= Solution
For the <Friedmann-Lemaître-Robertson-Walker metric> in Cartesian spatial coordinates, the only time-dependent components are $g_{ij}=a^2\delta_{ij}$. The <Christoffel symbol> formula therefore gives
$$
\Gamma^0{}_{ij}=a\dot a\,\delta_{ij},
\qquad
\Gamma^i{}_{0j}=\Gamma^i{}_{j0}
=\frac{\dot a}{a}\delta^i{}_j.
$$
Substituting these into the definition of the <Ricci tensor> gives
$$
R_{00}=-3\frac{\ddot a}{a},
\qquad
R_{ij}=(a\ddot a+2\dot a^2)\delta_{ij}.
$$
\b[Thus, with no summation over $i$, the requested components are $\Gamma^0{}_{ii}=a\dot a$, $\Gamma^i{}_{0i}=\Gamma^i{}_{i0}=\dot a/a$, and $R_{ii}=a\ddot a+2\dot a^2$.]
Back to article page
