リッチテンソルの展開

さあ、見よ! これが R00 の展開だ。

[
前の記事へ]  [相対性理論の目次へ]  [次の記事へ]


\[ \begin{align*} R_{00}\ =\ - \frac{1}{2} g^{11} \frac{\partial^2 g_{00}}{\partial x^{1} \partial x^{1}} + g^{11} \frac{\partial^2 g_{01}}{\partial x^{0} \partial x^{1}} - \frac{1}{2} g^{11} \frac{\partial^2 g_{11}}{\partial x^{0} \partial x^{0}} - g^{12} \frac{\partial^2 g_{00}}{\partial x^{1} \partial x^{2}} \end{align*} \]
\[ \begin{align*} + g^{12} \frac{\partial^2 g_{01}}{\partial x^{0} \partial x^{2}} + g^{12} \frac{\partial^2 g_{02}}{\partial x^{0} \partial x^{1}} - g^{12} \frac{\partial^2 g_{12}}{\partial x^{0} \partial x^{0}} - g^{13} \frac{\partial^2 g_{00}}{\partial x^{1} \partial x^{3}} \end{align*} \]
\[ \begin{align*} + g^{13} \frac{\partial^2 g_{01}}{\partial x^{0} \partial x^{3}} + g^{13} \frac{\partial^2 g_{03}}{\partial x^{0} \partial x^{1}} - g^{13} \frac{\partial^2 g_{13}}{\partial x^{0} \partial x^{0}} - \frac{1}{2} g^{22} \frac{\partial^2 g_{00}}{\partial x^{2} \partial x^{2}} \end{align*} \]
\[ \begin{align*} + g^{22} \frac{\partial^2 g_{02}}{\partial x^{0} \partial x^{2}} - \frac{1}{2} g^{22} \frac{\partial^2 g_{22}}{\partial x^{0} \partial x^{0}} - g^{23} \frac{\partial^2 g_{00}}{\partial x^{2} \partial x^{3}} + g^{23} \frac{\partial^2 g_{02}}{\partial x^{0} \partial x^{3}} \end{align*} \]
\[ \begin{align*} + g^{23} \frac{\partial^2 g_{03}}{\partial x^{0} \partial x^{2}} - g^{23} \frac{\partial^2 g_{23}}{\partial x^{0} \partial x^{0}} - \frac{1}{2} g^{33} \frac{\partial^2 g_{00}}{\partial x^{3} \partial x^{3}} + g^{33} \frac{\partial^2 g_{03}}{\partial x^{0} \partial x^{3}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{33} \frac{\partial^2 g_{33}}{\partial x^{0} \partial x^{0}} - \frac{1}{2} g^{00} g^{11}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{01}}{\partial x^{1}} + \frac{1}{4} g^{00} g^{11}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{0}} + \frac{1}{4} g^{00} g^{11}\left( \frac{\partial g_{00}}{\partial x^{1}} \right)^2 \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{00} g^{12}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{01}}{\partial x^{2}} - \frac{1}{2} g^{00} g^{12}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{02}}{\partial x^{1}} + \frac{1}{2} g^{00} g^{12}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{0}} + \frac{1}{2} g^{00} g^{12}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{00}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{00} g^{13}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{01}}{\partial x^{3}} - \frac{1}{2} g^{00} g^{13}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{03}}{\partial x^{1}} + \frac{1}{2} g^{00} g^{13}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{0}} + \frac{1}{2} g^{00} g^{13}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{00}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{00} g^{22}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{02}}{\partial x^{2}} + \frac{1}{4} g^{00} g^{22}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{0}} + \frac{1}{4} g^{00} g^{22}\left( \frac{\partial g_{00}}{\partial x^{2}} \right)^2 - \frac{1}{2} g^{00} g^{23}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{02}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{00} g^{23}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{03}}{\partial x^{2}} + \frac{1}{2} g^{00} g^{23}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{0}} + \frac{1}{2} g^{00} g^{23}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{00}}{\partial x^{3}} - \frac{1}{2} g^{00} g^{33}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{03}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + \frac{1}{4} g^{00} g^{33}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{0}} + \frac{1}{4} g^{00} g^{33}\left( \frac{\partial g_{00}}{\partial x^{3}} \right)^2 - \frac{1}{4} g^{01} g^{11}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{1}} + \frac{1}{2} g^{01} g^{11}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{01}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} + \frac{1}{4} g^{01} g^{11}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{11}}{\partial x^{0}} - g^{01} g^{11}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{01}}{\partial x^{1}} + \frac{1}{2} g^{01} g^{11}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{0}} - \frac{1}{2} g^{01} g^{12}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + g^{01} g^{12}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{01}}{\partial x^{2}} + \frac{1}{2} g^{01} g^{12}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{11}}{\partial x^{0}} - g^{01} g^{12}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{01}}{\partial x^{2}} - g^{01} g^{12}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{02}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} + g^{01} g^{12}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{0}} - \frac{1}{2} g^{01} g^{13}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{3}} + g^{01} g^{13}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{01}}{\partial x^{3}} + \frac{1}{2} g^{01} g^{13}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{11}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} - g^{01} g^{13}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{01}}{\partial