User:Z E U S/sandbox

$$ {d^2 x^\mu \over ds^2} =- \Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}\ $$

$$\Gamma^{\mu}{}_{\alpha\beta}=\frac{1}{2}g^{\mu\lambda}\left(\partial_{\beta}g_{\lambda\alpha}+\partial_{\alpha}g_{\lambda\beta}-\partial_{\lambda}g_{\alpha\beta}\right)$$

$$\Gamma^{\mu}{}_{\alpha\beta}=\frac{1}{2}g^{\mu \lambda} \left(\frac{\partial g_{\lambda\alpha}}{\partial x^\beta} + \frac{\partial g_{\lambda\beta}}{\partial x^{\alpha}} - \frac{\partial g_{\alpha\beta}}{\partial x^{\lambda}} \right) $$

$$ {d^2 x^\mu \over ds^2} =- \frac{1}{2}g^{\mu\lambda}\left(\partial_{\beta}g_{\lambda\alpha}+\partial_{\alpha}g_{\lambda\beta}-\partial_{\lambda}g_{\alpha\beta}\right){d x^\alpha \over ds}{d x^\beta \over ds}\ $$

$$ {d^2 x^\mu \over ds^2} =- \frac{1}{2}[g^{\mu\lambda1}\left(\partial_{\beta}g_{\lambda1\alpha}+\partial_{\alpha}g_{\lambda1\beta}-\partial_{\lambda1}g_{\alpha\beta}\right)+g^{\mu\lambda1}\left(\partial_{\beta}g_{\lambda1\alpha}+\partial_{\alpha}g_{\lambda1\beta}-\partial_{\lambda1}g_{\alpha\beta}\right)]{d x^\alpha \over ds}{d x^\beta \over ds}\ $$

$$ {d^2 x^\mu \over ds^2} =- \frac{1}{2}[g^{\mu0}\left(\partial_{\beta}g_{0\alpha}+\partial_{\alpha}g_{0\beta}-\partial_{0}g_{\alpha\beta}\right)+g^{\mu1}\left(\partial_{\beta}g_{1\alpha}+\partial_{\alpha}g_{1\beta}-\partial_{1}g_{\alpha\beta}\right)+g^{\mu2}\left(\partial_{\beta}g_{2\alpha}+\partial_{\alpha}g_{2\beta}-\partial_{2}g_{\alpha\beta}\right)+g^{\mu3}\left(\partial_{\beta}g_{3\alpha}+\partial_{\alpha}g_{3\beta}-\partial_{3}g_{\alpha\beta}\right)]{d x^\alpha \over ds}{d x^\beta \over ds}\ $$

$$ \Gamma^\lambda {}_{\mu \nu} \dot x^\mu \dot x^\nu + \ddot x^\lambda = 0\ .$$

$$ {\ddot x^\mu} =- \frac{1}{2}[g^{\mu0}\left(\partial_{\beta}g_{0\alpha}+\partial_{\alpha}g_{0\beta}-\partial_{0}g_{\alpha\beta}\right)+g^{\mu1}\left(\partial_{\beta}g_{1\alpha}+\partial_{\alpha}g_{1\beta}-\partial_{1}g_{\alpha\beta}\right)+g^{\mu2}\left(\partial_{\beta}g_{2\alpha}+\partial_{\alpha}g_{2\beta}-\partial_{2}g_{\alpha\beta}\right)+g^{\mu3}\left(\partial_{\beta}g_{3\alpha}+\partial_{\alpha}g_{3\beta}-\partial_{3}g_{\alpha\beta}\right)]{\dot x^\alpha }{\dot x^\beta}\ $$

$$ {\ddot x^\mu} =- \frac{1}{2}[g^{\mu0}\left(g_{0\alpha,\beta}+g_{0\beta,\alpha}-g_{\alpha\beta,0}\right)+g^{\mu1}\left(g_{1\alpha,\beta}+g_{1\beta,\alpha}-g_{\alpha\beta,1}\right)+g^{\mu2}\left(g_{2\alpha,\beta}+g_{2\beta,\alpha}-g_{\alpha\beta,2}\right)+g^{\mu3}\left(g_{3\alpha,\beta}+g_{3\beta,\alpha}-g_{\alpha\beta,3}\right)]{\dot x^\alpha }{\dot x^\beta}\ $$