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Visualization of Four-Dimensional Spacetimes

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 radiosity, texture-based relativistic rendering, and image-based relativistic rendering are proposed as new rendering methods. Relativistic radiosity can visualize effects on illumination 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 relativistic 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 threedimensional modeling and achieves photo-realism at high rendering speeds. Imagebased 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 snapshots for both entertainment and educational purposes. Interactive virtual environments for the exploration of special relativity are introduced. The first environment is a simple “relativistic flight simulator” which runs on a standard PC or graphics workstation. The second system is a sophisticated immersive virtual environment which exploits multi-pipe and multi-processor architectures. Parallelization of the relativistic transformation results in the same frame rates for relativistic rendering as for standard non-relativistic rendering. The relativistic-vehiclecontrolmetaphor 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 general 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 visualization of a wormhole. Ray tracing is applied to the field of gravitational lensing. It is shown how the visualization of standard lensing can be included in a ray tracing system. Furthermore, ray tracing allows to investigate deflecting objects beyond the approximations of standard 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 visualization means the visual representation of data from relativistic simulations without taking into account the curvature of spacetime. An algorithm for the adaptive triangulation 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.