Relativity and Scientific Computing

Color Plates.- I: Numerics.- 1. Numerical Relativity and Black-Hole Collisions.- 1. Astrophysical Motivation and Mathematical Formulation for Numerical Relativity.- 1.1 Overview and Motivation.- 1.2 Mathematical Formulation of the Equations.- 2. Numerical Techniques and Supercomputing.- 2.1 Finite Difference Techniques.- 2.2 Treating Elliptic Equations.- 2.3 Treating Evolution Equations.- 2.4 Testbeds and Convergence of Numerical Solutions.- 2.5 Coding and Parallel-Computing Issues.- 2.6 General Code Strategies.- 3. Black-Hole Initial-Data Sets, Tools for Analysis, and Techniques for Evolution.- 3.1 Basic Theory and Initial-Data Sets.- 3.2 Tools for Numerical Black-Hole Spacetimes.- 3.3 Evolution.- 4. Present Research Status in Black-Hole Studies.- 4.1 Spherical BH — 1D.- 4.2 Distorted BH — 2D.- 4.3 Rotating BH — 2D.- 4.4 Colliding BH — 2D.- 4.5 Black-Hole Horizon Studies.- 4.6 3D Black-Hole Studies.- References.- 2. Four Lectures on Numerical Relativity.- 1. The Causal Structure of Einstein’s Field Equations.- 1.1 The Space-Plus-Time Decomposition.- 1.2 Invariant Algebraic Slicing.- 1.3 The Evolution System.- 1.4 Causal Structure of the Evolution System.- 2. First-Order Flux-Conservative Systems.- 2.1 Linear Systems.- 2.2 Nonlinear Systems.- 2.3 Einstein’s Evolution Equations.- 3. Standard Numerical Methods.- 3.1 Flux-Conservative Equations.- 3.2 Boundary Conditions.- 3.3 Nonsmooth Data.- 4. Total Variation Diminishing Methods.- 4.1 Flux-Conservative Methods.- 4.2 The 1D Black-Hole Test.- References.- 3. Alternatives to Finite Difference Methods in Numerical Relativity.- 1. Introduction.- 2. The 3+1 Formalism.- 3. The Initial-Data Problem.- 3.1 Multiquadrics.- 3.2 Finite Elements.- 4. Matter Evolution in Curved Spacetimes.- 4.1 Particle-Mesh Methods.- 4.2 Smoothed Particle Hydrodynamics in Curved Space.- 5. Conclusions.- References.- 4. Temporal and Spatial Foliations of Spacetimes.- 1. Introduction.- 2. Time Slicings.- 2.1 Maximal Slicing.- 2.2 Harmonic Slicing.- 2.3 Results for the Oppenheimer-Snyder Spacetime.- 3. Spatial Foliations.- 3.1 Constant Mean Curvature Foliations.- 3.2 Examples of CMC Foliations.- References.- 5. Rotating and Oscillating Neutron Stars.- 1. Introduction: Nonrotating Neutron Stars.- 2. Rotating Neutron Stars.- 2.1 Basic Formulation.- 2.2 Numerical Solution Method.- 2.3 Results.- 3. Oscillations of Neutron Stars.- 3.1 Basic Formalism.- 3.2 Results for Quasinormal Mode Frequencies.- References.- 6. Rotating Boson Stars.- 1. Introduction.- 2. Field Equations.- 3. Spherically Symmetric Boson Star.- 4. Field Equations of a Rotating Boson Star.- 5. Particle Number, Mass, and Angular Momentum.- 6. Differential Rotation.- 7. Numerical Solution.- 8. Remarks.- References.- 7. Numerical Investigation of Cosmological Singularities.- 1. Introduction.- 2. Symplectic Methods.- 3. Mixmaster Model.- 4. Gowdy Model on T3 × R.- 5. U(1) Models.- 6. Conclusions.- References.- II: Computer Algebra.- 8. Overview of Computer Algebra in Relativity.- 1. Introduction.- 2. General-Purpose Systems.- 2.1 Computer Algebra.- 2.2 Modern Systems.- 2.3 Evaluation.- 2.4 Simplification.- 2.5 Programming.- 2.6 Data Types.- 3. General Relativity Systems.- 3.1 Requirements: Riemannian Geometry.- 3.2 Requirements: GR Library.- 3.3 Requirements: Generalizations and Applications.- 3.4 Efficiency.- 3.5 Dummy Indices.- 4. Applications.- 5. Summary.- References.- 9. Two-Loop Quantum Gravity with the Computer Algebra Program FORM.- 1. Introduction.- 2. Covariant Quantization of Gravity.- 2.1 Noncovariant Method.- 2.2 Covariant Method.- 3. FORM.- 3.1 Annotated Yang-Mills Program.- 3.2 Quantum Gravity.- References.- 10. The Mathematic Packages CARTAN and MathTensor for Tensor Analysis.- 1. Mathematica.- 1.1 The Front End.- 1.2 The Kernel.- 2. Tensor Calculations.- 3. CARTAN.- 3.1 General Features.- 3.2 A Charged Black Hole.- 3.3 A Spin-Polarized Cosmic String.- 4. MathTensor.- 4.1 Tensor Indices.- 4.2 Metric Variation of R2.- 4.3 Differential Forms with MathTensor.- 5. Conclusions.- References.