The gravitational analogue to the hydrogen atom

This paper 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 spacetime. 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