Leopold-Franzens University Innsbruck Institute of Computer Science Computing Geodesics in Numerical Space Times De computatione distantiarum brevissimarum in spatiis quattuor dimensionum numeris descriptis Master Thesis Marcel Ritter Supervisor: O. Univ. Prof. Dr. Sabine Schindler March 30, 2010 * In memoriam: O. Univ. Prof. i.R. Dr. sub ausp. Manfred Ritter Who never lost his curiosity, fascination and deep passion for science and teaching even during difficult circumstances. Although he was unable to support my work in this world I know his guiding spirit was always by my side. 2 Figure 1: Geodesics in a Kerr metric. Abstract: Traditionally, tools to visualize geodesics in curved spacetimes of general rel- ativity have been specialized solutions, either tailored to a certain spacetime or limited to certain kinds of numerical data. Utilizing a Fiber Bundle data model and the Vish visualization environment, this thesis aims to solve this problem by developing an approach that is independent of the underlying numerical data. My approach allows the combination of several visualization modules, and opens the possibility of applying the computation and visu- alization of integral lines more readily to other scientific domains. These domains include computational fluid dynamics and medical imaging. Contents 1 Introduction 6 2 Theoretical Background 10 2.1 Differential Geometry . 10 2.1.1 Manifolds and Charts . 10 2.1.2 Curve . 12 2.1.3 Tangential Vector . 12 Transformation . 13 2.1.4 Covector . 13 2.1.5 Tensor Field . 14 2.1.6 Metric . 15 2.1.7 Geodesic Equation and Christoffel Symbols . 16 Geodesic Equation . 16 Christoffel Symbols . 19 2.1.8 Geodesic Deviation and Riemann Tensor . 20 2.1.9 Ricci Tensor and Scalar . 21 2.2 General Relativity . 22 2.2.1 Einstein Field Equation . 22 2.2.2 Schwarzschild Metric . 23 2.2.3 Kerr Metric . 24 2.3 Fluid Dynamics . 26 2.4 Medical Imaging . 28 3 Implementation Concepts 29 3.1 Type Traits . 29 3.2 STL Encapsulation . 33 3.3 Reference Pointers . 33 4 Modeling of Scientific Data 36 4.1 Fiber Bundle Data Model . 37 4.2 The Hierarchy Levels . 38 3 CONTENTS 4 4.2.1 Fiber Bundle . 38 4.2.2 Fiber Slice . 39 4.2.3 Fiber Grid . 40 4.2.4 Fiber Topology . 41 4.2.5 Fiber Representation . 42 4.2.6 Fiber Field . 43 4.3 Simplified Access via Selectors . 46 4.4 Data Examples . 47 4.4.1 Uniform, procedural . 48 4.4.2 Multiblock, Curvilinear . 49 4.4.3 Lines . 50 5 Vish - The Vis(h)ualization Environment 52 5.1 Development Quick Start . 53 5.1.1 Availability and Installation . 53 5.1.2 Source Code Organization and Naming Conventions . 54 5.1.3 Make Files and Compilation . 56 5.2 Scene Network . 57 5.2.1 Modules . 58 5.2.2 Data Transport and Access . 62 Creating new Parameter Types . 62 Fiber Bundle Data Access . 63 5.2.3 Rendering Modules . 65 Geometric Algebra . 66 OpenGL . 67 Using OpenGL in a Rendering Module . 67 5.2.4 Vish Scripts . 72 5.3 Caching . 75 5.4 Data Field Interpolation and Finding Local Coordinates . 77 5.4.1 UniGrid . 79 5.4.2 Multiblock . 80 5.4.3 Curvilinear . 85 Local Coordinates in one Hexahedral Cell . 86 Finding Candidates in the Grid . 89 Summarizing the Steps . 91 5.5 Basic Visualization Modules . 93 5.5.1 Coordinate Grid . 93 5.5.2 Coordinate Grid Box . 94 5.5.3 Uniform Grid Lines . 95 5.5.4 Color Map Legend . 95 5.5.5 Multiblock Outlines . 97 CONTENTS 5 6 Computation and Visualization 99 6.1 Defining Initial Conditions for Integral Lines . 100 6.1.1 Initial Positions, Seed Points . 100 Geometric Point Distributions . 101 Random Point Distribution . 105 Grid Union, Convolution and Transformation . 107 6.1.2 Defining Initial Directions . 111 Grid Subtraction . 111 6.2 Computation . 112 6.2.1 Computing First Order Integration Lines . 121 6.2.2 Computing Second Order Integration Lines . 126 6.3 Rendering Lines . 134 7 Applications 141 7.1 Visualizing Flow of CouetteFlow . 142 7.2 Visualizing Flow and Pressure in a Stirred Fluid Tank . 149 7.3 Visualizing Geodesics in a sampled Schwarzschild Metric . 160 7.4 Visualizing Geodesics in a sampled Kerr Metric . 179 7.5 Fiber Tracking in MRI Data . 205 8 Future Work 209 9 Conclusion 211 10 Fazit 213 A Acknowledgements 221 B Definition of Fiber Bundles 222 C Related Publications 223 Chapter 1 Introduction The objective of this thesis is to provide a visualization framework suitable for exploring numerical spacetimes originating from astrophysical numerical relativity. Numerical relativity is a very active research area. Having immense com- puting power available, numerical methods allow to solve the Einstein field equations, chapter 2.2.1, as is was not possible before. One specific applica- tion is the detection of gravitational waves. Gravitational waves have not yet been directly measured, but a Nobel Prize was given to Hulse and Taylor, [44], for finding a convincing indirect evidence. They measured timings in a binary pulsar. The observed frequency increase can only