- 11. A Quadratic Curvature Lagrangian of Paw?owski and Raczka: A Finger Exercise with MathTensor.- 1. Introduction.- 2. Riemann Tensor and its Irreducible Pieces.- 3. The Topological Euler Density.- 4. Bach Tensor.- 5. The Bach Tensor Streamlined.- 6. Gravitational Field Equation of the P&R Model.- 7. Discussion.- References.- 12. The Program CRACK for Solving PDEs in General Relativity.- 1. Introduction.- 2. Contents of CRACK.- 2.1 General Remarks.- 2.2 Decoupling.- 2.3 Integrating Exact PDEs.- 2.4 Separation of PDEs.- 2.5 Solving Standard ODEs.- 3. The Calculation of Spacetime Symmetries.- 3.1 General Remarks.- 3.2 Generation of the Conditions.- 3.3 Post Processing of Results.- 3.4 An Example.- 4. Symmetries of a Field Theory.- 5. Applying Symmetries of Differential Equations.- 6. Future Work.- 6.1 General Remarks.- 6.2 Extending Capabilities in Dealing with DEs.- 6.3 Advanced Symmetry Investigations.- 7. Availability.- References.- 13. Algebraic Programming in the Hamiltonian Version of General Relativity.- 1. Introduction.- 2. The Local Form of the Canonical Formalism of Gravity.- 3. The Computer Procedures.- 4. About the Concrete Results.- 4.1 Spacetime Model with a Three Subspace in Form of a 3-Torus.- References.- 14. Causal Structure and Integrability in Moving Frames with Reduce.- 1. Introduction.- 2. Exterior Differential Systems.- 3. EDS Package.- 4. Ricci-Flat Immersions.- 5. Twisting Type-N Solutions.- References.- III: Visualization.- 15. Four Lectures on Computer Graphics and Dkta Visualization.- 1. Introduction.- 2. Computers, Humans, and Interaction.- 2.1 Computer Graphics Defined.- 2.2 Computer Displays.- 2.3 Graphics Hardware.- 2.4 Human Visual System.- 2.5 History of Computer Graphics.- 3. Visual Simulation.- 3.1 Visibility Calculation.- 3.2 Simulated Optics.- 3.3 Ray-Tracing.- 3.4 Rasterization.- 3.5 Graphics Pipeline.- 4. Data Visualization.- 4.1 Seeing the Invisible.- 4.2 The Analysis and Visualization Cycles.- 4.3 History of Visualization.- 4.4 Data Sources and Structure.- 4.5 Data Interpolation.- 5. Visualization Techniques.- 5.1 Scalar Fields.- 5.2 Vector Fields.- 5.3 Tensor Fields.- 5.4 Differential Geometry Applied to Height Maps.- 5.5 Second Derivatives and Curvature.- 5.6 Visualizing Higher Dimensions.- 5.7 How to Lie and Confuse with Visualization.- 6. Interactive Visualization Environments.- 6.1 Data Structures.- 6.2 Modules.- 6.3 Visual Programming Language and Execution Model.- 6.4 Programmers’ Tools.- 7. Virtual Environments in Visualization.- 7.1 Visualization over the World Wide Web.- 8. Applications and Demonstrations.- 8.1 Case Study: Structural Dynamics Using DYNA3D.- 8.2 Case Study: Helicopter Rotor Wake Visualization.- 8.3 An Example of Remote Collaboration.- 9. Conclusions.- References.- 16. Visualization in Curved Spacetimes. I. Visualization of Objects via Four-Dimensional Ray-Tracing.- 1. Introduction.- 2. Ray-Tracing Special Relativity.- 2.1 Geometrical Appearance.- 2.2 Lighting.- 3. Ray-Tracing General Relativity.- 3.1 Requirements.- 3.2 Geodesic Equation.- 3.3 Camera.- 3.4 Realizing Relativistic Ray-Tracing with Conventional Ray-Tracing Programs.- 3.5 Astrophysical Examples.- 3.6 Astrophysical Application: Light Curves of X-Ray Pulsars.- 4. Discussion.- References.- 17. Visualization in Curved Spacetimes. II. Visualization of Surfaces via Embedding.- 1. Introduction.- 2. Finding the Parametric Representation of the Embedding in R3.- 2.1 Useful Results from Differential Geometry of Surfaces.- 2.2 The System of Equations.- 2.3 An Alternative: The Darboux Equation.- 3. Examples.- 3.1 Kerr Horizon.- 3.2 Colliding Black Holes.- 4. Direct Construction of Wire Frames.- 4.1 Triangular Wire Frames.- 4.2 Surfaces with Spherical Topology.- 4.3 Procedure.- 4.4 Examples.- 4.5 Future Developments.- References.- IV: Exotica.- 18. Exotic Smoothness on Spacetime.- 1. Introduction.- 2. Differential Topology.- 3. Early Exotica of Various Kinds.- 4. Gauge Theory and Moduli Spaces.- 5. The Road to R4?.- 6. Some General Properties of R4?.- 7. Some Geometry and Physics on R4?s.- 8. Conclusions.- References.- List of Figures.- List of Tables.