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I Mistake in derivation of the Schwartzschild solution

  1. Oct 19, 2016 #1
    Hi, I've been unsuccesfully trying to derive the Schwartzschild solution, narrowing my mistake down to a single component of the Ricci tensor. The problem is the following(taken from Weinberg's book, with grr = A , gtt = -B and the angular part being the same as in spherical coordinates):
    t2HFlqk.png
    I know it must be a simple mistake, but I must have spent three hours playing around with it and I still don't see it. Could someone please calculate the Rθθ component step by step so that i can see where I went wrong?

    Thanks in advance.
     
  2. jcsd
  3. Oct 19, 2016 #2

    PeterDonis

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    It would be better for you to try posting your steps. The first step is to take equation 8.1.12 and simply fill in ##\mu = \theta## and ##\kappa = \theta##, leaving all other indexes free. Do that and post what you come up with, and we can go from there.
     
  4. Oct 19, 2016 #3

    PeterDonis

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  5. Oct 19, 2016 #4
    I'm sorry it's not in LaTeX, I thought i was gonna be able to convert it easily from Word, but I didn't succeed.

    KVnqugp.png
     
  6. Oct 19, 2016 #5

    PeterDonis

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    I think you missed a term in the third line of your derivation. We have:

    $$
    - \frac{\partial \Gamma^r_{\theta \theta}}{\partial r} = - \frac{\partial}{\partial r} \left( \frac{- r}{A} \right) = \frac{1}{A} - \frac{r A'}{A^2}
    $$

    I don't see the ##- r A' / A^2## anywhere in your third line. The rest of the third line looks OK. If you put in that missing term in the third line, it should fix the fourth line to look like Weinberg's result.
     
  7. Oct 20, 2016 #6
    Thank you very much, it finally makes sense.
     
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