x^{3}} - g^{01} g^{13}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{03}}{\partial x^{1}} + g^{01} g^{13}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{0}} - \frac{1}{2} g^{01} g^{22}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + \frac{1}{4} g^{01} g^{22}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{1}} + \frac{1}{2} g^{01} g^{22}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{02}}{\partial x^{2}} - \frac{1}{4} g^{01} g^{22}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{22}}{\partial x^{0}} + \frac{1}{2} g^{01} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{01}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{01} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{02}}{\partial x^{1}} + \frac{1}{2} g^{01} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{12}}{\partial x^{0}} - g^{01} g^{22}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{02}}{\partial x^{2}} + \frac{1}{2} g^{01} g^{22}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{01} g^{23}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{3}} - \frac{1}{2} g^{01} g^{23}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{2}} + \frac{1}{2} g^{01} g^{23}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{1}} + \frac{1}{2} g^{01} g^{23}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{02}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{01} g^{23}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{2}} - \frac{1}{2} g^{01} g^{23}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{23}}{\partial x^{0}} + \frac{1}{2} g^{01} g^{23}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{01}}{\partial x^{3}} - \frac{1}{2} g^{01} g^{23}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{03}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{01} g^{23}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{13}}{\partial x^{0}} + \frac{1}{2} g^{01} g^{23}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{01}}{\partial x^{2}} - \frac{1}{2} g^{01} g^{23}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{02}}{\partial x^{1}} + \frac{1}{2} g^{01} g^{23}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{12}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} - g^{01} g^{23}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{02}}{\partial x^{3}} - g^{01} g^{23}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{03}}{\partial x^{2}} + g^{01} g^{23}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{0}} - \frac{1}{2} g^{01} g^{33}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + \frac{1}{4} g^{01} g^{33}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{1}} + \frac{1}{2} g^{01} g^{33}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{3}} - \frac{1}{4} g^{01} g^{33}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{33}}{\partial x^{0}} + \frac{1}{2} g^{01} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{01}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{01} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{1}} + \frac{1}{2} g^{01} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{13}}{\partial x^{0}} - g^{01} g^{33}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{03}}{\partial x^{3}} + \frac{1}{2} g^{01} g^{33}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{4} g^{02} g^{11}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{2}} - \frac{1}{2} g^{02} g^{11}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{1}} - \frac{1}{2} g^{02} g^{11}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{01}}{\partial x^{2}} + \frac{1}{2} g^{02} g^{11}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{02}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{02} g^{11}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{12}}{\partial x^{0}} + \frac{1}{2} g^{02} g^{11}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{01}}{\partial x^{1}} - \frac{1}{4} g^{02} g^{11}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{11}}{\partial x^{0}} - g^{02} g^{11}\frac{\partial g_{01}}{\partial x^{1}} \frac{\partial g_{02}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{02} g^{11}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{0}} - \frac{1}{2} g^{02} g^{12}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{1}} + \frac{1}{2} g^{02} g^{12}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{22}}{\partial x^{0}} + g^{02} g^{12}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{02}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} - g^{02} g^{12}\frac{\partial g_{01}}{\partial x^{2}} \frac{\partial g_{02}}{\partial x^{0}} - g^{02} g^{12}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{02}}{\partial x^{1}} + g^{02} g^{12}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{0}} - \frac{1}{2} g^{02} g^{13}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{02} g^{13}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{2}} - \frac{1}{2} g^{02} g^{13}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{1}} + \frac{1}{2} g^{02} g^{13}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{02}}{\partial x^{3}} - \frac{1}{2} g^{02} g^{13}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{02} g^{13}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{23}}{\partial x^{0}} + \frac{1}{2} g^{02} g^{13}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{01}}{\partial