incorporate the effect of the new metric on the Spinor space. So the situation is more difficult than in the case of the Klein-Gordon equation where the new metric has just to be inserted. Therefore we will have to derive the general form of the Dirac equation in curved space time by considering the gauge formulation of general relativity as mentioned above. This is the subject of the second section. After this we will explore the special case of the Schwarzschild metric and look for a suitable representation to get an equation which is as simple as possible.

However the changing of the metric must have an effect similar to the coupling of a potential. Indeed, as will be shown below, the Dirac equation takes the form of the free Dirac equation with an additional perturbative term in the Hamiltonian when a certain representation is chosen. It remains the task to solve the equation. The main reason why it is more difficult to solve this problem compared to the hydrogen atom is the singularity at the event horizon in addition to the singularity at the origin. However, using a series expansion it turns out that a solution can be found 