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Schwarzchild metric - rescaled coordinates
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[QUOTE="radiogaga35, post: 2619289, member: 79570"] [b]Schwarzschild metric - rescaled coordinates[/b] 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 [tex]{\textstyle{M \over r}} \ll 1[/tex] (weak gravitational field), the standard Schwarzschild metric can be written [tex]d{s^2} = - (1 - {\textstyle{{2M} \over r}})d{t^2} + (1 + {\textstyle{{2M} \over r}})d{r^2} + {r^2}d{\Omega ^2}[/tex] in units where [tex]G = c = 1[/tex], and where [tex]d{\Omega ^2} = d{\theta ^2} + {\sin ^2}\theta d{\phi ^2}[/tex]. Lightman states that, in an "appropriate" coordinate system, the Schwarzschild metric (again to lowest order in [tex]{\textstyle{M \over r}} \ll 1[/tex]) can be written [tex]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})[/tex], where [tex]{r^2} = {x^2} + {y^2} + {z^2}[/tex]. 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. [/QUOTE]
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