Using the metric above, we find the Christoffel symbols, where the indices are (1,2,3,4)=(r,\theta,\phi,t).
\Gamma^3_{ik} = \begin{bmatrix} 0 & 0 & 1/r & 0\\ 0 & 0 & \cot\theta & 0\\ 1/r & \cot\theta & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} .