized electromagnetic contribution and the dashed red line the normalized contribution from coupling terms. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 175 t=77.11 1.00e-07 1.00e-05 F i g u r e 4.80: Displayed is the normalized total energy density at t = 77.11 in a = 0, M = 5 Kerr spacetime for e = 0.6, m<j> = 0.3 perturbations. Observe how closely the a = 0 data matches that of a = M shown in Figure 4.72. The amplitude is scaled linearly and the colour gradient is scaled logarithmically. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 176 -0.5 so o - 1 . 5 200 F i g u r e 4.81: Integrated normalized energy density for m$ = 0.3, e = 0.6 perturbations in M = 5 Kerr spacetime. Plotted is log(T), the logarithm of the total energy contained within 5 < r < 5 0 , 0 < # < 7 r and 0 < <j> < 2TT, as a function of coordinate time t. The dashed red line represents data for a/M = 0, the dotted red line data for a/M = 0.8, and the dashed black line data for a/M = 1. Observe how black hole angular momentum decreases the overall decay rate of electromagnetic perturbations. Again, the decrease of energy over time results from the flux across surfaces r = 5 and r = 50, and thus out of the computational domain. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 177 -1.00e-03 l.OOc-03 t= 154.22 -1.00c-03 1.00c-03 F i g u r e 4.82: Angular momentum density at t = 0,77.11,154.22 for e = 0.6, m<s> = 0.3 per-turbations in a = M = 5 Kerr spacetime. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 6 < TT, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude and colour gradient are both scaled linearly. A n M P E G animation showing evolution over 0 < t < 154 is available at http://$WWW/People/petryk/animations/7.mpeg where $WWW is cur-rently laplace.physics.ubc.ca. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 178 F i g u r e 4.83: Angular momentum density at t = 77.11 for e = 0.6, m$ = 0.3 perturbations in a = 0, M = 5 Kerr spacetime. The directions of coordinate vectors r ° ~ (0,1,0,0) and 6a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 0 < 7r, and the corners of the domain are labelled with their (r, 6) ordinate values. The amplitude and colour gradient are both scaled linearly. Observe how small these angular momentum density gradients are in com-parison to those at t = 77.11 in a = M Kerr spacetime as shown in Figure 4.82. Examining the charge density depicted in Figure 4.85 for time steps t = 0,77.11 and t = 154.22 for the e = 0.6, m$ = 0.3 perturbations in a = M = 5 spacetime, I observe nontrivial evolution beginning from a zero net charge density initial data profile and leading to dynamic charge separation for all t > 0. While somewhat more intricate than the corresponding evolution of massless e = 0.6 perturbations (cf. Figure 4.55), the dynamics are certainly not as rich as those of massless e = 2.4 perturbations (cf. Figures 4.69 and 4.70). It therefore appears that addition of a nonzero mass parameter has a significant effect on the details of the evolution of charge density. Comparing with the t = 77.11 time step of e = 0.6, m$ = 0.3 perturbation evolution in a = 0, M = 5 spacetime as shown in Figure 4.86, it is clear that while slightly different in detail, the degree of charge separation is comparable in rotating and nonrotating black hole backgrounds. The comparison is further investigated in Figure 4.87 where I display the total charge integrated over the computational domain for 0 < t < 200 in a = 0 ,0 .8M and a = M backgrounds. This figure shows how the total charge is in very close agreement for each of the spacetimes throughout evolution. It is thus obvious that while a modest change in the value of m$ has a substantial effect on charge separation, the evolution of charge density from the given initial conditions is only slightly dependent on black hole angular momentum for e = 0.6 perturbations (cf. Figures 4.55, 4.56 and 4.57). C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 179 300 -i 1 r ~i 1 1 1 1 r 200 h 100 1 I! II I! 1 1 Vh m — AS \ 50 100 t _ l I I I I I— 150 200 F i g u r e 4.84: Integrated angular momentum density for e = 0.6, m$ = 0.3 perturbations in M = 5 Kerr spacetime. Plotted is 5, the total angular momentum contained within 5 < r < 5 0 , 0 < 9 < TT and 0 < <f> < 2TT, as a function of coordinate time t. The dashed red line represents data for a/M = 0, the dotted red line data for a/M = 0.8, and the dashed black line data for a/M = 1. Clearly, perturbation angular momentum is especially variable in rotating black hole spacetimes over 50 < t < 80, but tends towards S = 0 at late time. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 180 -1.00e-03 1.00e-03 t=154.22 -1.00e-03 l.OOc-03 F i g u r e 4.85: Charge density at t = 0,77.11,154.22 for e = 0.6, m$ = 0.3 perturbations in a = M = 5 Kerr spacetime. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 6 < 7r, and the corners of the domain are la-belled with their (r, 6) ordinate values. The amplitude and colour gradient are both scaled linearly. A n M P E G animation showing evolution over 0 < t < 154 is avail-able at ht tp:/ /$WWW/People/petryk/animations/8.mpeg where $WWW is currently l a p l a c e . p h y s i c s . u b c . c a . C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 181 -1.00e-03 l.OOc-03 F i g u r e 4.86: Charge density at t = 77.11 for e = 0.6, m$ = 0.3 perturbations in a = 0, M = 5 Kerr spacetime. The directions of coordinate vectors ra ~ (0,1,0,0) and 9A ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 0 < IT, and the corners of the domain are labelled with their (r, 6) ordinate values. The amplitude and colour gradient are both scaled linearly. Comparing with Figure 4.85, it is clear that charge density distributions at t = 77.11 are affected by the value of black hole rotation parameter a, but not very strongly. As could have been deduced from the preceding study of angular momentum evolution—and also from earlier investigation of angular momentum for the massless e = 0.6 perturbations—the effects of black hole rotation on the axial component of current, J^, are significant. Figure 4.88 displays at t = 0,77.11 and t = 154.22 for the e = 0.6, m$ = 0.3 perturbations in a = M = 5 spacetime. From a relatively simple profile at t = 0, the axial component evolves in a nontrivial manner. The current oscillates in direction and amplitude, indicating dynamo-like behaviour. Furthermore, the distribution displays large amplitude and steep gradients at t = 77.11 before relaxing at late time. As shown in Figure 4.82, the angular momentum density of the e = 0.6, m$ = 0.3 perturbations also evolved from a simple distribution at t = 0 to a state with steep gradients in the ergoregion at t = 77.11 before settling down to some degree at late time. To determine the effects of black hole rotation on the axial component of the current, I evolve the e = 0.6, m$ = 0.3 perturbations evolved in an o = 0, M = 5 Kerr spacetime. The distribution of at t = 77.11 is displayed in Figure 4.89. Comparing the small amplitudes and gradients of J$ at t = 77.11 for the a = 0, M = 5 spacetime to the profile of at t = 77.11 for the a = M = 5 Kerr spacetime of Figure 4.82 it is clear that black hole rotation leads to significant amplification of in the ergoregion and surrounding space. To examine the effects of black hole rotation in more detail, I display the global maxima of evolved in a = 0 and a = M spacetimes in Figure 4.90. The corresponding global minima are shown in Figure 4.91. As in the data of massless e = 0.6 perturbations (cf. Figures 4.61 and 4.60), max(J^) for a = M dominates over that of a = 0 after an early phase of evolution. The same is observed in the development of min( J$). Therefore, while black hole rotation initially suppresses 3$, its effect is to amplify the axial component of current after t w 20. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 182 100 5 0 h - 5 0 h -100 200 F i g u r e 4.87: Integrated charge density for e = 0.6, ra<j> = 0.3 perturbations in M = 5 Kerr spacetime. Plotted is Q, the total charge contained within 5 < r < 50, 0 < 9 < n and 0 < <f> < 2TT, as a function of coordinate time t. The dashed red line represents data for a/M = 0, the dotted red line data for a/M = 0.8, and the dashed black line data for a/M = 1. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 183 -l.OOc-03 l.OOc-03 F i g u r e 4.88: Axia l current component, J^, at t = 0,77.11,154.22 for e = 0.6, m$ = 0.3 per-turbations in a = M = 5 Kerr spacetime. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude and colour gradient are both scaled linearly. A n M P E G animation showing evolution over 0 < t < 154 is available at ht tp:/ /$WWW/People/petryk/animations/9.mpeg where $WWW is cur-rently l a p l a c e . p h y s i c s . u b c . c a . C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 184 1=77.11 F i g u r e 4.89: Axia l current component, J ^ , at t = 77.11 for e = 0.6, m<j> = 0.3 perturbations in a = 0, M = 5 Kerr spacetime. The directions of coordinate vectors ra ~ (0,1,0,0) and 9A ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < IT, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude and colour gradient are both scaled linearly. Observe how small the amplitude and gradients are when compared to those at t = 77.11 in a = M Kerr spacetime shown in Figure 4.88. Before concluding discussion of the results, a brief analysis of electric and magnetic field am-plitudes is in order. Again, the details can be deduced from preceding results. As in the case of massless e = 0.6 perturbations, the magnetic field square, BABA, is very small in the equatorial plane (9 = TT/2) and also along the axis of black hole rotation (9 = 0 and 9 = 7r) as depicted in Figure 4.92. Meanwhile, Figure 4.93 shows how the electric field square EAEA begins evolution from a state of zero initial amplitude and displays significant amplitude along both the axis of rotation and equatorial plane at later time. The integrals of the magnetic and electric field norms (respectively B2 and E2) are shown in Figure 4.94. After an initial phase of rapid change the amplitudes take on comparable magnitudes. Also note how the most significant increases in B2 and E2 take place well before time t « 40 at which collimation of energy density becomes apparent (cf. Figures 4.71 and 4.72). This suggests electromagnetic fields likely play a secondary role in collimating the perturbations along the axis of black hole rotation (8 = 0 and 9 = IT) . Also note how the late time decay of B2 and E2 can be approximated by log(B 2 ) ~ -kt , (4.198) and l o g ( £ 2 ) ~ -kt , (4.199) where k re 1 x 10~ 2 . This decay rate is comparable to those for the massless cases with e = 0.0 and e = 0.6 (cf. Figures 4.33 and 4.51). Finally, the evolution of BABA, EAEA, B2 and E2 in the a = 0 spacetime (figures not provided) are qualitatively similar to those in the a = M black hole spacetime. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 185 F i g u r e 4.90: Global maxima of J^, in the computational domain, max(J^), as a function of t for e = 0.6 coupled massive perturbations in M = 5 Kerr spacetime. The dotted black line represents data for a/M = 0 and the dashed black line data for a/M = 1. The global maximum value of for a/M = 1 spacetime is substantially larger than those of a/M = 0 spacetime at intermediate and late time, providing evidence that black hole rotation amplifies the axial current. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 186 -0.5 a 6 - 11 i / •1.5 _l I I I I I I I I I I I I L 50 100 t 150 200 F i g u r e 4.91: Global minima of in the computational domain, min(J^,), as a function of t for e = 0.6 coupled massive perturbations in M — 5 Kerr spacetime. The dotted black line represents data for a/M = 0 and the dashed black line data for a/M = 1. The global minimum value of J^, for a/M = 1 spacetime is substantially larger than those of a/M = 0 spacetime at intermediate and late time, providing further evidence that black hole rotation amplifies the axial component of the electric current. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 187 1.00e-05 1.00e-03 1.00e-05 l.OOc-03 F i g u r e 4.92: The figure displays the square of the magnetic field, BaBa, at t = 0,77.11,154.22 for e = 0.6, m<j> = 0.3 perturbations in a = M = 5 Kerr spacetime. The directions of coordinate vectors ra ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 ^ v < 50 and 0 ^ 9 ^ 7r, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude is scaled linearly, while the colour gradient is logarithmic. Observe how the magnetic field square is relatively small in the equatorial plane (9 = TT/2) and along the axis of black hole rotation (9 = 0 and 9 = ar). C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 188 1.00e-05 1.00e-03 t= 154.22 1.00e-05 1.00e-03 Figure 4.93: The figure displays the square of the electric field, EaEa, at t = 0,77.11,154.22 for e = 0.6, m<j> =0.3 perturbations in a = M = 5 Kerr spacetime. The directions of coordinate vectors r° ~ (0,1,0,0) and 9a ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 5 < r < 50 and 0 < 9 < n, and the corners of the domain are labelled with their (r, 9) ordinate values. The amplitude is scaled linearly, while the colour gradient is logarithmic. Note how the electric field square in the equatorial plane (8 = TT/2) and along the axis of black hole rotation (9 = 0 and 9 = w) are comparable to the square along other surfaces of constant 9. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 189 3.5 O 2.5 V. V S-. " Y - . _i i i i i i i_ _j i L _i i i L. J i_ 50 100 t 150 20C Figure 4.94: Plotted are.log(B2) and log(.E2), the logarithm of the integrated squares contained within 5 < r < 50, 0 < 8 < n and 0 < <j> < 2ir, as a function of coordinate .time t: The .results are for e = 0.6, ra$ = 0.3 perturbations in a = M = 5 Kerr spacetime. The dashed black line represents log(jB2), and the dotted black line \og(E2). Initially, the field amplitudes undergo substantial oscillation, but the decay of integrated electric and magnetic amplitudes become closely matched at intermediate and late time. 4.3.7 Convergence of Conserved Quantities I examine the convergence and accuracy of my code by examining the evolution of e = 0.6, m$ = 0.3 perturbations (initial data specified as above) in the a = M = 5 spacetime. As discussed, the implemented solution scheme incorporates second-order finite difference methods, and the results, I have presented are typically accurate to « 1% over the period 0 < t < 12—the time at which perturbations begin crossing the excision surface at r = 5—as demonstrated by the invariance of conserved quantities over this period. The conserved quantity values are expected to vary after t « 10 as perturbations cross the ergosphere and then flow out of the computational domain at the r = 5 excision surface. After t w 17 there will be additional change as they also begin flowing through the r = 50 outer boundary. Figures 4.95, 4.96 and 4.97 respectively display the total normalized energy, total angular momentum and total charge within the computational domain 5<r<50, 0 < 0 < ir over the period 0 < t < 10 for three levels of grid resolution. The lowest resolution grid has J = 801 radial points and K = 401 angular points, the intermediate resolution grid has J = 1601 radial points and K = 801 angular points, and the grid with highest resolution has J = 3201 radial and K = 1601 angular points. All data was computed in parallel across 64 C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 190 0.9995 h 0.999 h 0.9985 h 0.998 Figure 4 . 9 5 : Convergence of total normalized energy, T, for e = 0.6, m<j> = 0.3 perturbations in a = M = 5 Kerr spacetime. The dashed red line represents data for a finite difference grid with J = 801 radial points and K = 401 angular points, the dotted red line represents data for a grid with J = 1601 radial and K = 801 angular points, and the dashed black line represents data for a grid with J = 3201 radial and K = 1601 angular points. processors. Clearly, the conserved quantities display convergence.2 2 Furthermore, the convergence factor, time averaged over the period 0 < t < 24, is C « 7.7 for the total normalized energy, C sw 3.7 for the total angular momentum and C sa 8.6 for the total charge of the system. This is somewhat better than the C ~ 4 I would expect for a completely centered second-order convergent scheme. However, some deviation is expected, as the operators I use near the computational boundaries are not completely centered. At any rate, the convergence factors may still approach values closer to 4 as resolution is further increased. While the second-order convergence of a conserved density implies the second order convergence of its integral (providing the numerical integration scheme is itself at least second-order), it may still be useful to display the convergence factor for the density itself. I calculate the convergence factors for the total angular momentum density ps, the total energy density pr and the charge density PQ as displayed in Figure 4.98. It is there shown how the convergence factors for energy density and charge remain near C ~ 4 for the evolution. However, while still convergent, the result is not as good for the angular momentum density. Averaged over time, C sa 4.9 for the energy density, C sa 4.1 for the angular momentum density and C sa 5.0 for the charge density. 