x^{3}} + \frac{1}{2} g^{02} g^{13}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{03}}{\partial x^{1}} - \frac{1}{2} g^{02} g^{13}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{13}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{02} g^{13}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{01}}{\partial x^{2}} + \frac{1}{2} g^{02} g^{13}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{02}}{\partial x^{1}} + \frac{1}{2} g^{02} g^{13}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{12}}{\partial x^{0}} - g^{02} g^{13}\frac{\partial g_{01}}{\partial x^{3}} \frac{\partial g_{02}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} - g^{02} g^{13}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{03}}{\partial x^{1}} + g^{02} g^{13}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{0}} - \frac{1}{4} g^{02} g^{22}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{2}} + \frac{1}{2} g^{02} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{02}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + \frac{1}{4} g^{02} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{22}}{\partial x^{0}} - g^{02} g^{22}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{02}}{\partial x^{2}} + \frac{1}{2} g^{02} g^{22}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{0}} - \frac{1}{2} g^{02} g^{23}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + g^{02} g^{23}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{02}}{\partial x^{3}} + \frac{1}{2} g^{02} g^{23}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{22}}{\partial x^{0}} - g^{02} g^{23}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{02}}{\partial x^{3}} - g^{02} g^{23}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{03}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + g^{02} g^{23}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{0}} - \frac{1}{2} g^{02} g^{33}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{3}} + \frac{1}{4} g^{02} g^{33}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{2}} + \frac{1}{2} g^{02} g^{33}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{03}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} - \frac{1}{4} g^{02} g^{33}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{33}}{\partial x^{0}} + \frac{1}{2} g^{02} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{02}}{\partial x^{3}} - \frac{1}{2} g^{02} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{2}} + \frac{1}{2} g^{02} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{23}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} - g^{02} g^{33}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{03}}{\partial x^{3}} + \frac{1}{2} g^{02} g^{33}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{0}} + \frac{1}{4} g^{03} g^{11}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{3}} - \frac{1}{2} g^{03} g^{11}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{03} g^{11}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{01}}{\partial x^{3}} + \frac{1}{2} g^{03} g^{11}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{1}} + \frac{1}{2} g^{03} g^{11}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{13}}{\partial x^{0}} + \frac{1}{2} g^{03} g^{11}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{01}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} - \frac{1}{4} g^{03} g^{11}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{11}}{\partial x^{0}} - g^{03} g^{11}\frac{\partial g_{01}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{0}} + \frac{1}{2} g^{03} g^{11}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{0}} + \frac{1}{2} g^{03} g^{12}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{03} g^{12}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{2}} - \frac{1}{2} g^{03} g^{12}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{1}} - \frac{1}{2} g^{03} g^{12}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{02}}{\partial x^{3}} + \frac{1}{2} g^{03} g^{12}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{03} g^{12}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{23}}{\partial x^{0}} - \frac{1}{2} g^{03} g^{12}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{01}}{\partial x^{3}} + \frac{1}{2} g^{03} g^{12}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{03}}{\partial x^{1}} + \frac{1}{2} g^{03} g^{12}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{13}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{03} g^{12}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{01}}{\partial x^{2}} + \frac{1}{2} g^{03} g^{12}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{02}}{\partial x^{1}} - \frac{1}{2} g^{03} g^{12}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{12}}{\partial x^{0}} - g^{03} g^{12}\frac{\partial g_{01}}{\partial x^{2}} \frac{\partial g_{03}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} - g^{03} g^{12}\frac{\partial g_{02}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{0}} + g^{03} g^{12}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{0}} - \frac{1}{2} g^{03} g^{13}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{1}} + \frac{1}{2} g^{03} g^{13}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{33}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + g^{03} g^{13}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{1}} - g^{03} g^{13}\frac{\partial