Deriving The Schwarzschild Solution - Calculating The Christoffel Symbols

Calculating The Christoffel Symbols

Using the metric above, we find the Christoffel symbols, where the indices are . The sign denotes a total derivative of a function.

\Gamma^0_{ik} = \begin{bmatrix} A'/\left( 2A \right) & 0 & 0 & 0\\ 0 & -r/A & 0 & 0\\ 0 & 0 & -r \sin^2 \theta /A & 0\\ 0 & 0 & 0 & -B'/\left( 2A \right) \end{bmatrix}
\Gamma^1_{ik} = \begin{bmatrix} 0 & 1/r & 0 & 0\\ 1/r & 0 & 0 & 0\\ 0 & 0 & -\sin\theta\cos\theta & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}
\Gamma^2_{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}
\Gamma^3_{ik} = \begin{bmatrix} 0 & 0 & 0 & B'/\left( 2B \right)\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \\ B'/\left( 2B \right) & 0 & 0 & 0\end{bmatrix}

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