2 2 Convergent invariance of conserved quantities indicates I am numerically approximating the true solutions of the system. Technically, it is then not necessary to perform independent residual evaluations as discussed in Appendix C. I can for this reason justifiably omit independent residual testing of the code. C H A P T E R 4 . C H A R G E D S C A L A R A C C R E T I O N 191 0.3 H 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 0.2 0.1 c o \ / / -/ / 7 " y -o.i -0.2 _i i i i_ JL ± —1 I I I L_ 4 6 t 1( Figure 4.96: Convergence of total angular momentum, S, for e = 0.6, m<j> = 0.3 perturbations in a = M = 5 Kerr spacetime. The dashed red line represents data for a finite difference grid with J = 801 radial points and K = 401 angular points, the dotted red line represents data for a grid with J = 1601 radial and K = 801 angular points, and the dashed black line represents data for a grid with J = 3201 radial and K = 1601 angular points. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 192 o- l F i g u r e 4 .97: Convergence of total charge, Q, for e = 0.6, m $ = 0.3 perturbations in a = M = 5 Kerr spacetime. The dashed red line represents data for a finite difference grid with J = 801 radial points and K = 401 angular points, the dotted red line represents data for a grid with J = 1601 radial and K = 801 angular points, and the dashed black line represents data for a grid with J = 3201 radial and K = 1601 angular points. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 193 8 i — i — i — i — i — i — i — i — i — i — i — r -o 4 H — i — i — i — I — i — i — r \ / , _i i i i_ _i i i i_ 10 15 20 Figure 4.98: Convergence factors for energy density (dashed red line), angular momentum density (dotted red line) and charge density (dashed black line) for e = 0.6, m$ = 0.3 perturbations in a = M = 5 Kerr spacetime. The convergence factors are based on a comparison of functions evolved on finite difference grids with J = 801 radial points and K = 401 angular points, J = 1601 radial and K = 801 angular points, and J = 3201 radial and K = 1601 angular points. Again, deviations from the expected value of C ~ 4 are likely due to points near the computational boundary where non-centered operators are employed. Since, given the initial conditions, at least some amount of perturbation should evolve to the computational domain boundaries by t w 24, convergence testing over the period 0 < t < 24 should be adequate. To verify the results, testing over a longer period of evolution could be performed at a later date. 4.3.8 Independent Residual Testing Dynamical Variables Having shown convergence of conserved quantities over the period 0 < t < 24, further validation of the code is provided by independent residual testing as described in Appendix D. Recall that a properly implemented finite difference scheme will not only be convergent, but also drive the residual rh = Lhuh - fh (4.200) to zero, where L is the discretized difference operator, / is the discretized source function, and uh is the approximate solution obtained by the scheme. In an independent residual test a second (independent) discretization Lh is applied to the approximate solution, and it is verified whether the residual-like quantity '•h~h j it (4.201) C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 194 approaches zero as the scale of discretization h approaches zero. If it does, then quantity uh has been verified a true approximation of the solution u to differential equation Lu-f = 0. (4.202) In obtaining solutions presented in this chapter, equations of motion for $, Ar, Ag and A^ are discretized using the Crank-Nicholson scheme with centered spatial derivatives when rm-m + Ar < T < r m a x — A r and 2A9 < 6 < TT — 2A0. Approximate solutions of auxiliary variables 4>r, &g, Arr, Agr, A^r, Ar$, Agg and A^g are thus obtained. A suitable independent residual test then involves substituting the approximate solutions into respective equations (4.75), (4.76), (4.84), (4.85), (4.86), (4.87), (4.88) and (4.89), where partial derivatives in time and space are replaced by independent difference approximations. Formally, if u = u(t, r, 6) represents an auxiliary variable and R = Rit, r, 9) the operand on the right hand side of differential equation drR , (4.203) is 1 fu(t + At, r + Ar, 9) - u(t - At, r + Ar, 9) . • u(t + At, r, 9) - u{t - At, r, 9)' then a suitable independent difference approximation for (4.203) 2 V 2At 2At _ R(t, r + A r , 9) - R(t, r, 9) Ar (4.204) where a leap-frog style centered difference scheme is used to approximate the derivative in t, and the difference scheme approximating the derivative in r is centered about the virtual spatial point r + i A r in analogy to Crank-Nicholson. Observe how the left hand side of (4.204) has been averaged in space'between r and r + A r so that it too is centered about virtual point r + ^ A r . Similarly, a suitable independent difference approximation for u = d6R, (4.205) 1 (u(t + At,r,9 + A9) - u(t - At,r,9 + A9) - u(t + At,r, 9) - u(t - At,r, 9)^ (4.206) 2 V 2At 2A£ R(t,r,9 + A9) - R(t,r,9) A9 which is now centered about the virtual point 9 + \ A9. Rewriting (4.205) and (4.206) in the form (4.201), and substituting in approximate solutions u and R, I thus compute independent residuals from the equations for Arr, Arg, Agr, Agg, A,pr, A^g, 4>r and and then calculate the h norms | | f ' l | | 2 . The independent residuals are computed in parallel across 64 processors on difference meshes of J = 210 radial and K = 101 angular points, J = 401 radial and K = 201 angular points, J = 801 radial and K = 401 angular points, and J = 1601 radial and K = 801 angular points. Figure 4.99 displays \ \fh\\2 from the equations Arr, Agr and A^r at three subsequent resolutions. Similarly, Figure 4.100 displays \\fh\\2 from the equations for <&th at three different subsequent levels of resolution. The decrease of \\rh\\2 with h is proof the solutions obtained are true approximations of the system of differential equations. 4.3.9 Testing The Sommerfeld Boundary Condition The Sommerfeld boundary condition is only approximate for the fields evolved in my computations. A test of its suitability is therefore required, especially for the case when the fields have both nonzero C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 195 0.0015 0.001 0.0005 -: . i i i ! r 1 y\l VJ 0 1 1 1 1 1 1 1 1 1 0.015 0.01 ll?hll2 0.005 -1 1 1 1 1 1 1 1 1 ^ -^sf\i / -r -/ 0 — i i i i i i i • • 0.004 ~- i i i I I i i i i \ ii 0.003 '- fi 1 : J -^ / ... •-_ 0.002 0.001 ' -0 i i i i i i i i i i i (b) 0 20 40 60 t Figure 4.99: Independent residual norms \ \rh\\2 from the equation (4.84) for ATr, equation (4.85) for Agr and equation (4.86) for A^r. Frame (a) represents the independent residual norm associated with Arr, frame (b) the independent residual norm associated with Agr, and (c) represents the same for A^r- The fh have been computed for difference grids with J = 401 radial and K = 201 angular points (dashed red line), J = 801 radial and K = 401 angular points (dotted red line), and J = 1601 radial and K = 801 angular points (dashed black line). As expected of a correctly implemented scheme, the independent residual norm decreases with increased resolution. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 196 0.1 0.01 0.001 0.0001 11?% 1fc 0.1 n 1 1 1 1 1 1 r r- v . / •J (a) H—I—h H — I — I — I — — I — I — I — I — — I — h A 0.01 , . , / \ TJ v 0.001 0.0001 !• _ l I I I I I I I I I I I I I I 1_ (b) 50 100 t 150 200 F i g u r e 4.100: Independent residual norms | | f H2 from equation (4.76) for 4?g. Recall that 4> = (j>i+i<j>2, and so $0 = <fiie+i<p2e- Frame (a) displays the independent residual norm associated with fce while frame (b) displays the same for foe- Note the logarithmic scaling of the vertical axes. The fh have been computed for difference grids with J = 201 radial and K = 101 angular points (dashed red line), J = 401 radial and K = 201 angular points (dotted red line), and J = 801 radial and K = 401 angular points (dashed black line). The decrease in | | f h | |2 with h provides evidence the numerical implementation is correct. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 197 mass and electromagnetic coupling parameters. I test this suitability by comparing the results of evolution for m$ = 0.3, e = 0.6 perturbations in a = M '= 5 Kerr spacetime on a finite difference grid with resolution and initial conditions identical to these used in above, but now covering the larger computational domain 5 < r < 9 5 , 0 < 6 * < 7 r : The grid of this larger computational domain is therefore composed of J = 1601 radial and K = 401 angular points. I again compute across 64 processors, but division of the computational domain across processors now differs from that used for the smaller g r id . 2 3 In detail, comparison is through the relative errors of conserved quantities. For example, the relative error of energy density, Spr/pr, is given by Spr/PT = R P T - P * H , (4.207) | | P r | | 2 where pr is the solution for energy density on the large computational domain, and 11 • | (2 represents the l2 norm. While the solution px is treated as exact, it is known to deviate from the ideal solution after perturbations evolve to its boundary. This is because the approximate Sommerfeld condition is also applied on the larger computational domain. For this reason, calculation of Spx/pr should be limited to the period 0 < t < 65. However, I extend calculation to 0 < t < 77.11 for good measure. I similarly calculate relative errors for the angular momentum density, Sps/ps, and charge density, 5PQ/PQ. The results are displayed in Figure 4.101. It is there apparent how relative errors of all three conserved densities are on the order of a percent. Considering that these densities are known to be conserved to « 1% for the early stages of evolution, the Sommerfeld boundary condition has produced surprisingly good results. So while only approximate, the Sommerfeld condition is likely sufficient for the results presented above. While a quantitatively better approximation is desirable, I have not implemented one for the evolution of Maxwell-Klein-Gordon fields on Kerr spacetime. However, a different treatment is included in Appendix D, where I provide results on Minkowski spacetime for comparison. A n alternate test (the one I use in Appendix D.) is through the error of the integrated conserved densities. For example, the error in total normalized energy, 5T, is given by 5T=\\f-T\\2, (4.208) where T is the total charge in the smaller computational domain, T is that in the large domain, and || • H 2 again represents the l2 norm. T and T are obtained through integration over a common region of space. I again calculate this quantity over the period 0 < t < 77.11. The result is displayed in Figure 4.102 along with the analogous errors for angular momentum (SS) and charge (SQ). Comparing these results with the relative errors presented above, it is apparent how the true quality of the boundary condition is obfuscated by the amplitudes of the functions under considered. 2 3This comparison will therefore also test whether parallel distribution between the processors has a significant influence on the computed results. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 198 0.04 0.03 h Q. •a 0.02 0.01 n 1 1 1 1 1 1 i 1 1 1 j 1 1 r 1 r r i \ I / s A _o oi 1 1 1 ' ' 1 1 1 1 1 1 1 1 L 0 20 40 60 t J _ Figure 4.101: Displayed are the relative errors for energy density {Spr/pT, dashed red line), angular momentum density (Sps/ps, dotted red line), and charge density [SPQ/PQ, dashed black line) for e = 0.6, m$ = 0.3 perturbations in a = M = 5 Kerr spacetime. The relative errors are based on the comparison of functions evolved on domains covering ranges 5 < r < 50 and 5 < r < 95. C H A P T E R 4. C H A R G E D S C A L A R A C C R E T I O N 199 T 1 1 1 1 1 1 1 r Cf 4 OQ E -2 h /I / / i / -/ / r 1 J \J 20 _ L _ 40 t 60 F i g u r e 4.102: Displayed are the errors for normalized integrated energy (6T, dashed red line), integrated angular momentum (<5S, dotted red line) and integrated charge (6Q, dashed black line) of e = 0.6, m$ = 0.3 perturbations in a = M = 5 Kerr spacetime. The errors are based on the comparison of functions evolved on domains covering ranges 5 < r < 50 and 5 < r < 95. C H A P T E R 5. C O N C L U S I O N S A N D F U T U R E W O R K 200 C H A P T E R 5 C O N C L U S I O N S A N D F U T U R E W O R K I have thus investigated critical solutions for massive charged scalar fields and the detailed evolu-tionary dynamics of Maxwell-Klein-Gordon perturbations on a Kerr spacetime background. The studies were numerical, based on finite difference techniques, and have made use of adaptive mesh refinement and parallel computation. In Chapter 3, I have obtained the critical solutions at black hole threshold for massive charged scalar fields in spherical symmetry. For the case of sufficiently small mass parameter, the solutions are discretely self-similar and exhibit Type II scaling in both the black hole mass and charge. When the mass parameter is sufficiently large, a new critical solution is obtained. In this case it is periodic and resembles a perturbed charged boson star. These new periodic solutions exhibit behaviour typical of Type I phenomena, and their lifetimes exhibit a well-defined scaling relation. In Chapter 4, I have studied Maxwell-Klein-Gordon perturbations on a Kerr spacetime back-ground and have discovered that: (1) gravitational interaction effects are strongly manifest as orbiting resonances for weakly coupled perturbations; (2) massive Klein-Gordon perturbations dis-play mode trapping and a tendency for scattering along the axis of symmetry for intermediate mass parameter values, but are quickly overcome by attraction to the black hole when the values are large; (3) there is evidence for superradiant scattering of purely electromagnetic perturbations in rotating black hole spacetimes; (4) electromagnetic coupling introduces the phenomena of charge separation and dynamo-like behaviour; (5) the complexity of the observed dynamics tends to in-crease with charge coupling parameter and black hole rotation; and finally, (6) there is strong evidence for energy collimation along the axis of rotation for an appropriate choice of mass and charge coupling parameters, but it is not clear that either rotational or electromagnetic effects are the strongest contributors to this phenomenon. For comparison, Appendix D provides results for the dynamics of Maxwell-Klein-Gordon fields in Minkowski spacetime. That appendix also describes the utility of Monotonically Increasingly Boosted coordinates in treating outer boundary conditions during numerical evolution of the field equations. In future study, it would be useful to complete the investigation of critical phenomena, specifi-cally investigating the possibility of new solutions details with a sufficiently large charge coupling. In particular, it would be interesting to see if Type I solutions can be obtained for the massless Maxwell-Klein-Gordon system when e ~S> 1. It would also be interesting to study critical phenom-ena and compact object formation for a plasma-like matter (i.e., one where positive charge can be associated with a distribution of energetic heavy particles and negative charge associated With a corresponding distribution of light particles). The Einstein-Vlasov-Maxwell system (see [109], [110], [111], [112], [113]) appears to be a good candidate and should be investigated in detail for this purpose. As well, the study of Maxwell-Klein-Gordon perturbations in axial symmetry should be ex-tended to include evolution of the charged case when the mass parameter m$ is very large—it seems possible that electromagnetic interactions could balance or overcome the tendency for the otherwise rapid plunge into the black hole. Furthermore, investigation should be extended to the case with very large electromagnetic coupling parameter e where new physics like mode trapping is likely to occur. It would also be worth investigating initial conditions more closely resembling a stationary accretion disk. From such initial conditions it would then be possible to study the possible destabilization of stationary disks through interaction with electromagnetic waves. 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G L O S S A R Y O F N O T A T I O N 211 A P P E N D I X A G L O S S A R Y O F N O T A T I O N This appendix tabulates some notation used throughout the dissertation, if not explicitly defined in the text. notation A = B A=> B A->B A e B {A} \A\ 1141 U \ \ 2 U\\oo T r A Q on z z+ definition A defined to be B A implies B A approaches B A is a member of B set of objects similar to A modulus of object A arbitrary norm of object A 1-2 norm of object A loo norm of object A component m of vector Aa component mn of matrix Aa trace of matrix A n-dimensional domain ft n — 1-dimensional boundary of domain Q, the set of integers the set of positive integers Table A . l : Mathematical miscellanea. notation definition rpabc... 1def... components of tensor T rpa(bc)... 