be explained by energy loss due to the emission of gravitational waves. Thus, the waves them- selves have not yet been measured, but their effect on the emitting systems has been observed. Gravitational waves are ripples in spacetime curvature propagating with the speed of light. \Any mass in nonspherical, nonrecti- linear motion produces gravitational waves (...), but gravitational waves are produced most copiously in events such as the coalescence of two compact stars, the merger of massive black holes, or the big bang." [36] Detectors for gravitational waves have been built in the United States in Livingston, Louisiana, in Hanford, Washington [39], in Europe in Hannover, Germany [29], and near Pisa, Italy [24]. In order to detect a gravitational wave, relative distance changes in the order of 10−22 have to be measured. Numerical analysis of situations involving strong gravity, such as the merging of black holes is of great importance. These simulations can be used to match the actually measured data of the gravitational wave detec- tors. Interactive 3D visualization techniques based purely on numerical data helps to analyze these simulations. This thesis focuses on the visualization of geodesics, the shortest (or longest) path between points in space (or space- time). They are important indicators of the structure of spacetime. 6 CHAPTER 1. INTRODUCTION 7 Geodesics in curved spacetimes have been studied before, but most work is related to the visualization of analytic spacetimes, 1992 [2], 1997 [26], 1999 [22], 2001 [15] and 2004 [28]. Computing geodesics is required for ray-tracing black holes. In 1991 Corvin Zahn at the University of T¨ubingenimplemented four dimensional ray-tracing in an analytic Schwarzschild space time, [63]. The most similar work was done already in 1992 by Steve Bryson, who im- plemented geodesic visualizations for exploration using a \boom mounted six degree of freedom head position sensitive stereo CRT system", [17]. For his setup the curved spacetimes could be given by closed formulas or also on simple uniform grids. Geodesics were also analyzed in numerical spacetimes in the 2D (axisym- metric) era. The famous \pair of pants" picture contains the event horizon in a head-on collision, along with some geodesics. The corresponding movies were made in 1995 with great effort in TV resolution, see e.g. [43]. Werner Benger at the University of Innsbruck simulated a black hole by raytracing in 1996 [4]. Andrew Hamilton implemented a real time flight simulator for a charged black hole. He uses a projective technique to compute the paths of geodesics, [35]. Spacetimes visualized in this thesis are sampled on uniform grids. How- ever, the developed infrastructure used in this work easily extends to adaptive mesh refinement (AMR) grids, which are currently used for numerical simu- lations of merging and colliding black holes, without changing the computa- tion and visualization algorithms. Moreover, the rudimentary visualization of geodesics can be enhanced with other visualization modules, for instance to show the coordinate-acceleration, equation (2.53), on the geodesics. Inspired by Bryson's seeding methods for geodesics I developed a flexible technique for the creation of seeding geometries using basic operating blocks based on the theory of fiber bundles, that can be combined to a huge variety of seeding geometries. In addition to the visualization and computation, other aspects have been addressed in this work: the data model high code re-usability high modularity provide an introduction to utilized software environments and libraries The modern scientific world uses many computational methods in different scientific domains. The requirement of collaborations increases and thus CHAPTER 1. INTRODUCTION 8 exchanging data becomes important. During my research visit at Louisiana State University I was taught that in 2005 hurricane Katrina forced scientific groups to exchange their data sets and couple different kinds of numeric simulations to predict the path of Katrina to provide warning and rescue to the public. For example, actual satellite data was coupled with simulation of the motion of air of the hurricane, which was then coupled with a simulation for wave propagation on the ocean. Such coupling is only possible if data can be exchanged without time-consuming data conversion processes between the different scientific groups. The simulation results were later gathered in a visualization combining several layers stemming from all the different domains, [14]. The thesis starts with a short theoretical part providing the necessary mathematical and physical background of general relativity, computational fluid dynamics and magnetic resonance imaging needed for the visualization and interpretation of the resulting illustrations, chapter 2. Thereafter, some C++ programming techniques used for implementation are presented in chapter 3.