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?

  • $\begingroup$ 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? $\endgroup$
    – Everiana
    Dec 19, 2018 at 8:04
  • $\begingroup$ Hmm, no I want to stay in the PG basis ;) $\endgroup$
    – Armani42
    Dec 19, 2018 at 12:24
  • $\begingroup$ @Armani42 It is not clear to me exactly what you want to do? Do you want to find the Killing vectors? Find conserved quantities? $\endgroup$
    – magma
    Dec 24, 2018 at 12:18


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