What is the Schwarzschild metric with proper radial distance?

Question

Reading the marvellous book "The Membrane Paradigm" I stumbled upon a suggested change of variable that I'm not able to deal with. Starting with the usual Schwarzschild metric for the spatial 3-geometry

$$ ds^2 = \frac{1}{f(r)} dr^2 + r^2 \left( d\theta^2 + \sin ^2 \theta \ d\phi\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1) $$

where $ f(r) = 1-\frac{2M}{r}$ (that is simply Schwarzschild metric without time element $-f(r)\ dt^2 $) they suggest a new radial coordinate $R$ with the "property that $R-2M$ measures the proper radial distance outward from the horizon". This new variable is

$$ R= 2M + \sqrt{r(r-2M)}+\ln \left[ \sqrt{\frac{r}{2M} -1} + \sqrt{\frac{r}{2M}} \right].\ \ \ \ \ \ \ \ \ \ (2) $$

I'd like to see the new metric but I'm not able to invert (2) (namely find $r(R)$) to perform the coordinate change $$ ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=g_{\mu\nu}\frac{\partial x^{\mu}}{\partial x'^{\rho}}\frac{\partial x^{\nu}}{\partial x'^{\sigma}}dx'^{\rho}dx'^{\sigma}=g'_{\rho\sigma}dx'^{\rho}dx'^{\sigma} $$

I tried also to use the inverse of the Jacobian as suggested in https://physics.stackexchange.com/a/43084 but at the end I always need at least to change the variable $r^2$ in the angular part of (1).

Do you have any idea on how to deal with it or how to invert (2) ?

Answer 1

If I take the (corrected) formula :

$$R= 2M + \sqrt{r(r-2M)}+2M\ln \left[ \sqrt{\frac{r}{2M} -1} + \sqrt{\frac{r}{2M}} \right] \tag{1}$$

You have (if no error) : $dR = \dfrac{dr}{\sqrt{ 1 - \dfrac{2M}{r}}}$, so $dR^2 = \dfrac{dr^2}{f(r)}$, and this simplifies your metrics.

However, you cannot invert the formula $(1)$ to get $r$ as an explicit function of $R$. You will simply write the metrics :

$$ds^2 = dR^2 + (r(R))^2 \left( d\theta^2 + \sin ^2 \theta \ d\phi\right) \tag{2}$$ where $r(R)$ is implicitely defined by the equation $(1)$