# 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 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 here

# Gallery

# Publications

<?xml version="1.0" encoding="utf-8"?> <b:Sources SelectedStyle="" xmlns:b="http://schemas.openxmlformats.org/officeDocument/2006/bibliography" xmlns="http://schemas.openxmlformats.org/officeDocument/2006/bibliography"> <b:Source> <b:Tag>prd08bmuelleta</b:Tag> <b:SourceType>JournalArticle</b:SourceType> <b:GUID /> <b:LCID>0</b:LCID> <b:Author> <b:Author> <b:NameList> <b:Person> <b:Last>Müller</b:Last> <b:First>Thomas</b:First> </b:Person> </b:NameList> </b:Author> </b:Author> <b:Title>Einstein rings as a tool for estimating distances and the mass of a Schwarzschild black hole</b:Title> <b:Year>2008</b:Year> <b:BookTitle>Phys. Rev. D</b:BookTitle> <b:Volume>77</b:Volume> <b:Pages>124042</b:Pages> </b:Source> </b:Sources>

# Further publications

Further publications can be found here.