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Homework Help: Schwarzchild metric - rescaled coordinates

  1. Mar 11, 2010 #1
    Schwarzschild metric - rescaled coordinates


    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.
    Last edited: Mar 11, 2010
  2. jcsd
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