TITLE: Help with the Ricci tensor and Chistoffel Symbols
QUESTION [1 upvotes]: I really am confused with certain notations of the Ricci tensor and the Christoffel symbols. 
I'm looking to evaluate $R_{00}$ from my lecture notes, but I'm a little stuck at one point. The Ricci tensor, $R_{\mu\nu}$ is quoted as:
$$
R_{\mu\nu}=\partial_{\lambda}\Gamma^{\lambda}_{\mu\nu}-\partial_{\nu}\Gamma^{\lambda}_{\mu\lambda}+\Gamma^{\lambda}_{\lambda\rho}\Gamma^{\rho}_{\mu\nu}-\Gamma^{\rho}_{\mu\lambda}\Gamma^{\lambda}_{\nu\rho}
$$
Now, for $R_{00}$ I evaluate:
$$
R_{00}=\partial_{\lambda}\Gamma^{\lambda}_{00}-\partial_{0}\Gamma^{\lambda}_{0\lambda}+\Gamma^{\lambda}_{\lambda\rho}\Gamma^{\rho}_{00}-\Gamma^{\rho}_{0\lambda}\Gamma^{\lambda}_{0\rho}
$$
Now, all the terms with subscript $00$ vanish, leaving:
$$
R_{00}=-\partial_{0}\Gamma^{\lambda}_{0\lambda}-\Gamma^{\rho}_{0\lambda}\Gamma^{\lambda}_{0\rho}
$$
Exchanging $\lambda=i$ and $\rho=j$, gives:
$$
R_{00}=-\partial_{0}\Gamma^{i}_{0i}-\Gamma^{j}_{0i}\Gamma^{i}_{0j}
$$
My metric is given as a straight forward FRW metric, i.e., $g_{ij}=\mathrm{diag}[1,-a^{2}(t),-a^{2}(t),-a^{2}(t)]$. 
The lecture notes then skip a few steps and give:
$$
R_{00}=-\frac{d}{dt}\,\,3\left(\frac{\dot{a}}{a}\right)-3\left(\frac{\dot{a}}{a}\right)^{2}.
$$
How would these steps be filled in for the metric given?
I'm equally frustrated as I can't seem to evaluate the Christoffel symbols $\Gamma^{i}_{0j}$ and $\Gamma^{j}_{0i}$ to match up with the given value for $R_{00}$. Maybe that would help to start by running through the correct calculation for $\Gamma^{i}_{0j}$ and $\Gamma^{j}_{0i}$. This is the part that I think I'm not getting right.
Thank you.

REPLY [3 votes]: The Christoffel symbols for your metric are 
$$
\begin{split}
&\Gamma^0_{ij}=a\dot{a}\gamma_{ij}\\
&\Gamma^i_{0j}=\frac{\dot{a}}{a}\delta^i_j\\
&\Gamma_{jl}^{i}=\tilde\Gamma^i_{jl}
\end{split}
$$
then 
$$
R_{00}=-\partial_{0}\Gamma^{i}_{0i}-\Gamma^{j}_{0i}\Gamma^{i}_{0j}=-\frac{d}{dt}\left(\frac{\dot{a}}{a}\delta^i_i\right)-\left(\frac{\dot{a}}{a}\delta^i_j\right)\left(\frac{\dot{a}}{a}\delta^i_j\right)
$$
Then using $\delta^i_i=3$ you obtain the desired result.
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