How do I correctly find the Christoffel symbol for a specific component?

[jordy1113]

Homework Statement

Find Christoffel symbol

Relevant Equations

$$
\Gamma^l_{ki} = \frac{1}{2} g^{lj} (\partial_k g_{ij} + \partial_i g_{jk} - \partial_j g_{ki})
$$
\begin{eqnarray}
g_{00}(x,t)=1\\
g_{0i}(x,t)=0\\
g_{ij}(x,t)=a^{2}(t)[\delta_{ij}+h_{ij}(x,t)]\\
\end{eqnarray}

I was not given a formal teaching on christoffel symbols and how to find them so I just need some help.
I'm trying to find the cristoffel symbol:
\begin{equation}
\Gamma^{i}_{00}
\end{equation}
I set my equation up as:
\begin{equation}
\Gamma^i_{00} = \frac{1}{2} g^{ij} (\partial_0 g_{0j} + \partial_0 g_{0j} - \partial_j g_{00})
\end{equation}
Am I correct in getting that this christoffel symbol is equal to zero? If not what am I doing wrong? Many thanks in advance (sorry I am still trying to figure out the latex on the forum)

 

[anuttarasammyak]

From (5) you seem to be right. Why do not you calculate general jk components not only 00 ?
If not all components are zero, you would have some confidence.

 

[haushofer]

You're right, but what kind of metric is this?

 

[jordy1113]

You're right, but what kind of metric is this?

cosmological synchronous gauge according to Dodelson and Schmidt

 

[jordy1113]

From (5) you seem to be right. Why do not you calculate general jk components not only 00 ?
If not all components are zero, you would have some confidence.

thanks, I calculated some of the other christoffel symbols already and my professor checked them, we just forgot to do this one so I needed some reassurance I was doing it correctly