1def... symmetrization of tensor T in components b and c rpabc... 1d(ef)... symmetrization of tensor T in components e and / rpa[bc]... 1def... antisymmetrization of tensor T in components b and c Tabc... 1d[ef}... antisymmetrization of tensor T in components e and / Kronecker delta where 5% = 1 if a = b and 5% = 0 if a ^  b tabcd totally antisymmetric tensor of positive orientation where 60123 = 1 Table A.2: Tensor notation. A P P E N D I X A. GLOSSARY OF NOTATION 212 notation d rr-igbc... dxs^def... p qnabc... -L-A-I-def... V y v ^ ; ; ; 1def.. rpabc... 1def...,g definition partial derivative of tensor with respect to coordinate x9 Lie derivative of tensor T^jy along vector A cpvariant derivative of tensor T^fy with respect to coordinate x9 the Laplacian operator Table A.3: Derivative notation. notation M DM (M,gab) Rabcd Rab R definition n-dimensional manifold Ai n — 1-dimensional boundary of manifold M. n-dimensional manifold M. with metric gab and Christoffel connection Riemann tensor Ricci tensor curvature scalar Table A .4: Manifolds and Riemannian spaces. notation definition 3- past null infinity 3+ future null infinity i~ past timelike infinity. i° spatial infinity i+ future timelike infinity Table A.5: Causal structure of spacetime. APPENDIX B. DIFFERENCE AND DISCRETE OPERATORS 213 APPENDIX B DIFFERENCE AND DISCRETE OPERATORS This appendix tabulates the F D A s and other discrete operators used in the dissertation. B.l Finite Difference Operators This section tabulates the F D A s used in the thesis. In the tables below I use the same notation as introduced in Appendix C, except now all difference operators are given for the case of two spatial dimensions and one temporal dimension, and spacetime coordinates given by the ordered triple (t, r, 9). That is, it™ • labels a point in the'finite difference mesh with ordinates (t, r, 9), uf+ij labels a point with ordinates (t,r + Ar,9), et cetera. Furthermore, r j _ i and rj+i respectively designate coordinate values r — Ar and r + A r . Also as before, (M"^ ) designates the partial derivative of u(t, r, 9) with respect to coordinate r and evaluated at (t,r, 9), (ufj) r r the second partial derivative of u(t,r,9) with respect to r at (t,r,9), and so on. The special case is (w™^ 2 ) i which designates the partial derivative of u(t,r,9) with respect to t and is centered at the virtual time step t + \. A l l the listed F D A s are 2-nd order accurate in grid spacings A r , A9 and time interval A i , and approach the corresponding continuum limit P D E as grid spacings and time interval approach zero. P D E ,3/ ,t F D A Ui+l,j Ui-l,j 2 A r "? + i , . , -2< 7 -+<- 1 , . 7 -(Ar)2 ri+l^rj_x ui+i,j-ui-i,j t t j j + l ~ M i ! j - l 2A6» Xj+r2<iKj-i (A9)2 „.n _..n ai,j+l ui,j-l COS0j+i—COs9j-i u ™ + 1 - u n ~ 1 "»,3 Ui,j 2At u n + l - u n • At Table B . l : Centered difference operators. A P P E N D I X B . D I F F E R E N C E A N D D I S C R E T E O P E R A T O R S 214 P D E F D A ( u n ) . - " ^ + 4 ^ 1 ^ - 3 ^ \Ui,j),r ^ 2 A r , „ , - < , + 2 + 4 ^ + 1 - 3 < , \ui,j),e ^ 2A6> Table B . 2 : Forward difference operators. P D E F D A \Ui,j),r ^ 2 A r 3 ^ J - 4 u ? J - _ 1 + ^ j _ 2 2A6> Table B . 3 : Backward difference operators. B.2 Other Discrete Operators In this section of the appendix I list other discrete operators used in the finite difference schemes of the dissertation. B.2.1 Quadratic Fix Operators Quadratic fix operators are used on functions in regions of the finite difference grid which correspond to locations in space where those functions are known to be quadratic. Typically, these regions will be at the point of spherical symmetry or along the line of axial symmetry. The quadratic fix operators are derived from the Forwards and backwards difference operators for u>r or utg. If a function is quadratic at the point of spherical symmetry, r = 0, then its first derivative is known to vanish there. Using the forwards difference operator for utT obtains the expression 0 = - < + 2 j + 4 < + l j J - 3 < i , which is used to update the value of the function at the point of spherical symmetry. When the function is quadratic about the line of axial symmetry, the forwards and backwards difference forms for Ufi are used to obtain the quadratic fix operators 0 = - < j + 2 + 4 < J + 1 - 3<, . (B.2) to be applied along axis when 9 = 0, and 0 = 3 « ^ - 4 « ^ . _ 1 + « ^ _ 2 (B.3) which is used to update points along 8 = n. These results are tabulated below. (B. l ) A P P E N D I X B . D I F F E R E N C E A N D D I S C R E T E O P E R A T O R S 215 quadratic fix operator = \ 4- Ann = \ ~uli,3+2 = 5 ( - 4 < i -Table B .4: Quadratic fix operators. Alternately, it is'sometimes useful to use the (B . l ) , (B.2) and (B.3) to update grid points adjacent to the those at which the functions are known to be quadratic in coordinate 9. Under these circumstances, the quadratic fix operators wil l be expressed as in Table B.5. quadratic fix operator ^ • = 3 ( 3 ^ + 1 + ^ - 1 ) Table B . 5 : Adjacent quadratic fix operators. B.2.2 Interpolation and Extrapolation Operators For function u^j, the mth degree Lagrange polynomial in coordinate r, P " r ( r , 9), is rri P £ > , c ? ) = ] T p f ( r , 0 ) , . (B.4) i=l where m i?M)Stt?Jni^i. (B.5) « = i r % n Similarly, the mth Lagrange polynomial in coordinate 9, P™ s(r,9), is m P ^ M ) ^ P ; M ) , (B.6) i=i where m p?fr 0) = 7 / . n . TT 9j-9t m 9 - 0, 1=1 At times I use these Lagrange polynomials to update values of u™ • from its values at neigh-bouring points by extrapolating in coordinate r or interpolating in coordinate 9. Since high degree Lagrange polynomials generally produce poor results when used as extrapolants, especially for A P P E N D I X B . D I F F E R E N C E A N D D I S C R E T E O P E R A T O R S 216 points far from those upon which they are based, I use only the 2nd degree Polynomial to extrap-olate for smaller r as per <i = « + 3 , i - 3 < + 2 j - + 3 < + 1 ,j) • (B.8) Interpolation from Lagrange polynomials is, however, much more reliable. When I interpolate in coordinate 8 I use 3rd degree Lagrange polynomials. I alternately use the Lagrange polynomial in a centered fashion as per 1 n 2 n 2 n 1 n (B.9) and at other times in an off-center fashion as per 1 % j + 3 (B.10) or ^ - 2 + 4 W i ! j - 3 j ' ( B . l l ) depending on the distribution of available grid points. Finally note the time-averaging operator of the Crank-Nicholson scheme + < ; ) (B.12) i n + * = - (un+1 is itself an interpolant based upon a 2nd degree Lagrange polynomial in temporal coordinate t. These results are all summarized in Table B.6. interpolation/extrapolation operator < i = « + 3 J - - 3 « ? + 2 j - + 3 U ? + l j . ) — A 6Ui,j+2 + 3 U i , j + l + 3 U i , J - l 6Ui,j-2> UZj = ( i U i j ' + 3 _ U"i,i+2 + f U i j + l + U?J — (\u?,j+l + 2 U i ! i - l u i J - 2 + | u i ^ j - 3 ) 1 („n+1 4- , ,n 1 2 + Table B . 6 : Interpolation and extrapolation operators.. B.2.3 Kreiss-Oliger Dissipation Operators I designate the Kreiss-Oliger dissipation operator in coordinate r for an F D A acting on u™ • by s r i t " j , the dissipation operator in coordinate 8 by sgu^j, and dissipation operator in both coordinates r and 8 by srou™j. The following table provides a listing of these operators. dissipation operator Wl, = ft « - 2 j - " 4 < - i j + - 4 < + l j -= ft - K j - i + K > - K i + i + < - + 2 ) + 2 , j J Cr8 16 (0 •a; - 2 , J •4<+u+<+2J Table B.7: Kreiss-Oliger dissipation operators. A P P E N D I X B. D I F F E R E N C E A N D D I S C R E T E OPERATORS 217 B.2.4 Integration Rules The following table lists two integration rules for equally spaced abscissae. The second in the list is the trapezoidal rule used in calculations presented throughout this dissertation. integration rule ' • XT/ u(r)dr = Y.U u>Ar + ° ((Ar)2) /;/ u(r)dr = E • {fa + K+i ) A r + O ( ( A r ) 3 ) Table B.8: Integration rules for equally spaced abscissae. APPENDIX C. NUMERICS 218 A P P E N D I X C N U M E R I C S C.l Finite Difference Techniques and Approximations Of the numerical techniques available, finite difference methods are chosen to solve the equations of constraint and evolution presented in this thesis. Finite difference methods are well-established and understood, and the notion of the finite difference approximation on which they are based is simple and intuitive. This makes them a natural choice for numerical relativists, especially when solving the systems similar to those in this thesis:1 Good references for the topics in this section are [91], [103], [61], [134], along with the summary [30]. Furthermore, [21] and [22] are useful references for finite difference techniques as applied to the equations of general relativity. The key element of finite difference methods is the finite difference approximation (FDA) . Essentially, an F D A is what one obtains when a continuous differential equation is replaced with a discrete difference equation. The discrete difference equation is based on the notion of a grid of points at which functions are evaluated, and thus is an approximation to the continuum solution one aims to solve. In the subsections that follow the concept of the F D A is discussed in the context of consistency, accuracy, stability and convergence of a finite difference scheme. Another topic covered is dissipation in finite difference methods, as it plays a vital role in the computation of solutions to equations presented in later chapters. C . l . l Basic Definitions and Concepts This dissertation considers F D A s to systems of partial differential equations (PDEs) on a mesh or grid of spacing h, and will therefore be vitally interested in the behaviour of the F D A s and their solution schemes in the continuum limit as the grid spacing tends to zero, h —> 0. It is expected that, for a correctly implemented finite difference scheme, the error of the approximation to approach zero in the continuum limit. Following discussion wil l begin with the introduction of some notation. Let a differential equation be denoted by Lu = f, (G. l) where L is a difference operator, u is the unknown function to be solved, and / is some source function, f wil l generally be a function of the variables on which u also depends. The discretized operators, variables and functions will be denoted with a superscript h . The discretization of the system (C. l ) is thus denoted Lhuh = fh, (C.2) or Lhuh _ jh = o ( C 3 ) where Lh is the discretized difference operator, uh the discrete unknown function, and fh the discretized source function. A l l functions of the discrete system are evaluated at points on the finite difference mesh. If h is the only discretization scale in the problem the F D A is said to be 1Other commonly used methods are spectral and finite element, and they should be used where appropriate. However, finite difference methods are typically simpler, especially when solving systems of equations with regular domain boundaries, as is the case in this thesis. A P P E N D I X C . N U M E R I C S 219 convergent if and only if uh -> u as h 0. . (C.4) For obvious reasons, all F D A s are desired to be convergent. The solution error, eh, of the F D A is defined eh = u - u h . (C.5) So if the F D A is convergent, then eh —> 0 as h —> 0. Another type of error in the subject of F D A s will now be developed. The truncation error, r h , of the F D A is given by T h = L h u - f h : (C.6) So while the solution error is a measure of how well the F D A solution approximates the solution of the P D E , the truncation error is a measure of how closely the finite difference equation approximates the P D E whose solutions are sought. Assume the F D A again has a single discretization scale h. Then, if for the F D A r h ^ 0 as h->0 (C.7) holds, it is called consistent. Consistency is therefore required if the F D A is to be convergent. If, furthermore, l im rh = 0 (hp) (C.8) for positive integer p (i.e., p £ Z + ) , then the F D A is said to be pth order accurate. A useful result pertaining to F D A truncation errors is the observation that rh = O (hp). (C.9) This was first noted in [133] and is based on the Richardson ansatz, that if an F D A is completely centered2 and is based upon a mesh uniform in a single scale parameter h, then uh has the expansion uh = u + h2e2 + h4e4 + ••• ' (CIO) in the limit as h —> 0, where e2, ... are functions independent of h.3 Finally, the concept of the residual of an F D A is defined. The form (C.3) leads us to the definition that if uh is an approximation to the true solution uh of (C.3), then the residual of the F D A , rh, is given by rh = L h u h - f h . (C.12) Essentially, the residual is the degree to which finite difference solutions fail to satisfy the F D A . The aim therefore is to find a convergent process by which the residual of the F D A is driven to zero. C.l.2 Convergence Testing and Independent Residual Evaluation From (C.10) the notion of the convergence test can be developed. Suppose an F D A is solved three times, each time beginning from the same initial data set, but each time using a different grid spacing. For the sake of argument, assume the grid spacings are h, 2h, and Ah. Furthermore, assume the three finite difference grids overlap in the sense that the Ah grid points are subsets of the 2h grid points, and that the 2h grid points are a subset of the h grid points. Finally, assume the F D A s are completely centered, and that h, 2h and Ah are respectively the only scale parameters 2 A n FDA is said to be centered if Lh is symmetric about the grid point where uh is evaluated. 3 If the scheme is not completely centered then (C.10) is modified as per uh = u + he1 + h2e2 + -- - . (C. l l ) A P P E N D I X C . N U M E R I C S 220 in the problem. Then investigating how solutions uh, u2h, and u4h differ provides information about the convergence of the chosen F D A . The Richardson ansatz (C.10) for a completely centered scheme then provides uh = u + h2e2 + h4eA + • • • , (C.13) u2h = u + (2h)2 e2 + (2h)4 e 4 + • • • , (C.14) and u4h = u + (Ah)2 e 2 + (4/i) 4 e 4 + • • • . (C.15) Then for some suitable spatial norm 11 • j | such as the l2 norm (j-'EK)2)2. (c-16) 1^1 being the ith component of vector uh, a quantity called the convergence factor, C (t), is defined c ( t ) _ \\u4h-u2»\\ where argument t designates comparing uh, u2h and u4h at the same time, or step, in the evolution. Using (C.13), (C.14) and (C.15), and providing the system is convergent, it should then observed lim C (t) = 4. (C.18) A convergence test using discretizations h, 2h and Ah would then amount to verification that (C.18) holds. Furthermore, having confirmed convergence of the F D A , the solution error between any two discretizations can explicitly be calculated as per u 2 h - uh = (u + (2h)2 e2 + (2h)4 e 4 + • • • ) - (u + h2e2 + h4eA + • • • ) = 3^ 2 e 2 + O (h4) (C.19) or u4h _ u2h =(u+ ( 4 / l ) 2 e 2 + ( 4 f t ) 4 ^ + . . . j _ ^ + ( 2 / l ) 2 g 2 + ^ 2 ^ 4 g 4 + . . . ^ = Uh2e2 + Q ^ _ (C.20) Suppose a convergence test has been performed as outlined above and find the finite difference evolution is in fact converging to some continuum solution. The problem then to resolve is whether the evolution is approaching the correct continuum solution. For example, it is entirely possible the F D A is convergent, but incorrectly implemented, and so a closely related (but different) system is now being solved. If the system possesses conservation laws, perhaps the best way to test whether ( C l ) is being correctly solved is to ensure the associated conserved quantities are converging to constant values in the expected manner. In the absence of conservation laws, or simply to check the F D A in another manner, an independent residual test can be used. This test is also based upon the Richardson ansatz (C.10) for uh, but additionally upon the observation that for a pth order F D A Lh of L, Lh can always be expanded in powers of the mesh spacing according to Lh = L + hpEp + O (h2p) , (C.21) where Ep is a differential operator involving higher order derivatives in L. Then if Lh is another F D A for L, but this time is 2nd order, expression /> = L + h2E2 + O (hA) (C.22) A P P E N D I X C. NUMERICS 221 is obtained, where E2 is another differential operator involving higher order derivatives in L. Since uh =u + h2e2 + O (ft4) , (C.23) applying Lh to uh expression Lhuh = ( ( L + h2E2 + O (ft4)) (u + h2e2 + O (ft4))) = Lu + O (h2) (C.24) is obtained. It is then concluded Lhuh is of the form of a residual and converges to Lu as ft —> 0. Now suppose the solution scheme has been incorrectly implemented but convergence is still observed as per lim C ( i ) = 4 . . " (C.25) h—>0 The expansion of uh instead would have to be of form uh = u + e0 + h2e2 + O (h2) , (C.26) and thus would have an O (ft0) error, eo being an independent function of ft. Calculating Lhuh now would yield Lhuh =((•£ + h2E2 + O (ft4)) (u + e 0 + h2e2 + O (/i 2))) = Lu + O (h°) , (C.27) and so the independent residual, rh, • fh = Lhuh (C.28) would approach a finite value as h — 0 . Such a result would indicate that either the implementation of the solution scheme, or the independent residual evaluation, is somehow flawed. Since this sort of independent residual test is applied after the finite difference solution is obtained, it typically adds negligible computational work overall. C . l . 