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.


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.


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