Geometry Around Black Holes

Michael Cramer Andersengif

June 1996

In this paper I will investigate the geometry around Black Holes, and how this affects freely falling relativistic particles along geodesics which is certainly not straight lines as in normal flat spacetime. Stellar Black Holes are the relicts of collapsed massive stars which provides extreme mass-/energydensities. Nothing can escape from the Black Hole, not even light. This is because of the extreme curvature of space. The detailed description of Black Holes is included in The Einstein Field Equations. Historically one of the first exact solutions to these equations was that of Schwarzschild 1916 describing a spherical symmetric point mass, later identified as a Black Hole.
I will concentrate on the Schwarzschild-solution, describing a non-rotating Black Hole and the Kerr solution, describing Black Holes with angular momentum. There exists of course more general - and complicated - solutions including charge, and electromagnetic fields around the Black Holes. But I will not go into this exciting area.
The aim of this work is to use some of the fundamental results to get a view of the geometry around a Black Hole. Curvature is one of the most remarkable geometrical properties, but some other basic concepts has to be introduced, these are: world lines, geodesics and metric tensors. Through out this text, I will use the space-like sign convention: (-,+,+,+) for the metrics considered; indices going from 1 to 3 are written in latin (i,k,..) denoting 3 space dimensions while indices going from 0 to 3 are written in greek ( tex2html_wrap_inline1060 ,..) denoting one time coordinate and 3 space coordinates.

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Michael Cramer Andersen
Fri Jun 14 12:07:09 MDT 1996