3 Deriving Finite Difference Formulae The basic element of a finite difference scheme is the F D A . F D A s for a P D E are obtained rather simply using Taylor series expansions of the function u on the finite difference grid. It is now out-lined how to obtain F D A s for P D E s in one spatial and one temporal dimension. The generalization to greater than one spatial dimension is straightforward and so wil l be omitted from the discussion. Finally, Appendix B tabulates all F D A s used in the dissertation, including all those derived in this subsection. Consider the finite difference grid shown in Figure C . l . Here, spacetime coordinates are given by the ordered pair (t, r), grid spacing in the spatial dimension is characterized by a single parameter A r = h, and the temporal interval is characterized by parameter A t . Furthermore, grid points are labelled by spatial indices ...i — 1, i, i + ... and temporal indices ...n — 1, n, n + 1,.... The discretized function uh at ordinates (t, r — A r ) will be labelled « n - i ! a t ordinates (t,r) by u n , at (t,r + A r ) by uf+1, at (t + At,r) by and so on. The Taylor expansions of u(t,r — 2 A r ) , u(t,r — Ar), u(t, r) u(t, r + Ar), and u(t,r + 2Ar) evaluated at (t,r) are thus < _ 2 = < - 2 A r « ) _ r + \ ( 2 A r ) 2 « ) > r r - \ ( 2 A r ) 3 « ) , _ + O ( ( A r ) 4 ) , (C.29) = < - A r « ) i r + \ ( A r ) 2 « ) , r r - \ ( A r ) 3 « ) , _ . + O ( ( A r ) 4 ) , (C.30) (C.31) APPENDIX C. NUMERICS 222 2 - 1 i + 1 A n + 1 At Ar n n r F i g u r e C . l : A schematic representation of the finite difference grid in one spatial and one tem-poral dimension. Spacetime coordinates are given by the ordered pair (t,r), grid spacing in the spatial dimension is characterized by a single parameter A r , and the temporal interval is characterized by parameter At. Gr id points are labelled by spatial indices ...i — 1, i, i + 1,... and temporal indices ...n — 1, n , n + 1, = < + A r ( < ) i r + i ( A r ) 2 « ) , r r + ± ( A r ) 3 « ) , _ + O ( ( A r ) 4 ) , (C.32) < + 2 = < + 2 A r « ) i r + I ( 2 A r ) 2 « ) > r r + i ( 2 A r ) 3 « ) > r r p + O ( ( A r ) 4 ) , (C.33) where subscript notation (tt™) r designates the partial derivative of u(t, r) with respect to coordinate r evaluated at point (t,r), (uf) r r the second partial derivative of u(t,r) with respect to r at (t,r), and so on. Then, as an example, to find the centered O ( ( A r ) 2 ) approximation for the F D A (u") r , solve the linear system of three equations (C.30), (C.31), (C.32). The solution in this case is simply 2Ar + o ( ( A r ) 2 ) . (C.34) Using this technique, the centered O ( ( A r ) 2 ) approximation for F D A («") r r could also be ob-tained. It is given by K),r=<+1~(2^f+UU+0((Arf). (C.35) Analogous expressions hold for F D A s involving derivatives of t. For example, the centered F D A for the first derivative in time is 2At + o ((AO 2 ) • (C.36) APPENDIX C. NUMERICS 223 Notice the similarity between these F D A s and a centered limit definition of the derivative, du • tt(r + A r ) - t , ( r - A r ) dr Ar-^o 2 A r As another example, consider the one- sided o ' ( ( A r ) 2 ) approximation for (uf) r which makes use of Jt™, u™ + 1, and u™+2- Solving (C:31), (C.32), and (C.33) obtains ( « ? ) . r = " < + 2 +TIr+1 " 3 < + ° ((Ar)2) • ( C - 3 8 ) This so-called forwards difference operator allows me to evaluate (u™) r at the point on the finite difference grid with smallest value of coordinate r. Similarly, the backwards difference operator for (w n) can be obtained by solving the system (C.29), (C.30) and (C.31). This operator allows me to evaluate (uf) r at the largest value of coordinate r and has form ( < ) ^ < " 2 " S 1 + 3 < + 0 ( ^ 2 ) - <c-39> C. l .4 The Leap-Frog and Crank-Nicholson Schemes As an example P D E consider the advection equation Ov On . ( C - 4 0 ) which represents the passive advection of scalar field u carried along by a flow of constant speed v. Since later chapters of this dissertation deal almost exclusively with P D E s first order in both time and space, the advection equation is a useful prototype. Using the results of the previous subsection, this P D E could be represented with F D A < + 1 -2 A i 0\[(Atf) = v (< + 1 2 A r <-1) + O ( (Ar-) 2 ) . . (C.41) The stencil of an F D A is defined as the discrete set of spacetime points on which the F D A depends. Since this F D A employs values uri_1, w™+1 and un+1, it has a four-point stencil as represented in Figure C.2. This scheme is known as leap-frog and is one of the more common finite difference schemes used in solving P D E s . Since the value of u in the future, u n + 1 , is determined entirely from values of u in its past, this scheme is explicit. Being explicit, the value of can be obtained in a single step. Instead of the leap-frog scheme, iterative Crank-Nicholson wil l be used to obtain the majority of results presented in this dissertation. The principle reasons for this alternate choice are initial data and stability. While the leap-frog scheme must be initialized with data from two time steps, u™+ 1 and i t " - 1 , the Crank-Nicholson scheme is initialized from values at a single time step, and w™+1. Second, the Crank-Nicholson scheme is known to be stable 4 for a wide range of F D A s , given grid spacing A r and temporal interval A i . The Crank-Nichols on scheme is defined by modifying the temporal derivative operator to be centered at the virtual time step t + \At. A straightforward derivation obtains t = +0 ( ( A i ) 2 ) , (C.42) 4The issue of stability of a solution scheme is briefly discussed in subsection C.1.5. APPENDIX C. NUMERICS 224 i - 1 At Ar n + 1 n n — 1 r F i g u r e C.2: The four-point stencil of the leap-frog scheme for the advection equation. The stencil is marked by dashed lines and circles. Gr id points are labelled by spatial indices ...i — 1,... and temporal indices ...n — l,n,n + 1,.... The scheme employs values and of function u. Spacetime coordinates are given by the ordered pair (t,r), grid spacing in the spatial dimension is characterized by a single parameter A r , and the temporal interval is characterized by parameter A i . APPENDIX C. NUMERICS 225 i - 1 i + 1 n + 1 At Ar n n — 1 F i g u r e C . 3 : The four-point" stencil of the Crank-Nicholson scheme for the advection equation. The stencil is marked by dashed lines and circles, and is centered about the virtual point marked with a dashed square. Gr id points are labelled by spatial indices ...i — 1, i , i + 1,... and temporal indices ...rc — 1, n, n + 1,.... The scheme employs values v-i'-i and of function u: Spacetime coordinates are given by -the ordered pair (t,r), grid spacing in the spatial dimension is characterized by a single parameter A r , and the temporal interval is characterized by parameter A i . n-\- — where ut 2 designates the value of u centered spatially at point labelled i and temporally at a virtual point between rc and rc +1 . The expression of advection equation (C.40) as an F D A is then accomplished by evaluating spatial derivatives at this intermediate time step through a temporal average. The F D A thus becomes , n + l A i + o (At)2) V / u. i+1 2 A r vn+1 ui+i vn+1 ui-l 2 A r 0 ( ( A r ) 2 ) . (C.43) This scheme has a four-point stencil as displayed in Figure C.3. Since this scheme couples values at the future time step, w™^1 and , it is called implicit. Being implicit, values of u in the future cannot be obtained in a single step. Rather, they must be obtained with an iterative procedure. C.1.5 Stability As mentioned in the previous subsection, one reason for choosing the implicit Crank-Nicholson scheme is because it tends to minimize instabilities for many PDEs . Since a detailed discussion of stability for finite difference schemes is beyond the scope of this dissertation, the conditions for stability wil l not be explicitly calculated for the P D E s used in this thesis, but instead wil l introduce the concept of stability at a heuristic level and mention an important result known as the C F L condition. Technical aspects and detailed discussions of stability can be found in [91], [103], [61], [134], and [30]. A P P E N D I X C . N U M E R I C S 226 It is entirely possible to correctly implement a finite difference scheme for a time dependent problem, yet have the solutions plagued by floating point overflows at some point in the evolution. Providing there are no instabilities in the P D E being solved, the overflow wil l be as a consequence of the nature of the finite difference scheme itself—some schemes are inherently unstable for a given class of problem while others can be conditionally unstable. As an example, consider the previously discussed advection equation (C.40) with periodic boundary conditions. A property of the advection equation is that a norm of the solution wil l not change significantly in time. That is to say, \\u(t,r)\\ ~ ||u(t + A t , r ) | | ~ | |u(i + 2 A i , r ) | | ~ ••• . (C.44) Equation (C.44) wil l be the working definition of stability. In fact, a system would be considered stable even if solutions grew as a polynomial in | |u(t,r)| | . However, exponential growth in the norm is certainly considered unstable. If the F D A to the P D E exhibiting property (C.44) is convergent, the discrete solution is expected to exhibit the analogous behaviour I K I M K + 1 i l ~ l l < + 2 l l ~ - - - - (C45) It would then be hoped solutions obtained from a scheme such as leap-frog, < + 1 = v £ K + i - + < _ 1 + O ( ( A i ) 2 ) + O ( ( A r ) 2 ) , (C.46) would exhibit property (C.46). As it turns out, there are values of the Courant factor, for which (C.45) is violated. In fact, (C.46) is unstable for all A > n - , (C.48) Now, if instead of leap-frog, the Crank-Nicholson scheme U i ~ 4 Ar H+i + u ? £ - u ^ ) + u ? + 0((At)2)+0((Ar)2) (C.49) is implemented, stability will be observed for all values A. It is in this sense that Crank-Nicholson is considered unconditionally stable, in support of the previous claim that the Crank-Nicholson scheme tends to result in an F D A with improved stability. To obtain the preceding stability conditions one needs to consider more detailed study of the heuristic stability condition, or alternately perform a von-Neumann stability analysis. Such analyses become complicated for nontrivial (i.e., nonlinear) P D E s like those used in later chapters of this thesis. The complexity of the analyses is further increased when nontrivial boundary conditions beyond periodic or Dirichlet, typical in systems of interest to numerical relativity, are applied to the PDEs . The reader is again referred to [91], [103], [61], [134], [30], [21] and [22]. In closing this discussion on the stability of finite difference schemes, it is worth mentioning one important result extracted from the von-Neumann stability analysis of (C.46)—the CFL condition, as derived by Courant, Friedrichs and Lewy in 1928. The condition is a statement that all A > 1 are unstable for an explicit scheme, and carries the interpretation that the physical domain of dependence of a P D E must be entirely contained within the numerical domain of dependence if the scheme is to be stable. That is, Courant factor, A, must be adjusted for the P D E under consideration, and it must certainly be less than unity. Note that since an implicit scheme couples values at the future time step, they do not possess similar interpretation for a single iteration. A P P E N D I X C . N U M E R I C S 227 F i g u r e C . 4 : The C F L condition for a stable leap-frog scheme. Gr id points are labelled by spatial indices ...i — 1, i, i + 1,... and temporal indices ...n — 1, n, n + 1,.... Spacetime coor-dinates are given by the ordered pair (t, r), grid spacing in the spatial dimension is characterized by a single parameter A r , and the temporal interval is characterized by parameter At. The physical domain of dependence is shaded dark grey, while the numerical domain of dependence is shaded light grey. Note the physical domain of dependence is entirely contained by the numerical domain of dependence. The C F L condition is represented pictorially for the leap-frog scheme of the advection equation in Figure C.4. C . l . 6 Kreiss-Oliger Dissipation Operators This section on finite difference techniques will now be ended with discussion of Kreiss-Oliger dissipation operators. To start, note that the advection equation (C.40) admits normal mode solutions u{t,r) ~ e i k ( r + v t ) (C.50) with wavenumber k, which do not change in time. The solutions thus are said to be non-dissipative. Some F D A s possess similar non-dissipative character, while others do not. The leap-frog scheme in particular is non-dissipative. The dissipative character of finite difference schemes is considered because, especially for nonlinear PDEs , propagation and growth of high frequency components of a difference solution can lead to instabilities. It is therefore advantageous to ensure the scheme employs some form of dissipation to stabilize the solutions. The key point is to do so in a way that affects only the high frequency components, thereby mimicking the behaviour of a low-pass filter. Again revert to discussion of the leap frog-scheme (C.46). If (C.46) is modified as per <+1 = « + 1 - utj+u-r1-^ « + 2 - 4 < + 1 + 6 < - 4 < _ ! + uU)+0 ((At)2)+0 ((Ar)2) (C.51) A P P E N D I X C . N U M E R I C S 228 for adjustable parameter 0 < e < 1, the additional terms do not modify the truncation error to leading order. The new scheme, does, however modify the amplification of modes with wave number f = kAx to be l - e s i n 4 | . (C.52) The addition of Kreiss-Oliger dissipation operator - ^ « + 2 - 4 < + 1 + 6 < - 4 < _ 1 + < _ 2 ) , (C.53) to the scheme thus acts as a low-pass filter with relatively sharp cutoff as £ —> TT. For brevity, Kreiss-Oliger dissipation will not be discussed in more detail, but will note that since it does not modify truncation error at second order, it can be safely employed for all second-order accurate schemes. In particular, Kreiss-Oliger dissipation is used everywhere P D E s are solved in this dissertation with the Crank-Nicholson scheme, solve are nonlinear, there is significant likelihood that instability from high frequency modes would be an issue. Instead of explicitly determining the conditions for stability of the chosen F D A s , an experimental approach as adopted where some choice is made for A < 1 and e < 1 and evolve the system. If the solutions blow up then A and e are adjusted until the system evolves in a stable manner. Once stability is achieved the values of these adjustable parameters are left unchanged for future evolutions. The advantage to using adjustable dissipation, as per Kreiss-Oliger, is that the amount of dissipation can be increased to the desired amount without applying so much dissipation that physical (i.e., lower-frequency) modes of the solution wil l be affected. C.1.7 Iterative Solution Scheme for Evolutionary Equations A n explicit scheme such as leap-frog is soluble in a single step. Implicit schemes, however, couple functions at adjacent spatial-grid points at the future time step and must therefore be solved it-eratively. The Crank-Nicholson scheme is an example. For demonstrative purposes, consider the Crank-Nicholson scheme for advection equation (C.40) with the addition of Kreiss-Oliger dissipa-tion. The O ((A*)2).. O ( ( A r ) 2 ) F D A is < + 1 = ^ ( - < + 2 + 4 < + 1 - 3 < - < + 2 1 + 4 < + 1 1 - 3 < + 1 ) + < fori = l , (C.54) < + 1 = ^ « + 1 - < _ 1 + < + Y - < - + i 1 ) + < f o r i = 2, (C.55) <+1 = \i% ~ + U ^ - U ^ + u * ~ h -4<++i1+6<+1 ~ A u ^ + U ^ - g | « + 2 - 4 < + 1 + 6 < - 4uj_i + < - 2 ) for 3 < % < J - 2, (C.56) < _ 1 + < + + 1 1 - < _ + 1 1 ) + < for i = J - l , (C.57) 4 < _ 1 + 3 < + < + 2 1 - 4 < + 1 + 3 < + 1 ) + < for i = J, (C.58) where the dissipation operator has been time averaged. The strategy of the iterative solution scheme begins with a specification of initial data for 1 < i < J, and an initial guess for where again 1 < i < J. (C.55) through (C.58);are then employed to obtain the first iterate of , " + 1 4 A r v At 4 A r + 1 ' ( < - 2 A P P E N D I X C . N U M E R I C S 229 solution Designating the fcth iterate of as u™ , a proper iterative scheme generates r ., , n+UO) n + l ( l ) n+l(2) n+l(fc) . , , n , . n+1 • a sequence of iterates ui , ^ , U j , u i , converging upon the solution m the limit k —» oo. Two issues need further discussion: (1) the initial guess; and (2) the generation of the sequence of iterates. For evolutionary equations it is usually valid to assume the solution undergoes little change between the nth and (n + l ) t h time steps. So without prior knowledge of the evolution, a reasonable choice is simply to use the values from the step in the immediate past as an initial guess. That is, u n + m = u „ ( C 5 9 ) • the iteration count of the past time step being omitted from the notation. This resolves the first issue. The second item now warrants discussion. In the most trivial scheme known as Jacobi iterative, the sequence of iterates is defined entirely from values of the preceding iterate. However, it can be shown that faster convergence can be obtained if the method uses the most recent iterates as they become