Deriving Schwartzschild Finding Mistake in Ricci Tensor

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abccdef125
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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):
Textbook page with Christoffel symbol components and Ricci tensor equations, with R_theta theta = -1 + r/(2A(r)) times the bracket (-A'(r)/A(r) + B'(r)/B(r)) + 1/A(r), highlighted in a red box.

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.
 

Discussion

abccdef125 said:
Could someone please calculate the Rθθ component step by step

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.
 
I'm sorry it's not in LaTeX, I thought i was going to be able to convert it easily from Word, but I didn't succeed.

Handwritten-style derivation of the Ricci tensor component R_θθ as a sequence of equalities involving Christoffel symbols, partial derivatives with respect to θ and r, and terms in A, B, r and cot θ, ending in R_θθ = -1 + (r/2A)(A'/A + B'/B) + 1/A.
 
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.