# PG-Metric dr/dt

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?

• PG coordinates are coordinates that are not adapted to the timelike Killing vector. The coordinates simplest for your purposes would be the Schwarzschild coordinates. Did you try that? Dec 19, 2018 at 8:04
• Hmm, no I want to stay in the PG basis ;) Dec 19, 2018 at 12:24
• @Armani42 It is not clear to me exactly what you want to do? Do you want to find the Killing vectors? Find conserved quantities? Dec 24, 2018 at 12:18