available. This is the essence of a Gauss-Seidel iterative scheme. For example, assuming the order of updates proceeds i = 3 , J — 2, the sequence of Gauss-Seidel iterates for equation (C.56) wil l be generated according to + 3^ ( 6 < + 1 ( f c ) ) = ^ ( ^ 1 - t 4 L 1 + C i 1 ^ 1 ) - « ( f c V < - h K+*1(fc-1) - 4<+"1(fc_1) - 4w"-1(fc)+<-1(fe)) - 3 ^ « + 2 - 4 < + 1 + 6 < - 4 < _ 1 + < _ 2 ) . (C.60) While alternatives such as Newton's method converge more rapidly, they typically require a more accurate initial guess u ™ + 1 ^ f c \ and are somewhat more difficult to implement. Gauss-Seidel is found to be a sufficient method and is used in obtaining all results presented in this thesis. This method has been implemented using the Rapid Numerical Prototyping Language (RNPL) [94], [95], [96],. [97]. In brief, R N P L is a high-level language which can be used to handle many details common to most numerical codes. For example, R N P L can be used to manage parameter fetching and memory allocation, along with input, output and update routines, allowing the user to focus efforts on aspects more interesting to physics and numerical relativity. After the user specifies (in the R N P L syntax) parameter names and types, a coordinate system, grid structure, grid function names and types, finite difference operators, evolutionary equations and an initialization scheme, R N P L can be used to generate an essentially complete F O R T A N or C code. In the case of an iterative scheme, R N P L automatically implements,Gauss-Seidel. The user thereby is saved from expending time on everything except for those aspe'cts of greatest interest to the typical numerical relativist. Refer to [94], [95], [96], [97] for details and examples of R N P L ' s capabilities and limitations. But one important issue to note by design is R N P L ' s lack of support for the solution schemes appropriate to elliptic equations. Therefore, users are required to implement their own methods for solving constraint equations. C.2 Constraint Equations Constraint conditions typically form systems consisting of a elliptic equation to be solved in a domain with boundary conditions and sources in its interior. Boundary conditions are typically Dirichlet (the boundary value of the unknown function is specified), Neumann (the normal deriva-tive of the function is specified) or Robin (an algebraic equation is given which relates the value of the unknown function to its normal derivative on the domain boundary). The canonical example A P P E N D I X C . N U M E R I C S 230 of a constraint condition is the Poisson equation i V 2 u = p, (C.61) where V 2 is the Laplacian operator, and u is to be solved on a- domain with specified source function p and where conditions for u are given on the boundary dfl. The Laplacian operator is a second-order P D E in the spatial coordinates. For example, the flat space Laplacian on u in spherical polar coordinates (r, 9, <j>) is given by 1 d ( 2 d u \ . 1 d ( . ndu\ ; 1 .d2u V 2 u = - 2 ^ [ r 2 ^ + - ^ - ^ s i n f l ^ + (C.62) r2 or \ or J r2 sin 9 89 \ 89 J r2 sin 9 oq>2 Finally, notice that the Poisson equation is linear in u. Depending on the symmetries involved, this linearity in u results in an F D A which can be solved by direct or indirect methods as mentioned below. [125] and [126] form the set of references for direct methods, and the primary set of references for discussion of iterative methods also includes [150], [143] and [146]. C.2.1 The Poisson Equation in Spherical Symmetry Restricting the system to spherical symmetry, u'=u(r), (C.62) becomes „ 2 1 d I' 2 0 u \ 2du d2u , n _ r2 dr \ dr J r dr dr2 Then using the F D A s of subsection C . l . 3 , the derivatives in (C.62) can be expressed to 2nd order accuracy using three-point stencils. Then, using the established notation, the one-dimensional finite difference equation ^-i+r-^r-o+o((^) ,«,«, is obtained. Substituting (C.64) into (C.61) the elliptic to be solved becomes n V 2 A r ) \ (Ar) Then if coordinate r in the computational domain Q ranges [7*1, rj]'over a grid of J points, (C.65) forms a tridiagonal system linear in u in the interior of Q,. However, boundary conditions still need to be specified before the full system can be solved. If the boundary conditions were Dirichlet at both r = r\ and r = rj, the complete system of equations would be = brl, (C.66) unj = bnj, (C.68) where 6 n and 6™ are specified boundary values. However, outside of cases where spacetime has been compactified so that computational bound-aries correspond to spatial infinity, systems of interest to numerical relativity seldom possess con-straints with Dirichlet boundary conditions. Neumann and Robin conditions are far more preva-lent. For instance, where constraint equation (C.61) represents the Gauss' law electromagnetic constraint, outside of a localized charge distribution, the electric potential wil l possess a Coulom-A P P E N D I X C . N U M E R I C S 231 bic dependence in spherical symmetry. This means the form of u will there be u=—-+u0, (C.69) r where u-\ and UQ are constants. The choice of UQ amounts to choosing a convenient location for the zero of the potential. Typically, UQ is chosen so that either u = 0 a t r = 0 o r u = 0 a t r = oo. Then if r = rj is external to sources, the choice UQ = 0 there amounts to setting the zero of potential at r = oo. Equation (C.69) then reduces to u = ^ i . (C.70) r Differentiating (C.70) with respect to r then yields du u-i 1 or rA r or, in finite difference form, '3u5-4«3_ 1 -+u3_ 2 = r t l , (C.72) 2 A r 7 TJ"3' where the backwards difference expression for the derivative has been used. Then solving (C.72) for Uj_2 obtains unJ_2 = - i y — + 3 J u j + AunJ_1,. (C.73) which, substituted into (C.65) for i = J — 1 yields f 2 2 4 2 \ „ / 4 2 \ „ r j r j _ i rjAr rj^Ar (Ar)2 I V rj^Ar. (Ar) PJ-I, (C74) the boundary condition to be applied at r = r j . To obtain a realistic boundary condition at r = r\, notice that for a smooth spherically sym-metric distribution, the charge density p will be expansible in powers of radial coordinate r near r = 0 with even leading order. That is, P = Po + Pir2 + pAr4 + O ( r 6 ) , % (C.75) where po, p2, and pA are appropriately chosen constants. Integrating over p to find qenci, the amount charge enclosed by a small sphere of radius R centered at r = 0 obtains pr=R p(j>~2'K r9=n qenci = / (p0 + p2r2+p4r4 + O(r6))r2 sin 0d6d<pdr Jr=0 J<j>=<j Je=o rr=R = 4TT / (p0+p2r2 + pAr4 + O(r6))r2dr Jr=0 = 4 7 r ( p o | U p 2 ^ + P 4 y + ^ 9 ) ) . (C.76) Then, using (C.75) or (C.76), straightforward integration obtains the amplitude of the electric field, E, at r = R as E = (Vof + P2^- + P ^ + 0 ( i ? 7 ) ) , (C.77) APPENDIX C. NUMERICS 232 and the value of the electric potential, u, at r = R as u = u 0 - [ P 0 ^ + p2^+p4^+O{R&)y (C.78) where UQ is a constant of integration, chosen so that (C.70) and (C.78) agree at the outer boundary of charge density p. However, it is not necessary to solve for UQ. Instead, the required boundary condition can be derived by simply by noting that u is an even function of r near r = 0, and so its derivative with respect to r must vanish. Thus, using the 2nd order forwards difference approximation, obtains - M g + 4 u g - 3 < 2 A r \ • i and thus uH = 4w n - 3w n . (C.80) Then, substituting (C.80) into (C.65) for i = 2 yields " ^ - M < + f ^KZ + 7 ^ ) « 5 = P2 • (C81) y r 2 A r ( A r ) 2 , / \ r 2 A r (Ar) In summary, the complete system of equations has become 4 + 7 ^ 2 ) = P5, (C82) : p" for 2 < i < J — 1, (C.83) r 2 A r ( A r ) 2 y < + 1 - 2 < + < _ ! ( A r ) 2 2 \ Tjrj-x rjAr rj-iAr (Ar) + t ^ + ( S J I j " 5 - = ' 5 - ! C M ) The elliptic system is again linear and still can be written in tridiagonal matrix form, even with nontrivial boundary conditions. However, while being able to rewrite the system in tridiagonal form wil l simplify computations presented in later chapters, the key feature here is the linearity of the system of equations. Being linear, direct solution schemes such as Gauss-Jordan elimination, Gaussian elimination or L U decomposition can be applied to solve for unknowns it™. Being elemen-tary techniques and well documented in the literature, The reader is simply referred to [125] and [126] for detailed discussion. Finally, it is sufficient to state that, after reducing the electromagnetic constraint equation of Chapter 3 to linear tridiagonal form, it is solved using Gaussian elimination 5 followed by back-substitution with routines provided by the L A P A C K software suite [2]. 5 A common source of error with systems requiring many operations is the accumulation or roundoff errors. While an elimination scheme will return a result under such circumstances, the solution will be incorrect by some degree, as can be checked by substituting the result back into the original system. To correct this problem the iterative improvement technique is used as outlined in [125], [126]. A P P E N D I X C . N U M E R I C S 233 C.2.2 The Poisson Equation in Axial Symmetry Now consider the Poisson equation in axial symmetry. Being dependent on just the two coordinate dimensions r and 8, it wil l be expressed 1 <9 / ndu\ 1 d ( . n d u \ _ , o r . ^( r VJ + r w a ? r ^ J = V t t = ^ (c-85) Off boundary, (C.85) can be written as the second order F D A p?tj for 2 < i < J - l , 2<j<K-l, (C.86) the indices i for coordinate r being a member of the positive integer set {1, 2, • • • , J } , indices j for 8 being a member of {1, 2, — , K}, the grid spacing in the direction of coordinate r being A r , and spacing in the direction of 9 being A8. A comma-separated subscript notation has been adopted for grid functions to avoid confusion in situations the likes of Ui12- If the boundary conditions (C.82) and (C.84) are assumed for coordinate direction r, and furthermore ufj is assumed to be an even function in 8 along the line of axial symmetry (8 = 0 and 8 = IT), the complete F D A can 2 H+l,j 2 A r 11171 • 1 / cos#,-smf 2A8 (Ar)2 - 2tiTl • (A8)2 A P P E N D I X C . N U M E R I C S 234 be expressed C0S<92\_1_ + 1 r2 U - ' s i n 0 2 > / (A9) 2 fcos92\ 1 1 \ „ n , 3 —TH T7i + 77772 <2 = P"2 for 1<*<J r 2 V V sin t92 7 AO (A9) at boundary j = 1, (C.87) -Ar + —^ 1 < 7 + ( —4—I —^2 | « 2 , - = P 2 , - for 2 < j < t f - l r 2 A r ( A r ) 2 / ' J I r 2 A r ( A r ) 2 / , J , J n \ 2Ar ) V ( A r ) 2 1 / c o s 0 A 1 ^ < i + 1 - 2 ^ - + ^ _ 1 2A69 / r? V (A6?)2 / r i J 2 / / cos 0K-at boundary i = 1, (C.88)~ for 2 < i < J - l , 2< j < J T - l , (C.89) - 2 I U n fl*-!,/ A6> (AO)2) i<K 2 / fcos6K-i\ 1 , 1 ^ „ „ • , . . . . 7 r? ^ \smOK-iJ AO (A9)2) ' at boundary j = K, (C.90) 2 + rjrj-i rjAr r j _ i A r (Ar) / 4 2 , - U™_ = p5_ for 2 < j < i f - l \^  r j _ ! A r (Ar)2) 3 1 , 3 3 1>J ~ at boundary i = J. (C.91) As displayed in Figure C.5, this F D A has a five-point stencil utilizing grid points •, •, •, and when 2 < i < J — 1 and 2 < j < K — 1. The stencils on boundary are two-point. Stencils for boundary conditions (C.87) and (C.88) are respectively depicted in Figures C.6 and C.7. This system of equations is amenable to any of a number of different solution schemes. Since it can be written in block tridiagonal form, it can be solved using line successive over-relaxation ( L S O R ) 6 , which was a standard iterative method until the 1970s. Again, because it is well doc-3 Line successive over-relaxation is sometimes called successive line over-relaxation (SLOR) by other authors. APPENDIX C. NUMERICS 235 i + 1 A9 0 r Ar F i g u r e C . 5 : The five-point stencil of the second order F D A for the Poisson equation off boundary in the two dimensions r and 9. The stencil is marked by dashed lines and circles. The stencil center lies on the range of indices 2 <i < J—l and 2 < j < K — l. The scheme employs values u™-, w"j_i> and ufj+1 of function u. Coordinates are given by the ordered pair (r,9), grid spacing in direction r is characterized by parameter A r and in direction 6 by parameter AO. APPENDIX C. NUMERICS 236 i - 1 A9 J = 2 J = 3 9 Ar F i g u r e C . 6 : The two-point stencil of the second order F D A for the Poisson equation in the two dimensions r and 9 at boundary 9 = 0. The stencil is marked by dashed lines and circles. Stencil points cover the range of indices 1 < i < J and 1 < j < 2. The scheme employs values u " ^ and u n 2 of function u. Gr id spacing in direction 9 is characterized by parameter A 0 . A P P E N D I X C . N U M E R I C S 237 i = 1 i = 2 i = 3 AO 6 Ar F i g u r e C .7 : The two-point stencil of the second order F D A for. the Poisson equation in the two dimensions r and 6 at boundary r = 0. The stencil is marked by dashed lines and circles. Stencil points cover the range of indices 1 < i < 2 and 2 < j < K — 1. The scheme employs values u n j , and u\\^ of function u. Gr id spacing in direction r is characterized by parameter A r . A P P E N D I X C . N U M E R I C S 238 umented in the literature, The reader deferred to standard texts such as [143] and [150] for the details. In summary, the electromagnetic constraint equation in Chapter 4 is solved using L S O R with zebra-stripe ordering and Chebyshev acceleration.7 Having concluded discussion of finite difference difference schemes for evolutionary and con-straint equations, the topics of adaptive mesh refinement for hyperbolic equations and parallel computation are now addressed, as they have played a crucial role in obtaining the results pre-sented in this dissertation. C.3 Adaptive Mesh Refinement Adaptive mesh refinement ( A M R ) is important for finite difference schemes whose solutions unfold on a wide range of spatial and temporal scales. A M R solves resolutional problems associated with finite computational resources when solving such systems by adding resolution where needed and removing resolution when no longer required. A M R is especially important when resolution re-quirements are not known a priori, as would be the case in discovering new phenomena numerically. In this thesis A M R is employed rather than a nonuniform mesh scheme, thereby circumvent-ing both the problem of complicated F D A stencils and the requirement of a priori knowledge on resolution requirements during evolution. A M R is employed in both spherical symmetry and axial symmetry. In spherical symmetry A M R is implemented using a version of the ad code, developed by Choptuik [25], and first used in [23]. In axial symmetry A M R is implemented using the Parallel Adaptive Mesh Refinement ( P A M R ) code developed by Pretorius [128], [129]. W i t h both the ad and P A M R codes, the user is provided the framework and utilities for A M R , but not the equations of motion for the systems of interest. In both cases it is up to the user to derive the equations and implement the associated F D A s . Both ad and P A M R use an adaptive algorithm based on that of Berger and Oliger first in-troduced in 1984 [7]. The Berger and Oliger ( B & O for short) algorithm is designed for general hyperbolic P D E s and based upon three key elements: (1) the A M R grid hierarchy; (2) a time step-ping algorithm; and (3) dynamical regridding via relative solution error estimate. The modified B & 0 algorithm is now described through its three key elements. C.3.1 The Grid Hierarchy A M R is implemented through a hierarchy of grids of different resolutions but uniform in each grid direction. Each level is assumed to be characterized by a single spatial discretization scale in each of these directions. 8 The mth level of the hierarchy, lm, is defined as the set of grids of resolution hm. In the set of levels, {lm}, where m is'a positive integer (i.e., m £ Z + ) , l\ is the coarsest level, 7For many classes of problem, LSOR was eventually replaced with generally more efficient methods such as multigrid [13]. However, due to the complexity of multigrid methods and issues pertaining to the choice restriction and prolongation operators [125], [126], it can at times be difficult to code a convergent multigrid solver. This is especially true for nonlinear equations or those with much stronger coupling in one coordinate direction than another. When faced with the situation of either obtaining a solution with a somewhat inefficient solver or not obtaining a solution at all, it is better to sacrifice efficiency. This is the approach adopted in obtaining results presented in this dissertation. While a pointwise Gauss-Seidel multigrid scheme was implemented to solve the Poisson equation in Kerr spacetime, the scheme displayed poor convergence for many choices of black hole parameters M B H and OBH-It was subsequently suggested by Choptuik [32] that the poor convergence was likely because the chosen pointwise relaxation scheme is an inadequate smoother for systems with a much stronger coupling in one coordinate direction than another. In such cases it is preferable to use a line iterative method, as in the line Gauss-Seidel multigrid algorithm implemented in [20]. Perhaps the first record of this issue were noted by Brandt in [13]. A complete discussion of the problem can be found in [146]. However, since a free evolution is performed for the electromagnetic variables, the Poisson equation needs to be solved just once during an evolution at the initial time step. Having written a demonstrably robust and relatively efficient LSOR solver, it was not necessary to revisit the multigrid problem. Implementing a line iterative multigrid scheme remains an avenue of future investigation. 