Question on Painlevé-Gullstrand coordinates

Question

A known set of coordinates used for the Schwarzschild metric is the Painlevé-Gullstrand coordinates. They consist in performing a change from coordinate time $t$ to the proper time $T$ of radially infalling observers coming from infinity at rest. The transformation is the following $$ dT=dt+\left(\frac{2M}{r}\right)^{-1/2} f(r)^{-1}dr $$

where $f(r)=1-2M/r$. I need to prove this, starting from the general linear relation $$ dT=A(r,t)dt+B(r,t)dr $$

What I have done until now:

Using that $E=1$ and $L=0$, I have $$ \dot{t}=\frac{dt}{dT}=f(r)^-1 $$ $$ \dot{r}=\frac{dr}{dT}=\sqrt{\frac{2M}{r}} $$

which helps me to arrive at the following relation $$ B=\left(f(r)^{-1}-A\right)\sqrt{\frac{r}{2M}} $$

Introducing $A=1$ in this last equation, I would have finished.

But how do I prove that $A=1$?

Could I use that $\frac{\partial A}{\partial r}=\frac{\partial B}{\partial t}$ in some way?

Thanks in advance, I'm having nightmares with this problem.

NOTE: I know there are several ways of proving this. I've read some of them that use concepts such as T=constant hypersurfaces, or the fact that in the Painlevé-gullstrand coordinates we have $g_{rr}=1$. Unfortunately, I am not allowed to use any of those methods. Thanks in advance again.

Answer 1

There is a beautiful trick I have modified from Martel & Poisson 2001. Take the 4-velocity of the radial observers, lower the indices to get the dual vector version, then express this in terms of the dual basis vectors $dt$ and $dr$ as required.

Another approach was used by Gautreau & Hoffmann 1978, see also the excellent pedagogical description in Moore's textbook. This uses first principles: consider the change $dT$ in proper time at fixed $r$ but changing $t$, and separately at fixed $t$ and changing $r$.

I am working on a paper which includes this information.

Answer 2

I guess that the most interesting answer is provided by Painlevé himself in his paper in 1921 Nov 10th published in the reports of the Academy of Science. He did not start from the Schwarzschild but from a generic form of a metric for a solution with spherical symmetry, then constraining it by the Einstein Equation which is that the Ricci tensor must vanish as it is a solution in vacuum, he gets a generic solution depending on two functions $f(r), g(r)$ of the coordinate $r$. This generic solution is most general solution in these coordinates $(t,r, \theta,\phi)$ as the Ricci tensor identically vanishes for any function $f(r), g(r)$. Therefore, all solutions (Scharzschild, Painlevé, Finkelstein, etc.) in these coordinates correspond to different function $f(r)$ and $g(r)$.

For more information, see http://www.astromontgeron.fr/Painleve-article-english.pdf, especially Chapter 7.