g_{01}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{0}} - g^{03} g^{13}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{03}}{\partial x^{1}} + g^{03} g^{13}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{4} g^{03} g^{22}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{3}} - \frac{1}{2} g^{03} g^{22}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{2}} - \frac{1}{2} g^{03} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{02}}{\partial x^{3}} + \frac{1}{2} g^{03} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{03}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{03} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{23}}{\partial x^{0}} + \frac{1}{2} g^{03} g^{22}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{02}}{\partial x^{2}} - \frac{1}{4} g^{03} g^{22}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{22}}{\partial x^{0}} - g^{03} g^{22}\frac{\partial g_{02}}{\partial x^{2}} \frac{\partial g_{03}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{03} g^{22}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{0}} - \frac{1}{2} g^{03} g^{23}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{2}} + \frac{1}{2} g^{03} g^{23}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{33}}{\partial x^{0}} + g^{03} g^{23}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} - g^{03} g^{23}\frac{\partial g_{02}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{0}} - g^{03} g^{23}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{03}}{\partial x^{2}} + g^{03} g^{23}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{0}} - \frac{1}{4} g^{03} g^{33}\frac{\partial g_{00}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{03} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{3}} + \frac{1}{4} g^{03} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{33}}{\partial x^{0}} - g^{03} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{03}}{\partial x^{3}} + \frac{1}{2} g^{03} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{4} (g^{11})^2\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{11}}{\partial x^{1}} - \frac{1}{2} (g^{11})^2\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{1}} + \frac{1}{4} (g^{11})^2\left( \frac{\partial g_{11}}{\partial x^{0}} \right)^2 + \frac{1}{4} g^{11} g^{12}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{11}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{11} g^{12}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{12}}{\partial x^{1}} + \frac{1}{4} g^{11} g^{12}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{11}}{\partial x^{1}} - \frac{1}{2} g^{11} g^{12}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{2}} - g^{11} g^{12}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{11} g^{12}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{1}} + g^{11} g^{12}\frac{\partial g_{11}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{0}} + \frac{1}{4} g^{11} g^{13}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{11}}{\partial x^{3}} + \frac{1}{2} g^{11} g^{13}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{13}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} + \frac{1}{4} g^{11} g^{13}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{11}}{\partial x^{1}} - \frac{1}{2} g^{11} g^{13}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{3}} - g^{11} g^{13}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{1}} - \frac{1}{2} g^{11} g^{13}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} + g^{11} g^{13}\frac{\partial g_{11}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{0}} + \frac{1}{2} g^{11} g^{22}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{12}}{\partial x^{2}} - \frac{1}{4} g^{11} g^{22}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{22}}{\partial x^{1}} - \frac{1}{4} g^{11} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{11}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{11} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{12}}{\partial x^{1}} - g^{11} g^{22}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{2}} + \frac{1}{2} g^{11} g^{22}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{1}} + \frac{1}{2} g^{11} g^{22}\left( \frac{\partial g_{01}}{\partial x^{2}} \right)^2 \end{align*} \]
\[ \begin{align*} - g^{11} g^{22}\frac{\partial g_{01}}{\partial x^{2}} \frac{\partial g_{02}}{\partial x^{1}} + \frac{1}{2} g^{11} g^{22}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{2}} - g^{11} g^{22}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{1}} + \frac{1}{2} g^{11} g^{22}\left( \frac{\partial g_{02}}{\partial x^{1}} \right)^2 \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{11} g^{22}\left( \frac{\partial g_{12}}{\partial x^{0}} \right)^2 + \frac{1}{2} g^{11} g^{23}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{12}}{\partial x^{3}} + \frac{1}{2} g^{11} g^{23}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{13}}{\partial x^{2}} - \frac{1}{2} g^{11} g^{23}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{23}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} - \frac{1}{4} g^{11} g^{23}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{11}}{\partial x^{3}} + \frac{1}{2} g^{11} g^{23}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{13}}{\partial