at the moment, I investigate some time in learning General Relativity and there I saw the Painlevé-Gullstrand Metric which is given as
$ds^2 = -dT^2 + \left(dr+\sqrt{\frac{r_s}{r}} dT\right)^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2$
and the time coordinate $t$ is replaced by
$T(t, r) = t + 2 \sqrt{rr_s} + r_s \ln \left|\frac{\sqrt{r/r_s}-1}{\sqrt{r/r_s} +1}\right|$
Now, I want to compute some properties for a massive particle which falls into a black hole from infinity, where I want to compute the velocity
$\dot{r} = dr/d\tau$.
So I would begin with the fact that the particle firstly rests at infinity and use the conservation of energy and momentum.
But how do I come then with the Killing Vector to the velocity?
