# 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? – Everiana Dec 19 '18 at 8:04
• Hmm, no I want to stay in the PG basis ;) – Armani42 Dec 19 '18 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? – magma Dec 24 '18 at 12:18