Schwarzschild geometry in Schwarzschild coordinates $(t,r,\theta,\phi)$ is time-symmetric \begin{equation} ds^2=-\left(1-\frac{2GM}{c^2 r}\right)c^2dt^2+\left(1-\frac{2GM}{c^2 r}\right)^{-1}dr^2+r^2\left(d \theta^2 +\sin^2 \theta d \phi^2\right) \;. \end{equation}
<PERSON> coordinate system is defined by a set of geodesic clocks. The coordinate clocks are freely falling from some maximal radius $r_m$ towards $r=0$, where $r_m$ is different for each clock. All clocks start falling at the same Schwarzschild time $t_0$ and they are synchronized in such a manner that each clock shows $0$ at $r_m$. 578
<PERSON> coordinate is defined to stay constant along the trajectory of each clock, while for time coordinate proper time is taken.
From now on the angular part metric will be omitted, since is stays the same. We also take $r_s=2M$ and $G=c=1$: \begin{equation}\label{eq:sch-met2} ds^2=-\left(1-\frac{r_s}{r}\right)dt^2+\left(1-\frac{r_s}{r}\right)^{-1}dr^2 \;. \end{equation}
Geodesics in Schwarzschild gometry
To get the equation of geodesics in Schwarzschild geometry we have to solve equations of motion of a free particle: \begin{equation}\label{eq:lagrangian} \mathcal{L}=\frac{1}{2}mg_{\mu\nu}\dot{x}^\mu\dot{x}^\nu \;, \end{equation} \begin{equation}\label{eq:dot} \dot{x}^\mu=\frac{dx^\mu}{d\tau}=u^\mu \;. \end{equation} \begin{equation}\label{eq:lagrangian2} \mathcal{L}=-\frac{m}{2}\left(1-\frac{r_s}{r}\right)\dot{t}^2+\left(1-\frac{r_s}{r}\right)^{-1}\dot{r}^2 \;, \end{equation} \begin{equation}\label{eq:EL} \frac{d}{d\tau}\frac{\partial\mathcal{L}}{\partial \dot{x}^\mu}-\frac{\partial\mathcal{L}}{\partial x^\mu}=0 \;, \end{equation} For $\mu=0$ we get a constant of motion \begin{equation}\label{eq:ConstOfMotion} \frac{\partial}{\partial\tau}\left[\left(1-\frac{r_s}{r}\right)\dot{t}\right]=0 \qquad \Rightarrow \qquad \left(1-\frac{r_s}{r}\right)\dot{t}=a \;, \end{equation}
For timelike geodesics: $ds^2=-d\tau^2$ the radial geodesic equation becomes \begin{equation}\label{eq:orbit} \left(\frac{d\tau}{dr}\right)^2=\frac{1}{a^2-\left(1-\frac{r_s}{r}\right)} \;. \end{equation} Maximal radius is ($dr/d\tau=0$) \begin{equation}\label{eq:maximal} r_m=\frac{r_s}{1-a^2} \;. 394