Geometry Around Black Holes

stud. scient Michael Cramer Andersen
Astronomical Observatory, NBIfAFG
June 1996

Abstract:

Black Holes provide an interesting geometry which affects the motion of particles close to the horizon of the Black Hole. Some effects are discussed including curvature and gravitational redshift (when a photon leaves the gravitational field). An introduction to metrics and metric tensors leads to the Schwarzschild metric and it's "big brother" the Kerr metric describing rotating Black Holes. A computer code is used to integrate a number of particle trajectories.

The illustration above shows a photon approaching a rotating Black Hole (with a=1) in the equator plane. The space is curved, and the photon is bended. The energy of the incoming photon corresponds to a red wavelength. As the photon comes closer and closer to the center, it is gravitational shifted to the blue (more energetic) end of the spectrum. The rotation of the hole affects the geometry in a way, so a radius (in the polar coordinate system) will rotate with the hole with an increasing angular velocity nearer the center. This explains that the (angular) direction of the photon changes from going anti-clockwise to clockwise like the Black Hole itself.

• Contents
• General Relativity and Spacetime
• Einsteins field equations and Black Holes
• Kerr's rotating Black Holes
• Conclusion
• References
• About this document ...

Please note that this text is partly written for the course Relativistic Astrophysics II. The subject is chosen in a way so it fits the visualization project as well. The following reading should be enough to understand the main ideas used in the visualizations on the other pages.
The visualization pages may be judged independently of this report (also available as ps.gz file).