  This article reports on a student summer project performed in 2006 at the
University of Frankfurt. It is addressed to undergraduate students familiar
with the basic principles of relativistic quantum mechanics and general
relativity. The aim of the project was to study the Dirac equation in curved
space time. To obtain the general relativistic Dirac equation we use the
formulation of gravity as a gauge theory in the first part. After these general
considerations we restrict the further discussion to the special case of the
Schwarzschild metric. This setting corresponds to the hydrogen atom, with the
electromagnetic field replaced by gravity. Although there is a singularity at
the event horizon it turns out that a regular solution of the time independent
Dirac equation exists. Finally the Dirac equation is solved numerically using
suitable boundary conditions.