x^{1}} - \frac{1}{4} g^{11} g^{23}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{11}}{\partial x^{2}} + \frac{1}{2} g^{11} g^{23}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{12}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} - g^{11} g^{23}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{3}} - g^{11} g^{23}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{2}} + g^{11} g^{23}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{1}} + g^{11} g^{23}\frac{\partial g_{01}}{\partial x^{2}} \frac{\partial g_{01}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} - g^{11} g^{23}\frac{\partial g_{01}}{\partial x^{2}} \frac{\partial g_{03}}{\partial x^{1}} - g^{11} g^{23}\frac{\partial g_{01}}{\partial x^{3}} \frac{\partial g_{02}}{\partial x^{1}} + \frac{1}{2} g^{11} g^{23}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{3}} - g^{11} g^{23}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} + g^{11} g^{23}\frac{\partial g_{02}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{1}} + \frac{1}{2} g^{11} g^{23}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{2}} - g^{11} g^{23}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{1}} + g^{11} g^{23}\frac{\partial g_{12}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{11} g^{33}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{13}}{\partial x^{3}} - \frac{1}{4} g^{11} g^{33}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{33}}{\partial x^{1}} - \frac{1}{4} g^{11} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{11}}{\partial x^{3}} + \frac{1}{2} g^{11} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{13}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} - g^{11} g^{33}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{3}} + \frac{1}{2} g^{11} g^{33}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{1}} + \frac{1}{2} g^{11} g^{33}\left( \frac{\partial g_{01}}{\partial x^{3}} \right)^2 - g^{11} g^{33}\frac{\partial g_{01}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{11} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{3}} - g^{11} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{1}} + \frac{1}{2} g^{11} g^{33}\left( \frac{\partial g_{03}}{\partial x^{1}} \right)^2 + \frac{1}{2} g^{11} g^{33}\left( \frac{\partial g_{13}}{\partial x^{0}} \right)^2 \end{align*} \]
\[ \begin{align*} + \frac{1}{2} (g^{12})^2\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{22}}{\partial x^{1}} + \frac{1}{2} (g^{12})^2\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{11}}{\partial x^{2}} - (g^{12})^2\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{1}} - \frac{1}{2} (g^{12})^2\left( \frac{\partial g_{01}}{\partial x^{2}} \right)^2 \end{align*} \]
\[ \begin{align*} + (g^{12})^2\frac{\partial g_{01}}{\partial x^{2}} \frac{\partial g_{02}}{\partial x^{1}} - (g^{12})^2\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{2}} - \frac{1}{2} (g^{12})^2\left( \frac{\partial g_{02}}{\partial x^{1}} \right)^2 + \frac{1}{2} (g^{12})^2\frac{\partial g_{11}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} (g^{12})^2\left( \frac{\partial g_{12}}{\partial x^{0}} \right)^2 + g^{12} g^{13}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{23}}{\partial x^{1}} + \frac{1}{2} g^{12} g^{13}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{11}}{\partial x^{3}} + \frac{1}{2} g^{12} g^{13}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{11}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} - 2g^{12} g^{13}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{1}} - g^{12} g^{13}\frac{\partial g_{01}}{\partial x^{2}} \frac{\partial g_{01}}{\partial x^{3}} + g^{12} g^{13}\frac{\partial g_{01}}{\partial x^{2}} \frac{\partial g_{03}}{\partial x^{1}} + g^{12} g^{13}\frac{\partial g_{01}}{\partial x^{3}} \frac{\partial g_{02}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} - g^{12} g^{13}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{3}} - g^{12} g^{13}\frac{\partial g_{02}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{1}} - g^{12} g^{13}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{2}} + g^{12} g^{13}\frac{\partial g_{11}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + g^{12} g^{13}\frac{\partial g_{12}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{0}} + \frac{1}{4} g^{12} g^{22}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{22}}{\partial x^{2}} + \frac{1}{2} g^{12} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{12}}{\partial x^{2}} + \frac{1}{4} g^{12} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{22}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{12} g^{22}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{2}} - g^{12} g^{22}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{2}} - \frac{1}{2} g^{12} g^{22}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{1}} + g^{12} g^{22}\frac{\partial g_{12}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{12} g^{23}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{22}}{\partial x^{3}} + g^{12} g^{23}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{13}}{\partial x^{2}} + \frac{1}{2} g^{12} g^{23}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{22}}{\partial x^{1}} - g^{12} g^{23}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + g^{12} g^{23}\frac{\partial g_{01}}{\partial x^{2}} \frac{\partial