I have two questions about time dilation near a black hole.
(I question) The relation $d\tau^2 = (1-\frac{r_s}{r}) dt^2$ between the proper time $d\tau$ of an observer near a B.H. and the time dt relative to an observer infinite far away , is true only in the static case?
(II question) Consider a free falling observer toward a Schwarzchild black hole along the radial direction $$ ds^2 = -(1-\frac{r_s}{r})dt^2 +(1-\frac{r_s}{r})^{-1}dr^2 \tag{1} $$ While $ds^2 = -d\tau^2$ is the metric in his proper frame. How can we get the proper time $\tau$ only knowing this informations?
Edit for the II question: I saw from another question (Free falling into the Schwarzschild black hole: two times doubt) that the proper time of the free falling observer depends on the initial condition of the velocity respect to position( but i don't understand where this information is hidden inside his calculation). In fact i don't understand how @Riemannium (in the post linked) computed the proper time ($\tau = \pi GM$) relative to an observer in free fall that starts with zero speed from the horizon.
