Schwarzchild metric - rescaled coordinates

[radiogaga35]

Schwarzschild metric - rescaled coordinates

Hi,

I've been working through a problem (no. 14 in ch. 9) of Alan Lightman's book of GR problems. I can't understand one of the results that are stated without proof. Basically it amounts to a rescaling of coordinates.

I know that to first order in $${\textstyle{M \over r}} \ll 1$$ (weak gravitational field), the standard Schwarzschild metric can be written

$$d{s^2} = - (1 - {\textstyle{{2M} \over r}})d{t^2} + (1 + {\textstyle{{2M} \over r}})d{r^2} + {r^2}d{\Omega ^2}$$

in units where $$G = c = 1$$, and where $$d{\Omega ^2} = d{\theta ^2} + {\sin ^2}\theta d{\phi ^2}$$.

Lightman states that, in an "appropriate" coordinate system, the Schwarzschild metric (again to lowest order in $${\textstyle{M \over r}} \ll 1$$) can be written

$$d{s^2} = - (1 - {\textstyle{{2M} \over r}})d{t^2} + (1 + {\textstyle{{2M} \over r}})(d{x^2} + d{y^2} + d{z^2})$$,

where $${r^2} = {x^2} + {y^2} + {z^2}$$.

I've been pulling my hair out trying to derive Lightman's form from the standard form I gave above. Apparently it's just a simple rescaling of the radial coordinate but I've had no luck. Any help would be appreciated!

Thanks in advance.

 

[ergospherical]

Indeed, for this problem it's convenient to introduce a re-scaling $r \rightarrow \tilde{r}$ which satisfies $r = \tilde{r} (1+M/2\tilde{r})^2$, which can also be re-arranged for $1-2M/r = (1-M/2\tilde{r})^2/(1+M/2\tilde{r})^2$. Take the derivation,\begin{align*}
dr = (1+M/2\tilde{r})(1-M/2\tilde{r}) d\tilde{r}
\end{align*}Consider the classic Schwarzschild metric\begin{align*}
g &= -(1-2M/r) dt^2 + (1-2M/r)^{-1} dr^2 + r^2 d\Omega^2 \\ \\
&= - \dfrac{(1-M/2\tilde{r})^2}{(1+M/2\tilde{r})^2} dt^2 + \dfrac{(1+M/2\tilde{r})^2}{(1-M/2\tilde{r})^2} dr^2 + r^2 d\Omega^2 \\ \\
&= - \dfrac{(1-M/2\tilde{r})^2}{(1+M/2\tilde{r})^2} dt^2 + (1+M/2\tilde{r})^4 d\tilde{r}^2 + \tilde{r}^2 (1+M/2\tilde{r})^4 d\Omega^2 \\ \\
&= - \dfrac{(1-M/2\tilde{r})^2}{(1+M/2\tilde{r})^2} dt^2 + (1+M/2\tilde{r})^4 \left\{ d\tilde{r}^2 + \tilde{r}^2 d\Omega^2 \right\}
\end{align*}To first order in $M/\tilde{r}$, one has simply\begin{align*}
g = (1-2M/\tilde{r}) dt^2 + (1+2M/\tilde{r})\{ d\tilde{r}^2 + \tilde{r}^2 d\Omega^2 \}
\end{align*}Defining new Cartesian-ish coordinates $(x,y,z) = (\tilde{r} \sin{\theta} \cos{\phi}, \tilde{r} \sin{\theta} \sin{\phi}, \tilde{r}\cos{\theta})$ puts the metric into this familiar form\begin{align*}
g = (1-2M/\tilde{r}) dt^2 + (1+2M/\tilde{r})\{ dx^2 + dy^2 + dz^2 \}
\end{align*}