8 In the original B&O algorithm, grids could be nonuniform in a coordinate direction and thus characterized by more than a single spatial discretization scale. A P P E N D I X C . N U M E R I C S 239 h < lm < lf-i are intermediate resolution levels, and If is the level with the finest resolution. Successive levels of resolution are related via hm = o-shm+i, (C.92) where as is called the spatial refinement ratio and has properties as £ Z + and as > 2. Additionally assume as is the same in each coordinate direction and for all successive levels. 9 A grid at level lm is called a parent grid and a grid at lm+i a child grid. A child grid must be entirely contained by a parent grid. And finally, note that grids within a level may overlap. The modified scheme used requires all grids to share the same coordinate system, and grid directions to be aligned with the coordinate directions. In addition, this modified scheme requires child grids be aligned so that, in their regions of overlap, all parent grid points are coincident with child grid points. 1 0 Finally, the modified algorithm assumes a single level l\ grid called the base grid covers the entire computational domain. Figure C.8 displays a prototypical grid hierarchy for the case 1/ = h and as =2. The next two subsections describe, in sequence, the time stepping algorithm used to advance solutions in the grid hierarchy, and the dynamical regridding procedure by which it is ensured solutions, are adequately resolved at each time step in the evolution. C.3.2 The Time-Stepping Algorithm The core of a B&O-type A M R scheme is a' recursive time-stepping algorithm in which functions on all parent grids are evolved before those on child grids. In this scheme,, once functions on a parent grid in level lm has been evolved into the future one time step A i T O , the functions on child grids of level lm+i are evolved into the future up to the same time through a sequence of at time steps, where ot is the temporal refinement ratio with properties crt € Z + and at > 2. A constant Courant factor across the hierarchy can be maintained by demanding Atm = atAtm+1. (C.93) This modified algorithm assumes at is the same between each successive level . 1 1 Note that in order to ensure the C F L condition is satisfied, at > as must be chosen. Since functions update on parent grids before those on its children, values on the child grid boundaries off those of the base grid can be obtained by interpolating from function values on the parent grid. This avoids having to supply special conditions off the boundary of the base grid, and allows evolution only with the interior finite difference equations and solution scheme. Also, since the solution error of a function on the parent level near its child grid boundaries should be comparable to that on the child grid itself near those boundaries, the procedure of interpolating values on the child grid boundary from the parent grid solution wil l not result in increased solution error at these locations. Finally, once all functions on child grids have been updated to the future time step of their parents, the child grid function values are injected into the parent grids where their points coincide. This final step ensures the most accurate solution is available on all grids at all times. C.3.3 Dynamical Regridding via Relative Solution Error In B&O-type schemes, resolutional demands are periodically assessed and grids in the hierarchy are appropriately modified. The procedure of modifying grids is known as regridding. The period 9Historically, in B&O the spatial refinement ratio between two successive levels could be non-integer and different between different successive levels. 1 0 The original B&O algorithm allows parent grids to be offset or rotated with respect to the children. 1 1 As with the spatial refinement ratio, in the original B&O algorithm, the temporal refinement ratio between successive levels could differ of be non-integer. APPENDIX C. NUMERICS 240 o e- -e- -e-O O O CD o o o ® • .® • ® C) O o • o o o o () §> o © o o o -e-O O © e e -Q e © F i g u r e C . 8 : A schematic representation of the A M R grid hierarchy in two spatial dimensions. Displayed are the base grid l\ (large circles), along with prototypical levels l2 (small circles) and 3^ (black dots) for the case as = 2. A P P E N D I X C . N U M E R I C S 241 between resolutional assessments should be small enough to track all features of the solution. Since light speed, c = 1, is the upper limit of the speed, at which information should propagate, requirements should be assessed at least every f = j time steps, as this is the time in which a physical signal could spatially propagate a single grid point. The key then is to set a criterion which determines when the resolutional requirements have changed. The approach outlined in [7] is employed by tying regridding to relative solution error estimates. 1 2 In order to establish the measure, each grid, g,.at level lm is first cloned and the clone labelled gm. A second copy of g is then made and its number of grid points coarsened by factor as. The second copy is labelled gm~c'3. The two new grids are evolved into the future until they are next coincident in time. Then comparing a discrete unknown solution on cloned grid copy gm to that on coarsened grid copy gm~as at the first coincident time the core of the measure becomes the relative solution error eh™(u) = (uh™ - ua°hm) + O (hp), (C.94) where u h m is the discrete solution on gm, u G s h m is the solution on gm~ai>, and p G Z+ again being the order of the F D scheme. The criterion whether to regrid is then based on the magnitude of an appropriate point-by-point norm of the relative solution errors of some set of grid functions on each grid g. This point-by-point norm of the set of relative solution errors is denoted r f f , and is formally T f l = X > h m ( u ) H - ^ (C95) u This is a measure that can be used dynamically during the time stepping procedure. Once Tg has been calculated for all grids in the hierarchy, the information is passed to a clustering algorithm. Essentially, the clustering algorithm begins by nagging points at which r g exceeds some predefined tolerance. The sets of flagged points then form the regions where child grids gm+(7'> are to be introduced. Conversely, the unflagged points (where rg falls below the tolerance value) define regions where the grid structure is to be coarsened by elimination of grids gm. It is by this method that the grid hierarchy is refined. For more detail on clustering algorithms, and related topics of buffer zones and filling factors see [127] and [8]. Finally, after the grid points have been appropriately clustered and the hierarchy refined, function values are transferred .from the old hierarchy to the new by appropriately interpolating grid functions from old parent grids, gm, to new children, gm~Gs, and conversely by injecting grid function values from eliminated children onto their parent grids. This concludes discussion of the B&O-type A M R algorithms used in this thesis. For additional details the reader is referred to [7] and [127]. C.4 Parallel Computing Programs traditionally have been written for serial computation defined as the sequential execution of an instruction set by a single central processing unit (CPU) . Since the serial model allows just a single one-instruction execution at any moment in time, large calculations inevitably suffer from both both limited computer memory and long total execution time. The goal of an alternate com-putational model, such as parallel computation, is to delay, if not eliminate, the user experiencing the effects of these limitations. In the most basic form, the definition of parallel computing is the simultaneous use of multiple computing resources to solve a problem. Where required, the P A M R framework ([128], [129]) is adopted to parallelize codes used to obtain results for this dissertation. 1 2What has here been defined as the relative solution error historically has been referred to as the local truncation error (cf. [7]). A P P E N D I X C . N U M E R I C S 242 C.4.1 Parallel Program Design P A M R provides support for managing load distribution and communication for codes dependent on a grid structure, thus providing a degree of automation for programmers wanting to parallelize finite difference codes. P A M R can divide a domain of grid points fl with boundary dfl into a specified number of subdomains fli, Q.2,--- fls with respective boundaries dfli, dfl2,--- dfls. Using this option for domain decomposition, P A M R assigns the subdomains in a manner that contain, as closely as possible, equal volumes of grid points with minimal area. 1 3 In the case where s > 1, P A M R attaches ghost regions of specified width to the interior subdomain grid boundaries. These ghost regions allow points within adjacent grids to maintain their couplings. For example, assuming the range of index i for coordinate rj on fl is 1 < i < J, the centered three-point finite difference stencil for F D A requires addition of a single grid point in the direction parallel to coordinate rj for each subdomain boundary having 2 < % < J — 1. A n example grid decomposition is displayed in Figure C.9. Then, after being supplied with a modest set of additional information, P A M R can be used to compile a code for use on a set of s compute nodes, each subdomain being uniquely assigned to one of these nodes. 1 4 Providing the program is not partitioned between so many nodes that inter-node communication becomes an execution bottleneck, considerable increase in performance can be obtained with a code parallelized as above. This concludes discussion of numerical methodology as it pertains to the work present in this dissertation. 1 3 For AMR codes, PAMR can either partition each grid of each level into s subgrids and distribute them among the s compute nodes, or partition the set of all grids into s subgrids for distribution across the s nodes. The efficiency of either option will depend on the complete grid structure at a given time. .14Note that while PAMR uses domain decomposition, an alternate division is functional decomposition where the objective is to equally partition the work among nodes based on the cost of each function within the program. A P P E N D I X C . N U M E R I C S 243 Q C) -e-o -e--e-O ^3 © Q--e- * — a •e o C ) da, O -e--e-o ft, O 4) O -Q © F i g u r e C .9: Schematic representation of domain decomposition. The global domain Q. is divided into four subdomains flij, ^ 2 , ^ 3 and flA. The boundaries dQi dQ.2 dfl^ dfl^ are indicated with arrows. Ghost regions of a single grid point have been added to interior subdomain boundaries. Regions where two subdomains overlap is shaded light grey, and regions where four subdomains overlap is dark grey. Each of the subdomains contains an equal number of grid points, and is of minimal area. A P P E N D I X D . M I B C O O R D I N A T E S 244 APPENDIX D MIB COORDINATES D.l The Monotonically Increasingly Boosted (MIB) Coordinate System In this section I study boundary conditions for the massive electromagnetically coupled complex scalar field in fiat axial symmetry. Again I employ the metric signature (—, +, +, + ) m accordance with Hawking and Ellis [62] and Wald [145]. Ultimately, to study a long numerical evolution of the electromagnetically coupled complex scalar field system on a finite coordinate domain, I must properly impose conditions on the computational outer boundary. Many techniques have been tried by others, but the most effective numerical treatment of the outer boundary on a finite computational domain perhaps is through the use of numerical dissipation in conjunction with the Monotonically Increasingly Boosted (MIB) coordinate system first introduced by Honda [77], and later published in [78]. The M I B coordinate system leaves the interior of the computational grid unchanged, but outwardly boosts the coordinates of points in an exterior region to approximately the speed of light with respect to the interior. While the M I B coordinates work to effectively freeze any radiation, numerical dissipation of the Kreiss-Oliger style can be added.to quench this radiation. The problem of solution contamination by reflections off an outer boundary is thereby eliminated. The Minkowski metric in the usual (t, p, <f>, 2 ) cylindrical coordinate system is summarized by ds2 = -dP + dp2 + p2dj>2 + dz2 . (D.l) The axially symmetric M I B coordinate transformations employed in [77] and [78] are i = t , p = p+f(p)t , 4> = <j> , z = z+g{z)t , (D.2) where f(p) is a monotonic function that interpolates between approximately 0 and 1 near a char-acteristic cutoff Pc, and g(z) is a monotonic function that interpolates between approximately 0 and 1 near characteristic cutoff zc and between approximately 0 and —1 near characteristic cutoff —zc. That is, ><H! £ '>>':) <D3) and {—1 for z -C — zc 0 for -zc < z < zc > . (D.4) 1 for z 3> zc J APPENDIX D. MIB COORDINATES 245 In these M I B coordinates the metric becomes dS2 = (-l + f(p)2+g(Z)2)dt2 +2f(p){l + dpf(p)t)dtdp +2g(z)(l + dzg(z)t)dtdz +(l + dpf(p)t)2dp2 + {p + f(p)tfd4>2 +(1 + dzg{z)t)2dz2 , (D.5) or, in the language of the 3 + 1 formalism, ds2 = (-a2 + y2pP"2 + ^2)dt2 +212pf3pdtdp +2-y2zz/3zdtdz . +l2Ppdp2 2 JJ2 where +iUdr Wzzdz2 , (D.6) a(t,p) = l, Pp(t,p)~ f { p ) 1 + dpf(p)t pZ{t'z)=lTd~gTzjt' lPP{t,p) = l + dpf(p)t, (D.7) -r<p<t>{t,p) = P + f{p)t , lzz{t,z) = 1 + dzg{z)t . The unit-norm vector to spacelike hypersurfaces is n ° = ( l , - / ? " , 0 , - / 3 z ) , (D.8) and the spatial metric is described by the hypersurface line element da2=12dp2 + 1l4>d4>2 + 12zzdz2. (D.9) The timelike Ki l l ing vector field of Minkowski spacetime is now ta = na, = (0,0,1,0) is the axial Ki l l ing vector field, and sz = ( 0 , 0 , 0 , 7 J 1 ) is the spacelike Ki l l ing vector field in the direction of coordinate z. Furthermore, I make use of the same interpolating functions as in [77] and [78], f(p) _ [ 1 + t a n h ~ p c ) , 6 p ) ] [ 1 + t a n h ; (D.10) tanh((z-z c ) /<5 z ) , tanh ((z + zc)/5z) 9(z) = 2 h 2 ' (D-n) where parameters Sp and 5Z set the length scales over which interpolation takes place. The in-terpolating functions f(p) and g(z) for this system are displayed in Figure D . l and Figure D.2 respectively. Clearly, the radially outgoing and ingoing null geodesies along z = 0 for spacetime metric (D.6) A P P E N D I X D . M I B C O O R D I N A T E S 246 F i g u r e D.l: M I B interpolating function f(p). The shaded region represents pc ± 5P. are given by Xp±(t,p) = -0"±— , (D.12) 7pp while the axially outgoing and ingoing null geodesies along p = 0 are given by \'±(t,p) = -P'±— • CD-IS) *Jzz The null geodesies with f(p) and g(z) given by (D.10) and ( D . l l ) are displayed in Figure D.3 through Figure D.6 below. Observe that, as desired, £ ( D - u ) and 0 for z ~ — 2 C A | ( i , / 9 ) ~ - ( ± l for -zc<z<zc) . (D.15) 0 for z ~ zc A P P E N D I X D . M I B C O O R D I N A T E S 247 F i g u r e D . 2 : M I B interpolating function g(z). The shaded regions represent zc±Sz and — zc±6z. A P P E N D I X D . M I B C O O R D I N A T E S 248 F i g u r e D . 3 : Radially outgoing geodesic \p+(t,p) in the M I B coordinates at times t = % i — 1, t = 10, and i = 100. The shaded region represents pc ± 5P. A P P END IX D. MIB COORDINATES 249 Figure D .4 : Radially ingoing geodesic Xp_(t,p) in the M I B coordinates at times t = 0, t = 1, i = 10, and t = 100. The shaded region represents pc±6p. APPENDIX D. MIB COORDINATES 250 Figure D.5: Axially outgoing geodesic Xz+(t,z) in the MIB coordinates at times t = 0, t = 1, t = 10, and t = 100. Shaded regions represent zc ± 5Z and —zc±8z. A P P E N D I X D . M I B C O O R D I N A T E S 251 Figure D.6: Axially ingoing geodesic \z_(t,z) in the M I B coordinates at times t = 0, t = 1, t = 10, and t = 100. Shaded regions represent zc ± Sz and — zc ± Sz. A P P E N D I X D. MIB COORDINATES 252 D.2 Equations of Motion The system described by Lagrangian CM = v ^ H V a S - zeAa$)(Va$* + ieAa$*) - ^FabFab - m$2$*$] , (D.16) yields equations of motion V a (V0$ - ieAa$) - ieAa (Va$ - ieAa$) - m£<f> = 0 (D.17) for complex scalar field 4? = ®(t, p, z) (i.e., <3> = 4>i+i<f>2, where cf>i = 4>i(t, p, z) and fo = fait, p, z)), and evolutionary equations V a F o b - ie($*Vfc$ - $Vb$*) - 2e2$$*Ab = 0 (D.18) for four-vector potential A a = Aa(t, p, z). The Lorentz gauge condition V a A a = 0 (D.19) can be used to evolve the temporal component A t . I continue to used the established notation where rn$ is the scalar field mass parameter, e the electromagnetic coupling parameter, the symbol * denotes complex conjugation, and Fab is the antisymmetric electromagnetic field strength tensor. Explicit calculation of (D.17), (D.18) and (D.19) for metric (D.5) obtains a fairly complicated set of expressions. Defining auxiliary variables, II* (t, p, z) = n a l p p l z z (Va$ - ieAa<$>) , ' (D.20) $p(t,p,z) = dp$ , (D.21) $z(t,p,z) = dz$ , (D.22) and function f^(t,p,z) = ^ , (D.23) equation (D.17) can be recast in the first-order form1 l i , = ~ 1 9 1 H ( + ^ £ ($„ - ieAp$) + | - (Vn* + ^  (4>z - ieAz3>) oz \ T 2 2 -ie f ^ i ($p _ ieAp$) A p + ^ ($2 - ieAz$) A z - II* (At - f3"Ap - [3Z Az) \lpp Izz -iel£pglA*$\ _ JH^ _ 7 P P 7 „ M | $ , ( D . 