coefficients of the connection $\omega_\mu^{mn}$ with the Christoffel symbols expressed in terms of the local coordinates, because the connection between the tangential vector spaces of different points on the space time manifold is independent of the special space by which it is represented. If one recalls now that the vierbein $e^\mu_m$ describes the transformation between global and local coordinates, one can express the coefficients by the vierbein. Using Eq. $(\ref{28})$ in Eq. $(\ref{41})$, inserting it to Eq. $(\ref{46})$ and transforming the obtained expression a little bit, we arrive at

$$\omega_\mu^{mn}=2e^{\nu [m}\partial_{[\mu}e_{\nu]}^{n]}+e_{\mu p}e^{\nu m}e^{\sigma n}\partial_{[\sigma}e_{\nu]}^p.
\label{48}$$

So we have expressed the connection coefficients in terms of the vierbein. (The derivation of this formula can also be found in [@[book1]]. For advanced aspects concerning the relation between the usual formulation of general relativity and the vierbein formalism we suggest Refs. [@[book3]],[@[book1]] and [@[book5]].)

Solution of the Dirac equation in a Schwarzschild background
============================================================

The choice of a suitable formulation of the Dirac equation in a Schwarzschild background
----------------------------------------------------------------------------------------

So far we have formulated the Dirac equation in a general space time with an arbitrary metric tensor and an arbitrary connection between different space time points. To be specific we will from now 