# Visualization in Special and General Relativity

## Introduction:

Einstein's Special and General Relativity describe space and time not as two separate, absolute entities, but they combine space and time to one single entity - the so called spacetime. Special Relativity concentrates on relative motion between reference frames close to the speed of light in flat Minkowskian spacetime. General Relativity describes gravity as a geometric property of spacetime itself.

The resulting consequences of both theories are far from our daily experiences. Even with the most modern propulsion systems we move with velocities far below the speed of light; and the mass of the Earth curves the spacetime only such that we stay on the ground. The effects of general relativity become apparent only if big masses are concentrated in small regions. The most extreme example is a black hole, where the curvature of spacetime is so strong that even light cannot escape.

While Special Relativity is quite simple in the mathematical sense and can be discussed on high school level, General Relativity is quite demanding. In particular, a freshman has difficulties with the mathematics of General Relativity and the physical contents.

Here, computer simulations offer a visual access to both theories.  Phenomena like aberration of light, length contraction or  bending of light can be experienced. Hence, we obtain an intuitive access to Special and General Relativity.

## Visualization techniques

The most natural technique for relativistic visualization is the ray tracing method. Particularly, the finitness of the speed of light has to be taken into account when extending the ray tracing methog from three to four dimensions. Additionally, we have to consider the frequency-shift caused by the Doppler effect and gravitational fields, bending of light due to the curved spacetime, as well as light amplification. Four-dimensional relativistic ray tracing delivers high-quality images, however, it is very time consuming and thus, in general, not suitable for interactive visualizations.

In Special Relativity, the image-based method and the polygon-rendering are two techniques that make interactive visualization feasible. Both methods make use of modern graphics hardware for either an image space or an object space transformation. However, there are limitations concerning the scene and the motion of objects with both techniques. A recently developed third method combines polygon-rendering with local ray tracing and thus circumvents the shortcomings of the underlying methods.

In General Relativity, interactive visualizations are possible only by means of analytic solutions of the equations of motion for light or particles. In case of a highly symmetric spacetime geometry, simple scenaries can be visualized with interactive frame rates by means of precalculations and tabulating.

# Projects

### Visualization of light- and timelike geodesics in multi black hole spacetimes in the general theory of relativity

A detailled description can be found here. (Coming soon)

### Interactive visualization of a thin disk around a Schwarzschild black hole

A detailed description can be found here.

### GPU-based four-dimensional general-relativistic ray tracing

A detailed description can be found here.

### Detailed study of null and time-like geodesics in the Alcubierre Warp spacetime.

A detailed description can be found here.

### Studying null and time-like geodesics in the classroom.

A detailed description can be found here.

### GeodesicViewer - A tool for exploring geodesics in the theory of relativity

A detailed description can be found here.

### Visualizing circular motion around a Schwarzschild black hole

A detailed description can be found here.

### Distortion of the stellar sky by a Schwarzschild black hole

A detailed description can be found here.

### A trip to the end of the universe and the twin "paradox"

A detailed description can be found here.

### Visualization of the General Relativistic Disk of Dust

A detailed description can be found here.

#### Visualization in the Einstein Year 2005: A Case Study on Explanatory and Illustrative Visualization of Relativity and Astrophysics

A detailed description can be found here.

### Minkowski Diagrams

A detailed description of the Java2 application can be found here.

### Relativistic Movement Viewer

A detailed description can be found

# Publications

## 2016

 Müller, Thomas; Fechtig, Oliver: Empirical exploration of timelike geodesics around a rotating wormhole. In: : Vol. 84, No. 5 (2016) (to appear). [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2015

 Müller, Thomas: Image-based general-relativistic visualization. In: (2015), p. 065019. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 Müller, Thomas; Boblest, Sebastian; Weiskopf, Daniel: Visualization Showcase: General-Relativistic Black Hole Visualization. In: Eurographics Symposium on Parallel Graphics and Visualization, 2015. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2014

 Müller, Thomas: GeoViS – Relativistic ray tracing in four-dimensional spacetimes. In: Computer Physics Communications (2014), pp. 2301-2308. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas; Boblest, Sebastian: Visual appearance of wireframe objects in special relativity. In: European Journal of Physics (2014), p. 065025. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2012