g_{02}}{\partial x^{3}} - g^{12} g^{23}\frac{\partial g_{01}}{\partial x^{2}} \frac{\partial g_{03}}{\partial x^{2}} - 2g^{12} g^{23}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{2}} - g^{12} g^{23}\frac{\partial g_{02}}{\partial x^{1}} \frac{\partial g_{02}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + g^{12} g^{23}\frac{\partial g_{02}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{2}} - g^{12} g^{23}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{1}} + g^{12} g^{23}\frac{\partial g_{12}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{0}} + g^{12} g^{23}\frac{\partial g_{13}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{12} g^{33}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{23}}{\partial x^{3}} - \frac{1}{4} g^{12} g^{33}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{33}}{\partial x^{2}} + \frac{1}{2} g^{12} g^{33}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{13}}{\partial x^{3}} - \frac{1}{4} g^{12} g^{33}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{33}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{12} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{12}}{\partial x^{3}} + \frac{1}{2} g^{12} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{13}}{\partial x^{2}} + \frac{1}{2} g^{12} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{23}}{\partial x^{1}} - g^{12} g^{33}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{12} g^{33}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{2}} + g^{12} g^{33}\frac{\partial g_{01}}{\partial x^{3}} \frac{\partial g_{02}}{\partial x^{3}} - g^{12} g^{33}\frac{\partial g_{01}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{2}} - g^{12} g^{33}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{12} g^{33}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{1}} - g^{12} g^{33}\frac{\partial g_{02}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{1}} + g^{12} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{3}} - g^{12} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} - g^{12} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{1}} + g^{12} g^{33}\frac{\partial g_{03}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{2}} + g^{12} g^{33}\frac{\partial g_{13}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{0}} + \frac{1}{2} (g^{13})^2\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{33}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} (g^{13})^2\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{11}}{\partial x^{3}} - (g^{13})^2\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{1}} - \frac{1}{2} (g^{13})^2\left( \frac{\partial g_{01}}{\partial x^{3}} \right)^2 + (g^{13})^2\frac{\partial g_{01}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{1}} \end{align*} \]
\[ \begin{align*} - (g^{13})^2\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{11}}{\partial x^{3}} - \frac{1}{2} (g^{13})^2\left( \frac{\partial g_{03}}{\partial x^{1}} \right)^2 + \frac{1}{2} (g^{13})^2\frac{\partial g_{11}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{0}} + \frac{1}{2} (g^{13})^2\left( \frac{\partial g_{13}}{\partial x^{0}} \right)^2 \end{align*} \]
\[ \begin{align*} - \frac{1}{4} g^{13} g^{22}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{22}}{\partial x^{3}} + \frac{1}{2} g^{13} g^{22}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{23}}{\partial x^{2}} + \frac{1}{2} g^{13} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{12}}{\partial x^{3}} - \frac{1}{2} g^{13} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{13}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{13} g^{22}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{23}}{\partial x^{1}} + \frac{1}{2} g^{13} g^{22}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{12}}{\partial x^{2}} - \frac{1}{4} g^{13} g^{22}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{22}}{\partial x^{1}} + \frac{1}{2} g^{13} g^{22}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} - g^{13} g^{22}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{2}} - g^{13} g^{22}\frac{\partial g_{01}}{\partial x^{2}} \frac{\partial g_{02}}{\partial x^{3}} + g^{13} g^{22}\frac{\partial g_{01}}{\partial x^{2}} \frac{\partial g_{03}}{\partial x^{2}} - g^{13} g^{22}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + g^{13} g^{22}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{2}} - g^{13} g^{22}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{1}} + g^{13} g^{22}\frac{\partial g_{02}}{\partial x^{1}} \frac{\partial g_{02}}{\partial x^{3}} - g^{13} g^{22}\frac{\partial g_{02}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} - g^{13} g^{22}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{2}} + \frac{1}{2} g^{13} g^{22}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{1}} + g^{13} g^{22}\frac{\partial g_{12}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{0}} + \frac{1}{2} g^{13} g^{23}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{33}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{13} g^{23}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{33}}{\partial x^{1}} + g^{13} g^{23}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{12}}{\partial x^{3}} - g^{13} g^{23}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{2}} - g^{13} g^{23}\frac{\partial