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# Visualizing circular motion around a Schwarzschild black hole

### Thomas Müller, Sebastian Boblest

An observer who wants to move on a circular orbit around a Schwarzschild black hole with a constant but arbitrary velocity must compensate the gravitational and centrifugal acceleration to stay on this orbit. Depending on the radius of the orbit and the velocity, the local reference frame of the observer undergoes a geodesic precession. In this article, we will describe the details of such a circular motion and we will present an interactive application that shows what the observer could see.

 Qt application / data files
• Linux source files can be downloaded here (gzip-compressed tar-file, 1.1 MB, md5).
(Installation manual.)
(Installation manual.)
For each radial distance and type of torus definition, a set of three lookup tables is necessary:
The two different assignment tables for Planck spectra and their corresponding RGB colors can be found here:
 (binary data file, 49kB, md5). (binary data file, 49kB, md5).

As background image you can use e.g. the Milkyway panorama by ESO/S. Brunier.
As torus texture you can use e.g. torus_texture.png.

An example configuration file can be found here.

 System requirements
• At least 300MB GPU memory.
• The Linux version needs the free Qt SDK (>= 4.6) which can be found here.
The application was tested with the graphics board: NVidia GeForce 8800 GTS.
 Movies
The following movies show the circular motion for three different radii. Here, the black hole is indicated as gray region.

 r = 2rs, beta=0.5.Viewing direction is fixed.(1024x512, mp4, 18MB, 12sec) r = 1.5rs, beta=0.5.Viewing direction is fixed.(1024x512, mp4, 18MB, 12sec) r = 1.15rs, beta=0.5.de Sitter precision is shown.(1024x512, mp4, 18MB, 12sec) r = 1.15rs, beta=0.01, 4pi-projection.Viewing direction is fixed.(1024x512, mp4, 10MB, 14.4sec) r = 1.15rs, beta=0.5, 4pi-projection.Viewing direction is fixed.(1024x512, mp4, 10MB, 14.4sec) r = 1.15rs, beta=0.99, 4pi-projection.Viewing direction is fixed.(1024x512, mp4, 10MB, 14.4sec)
The left column is rendered using the spectrum scaled to Y_1500 luminance.
The right column uses the spectrum scaled to maximum luminance.

All the movies take either 12 or 14.4 seconds when played with 25 frames per second. The proper times experienced by an observer on these circular orbits, however, would be different:

 Images
All images from the article can be downloaded as eps-files in the original size:

• T. Müller, S. Boblest
Visualizing circular motion around a Schwarzschild black hole
American Journal of Physics 79, 63-73 (2011).
DOI: 10.1119/1.3492722
• Images:

• others:
 Contact
Visualisierungsinstitut der Universität Stuttgart (VISUS)
Allmandring 19
70569 Stuttgart, Germany
Email: Thomas.Muellervis.uni-stuttgart.de

Universität Stuttgart
1. Institut für Theoretische Physik (ITP1)
Pfaffenwaldring 57 // IV
70550 Stuttgart, Germany
Email: sebastian.boblestitp1.uni-stuttgart.de