Deriving Schwartzschild Finding Mistake in Ricci Tensor

[abccdef125]

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 g_rr = A , g_tt = -B and the angular part being the same as in spherical coordinates):

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 R_θθ component step by step so that i can see where I went wrong?

Thanks in advance.

 

[PeterDonis]

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.

 

[PeterDonis]

You should also take this opportunity to learn how to use the PF LaTeX feature; the howto is here:

https://www.physicsforums.com/help/latexhelp/

 

[abccdef125]

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.

 

[PeterDonis]

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.

 

[abccdef125]

Thank you very much, it finally makes sense.