all representations

$$g_{00}=(1-\frac{2M}{r}).
\label{51}$$

Besides this, the vierbein is still under-determined if a special time coordinate is chosen because of the possibility to apply a radial dependant Lorentz boost to the system connected to the choice of a global gauge.

This means that two degrees of freedom remain. However, two constraints can be formulated. The first constraint can be derived from the fact that the new objects $e^0_\mu\gamma^\mu$ and $e^r_\mu\gamma^\mu$ have to fulfil the same Lie-Algebra as $\gamma^0$ and $\gamma^r$ which means that

$$[\gamma^0,\gamma^r]_-=[e^0_m \gamma^m,e^r_n\gamma^n]_-.
\label{54}$$

And the second constraint arises from the invariance of the $g_{00}$-component of the metric. By applying the covariant metric tensor $g_{\mu\nu}$ to Eq. $(\ref{36})$ we get two further equations

$$\begin{array}{ccc}
{[e^{\mu}_m\gamma^m,e^n_\nu\gamma_n]_+} &=& 2\delta^\mu _\nu,\\
{[e_{\mu}^m\gamma_m,e_{\nu}^n\gamma_n]_+} &=& 2g_{\mu\nu}.
\end{array}
\label{55}$$

So we can formulate the two following constraints

$$\begin{array}{ccc}
e^0_{\bar 0}e^r_{\bar r}-e^0_{\bar r}e^r_{\bar 0}&=&1,\\
(e_0^{\bar 0})^2-(e_0^{\bar r})^2&=&(1-\frac{2M}{r}).
\end{array}
\label{56}$$

It is very important to emphasise that all choices of the vierbein fulfilling the constraints $(\ref{56})$ represent the usual Schwarzschild metric but in different representations. It turns out that the Dirac equation in a Schwarzschild background becomes quite simple if we choose the coefficients as

$$e^0_{\bar 0}=1\quad ,\quad e^0_{\bar r}=0\quad ,\quad e^r_{\bar r}=1\quad ,\quad e^r_{\bar 0}=-\left(\frac{2M}{r}\right)^{\frac{1}{2}}.
