User:Z E U S/sandbox

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

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

${d^{2}x^{\mu } \over ds^{2}}=-{\frac {1}{2}}[g^{\mu 0}\left(\partial _{\beta }g_{0\alpha }+\partial _{\alpha }g_{0\beta }-\partial _{0}g_{\alpha \beta }\right)+g^{\mu 1}\left(\partial _{\beta }g_{1\alpha }+\partial _{\alpha }g_{1\beta }-\partial _{1}g_{\alpha \beta }\right)+g^{\mu 2}\left(\partial _{\beta }g_{2\alpha }+\partial _{\alpha }g_{2\beta }-\partial _{2}g_{\alpha \beta }\right)+g^{\mu 3}\left(\partial _{\beta }g_{3\alpha }+\partial _{\alpha }g_{3\beta }-\partial _{3}g_{\alpha \beta }\right)]{dx^{\alpha } \over ds}{dx^{\beta } \over ds}\$

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

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