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Transforming to Eddington-Finkelstein Coordinates for Schwarzschild Geometry in GR

  1. Oct 21, 2009 #1
    1. The problem statement, all variables and given/known data

    I'm having problems seeing how the transformation to Eddington-Finkelstein in the Schwarzschild geometry works. Any help would be great!

    2. Relevant equations

    So we have the Schwarzschild Geometry given by:

    [tex]
    ds^2 = -(1-2M/r)dt^2 + (1-2M/r)^-^1 dr^2 + r^2(d\theta^2+sin^2\theta d\phi^2)
    [/tex]

    and the Edd-Fink transformation assigns

    [tex]
    t = v - r -2Mlog\mid r/2M-1 \mid
    [/tex]

    The textbook says it is straight-forward to simply sub this into the S.G line element to get the transformed geometry, but I can't seem to get it.

    3. The attempt at a solution

    OK, so differentiating the expression for the new t coordinates above, I get:

    [tex]
    dt = dv - dr -2M. \frac{d}{dr} [log(r-2m)-log(2m)]
    [/tex]
    [tex]
    dt = dv - dr -2M. \left(\frac{1}{r-2m}\right)
    [/tex]
    [tex]
    dt = dv - dr - \left(\frac{2m}{r}-1\right)
    [/tex]


    ...but the book says the answer is

    [tex]
    dt = -\frac{dr}{1-\frac{2M}{r}} + dv
    [/tex]

    >where am I going wrong here? Given this last expression, it is fairly easy to sub it into the SG line element to get the transformed coordinates. Problem is, I can't seem to get that far! Any help much appreciated...
     
  2. jcsd
  3. Oct 21, 2009 #2

    gabbagabbahey

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    Gold Member

    Re: Transforming to Eddington-Finkelstein Coordinates for Schwarzschild Geometry in G

    Ermm... shouldn't this be

    [tex]dt = dv - dr -2M. \frac{d}{dr} [log(r-2m)-log(2m)]dr[/tex]

    ?:wink:

    Also, [tex]2m\left(\frac{1}{r-2m}\right)
    \neq \left(\frac{2m}{r}-1\right)[/tex]
     
  4. Oct 21, 2009 #3
    Re: Transforming to Eddington-Finkelstein Coordinates for Schwarzschild Geometry in G

    Damn it! Still not seeing it...
     
  5. Oct 21, 2009 #4

    gabbagabbahey

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    Gold Member

    Re: Transforming to Eddington-Finkelstein Coordinates for Schwarzschild Geometry in G

    [tex]dt=dv-dr-2m\left(\frac{1}{r-2m}\right)dr=dv-\left[1+2m\left(\frac{1}{r-2m}\right)\right]dr[/tex]

    Just simplify....it's basic algebra from here.
     
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