Graduate Dirac's coordinates ##(\tau, \rho)## for the Schwarzschild metric with ##r \le 2m##

[Kostik]

TL;DR

Do Dirac's coordinates $(\tau, \rho)$ for the Schwarzschild metric have any recognized name?

Dirac in his "GTR" (Chap 19, page 34-35) finds a coordinate system $(\tau, \rho)$ which has no coordinate singularity at $r=2m$. Explicitly, the transformation looks like (after some algebra):
$$\tau=t + 4m\sqrt{\frac{r}{2m}} + 2m\log{\frac{\sqrt{r/2m}-1}{\sqrt{r/2m}+1}}$$
$$\rho=t + \frac{2r}{3}\sqrt{\frac{r}{2m}} + 4m\sqrt{\frac{r}{2m}} + 2m\log{\frac{\sqrt{r/2m}-1}{\sqrt{r/2m}+1}}$$
The first coordinate $\tau$ is similar to Gullstrand–Painlevé coordinates, except for the first term with an $r^{3/2}$ factor.

Is there a specific name for these coordinates attributable to their original discoverer?

 

[Ibix]

I'm not familiar with it, but I can have a look round later. Your last term in both expressions reduces to zero, though - presumably a typo and probably worth correcting.

 

[Kostik]

I'm not familiar with it, but I can have a look round later. Your last term in both expressions reduces to zero, though - presumably a typo and probably worth correcting.

Thanks - yes, fixed it.

 

[PeterDonis]

Is there a specific name for these coordinates attributable to their original discoverer?

They might be Lemaitre coordinates, or at least related to them:

https://en.wikipedia.org/wiki/Lemaître_coordinates

 

[Kostik]

Thanks! That looks exactly right.

 

[Kostik]

I found a superb analysis of these coordinates in Landau & Lifshitz Vol 2.