2 5 ) 7,4,4 / 14>4> dp \lpplz ®p = —A n * + i e A t ® + Pp ( $ p - ieAp®) + Pz - i e A ^ ) ) > (D-26) $2 = 4- (——n* + i e A * $ + ^ p ( $ p - ieAp®) + Pz - i e A ^ ) ) » (D-27) \lpplzz ' *I use the chain rule l= 2 ^ ( a 2 4 ) to eliminate factors p 1 that would otherwise appear in expressions such as (D.25). A P P E N D I X D . M I B C O O R D I N A T E S 253 * = n * + ieAt$ + (3p ( $ p - ieAp$) + (3Z ( $ 2 - ieAz$) , (D.28) Ipplzz where the overdot, ' , again designates pa r t i a l differentiation w i t h respect to coordinate t. S imi la r ly , defining Jl„(t,p,z) = na~/pp%z ( V Q A P - V P A Q ) , Tl^(t,p,z) = n a j p p j z z (V a A^, - V<t,Aa) , Uz(t,.p,z) = n a l p p l z z { V a A z - V z A a ) , (D.29) Atp{t,p,z)=dpAt , (D.30) Atz(t,p,z)=dzAt , . ' (D..31) Apz(t,p,z) =dzAp , (D.32) A4>p(t,p,z) = dpA4> , (D.33) Alj>z(t,p,z)=dzA4> , (D.34) Azp(t,p,z) =dpAz , (D.35) (D.18) can be rewri t ten ii„ ^-'Ypp'JzzJp > (D.36) ^2 1,7PP7* -LTp + A t p - / 3 2 ( A 2 p - A p z ) , ^ P = Ipplz -Up + A t p - p z ( A z p - A p z ) \ (D.37) (D.38) 7zz / "f<t>4> d_ ''dp V7PP 92 (D.39) A ^ P Aw, d_ dp \lpplz d ( 1 dz V7pp7; 1 -II* + + f3zA4 Ipplz Tlx + pPA^p + pzA^z , (D.40) (D.41) (D.42) n 2 U<t> d{p2) iu ( /? p n z + ( A 2 p - Apz] Ipp + A( /3.n,) + n,^(^)-^n, (D.43) A P P E N D I X D . M I B C O O R D I N A T E S 254 Azp = ^ - ( ^ — U z + A t z - p p ( A p z - A z p ) ) , (D.44) dp \lpplz Az = -^—Uz + Atz - (3" (Apz - Azp) , (D.45) "fpplzz where the conserved four-current, again defined -Ja = i e ( $ * V a $ - $ V a $ * ) + 2 e 2 $ $ * A a , (D.46) has spat ia l components J p = - i e ( $ * $ p - $ $ * ) - 2 e 2 $ $ * A p , (D.47) = - 2 e 2 $ $ * A 0 , (D.48) Jz = -ie($*$z - $ $ * ) - 2e2®$*Az . (D.49) W h i l e (D.36), (D.39), (D.43) provide the equations of evolut ion of the spat ia l components of Aa, the value of At at future t ime can be obtained either by solving the electromagnetic constraint , or by solving the Loren tz gauge condi t ion . Def ining £lt(t,p,z) = na-ypp-yzzAa , (D.50) the Loren tz gauge condi t ion can be recast as n- - 2^(A^'+tA'))+U0'a'+1tA')-^n- (D-51) Atp = ^ - ( ^ — Q t + f3^Ap + /3zAz), . ( D . 5 2 ) dp \lpplzz J Atz = (—— Slt + (3pAp + f3zAz\ , (D.53) oz \ 7 p p 7 z z / At = -^—Qt + (3pAp + (3zAz. (D.54) Ipplzz D.2.1 Physical and Conserved Quantities W h i l e the preceding equations of mo t ion have been wr i t t en i n terms of the four-vector potent ia l Aa, the phys ica l fields are themselves the electric field Ea and the magnet ic field Ba. For an observer mov ing w i t h four-velocity na, the measured electric field components are Ep = — I T P , (D.55) Ipplzz Ej, = —17^ , (D.56) "fpplzz Ez = — L T Z , (D.57) Ipplzz A P P E N D I X D . M I B C O O R D I N A T E S 255 Bp = 7PP A . B,p = Ipplzz Bz = A(pp lppl<t><p while the observed magnetic field is " z , . (D.58) -Azp) , (D.59) (D.60) The temporal component of the current four-vector is Jt = -ie(—— ($*n* - + /?p0*$p + pz ($*$2-$$:) \ Ipplzz -2e2 (/3P$$*AP +f3z$$*Az) , (D.61) and the observed charge density is PQ = -ie—^— ($*LLj, - $174) • (D.62) Ipplzz The stress-energy tensor for the system described by (D.16) is Tab = i ( V a $ V 6 $ * + V f c $ V a $ * ) - ^ e [ ( $ V 6 $ * - $ * V 6 $ ) A a + ( $ V a $ * - $ * V a $ ) A 6 ] +e2$$*AaAb + \FacFbdgcd - i f f a b [ ( V c $ - i e A c $ ) ( V c $ * + i e A c $ * ) + j F c d F c d +m* 2$*$] . (D.63) Contracting Tafe with na = (1, — /3 P, 0, — /32) and i a = na yields the conserved energy density PT = PT(§) + PT(A) + PT($A) , (D.64) PT(<P) being the contribution from scalar field terms, PT(A) the contribution from electromagnetic field variable terms, and PT($A) being the contribution from terms involving the electromagnetic interaction of $ and Aa- Explicitly, PT(*> = e 2Q(/3 p) 2A 2+/? pr^ ie 1 . , , ™ , . 1 . 1 ] ipp (f3pAp + pzAz - A t ) mi - $*n*) + > 2 $<r + ^ ^ - $ p $ ; ^ Ipplzz 7  ^ "22 ^ Ipp Izz • . • • 1 _ \ _ 2 1 1 ^ 1 1 „ 2 1 . 1 ^ 1 1 A 2 P T ( A ) ~ 7 Z2~ZrAPZ ~ 0 ^2 ^2 A P 2 ^ 2 p + 7 2 - 2 A(4p + 4 2 2 A<4* + 4 2 ~2 Azp ^ /pp /22 ^ /pp '22 ^ /pp l(p(p ^ '22 /pp IZZ + i ^ n p + i T ^ ^ n ^ + i - ^ - n 2 , (D.ee) I PP I ZZ ipp i(f)(f> IZZ I pp I zz A P P E N D I X D . M I B C O O R D I N A T E S 256 and 2 5 Ci - M 1 A l - m'A'A>+'fA'A'+XAA%+-vf)Al i +/?*A Z A T - | A T 2 Ipplzz iefAfm-^P)+^ pp Izz {p»Ap + {3ZAZ - At) ( $ n | - $*IL*)) . (D.67) - "i ( i f ( $ $ p - $ * $ P ) + - $*<E>z) Z J l Similarly, contracting Tab with n a and = ( 0 , 0 , 1 , 0 ) yields the conserved angular momentum density PS4> = Ps<t>(A) + PS<j>(<i>A) , (D.68) where -1 _l_ . „ 1 1 ipp PS4>(A) = -^-T—-A4,pUp---—rArf>zILz ' • (D.69) /  Izz ^ Ipp Izz is the electromagnetic field contribution, and ^ ( * A ) = - ? - ^ - ( $ I I * - $ * n * ) ( D - 7 0 ) ^ Ipplzz is from the coupling of scalar and electromagnetic fields. Finally, contracting Tab with n° and s" = ( 0 , 0 , 0 , 7 ^ ) yields the conserved momentum density in the direction of coordinate z, PSz = PSz($) + PSz(A) + PSz($A) , ( D . 7 1 ) where ie 1 1 1 PSzm = - | — ( / ? P ^ P + / ^ z - A t ) - $ * $ 2 ) + r ( * z I I £ + *:n$) , ( D . 7 2 ) ^ 7zz ^ 1PP1Z  z Ipplzz * IPP 1</></> Izz and P S Z ( ^ ) = v (—(P"AP + PZAZ - A T ) ( $ $ : - $ 2 ) - -^ V($n| - <rn*)) . ( D . 7 4 ) ^ \ 7 z z lpplzz J From ( 2 . 1 0 4 ) , the total charge f O O /»27T / 'OO from ( 2 . 1 0 8 ) , the total energy f O O /•27T />00 / Z = — OO J 0 = 0 «< p: />O0 <9=/ / / PQ (lppl<t><t>lzz) dpdfidz , (D.75) Jz=-co J<j>=0 Jp=0 sT = / / 7 Pr (ippiulzz) dpdcpdz , (D.76) Jz=-oo Jd>=0 Jp=0 A P P E N D I X D . M I B C O O R D I N A T E S 257 and from (2.109), the total angular-momentum S<t>= / Ps<t> (ippl^lzz) dpd(j)dz Jz=-oo J<p=0 Jp=0 (D.77) and total momentum in direction of coordinate z S z = / / PSz("/ppl4>4>1zz)dpd(l>dz Jz=-cx> J<p=0 Jp=0 (D.78) all wil l be constant. D .2 .2 Regularity, Boundary Conditions and Dissipation From section 2.4 (cf. [6], [45]), regularity for the evolved functions requires <J?Z, LT$, A^, A$z, IT.,*, Az, IIZ, Jz, At, A t z , Qt, J<t>, Erf,, Ez, B^, B z , Jt, pQ, pT, Ps<j> and pSz be, to leading order, even in p on axis (i.e., along p = 0). Furthermore, A^, A$z, LT^, J^, E$ and B^ must also vanish on the axis of symmetry (i.e., they are at least quadratic in p on axis). Meanwhile, remaining variables $ p , Ap, Apz, ILP, Jp, A$p, Azp, Atp, Ep and Bp are on axis odd in p to leading order. For the leading-order-even variables, I enforce regularity on grid points corresponding to p = 0 by applying a quadratic fix as listed in Appendix C. Also, I apply interpolation in the direction of p on the next-to-axis points using a 3rd order Lagrange interpolant as discussed in Appendix C. The additional interpolation helps ensure smooth and stable evolution along p = 0. Variables A^, A<j>z, FT,*, J,*, E$ and B$ are the exceptional cases. For them, I enforce regularity by setting their values equal to zero on axis and imposing a quadratic fix on the next-to-axis points. Meanwhile, regularity of the leading-order-odd variables is enforced by setting the functions equal to zero at p = 0. As with the leading-order-even variables, I apply interpolation on the next-to-axis grid points. Because their counterparts in the massless electromagnetically uncoupled (i.e., e = 0) theory can be expressed as wave equations, I could always apply the (approximate) Sommerfeld outgoing radiation condition to 4>, Ap, A^, Az and At. The condition would be applied on the outermost grid points. The Sommerfeld condition for 4> would provide an algebraic equation for boundary values of LT$ as well. Fields Ep, E^ and Ez similarly can be rewritten as a set of wave equations. The Sommerfeld condition for the electric field components, in turn, could provide outer boundary conditions for LTp, LT^ , and LTZ. Meanwhile, backwards difference operators could be applied, as appropriate, to 4>p, $ 2 , A p z , A$p, A^z, Azp, Atp and Atz, while variables J p , J ^ , Jz, Qt, Ep, E^, Ez, Bp, B<p, Bz, Jt, PQ, PT, Ps<t> and psz would require no special treatment. However, by using the MIB coordinates in conjunction with Kreiss-Oliger dissipation, radiation will be quenched. before it propagates from the interior of the domain to the computational outer boundary. Providing the boundary conditions are appropriate to the initial value problem, their details should therefore be irrelevant. Thus, I choose to fix the boundary values of all the grid functions to what they were at initial time. As numerical evolution demonstrates in following sections, even such static boundary conditions work well in the M I B coordinate approach. Kreiss-Oliger dissipation is applied as discussed in Appendix C. It is added to all functions whose equations of motion explicitly involve partial derivatives with respect to temporal coordinate t. They are $, $ p , 4>z, n $ , A p , Apz, Up, A^, A^p, A$z, U^, Az, Azp, Uz and Q t - Dissipation in both coordinate directions r and 9 is applied throughout the bulk of the computational domain, where at least two grid points distant from the boundaries p = 0, p = p m a x, z = z m m and z = z m a x -However, because the operator stencils are five points wide, dissipation in coordinate p alone is applied z = z m m + Az and z = z m a x — A z . Meanwhile, dissipation in coordinate z alone is applied at p = p m a x — Ap. Formally, if u = u(t, p, z) is one of the functions whose equation contains an explicit derivative in time, then, using the Crank-Nicholson scheme without dissipation, its expression wil l A P P E N D I X D . M I B C O O R D I N A T E S 258 be u(t + At, p, z) — u(t, p, z) 1 r _ , . . - i ^ ^ - ^ = -[R(t + At,p,z) + R(t,p,z)] , (D.79) where the left hand side is—to O ( ( A i ) 2 ) accuracy—the time derivative of u centered between times i and i + A i , and the right hand side represents—to O ( ( A p ) 2 ) and O ( ( A 2 ) 2 ) — a l l other terms in the equation time averaged between steps i and i + A i . Then, Kreiss-Oliger dissipation is implemented by adding specific terms on the right hand side of equation (D.79). In this case, u(t + At,p,z)-u{t,p,z) _ l m + to,p,z)+R{t,P,z)] At 2 1 + 2 1 [Dp(t + At,p,z) + Dp{t,p,z)} 2 [Dz{t + At,p, 2),+ Dz{t,p,z)] (D.80) when 2Ap < p < Pmax — 2Ap and zm-m + 2Az < z < 2 m a x — 2A2 u(i + A i , p, 2) — u(i , p, z) 1 A i 2 1 + 2 [i?(i + A i , p, z) + R{t, p, 2)] [ D p ( i + A i , p , 2 ) + D p ( i , p ,2 ) ] (D.81) when 2Ap < p < p m a x — 2Ap and z = zmi„ + Az or z = 2 m a x — Az, and w(i + A i , p, 2) — w(i, p, 2) 1 . • ,. + i [ D 2 ( i + A i , p , 2 ) + D 2 ( i , p ,2) ] (D.82) when 2 m ; n + 2A2 < 2 < 2 m a x — 2A2 and p = p m a x — A p , where Dp{t,p,z) = -^-t[u(t,p+2Ap,z)-4u(t,p + Ap,z) + 6u(t,p,z) - 4 u ( i , p - A p , 2) + u(i, p - 2Ap, z)\., (D.83) D 2 ( i , p , z ) = - ^ | ^ [ u ( i , p , 2 + 2 A 2 ) - 4 u ( i , p , z + A2) + 6u(i,p,2) -Au(t,p,z - Az) + u(t,p,z - 2Az)} , (D.84) and 0 < tp < 1 and 0 < e2 < 1 are adjustable parameters. I use values ep = 0.5 and e2 = 0.5 for the adjustable dissipation parameters in directions p and 2, respectively. Using a Courant factor A = M ~ 0.25 , (D.85) ^\{{ApY + {AzY) this system of equations evolves stably under the Crank-Nicholson iterative solution scheme. APPENDIX D. MIB COORDINATES 259 D.3 Results D.3.1 Initial Data Again, the initial value problem for the Maxwell-Klein-Gordon system in a spacetime background requires specification of all the variables in a manner consistent with regularity and gauge condi-tions, and satisfying the electromagnetic constraint equation DaEa = pQ , (D.86) where Da is the spatial covariant derivative operator, and P Q , is, according to equation (D.62), . = - i e — ( $ * I I * - . ' (D.87) Ipplzz Following the procedure of section 2.2.1, I again decompose Ea into its longitudinal and trans-verse parts, respectively La and TA, to obtain Ea=La + TA. ' (D.88) Then introducing the scalar potential, U = U(r.,8), as per DaU = L a , (D..89) (D.86) becomes DaDaU = DaEa = p Q , (D.90) where the left hand side is explicitly 2 DaDaU 73 /yS pp IZZ -Ippllzidp^l^idpU) ~ {dplPp)llz(dPU) + Ipplzz -l3pp(dzlzz)(dzU) + i3ppizz(dzzU)- (D.91) After solving (D.90) using the methods of Appendix C, I reconstruct the longitudinal electric field components, which are, from (D.89), Lp = dpU , (D.92) Lz = 0ZU , (D.93) L<j, = Q . (D.94) I then specify divergenceless components for the transverse component TA. For simplicity, I again choose TP = 0 , (D.95) TZ = 0 , (D.96) TA = 0 . (D.97) A P P E N D I X D. MIB COORDINATES 260 The completely reconstructed electric field at initial time is therefore Ep = dpU , (D.98) Ez = dzU , (D.99) E+ = 0 . (D.100) Having solved for Ea, I then initialize n p , n 2 and in accordance with (D.55), (D.56) and (D.57), meaning n p = -jpplzzEp , (D.101) n 0 = --Ypp-YzzE,/, , (D.102) n 2 = -lpp~izzEz>. (D.103) Having satisfied the initial data constraint, I then freely specify the remaining variables in accordance with their definitions and regularity conditions. For simplicity I again choose Gaussian profiles for the complex components of <!>(£, p, z) = <f>i(t, p, z)+i<p2(t, p, z), U$(t, p, z) = Ui(t, p, z) + iU2(t,p,z), and the vector potential components Aa(t,r, 6).2 That is, 0i(O,p,z) = aiexp : - « p - Pi)/<W2) exp (--((* -z^/6^)2) , (D.104) 02(0, p,z) = a2 exp :-{{p- P2)/OP2)2) exp (-~{{z -Z2)/8z2)2) , (D.105) ni(o , P , z ) = azexp ; - ( ( p - P3)/Sp3)2) exp (-~{{z -z3)/Sz3)2) , (D.106) n 2(o ,p,z) = 0 4 exp P4)/opl)2) exp (--((z - ^ 4 ) / ^ 4 ) 2 ) , (D.107) At(0, p,z) = ar exp Pt)/6pt)2) exp (--((z- -zt)/5zt?) , (D.108) Ar(0,p,z) = arexp :-((*>- PP)/6pp)2) exp (-~{{z -zp)/8zp)2) , (D.109) Ae(0,p,z) = agexp :-«/>- Pz)/5pzf) exp (-~((z -zz)/5zz)2) , (D.110) A^,{Q,p,-z) = a^exp ( - ( ( p - P<P)I&P4>)2) e x P ( ~((z -^)/<5 2<A) 2) , ( D . l l l ) where 0 1 , 0 2 , a3, 0 4 , at, ap, az and a^ are constants setting the initial amplitudes, p i , P2, P3, p 4 , Pt; Pp; r z a n d P<A determine the Gaussian peaks in the radial direction, z±, z2, 2 3 , z 4 , zt, zp, zz and Zcj, determine the Gaussian peaks in the z coordinate direction, 5pi, 5P2, <5P3, 5P4, Spt, 6PP, 5pz and 5P0 determine the Gaussian widths in coordinate p, and Sz\, Sz2, 8z3, 8z4, Szt, 8zp, 6ZZ and Sztp determine widths in z. Finally, all remaining variables are initialized according to the definitions of section D.2. D.3.2 Testing the Boundary Conditions One of the principal reasons for studying evolution in the M I B coordinate system it to determine the utility of these coordinates in treating outer boundary effects (in specific, reflections). As discussed, grid points outside the M I B boosting region effectively propagate outwards at a rate approaching the speed of light. Any fields reaching this boosted region will be spatially compressed relative to the finite difference mesh. It is at that point that Kreiss-Oliger dissipation wil l quench these fields as their high-frequency components (relative to the mesh) grow. In principle, the quenching should 2From equations (D.47) and (D.49), nonzero electric current components Jp and Jz require e, $ and <£* to all be nonzero. The condition for nonzero Jp additionally requires that either Ap, $ p or <I>* be nonzero. Similarly, the condition for nonzero Jz additionally requires that either Az, 3>2 or be nonzero. Clearly, the chosen initial data (D.104)-(D.lll) and (D.112)-(D.116) satisfies the nonzero conditions for both Jp and Jz when e ^  0. Therefore, it can again be said charge separation begins at t = 0 for this specification of the Maxwell-Klein-Gordon system. A P P E N D I X D . M I B C O O R D I N A T E S 261 be sufficient to prevent any information from reaching the computational outer boundary. To test the M I B system I evolve m$ = 1.0, e = 1.0 Maxwell-Klein-Gordon fields in Minkowski spacetime with spatial domain 0 < p < 3, — 3 < z- < 3 and compare the evolution to that on the larger domain having standard cylindrical coordinate range 0 < p < 6, —6 < z < 6. The evolution with smaller computational domain using M I B coordinates employs parameter values pc = 2.50, Sp = 0.10, zc = 2.50 and 5Z = 0.10, and contains J = 201 points in the p coordinate direction and K = 401 goints in the direction of z. The larger computational domain has the same resolution and thus J = 401 points in direction p and K = 801 points in direction z. Initial data is specified as discussed above for the choice of values a>i = Q-2 = 03 — a,4 = o,t = dp = az = as = 1.00 , (D.112) Pi = P2 = P3 = P4 = Pt = PP = pz = ps = 0.75 , (D.113) zi = z2 = z3 = Zi = zt = zp = zz = zs = 0.00 , (D.114) SPi = SP2 = Sp3 = SP4 = Spt = Spp = Spz = Sps = 0.20 , (D.115) 8zl = 6Z2 = Sz3 = 5zA = 5zt = 5zp = Szz = 5Z& = 0.20 . (D.116) The analogous choice is made for initial data on the standard cylindrical coordinate domain. Figure D.7 shows the evolution of charge density pQ in a sequence of steps for the M I B coordinate system. Observe how the grid function is compressed and quenched in the boosting region, and how no signal reaches the outer boundary of the computational domain. A n appropriate comparison between the M I B results and those of the larger computational domain is through the error of integrated conserved quantities. For example, I compute the error of total charge, SQ, as per SQ = \\Q-Q\\2, (D.117) where Q is the total charge in the M I B domain, Q is that in the large domain, and || • H2 represents the l2 norm. Q and Q are obtained through integration over a common region of space. Because of the location and growth of the boosting region I limit the region of integration to 0 < p < 1.5, — 1.5 < z < 1.5. I display these errors for total normalized energy (ST), angular momentum (SS) and charge (SQ) in Figures D.8, D.9, and D.10. To compare, these figures also display errors for the same system but using approximate Sommerfeld condition on a domain 0 < p < 3 , — 3 < z < 3 . It is clear that M I B coordinates together with dissipation are as good as an approximate Sommerfeld condition on the boundary. However, the useful spatial domain of the M I B system is effectively reduced in size due to the boosting regions. The errors of the M I B system results wil l likely drop significantly below those for the Sommerfeld condition when its interior (i.e., unboosted) region is increased to a comparable size. Furthermore, the choice e = 1.0, m$ = 1.0 provides significantly stronger coupling than those used.in following discussions of Maxwell-Klein-Gorddn fields in Minkowski spacetime. Testing on a larger computational domain with weaker coupling would therefore be a useful future calculation. A P P E N D I X D . M I B C O O R D I N A T E S 262 -1.00e-03 l.OOc-03 I -1.00e-03 1.00c-03 -1.00e-03 1.00e-03 • -1.00e-03 1.00e-03 F i g u r e D . 