on focus on the special case of the Schwarzschild metric, because we are investigating a particle in a spherically symmetric gravitational field.

The Schwarzschild metric (first presented in [@Schwarzschild:1916uq]) is usually represented in the following way

$$g_{\mu\nu}=\left(\begin{array}{cccc}
(1-\frac{2M}{r}) & 0 & 0 & 0\\
0 & \frac{-1}{(1-\frac{2M}{r})} & 0 & 0\\
0 & 0 & -r^2 & 0\\ 
0 & 0 & 0 & -r^2 {\rm sin}(\theta)\end{array}\right).
\label{49}$$

A spherically symmetric object only influences the radial part of the metric. Therefore, we have to deal with the time and the radial part of the metric and do not consider the angular coordinate. This leads to

$$\begin{array}{ccc}
e^0_\mu\gamma^\mu &=& e^0_{\bar 0}\gamma^{\bar 0}+e^0_{\bar r}\gamma^{\bar r},\\
e^r_\mu\gamma^\mu &=& e^r_{\bar 0}\gamma^{\bar 0}+e^r_{\bar r}\gamma^{\bar r}.
\end{array}
\label{50}$$

The components of the Schwarzschild metric are not fully determined due to the fact that the metric can be expressed as a function of a new time coordinate according to a transformation of the form $t\rightarrow t+x(r)$ where $x$ denotes an arbitrary function of $r$. It is important to note that this transformation of the time coordinate indeed changes the appearance of the metric, but does not affect the $g_{00}$-component of the metric. This component is equal in 