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Visualization of Four-Dimensional Spacetimes

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Visualization of Four-Dimensional Spacetimes Daniel Weiskopf Visualization of Four-Dimensional Spacetimes Dissertation der Fakultat¨ f¨ur Physik der Eberhard-Karls-Universitat¨ zu T¨ubingen zur Erlangung des Grades eines Doktors der Naturwissenschaften vorgelegt von Daniel Weiskopf aus Reutlingen 2001 Tag der m¨undlichen Pr¨ufung: 16. Februar 2001 Dekan: Professor Dr. Gerhard J. Wagner 1. Berichterstatter: Professor Dr. Hanns Ruder 2. Berichterstatter: Professor Dr. Wolfgang Straßer 3. Berichterstatter: Professor Dr. Thomas Ertl Abstract In this thesis, new and improved methods for the visualization of four-dimensional spacetimes are presented. The first part of this thesis deals with the flat spacetime of special relativity. A unified physical basis for special relativistic visualization is established. Issues of illumination, color vision, transformation of properties of light, and the kinematics of accelerating bodies are discussed. In particular, a derivation of the transformation of radiance is included. Rendering techniques for special relativistic visualization are presented. Previously known techniques—special relativistic polygon rendering and special relativistic ray tracing—are described in a unified framework. It is shown how relativistic effects on illumination can be incorporated in these techniques and it is demonstrated that visual perception is dominated by the searchlight and Doppler effects. Relativistic radios- ity, texture-based relativistic rendering, and image-based relativistic rendering are pro- posed as new rendering methods. Relativistic radiosity can visualize effects on illumi- nation up to arbitrary accuracy for scenes made of diffuse materials. Radiosity is well suited for interactive walk-throughs, but also for high-quality images. Texture-based relativistic rendering utilizes the texture-mapping hardware to implement the relativis- tic transformations. It is most appropriate for interactive applications which visualize special relativistic effects on both geometry and illumination. Image-based relativistic rendering closes the gap between well-known non-relativistic image-based techniques and relativistic visualization. Image-based rendering does not require laborious three- dimensional modeling and achieves photo-realism at high rendering speeds. Image- based relativistic rendering allows to generate photo-realistic images of rapidly moving real-world objects with great ease and is a powerful tool to produce movies and snap- shots for both entertainment and educational purposes. Interactive virtual environments for the exploration of special relativity are intro- duced. The first environment is a simple “relativistic flight simulator” which runs on a standard PC or graphics workstation. The second system is a sophisticated immer- sive virtual environment which exploits multi-pipe and multi-processor architectures. Parallelization of the relativistic transformation results in the same frame rates for rel- ativistic rendering as for standard non-relativistic rendering. The relativistic-vehicle- control metaphor is introduced for navigating at high velocities. This metaphor contains a physics-based camera control and provides both active and passive locomotion. The second part of the thesis deals with curved four-dimensional spacetimes of gen- eral relativity. Direct visualization of what an observer would see in a general relativistic setting is achieved by means of non-linear ray tracing. A generic system is presented for ray tracing in spacetimes described by a single chart. The suitability of ray tracing as a visualization tool is demonstrated by means of two examples—the rigidly rotating disk of dust and the warp metric. Extensions to single-chart ray tracing are proposed to incorporate the differential-geometric concept of an atlas. In this way, spacetimes of complex topologies can be considered. An example is included, showing the visualiza- tion of a wormhole. Ray tracing is applied to the field of gravitational lensing. It is shown how the vi- sualization of standard lensing can be included in a ray tracing system. Furthermore, ray tracing allows to investigate deflecting objects beyond the approximations of stan- dard lensing. For example, large angles of deflections can be considered. The caustic finder is proposed as a numerical method to identify two-dimensional caustic structures induced by a gravitational lens. The inner geometry of two-dimensional spatial hypersurfaces can be visualized by isometric embedding in three-dimensional Euclidean space. A method is described which can embed surfaces of spherical topology. This embedding scheme supports sampled metric data which may originate from numerical simulations. Finally, a specific application in