today are black holes.

Although the situation is in a certain sense similar to the case of the hydrogen atom, to describe gravitation one has to use general relativity instead of Maxwell theory. Therefore, the problem becomes much more involved because of the fundamental difference between the framework of electromagnetism and general relativity. In the case of general relativity one cannot simply couple a potential to the Hamiltonian but the structure of space time itself is changed by the presence of a massive body. This leads to a metric tensor $g^{\mu\nu}(x)$ which differs from the usual metric of flat Minkowski space time $\eta^{\mu\nu}={\rm diag}(+1,-1,-1,-1)$. The structure of space time near a spherical symmetric body is described by the Schwarzschild metric. In the case of the Klein-Gordon equation the metric appears explicitly in the equation and one has to replace the Minkowski metric by the Schwarzschild metric to obtain the correct equation adapted to the new situation. The situation is very different in the case of the Dirac equation, because the metric tensor does not appear explicitly in the Dirac equation. Besides this the Dirac equation does not describe a scalar valued state but a Spinor. I.e. that we have to 