7 : M I B coordinates and dissipation acting on m$ = 1.0, e = 1.0 Maxwell-Klein-Gordon fields. The figure displays the charge density in a sequence of steps. The directions of coordinate vectors pa ~ (0,1,0,0) and za ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 0 ^ p ^ 3 and —3 ^ z < 3, and the corners of the domain are labelled with their (p, z) ordinate values. The amplitude and colour gradient are both scaled linearly. A P P E N D I X D . M I B C O O R D I N A T E S 263 0.15 0.1 0 1 1 1 1 1 1 1 1 •' 1 1 1 i i i | i -/ / / / • / 1 / \ \ / \ \ \ \ ! W M -1 / / . / I / / _._ /../ i \ 1 \ \ 1 1 1 1 1 1 , 1 ,• ' , , 1 0 2 4 6 8 t F i g u r e D . 8 : Displayed is the error in total normalized energy for. e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields in the M I B coordinate system (dashed black line), and in the standard cylindrical coordinates using an approximate Sommerfeld boundary condi-tion (dotted black line). D.3.3 Maxwell-Klein-Gordon Fields on a Flat Background Having shown that M I B coordinates are as good as (and likely better than) the approximate Sommerfeld condition in treating the outer boundary of the computational domain, I now provide results which compare the Maxwell-Klein-Gordon dynamics on flat spacetime to those of chapter 4. Here and below I use a computational domain spanning the coordinate range 0 < p < 90 and —90 < z < 90, and containing 401 points in the direction of coordinate p and 801 points in the direction of z. The relevant M I B coordinate parameters are pc = 80, Sp — 4, zc = 80 and 5Z = 4. Finally, I specify initial data according to the outlined procedure for the choice of parameter values ai = a2 = a3 = a 4 = at = ap = az = as = 1.0 , ' (D.118) Pi = P2 = Pz = Pi = Pt = Pp = Pz= P<$>= 25.0 , (D.119) z\ = z2 = z3 = Z4 = zt = zp = zz = zA = 0.0 , (D.120) dpi •= SP2 = <5p3 = b~P4 = Spt = 5PP = 5pz = 5ps = 5.0 , (D.121) Szi = 5Z2 = 5z3 = 5Z4 = Szt — 5zp = 6ZZ = 5zs = 5.0 . (D.122) This provides initial conditions reasonably close to those used in evolving fields on the Kerr space-time. To compare and contrast, it should be sufficient to present just the conserved quantities of this system. A P P E N D I X D . M I B C O O R D I N A T E S 264 0.01 h 0.005 F i g u r e D.9: Displayed is the error in angular momentum for e = 1.0, m$ = 1.0 Maxwell-Klein-Gordon fields in the M I B coordinate system (dashed black line), and in the standard cylindrical coordinates using an approximate Sommerfeld boundary condition (dot-ted black line). A P P E N D I X D . M I B C O O R D I N A T E S 265 c? 0.08 0.06 0.04 h 0.02 -0.02 i/lK / 1 -I 1 r ft _J L _i i i_ F i g u r e D . 1 0 : Displayed is the error in charge for e = 1.0, ro$ = 1.0 Maxwell-Klein-Gordon fields in the M I B coordinate system (dashed black line), and in the standard cylindrical coordinates using an approximate Sommerfeld boundary condition (dotted black line). A P P E N D I X D . M I B C O O R D I N A T E S 266 Dynamics of Klein-Gordon Fields in Minkowski Spacetime Beginning with the massive electromagnetically uncoupled fields I find: (1) orbiting resonances and other manifestations of gravitational interaction are never present; (2) the rate of field dissipation from the computational domain always decreases with increasing mass parameter; and (3) rn$ = 0.3 fields strongly display scattering concentrated along the axis of symmetry. These three facts are substantiated by simple comparison of the normalized energy densities shown in Figures D. 11, D . l 2 and D.13, which respectively display the normalized energy densities for m$ =0.0. ro$ =0.3 and m$ = 2.4 Klein-Gordon fields. A P P E N D I X D . M I B C O O R D I N A T E S 267 1.00e-07 1.00e-05 F i g u r e D . l l : m$ = 0.0, e = 0.0 Klein-Gordon field dynamics in fiat spacetime in terms of the normalized energy density. The directions of coordinate vectors pa ~ (0,1,0,0) and za ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 0 < p < 90 and —90 < z < 90, and the corners of the domain are labelled with their (p, z) ordinate values. The amplitude is scaled linearly while the colour gradient is scaled logarithmically. A P P E N D I X D . M I B C O O R D I N A T E S 268 1.00e-05 F i g u r e D.12: m$ = 0.3, e = 0.0 Klein-Gordon field dynamics in flat spacetime in terms of the normalized energy density. The directions of coordinate vectors pa ~ (0,1,0,0) and za ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 0 < p < 90 and —90 < z < 90, and the corners of the domain are labelled with their (p, z) ordinate values. The amplitude is scaled linearly while the colour gradient is scaled logarithmically. A P P E N D I X D . M I B C O O R D I N A T E S 269 1.00e-07 1.00e-05 • 1.00e-07 1.00e-05 1.00e-07 l.OOc-05 F i g u r e D . 1 3 : m$ = 0.0, e = 2.4 Klein-Gordon field dynamics in flat spacetime in terms of the normalized energy density. The directions of coordinate vectors pa ~ (0,1,0,0) and za ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 0 < p < 90 and —90 < z < 90, and the corners of the domain axe labelled with their (/>, z) ordinate values. The amplitude is scaled linearly while the colour gradient is scaled logarithmically. A P P E N D I X D . M I B C O O R D I N A T E S 270 Dynamics of Maxwell Fields in Minkowski Spacetime Examining the conserved energy and angular momentum densities, respectively displayed in Figures D.14 and D.15, two facts are immediately clear: (1) effects resembling gravitational interaction are not present in flat spacetime; and (2) the electromagnetic angular momentum displays nontrivial evolution. The large amplitudes and steep gradients associated with scattering from the black hole spacetime are absent from the energy density. Furthermore, while the evolution of angular momentum density is nontrivial in Minkowski spacetime, the complexity observed in Kerr spacetime evolution is absent. A P P E N D I X D . M I B C O O R D I N A T E S 271 1.00e-07 1.00e-05 1 1.00e-07 1.00e-05 1.00e-07 1.00e-05 Figure D.14: Maxwell field dynamics in flat spacetime in terms of the normalized energy den-sity. The directions of coordinate vectors pa ~ (0,1,0,0) and za ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 0 < p < 90 and —90 < z < 90, and the corners of the domain are labelled with their (p,z) ordinate values. The amplitude is scaled linearly while the colour gradient is scaled logarithmically. A P P E N D I X D . M I B C O O R D I N A T E S 272 -1.00e-05 1.00e-05 F i g u r e D . 1 5 : Angular momentum of the Maxwell fields in flat spacetime. The directions of coordinate vectors pa ~ (0,1,0,0) and za ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 0 < p < 90 and —90 < z < 90, and the corners of the domain are labelled with their (p, z) ordinate values. The amplitude and colour gradient are both scaled linearly. A P P E N D I X D . M I B C O O R D I N A T E S 273 D y n a m i c s o f e = 0.6, m$ = 0.0 M a x w e l l - K l e i n - G o r d o n F i e l d s i n M i n k o w s k i Space t ime The conserved densities'show that massless e = 0.6 Maxwell-Klein-Gordon fields in flat space: (1) do not display effects in any way resembling gravitational interaction; (2) have nontrivial angular momentum evolution; and (3) generically display charge separation. The normalized energy density of Figure D.16, angular momentum density of Figure D.17 and charge density of Figure D.18 provide the required evidence. The large amplitudes and steep gradients associated with scattering from the black hole spacetime are absent, and while the evolution of angular momentum density is nontrivial, the complexity observed in Kerr spacetime evolution is absent in flatspace. Furthermore, while gravitational effects are absent, the charge separation exhibited in flat space is still quite complex. Finally, from the evolution of angular momentum and charge density, dynamo-like behaviour can be deduced. It is sufficient to state this physics is manifest in the evolution of angular current J& (figures not provided). A P P E N D I X D . M I B C O O R D I N A T E S 274 1.00e-07 l.OOc-05 I 1.00e-07 1.00e-05 1.00e-07 1.00e-05 F i g u r e D . 1 6 : Massless e = 0.6 Maxwell-Klein-Gordon field dynamics in flat spacetime in terms of the normalized energy density. The directions of coordinate vectors pa ~ (0,1,0,0) and za ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 0 < p < 90 and —90 < z < 90, and the corners of the domain are labelled with their (p,z) ordinate values. The amplitude is scaled linearly while the colour gradient is scaled logarithmically. APPENDIX D. MIB COORDINATES 275 -1.00e-03 1.00e-03 1 -1.00e-03 1.00e-03 -1.00e-03 1.00e-03 Figure D.17: Angular momentum of the massless e = 0.6 Maxwell-Klein-Gordon fields in flat spacetime. The directions of coordinate vectors pa ~ (0,1,0,0) and za ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 0 < p < 90 and —90 < z < 90, and the corners of the domain are labelled with their (p, z) ordinate values. The amplitude and colour gradient are both scaled linearly. A P P E N D I X D . M I B C O O R D I N A T E S 276 -1.00e-03 l.OOc-03 F i g u r e D . 1 8 : Angular momentum of the massless e = 0.6 Maxwell-Klein-Gordon fields in flat spacetime. The directions of coordinate vectors pa ~ (0,1,0,0) and za ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 0 < p < 90 and — 90 < z < 90, and the corners of the domain are labelled with their (p, z) ordinate values. The amplitude and colour gradient are both scaled linearly. A P P E N D I X D . M I B C O O R D I N A T E S 277 D y n a m i c s o f e = 0.6, m$ = 0.3 M a x w e l l - K l e i n - G o r d o n F i e ld s i n M i n k o w s k i Space t ime Finally, four facts can be stated for m$ = 0.3, e = 0.6 (massive electromagnetically coupled) Maxwell-Klein-Gordon field dynamics in flat space: (1) effects resembling gravitational interaction are never observed; (2) there is only a slight preference for scattering along the axis of symmetry; (3) the evolution of angular momentum is again nontrivial; and (4) charge separation is'again a generic feature of evolution. Figure D.19 displays the normalized energy density in support of the first two facts. Notice that while a slight collimation of energy along the axis is present, it is far less than observed for the scattering in M = 5 Kerr spacetime. The angular momentum density is presented in Figure D.20. While its evolution is again nontrivial, it here displays amplitudes and gradients far less than for the equivalent evolution in Kerr spacetime. While the amplitudes and gradients of charge density separation are comparable to those on the black hole background, Figure D.21 shows no evidence for morphology typical of the orbital effects seen on the curved spacetime calculations. Finally, as in the case of massless e = 0:6 evolution, the presence of dynamo-like behaviour can be deduced. This physics is again manifest in the evolution of angular current (figures not provided). A P P E N D I X D . M I B C O O R D I N A T E S 278 l.()()e-07 1.00e-05 I 1.00e-07 1.00e-05 1.00e-07 1.00e-05 F i g u r e D . 1 9 : m$ = 0.3 e = 0.6 Maxwell-Klein-Gordon field dynamics in flat spacetime in terms of the normalized energy density. The directions of coordinate vectors pa ~ (0,1,0,0) and za ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 0 < p < 90 and —90 < z < 90, and the corners of the domain are labelled with their (p, z) ordinate values. The amplitude is scaled linearly while the colour gradient is scaled logarithmically. A P P E N D I X D . M I B C O O R D I N A T E S 279 -1.00e-03 l.OOc-03 I -1.00e-03 1.00e-03 -1.00e-03 1.00e-03 F i g u r e D . 2 0 : Angular momentum of the m$ = 0.3, e = 0.6 Maxwell-Klein-Gordon fields in flat spacetime. The directions of coordinate vectors pa ~ (0,1,0,0) and za ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 0 < p < 90 and —90 < z < 90, and the corners of the domain are labelled with their (p,z) ordinate values. The amplitude and colour gradient are both scaled linearly. A P P E N D I X D . M I B C O O R D I N A T E S 280 -1.00e-03 1.00e-03 F i g u r e D . 2 1 : Angular momentum of the m$ = 0.3, e = 0.6 Maxwell-Klein-Gordon fields in flat spacetime. The directions of coordinate vectors pa ~ (0,1,0,0) and za ~ (0,0,1,0) are shown in the first frame. The computational domain covers the range 0 < p < 90 and —90 < z < 90, and the corners of the domain are labelled with their (p, z) ordinate values. The amplitude and colour gradient are both scaled linearly. A P P E N D I X D . M I B C O O R D I N A T E S 281 0.7 0.5 1 i i i i i i i i i i , i , , , i i I . i , . x . •T^T.~ s. \ \ — \ - \ -\ - \ - \ — \ \ \ \ \ \ \ \ \ \ \ \ - \ ' i i i i 1 i i i i 1 i i i i 1 i i i i 1 i i i i 0 5 10 15 20 25 t F i g u r e D . 2 2 : Convergence of total normalized energy, T , for e = 0.6, m$ = 0.3 fields in the M I B flat spacetime. The dashed red line represents data for a finite difference grid with J = 101 points in direction p and K = 201 points in direction z, the dotted red line represents data for a grid with J = 201 points in p and K = 401 points in z, and the dashed black line represents data for a grid with J = 401 points in p and K = 801 in z. D.3.4 Convergence of Conserved Quantit ies In concluding this appendix I provide evidence that the M I B coordinate code is both convergent and accurate. This is done by displaying convergence of the total energy, T , angular momentum, S and charge Q for m * = 0.3, e = 0.6 perturbations. Figure D.22 shows the convergence of energy, Figure D.23 the convergence of angular momentum, and Figure D.24 displays the results for charge. In each of the figures I show the total integrated quantity contained within coordinate range 0 < p < 50, - 5 0 < z < 50 for the period 0 < t < 25. I use the reduced spatial domain because of the location and growth of the coordinate boosting regions, and a reduced period because fields wil l propagate past the reduced spatial domain at later times. The comparison is between grids with J = 101 points in p and K = 201 points in z, J = 201 points in p and JK" = 401 points in z, and J = 401 points in direction p and K = 801 in the direction of z. Based on the results of total energy conservation, the results for the massive charged fields are typically accurate to ~ 2% over the period 0 < t < 25 on the grid with 401 x 801 points. Finally, the convergence factor, time averaged over the displayed period, i s C « 3.7 for the total energy, C » 3.1 for angular momentum and C w 5.1 for the total charge of the system. A P P E N D I X D . M I B C O O R D I N A T E S 282 200 \-100 OT 0 - 1 0 0 - 2 0 0 ~i—i—i—r / v- r \ i - 1 W / \ / J I I 1L 1 0 1 5 20 25 F i g u r e D . 2 3 : Convergence of total angular momentum, S, for e = 0.6, m * = 0.3 fields in the M I B flat spacetime. The dashed red line represents data for a finite difference grid with J = 101 points in direction p and K = 201 points in direction z, the dotted red line represents data for a grid with J = 201 points in p and K = 401 points in z, and the dashed black line represents data for a grid with J = 401 points in p and K = 801 in z. A P P E N D I X D . M I B C O O R D I N A T E S 283 i—i—i—i—I—i—r -i—i—I—i—i—r 1000 cr 500 / / / / / / / _i I I I I I I I I ! I i I I I I I I L 10 15 t 20 25 F i g u r e D .24 : Convergence of total charge, Q, for e = 0.6, m* = 0.3 fields in the M I B flat spacetime. The dashed red line represents data for a finite difference grid with J = 101 points in direction p and K = 201 points in direction z, the dotted red line represents data for a grid with J = 201 points in p and K = 401 points in z, and the dashed black line represents data for a grid with J = 401 points in p and K = 801 in z. 

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