classical visualization is described. Classical visu- alization means the visual representation of data from relativistic simulations without taking into account the curvature of spacetime. An algorithm for the adaptive trian- gulation of height fields is developed in order to achieve a good mesh quality, even in areas where the underlying function has high gradients. Height field visualization is exemplarily applied to data from neutron star simulations. i ii Imagination is more important than knowledge. Knowledge is limited. Imagination encircles the world. “What Life Means to Einstein: An Interview by George Sylvester Viereck,” for the October 26, 1929 issue of The Saturday Evening Post. iii iv Contents 9 General Relativistic Ray Tracing 55 9.1 Theoretical Background ............ 55 9.2 Ray Tracing in a Single Chart Spacetime . 57 1 Introduction 1 9.3 Visualization of the Rigidly Rotating Disk of 1.1 Principles, Methods, and Techniques ..... 1 Dust...................... 58 9.4VisualizationoftheWarpMetric....... 60 1.2Goals..................... 2 9.5 Spacetimes of Non-Trivial Topology . 62 1.3 Outline .................... 2 10 Gravitational Lensing 65 10.1 Theoretical Background ............ 65 I Special Relativistic Visualization 3 10.2 Visualization in Standard Lensing Theory . 66 10.3DirectApplicationofRayTracing...... 67 2 Introduction to the Visualization of Special Relativ- 10.4CausticFinder................. 68 ity 5 2.1 Historical Remarks .............. 5 11 Visualization of Inner Geometry by Embedding 71 11.1 Embedding of a Triangulated Two-Surface . 71 2.2PreviousWork................. 6 11.2ConditionsonTriangleMesh......... 71 2.3 Outline .................... 6 11.3 Implementation and Results ......... 72 11.4Discussion................... 73 3 Color Vision and Illumination 7 3.1Radiometry.................. 7 12 Classical Visualization 75 3.2Colorimetry.................. 7 12.1 Triangulation of Height Fields ........ 75 3.3ReconstructionofthePowerSpectrum.... 11 12.2 Implementation and Results ......... 76 3.4LocalIllumination.............. 12 13 General Relativistic Visualization: Summary and Open Questions 77 4 Physical Basis of Special Relativistic Visualization 15 13.1 Ray Tracing and Gravitational Lensing . 77 4.1LorentzTransformation............ 15 13.2Embedding.................. 78 4.2 Aberration of Light and Doppler Effect . 16 13.3ClassicalVisualization............ 78 4.3SearchlightEffect............... 17 14 Conclusion 79 4.4TransformationofPhotonField........ 17 4.5Inter-FrameRelationshipsforEvents..... 18 A Special Relativistic Transformation of Radiance 4.6 Equivalence of Exocentric and Egocentric View 18 and Irradiance 81 4.7 Acceleration of a Point Particle ........ 20 A.1 Derivation of the Transformation of Radiance 81 A.2 Incident Irradiance .............. 82 5 Rendering Techniques for Special Relativity 23 5.1 Special Relativistic Polygon Rendering . 23 B Implemented Gravitational Lenses 85 B.1 Axially Symmetric Lenses .......... 85 5.2SpecialRelativisticRayTracing....... 27 B.2 Quadrupole Lenses .............. 86 5.3SpecialRelativisticRadiosity......... 29 5.4 Texture-Based Special Relativistic Rendering 31 C Program Documentation 87 5.5 Image-Based Special Relativistic Rendering . 35 C.1 Special Relativistic Polygon Rendering . 87 C.2 Texture-Based Special Relativistic Rendering 88 6 Virtual Environments for Special Relativity 41 C.3ExtensionstoRayViS............. 89 6.1PreviousandRelatedWork.......... 41 C.4IsometricEmbedding............. 92 C.5 Adaptive Subdivision of Height Fields . 92 6.2RelativisticFlightSimulator......... 41 6.3 Relativistic-Vehicle-Control Metaphor .... 42 D Supplementary Video Material 93 6.4 Immersive Virtual Environment ........ 43 D.1DetailedDescriptionoftheVideos...... 93 7 Special Relativistic Visualization: Summary and E Conventions and Notation 95 Open Questions 47 E.1 Variable Names ................ 95 7.1ComparisonofRenderingTechniques.... 47 Bibliography 97 7.2FutureWork.................. 49 II General Relativistic Visualization 51 8 Introduction to the Visualization of General Rela- tivity 53 8.1PreviousWork................. 53 8.2 Outline .................... 53 v vi Visualization of Four-Dimensional Spacetimes Chapter 1 Introduction We perceive our world by means of our senses. Interaction or color coding allowed to visualize scalar data in maps and with our surroundings is based on material and energy trans- atlases. Collections of arrows displayed vector fields, such as port, for example, by sound or light waves or by scents. The the magnetic field. The visualization techniques were essen- senses are our “detectors” for these transferring “substances”. tially restricted to two-dimensional fields. With the invention Visual perception is the most pronounced of these senses. We of the computer and its graphical abilities, more
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