When trying to derive the Schwarzschild metric from the Minkowski metric, we begin by assuming that the metric coefficients are all functions of $r$ independent of $t,\phi,\theta$. We obtain
$ds^2=-f(r)cdt^2+g(r)dr^2+r^2(d\theta^2+sin^2\theta d\phi^2)$
Then we calculate the connection coefficients , which turn out to be
$Γ^r_{rr}=\tfrac{1}{2g}\tfrac{dg}{dr}$
$Γ^r_{tt}=\tfrac{1}{2g}\tfrac{df}{dr}$
$Γ^t_{tr}=Γ^t_{rt}=\tfrac{1}{2f}\tfrac{df}{dr}$
$Γ^r_{\theta\theta}=\tfrac{-r}{g}$
$Γ^r_{\phi\phi}=\tfrac{-rsin^2\theta}{g}$
$Γ^\theta_{\theta r}=Γ^\theta_{r \theta}=Γ^\phi_{\phi r}=Γ^\phi_{r \phi}=\tfrac{1}{r}$
$Γ^\theta_{\phi\phi}=-sin\theta cos\theta$
$Γ^\phi_{\phi\theta}=Γ^\phi_{\theta \phi}=cot \theta$.
I have two questions, (1) about calculation , (2) about the number of coefficients.
(1) When I try to find the Riemann tensor entry $R_{r \phi r \phi}$, I get the following :
$R_{r\phi r\phi}=g_{rr} R^r_{r\phi r}$, as $g_{ab}$ is symmetric. Calculating $R^r_{r\phi r}$I get ,
$R^r_{r\phi r}=\partial_\phi Γ^\phi_{\phi\theta}-\partial_r Γ^r_{\phi r}+Γ^r_{\lambda r }Γ^\lambda_{\phi\phi}-Γ^r_{\lambda \phi}Γ^\lambda_{\phi r}=Γ^r_{\lambda r }Γ^\lambda_{\phi\phi}-Γ^r_{\lambda \phi}Γ^\lambda_{\phi r}$ ( as the first two terms go to zero as there is no $\phi$ dependence and because $Γ^r_{\phi r}=0$.
Einstein summation is assumed so we have :
$Γ^r_{\lambda r }Γ^\lambda_{\phi\phi}-Γ^r_{\lambda \phi}Γ^\lambda_{\phi r}=Γ^r_{r r }Γ^r_{\phi\phi}-Γ^r_{\phi \phi}Γ^\phi_{\phi r}$ (as all the rest go to zero ) This gives :
$R^r_{r\phi r}=(\tfrac{1}{2g}\tfrac{dg}{dr})(\tfrac{-r^2sin^2\theta}{g})-\tfrac{sin^2\theta}{g} $
However my lecture notes tell me $R^r_{r\phi r}=(\tfrac{1}{2g}\tfrac{dg}{dr})(\tfrac{-r^2sin^2\theta}{g})$ So I must be doing something wrong , could anyone please explain
2) Am I correct in believing this is the right number of connection coefficients and all the others go to zero ?