 Frutos-Alfaro, Francisco; Grave, Frank; Müller, Thomas; Adis, Daria: Wavefronts and Light Cones for Kerr Spacetimes. In: Journal of Modern Physics (2012), pp. 1882-1890. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Kuchelmeister, Daniel; Müller, Thomas; Ament, Marco; Wunner, Günter; Weiskopf, Daniel: GPU-based four-dimensional general-relativistic ray tracing. In: Computer Physics Communications (2012), p. 2282–2290. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas; Weiskopf, Daniel: Detailed study of null and timelike geodesics in the Alcubierre warp spacetime. In: General Relativity and Gravitation (2012), pp. 509-533. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas; Frauendiener, Jörg: Interactive visualization of a thin disc around a Schwarzschild black hole. In: European Journal of Physics (2012), pp. 955-963. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2011

 Boblest, Sebastian; Müller, Thomas; Wunner, Günter: Twin paradox in de Sitter spacetime. In: European Journal of Physics (2011), pp. 1117-1142. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas; Weiskopf, Daniel: General-Relativistic Visualization. In: Computing in Science & Engineering: Vol. 13, No. 6 (2011), pp. 64-71. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas; Weiskopf, Daniel: Special-Relativistic Visualization. In: Computing in Science & Engineering: Vol. 13, No. 4 (2011), pp. 85-93. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas; Frauendiener, Jörg: Studying null- and time-like geodesics in the classroom. In: European Journal of Physics (2011), pp. 747-759. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas; Boblest, Sebastian: Visualizing circular motion around a Schwarzschild black hole. In: American Journal of Physics (2011), pp. 63-73. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2010

 Müller, Thomas; Weiskopf, Daniel: Distortion of the stellar sky by a Schwarzschild black hole. In: American Journal of Physics: Vol. 78, No. 2 (2010), pp. 204-214. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas; Grave, Frank: GeodesicViewer - A tool for exploring geodesics in the theory of relativity. In: Computer Physics Communications (2010), pp. 413-419. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas; Grottel, Sebastian; Weiskopf, Daniel: Special Relativistic Visualization by Local Ray Tracing. In: IEEE Transactions on Visualization and Computer Graphics: Vol. 16, No. 6 (2010), pp. 1243-1250. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2009

 Grave, Frank; Buser, Michael; Müller, Thomas; Wunner, Günter; Schleich, Wolfgang P.: The Gödel universe: Exact geometrical optics and analytical investigations on motion. In: Physical Review D (2009), p. 103002. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Grave, Frank; Müller, Thomas; Dachsbacher, Carsten; Wunner, Günter: The Gödel Engine - An interactive approach to visualization in general relativity. In: Computer Graphics Forum: Vol. 28, No. 3 (2009), pp. 807-814. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas; Grave, Frank: Motion4D - A library for lightrays and timelike worldlines in the theory of relativity. In: Computer Physics Communications (2009), pp. 2355-2360. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2008

 Müller, Thomas: Analytic observation of a star orbiting a Schwarzschild black hole. In: Gen. Rel. Grav. (2008), pp. 541-558. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas: Einstein rings as a tool for estimating distances and the mass of a Schwarzschild black hole. In: Phys. Rev. D (2008), p. 124042. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas: Exact geometric optics in a Morris-Thorne wormhole spacetime. In: Phys. Rev. D (2008), p. 044043. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas: Falling into a Schwarzschild black hole - Geometric aspects. In: Gen. Rel. Grav. (2008), pp. 2185-2199. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Müller, Thomas; King, Andreas; Adis, Daria: A trip to the end of the universe and the twin paradox. In: Am. J. Phys (2008), pp. 360-373. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Ruder, Hanns; Weiskopf, Daniel; Nollert, Hans-Peter; Müller, Thomas: How Computers Can Help Us in Creating an Intuitive Access to Relativity. In: New Journal of Physics (2008), p. 125014(22pp). [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2002

 Kobras, D.; Weiskopf, Daniel; Ruder, Hanns: General relativistic image-based rendering. In: The Visual Computer: Vol. 18, No. 4 (2002), pp. 250-258. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Kraus, Ute; Ruder, Hanns; Weiskopf, Daniel; Zahn, C.: Was Einstein noch nicht sehen konnte. Schnelle Computer visualisieren relativistische Effekte. In: Physik Journal: Vol. 0, No. 7 (2002), pp. 77-83. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 Ruder, Hanns; Weiskopf, Daniel: Simulation und Visualisierung in der Astrophysik oder die wundersame Reise des Christoph Zenger mit der U.S.S. Enterprise. In: Numerische Simulation als interdisziplinäre Herausforderung: Beiträge zum 60. Geburtstag von C. Zenger, pp. 59-84, 2002. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2001

 Kobras, D.; Weiskopf, Daniel; Ruder, Hanns: Image-Based Rendering and General Relativity. In: Proceedings of WSCG'01, pp. 130-137, 2001. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

# Further publications

Further publications can be found here.