g_{01}}{\partial x^{3}} \frac{\partial g_{02}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + g^{13} g^{23}\frac{\partial g_{01}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{2}} - g^{13} g^{23}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{1}} + g^{13} g^{23}\frac{\partial g_{02}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{1}} - 2g^{13} g^{23}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{12}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} - g^{13} g^{23}\frac{\partial g_{03}}{\partial x^{1}} \frac{\partial g_{03}}{\partial x^{2}} + g^{13} g^{23}\frac{\partial g_{12}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{0}} + g^{13} g^{23}\frac{\partial g_{13}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{0}} + \frac{1}{4} g^{13} g^{33}\frac{\partial g_{00}}{\partial x^{1}} \frac{\partial g_{33}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{13} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{13}}{\partial x^{3}} + \frac{1}{4} g^{13} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{33}}{\partial x^{1}} - \frac{1}{2} g^{13} g^{33}\frac{\partial g_{01}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{3}} - g^{13} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{13}}{\partial x^{3}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{13} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{1}} + g^{13} g^{33}\frac{\partial g_{13}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{0}} + \frac{1}{4} (g^{22})^2\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{22}}{\partial x^{2}} - \frac{1}{2} (g^{22})^2\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + \frac{1}{4} (g^{22})^2\left( \frac{\partial g_{22}}{\partial x^{0}} \right)^2 + \frac{1}{4} g^{22} g^{23}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{22}}{\partial x^{3}} + \frac{1}{2} g^{22} g^{23}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{23}}{\partial x^{2}} + \frac{1}{4} g^{22} g^{23}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{22}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{22} g^{23}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{3}} - g^{22} g^{23}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{2}} - \frac{1}{2} g^{22} g^{23}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{2}} + g^{22} g^{23}\frac{\partial g_{22}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{22} g^{33}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{23}}{\partial x^{3}} - \frac{1}{4} g^{22} g^{33}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{33}}{\partial x^{2}} - \frac{1}{4} g^{22} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{22}}{\partial x^{3}} + \frac{1}{2} g^{22} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{23}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} - g^{22} g^{33}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{3}} + \frac{1}{2} g^{22} g^{33}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{2}} + \frac{1}{2} g^{22} g^{33}\left( \frac{\partial g_{02}}{\partial x^{3}} \right)^2 - g^{22} g^{33}\frac{\partial g_{02}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} g^{22} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{3}} - g^{22} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{2}} + \frac{1}{2} g^{22} g^{33}\left( \frac{\partial g_{03}}{\partial x^{2}} \right)^2 + \frac{1}{2} g^{22} g^{33}\left( \frac{\partial g_{23}}{\partial x^{0}} \right)^2 \end{align*} \]
\[ \begin{align*} + \frac{1}{2} (g^{23})^2\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{33}}{\partial x^{2}} + \frac{1}{2} (g^{23})^2\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{22}}{\partial x^{3}} - (g^{23})^2\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{2}} - \frac{1}{2} (g^{23})^2\left( \frac{\partial g_{02}}{\partial x^{3}} \right)^2 \end{align*} \]
\[ \begin{align*} + (g^{23})^2\frac{\partial g_{02}}{\partial x^{3}} \frac{\partial g_{03}}{\partial x^{2}} - (g^{23})^2\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{22}}{\partial x^{3}} - \frac{1}{2} (g^{23})^2\left( \frac{\partial g_{03}}{\partial x^{2}} \right)^2 + \frac{1}{2} (g^{23})^2\frac{\partial g_{22}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{2} (g^{23})^2\left( \frac{\partial g_{23}}{\partial x^{0}} \right)^2 + \frac{1}{4} g^{23} g^{33}\frac{\partial g_{00}}{\partial x^{2}} \frac{\partial g_{33}}{\partial x^{3}} + \frac{1}{2} g^{23} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{23}}{\partial x^{3}} + \frac{1}{4} g^{23} g^{33}\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{33}}{\partial x^{2}} \end{align*} \]
\[ \begin{align*} - \frac{1}{2} g^{23} g^{33}\frac{\partial g_{02}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{3}} - g^{23} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{23}}{\partial x^{3}} - \frac{1}{2} g^{23} g^{33}\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{2}} + g^{23} g^{33}\frac{\partial g_{23}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{0}} \end{align*} \]
\[ \begin{align*} + \frac{1}{4} (g^{33})^2\frac{\partial g_{00}}{\partial x^{3}} \frac{\partial g_{33}}{\partial x^{3}} - \frac{1}{2} (g^{33})^2\frac{\partial g_{03}}{\partial x^{0}} \frac{\partial g_{33}}{\partial x^{3}} + \frac{1}{4} (g^{33})^2\left( \frac{\partial g_{33}}{\partial x^{0}} \right)^2 \end{align*} \]