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Really quick algebra question -- Eddington-Finkelstein coordinates

  1. Apr 22, 2015 #1
    Probably a really stupid question..

    ##u=t+r+2M ln(\frac{r}{2M}-1) ##
    From this I get
    ##\frac{du}{dr}=(1-\frac{2M}{r})^{-1}##

    But, 1997 Sean M. Carroll lectures notes get ##\frac{du}{dr}=2(1-\frac{2M}{r})^{-1}## . (equation 7.71).

    No idea where this factor of 2 comes from.

    Thanks
     
  2. jcsd
  3. Apr 22, 2015 #2

    PeterDonis

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    2016 Award

    Staff: Mentor

    First, ##u## is not just a function of ##r##, so what you are doing here is taking a partial derivative ##\partial u / \partial r## with ##t## held constant. That is, you are looking at how ##u## changes with ##r## along a curve of constant ##t##. This is not the same as what Carroll is doing in the equation you refer to (see below).

    Second, you might want to check your algebra; the result you are getting for ##\partial u / \partial r## does not look right.

    That equation is a solution for an outgoing null geodesic; it is not the same thing as you were trying to derive in what I quoted above. It is derived by setting ##ds^2 = 0## in equation 7.69 (and also setting ##d\Omega^2 = 0## so the curve is purely radial) and rearranging the resulting equation into an equation for ##du / dr##. (Note that there are two ways of doing this, corresponding to whether ##dr## is positive or negative for a positive ##du##. If ##dr## is positive, we have an outgoing geodesic; if it is negative, we have an ingoing geodesic. That's why Carroll gives two solutions in equation 7.71.)
     
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