Schwarzchild metric - rescaled coordinates

1. Mar 11, 2010

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!