d 2 x μ d s 2 + Γ μ α β d x α d s d x β d s = 0 {\displaystyle {d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}\ =0}
or
x ¨ μ + Γ μ α β x ˙ α x ˙ β = 0 {\displaystyle {\ddot {x}}^{\mu }+\Gamma ^{\mu }{}_{\alpha \beta }{\dot {x}}^{\alpha }{\dot {x}}^{\beta }=0\ }
x ¨ μ + 1 2 g μ ρ ( g ρ α , β + g ρ β , α − g α β , ρ ) x ˙ α x ˙ β = 0 {\displaystyle {\ddot {x}}^{\mu }+{\frac {1}{2}}g^{\mu \rho }\left(g_{\rho \alpha ,\beta }+g_{\rho \beta ,\alpha }-g_{\alpha \beta ,\rho }\right){\dot {x}}^{\alpha }{\dot {x}}^{\beta }=0}
x ¨ μ + 1 2 ( g μ 0 ( g 0 α , β + g 0 β , α − g α β , 0 ) + g μ 1 ( g 1 α , β + g 1 β , α − g α β , 1 ) + g μ 2 ( g 2 α , β + g 2 β , α − g α β , 2 ) + g μ 3 ( g 3 α , β + g 3 β , α − g α β , 3 ) ) x ˙ α x ˙ β = 0 {\displaystyle {\ddot {x}}^{\mu }+{\frac {1}{2}}\left(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)\right){\dot {x}}^{\alpha }{\dot {x}}^{\beta }=0}
x ¨ μ + 1 2 ( g μ 0 ( g 0 α , β + g 0 β , α − g α β , 0 ) + g μ 1 ( g 1 α , β + g 1 β , α − g α β , 1 ) + g μ 2 ( g 2 α , β + g 2 β , α − g α β , 2 ) + g μ 3 ( g 3 α , β + g 3 β , α − g α β , 3 ) x ˙ α x ˙ β + g μ 0 ( g 0 α , β + g 0 β , α − g α β , 0 ) + g μ 1 ( g 1 α , β + g 1 β , α − g α β , 1 ) + g μ 2 ( g 2 α , β + g 2 β , α − g α β , 2 ) + g μ 3 ( g 3 α , β + g 3 β , α − g α β , 3 ) x ˙ α x ˙ β + g μ 0 ( g 0 α , β + g 0 β , α − g α β , 0 ) + g μ 1 ( g 1 α , β + g 1 β , α − g α β , 1 ) + g μ 2 ( g 2 α , β + g 2 β , α − g α β , 2 ) + g μ 3 ( g 3 α , β + g 3 β , α − g α β , 3 ) x ˙ α x ˙ β + g μ 0 ( g 0 α , β + g 0 β , α − g α β , 0 ) + g μ 1 ( g 1 α , β + g 1 β , α − g α β , 1 ) + g μ 2 ( g 2 α , β + g 2 β , α − g α β , 2 ) + g μ 3 ( g 3 α , β + g 3 β , α − g α β , 3 ) x ˙ α x ˙ β + g μ 0 ( g 0 α , β + g 0 β , α − g α β , 0 ) + g μ 1 ( g 1 α , β + g 1 β , α − g α β , 1 ) + g μ 2 ( g 2 α , β + g 2 β , α − g α β , 2 ) + g μ 3 ( g 3 α , β + g 3 β , α − g α β , 3 ) x ˙ α x ˙ β + g μ 0 ( g 0 α , β + g 0 β , α − g α β , 0 ) + g μ 1 ( g 1 α , β + g 1 β , α − g α β , 1 ) + g μ 2 ( g 2 α , β + g 2 β , α − g α β , 2 ) + g μ 3 ( g 3 α , β + g 3 β , α − g α β , 3 ) x ˙ α x ˙ β + g μ 0 ( g 0 α , β + g 0 β , α − g α β , 0 ) + g μ 1 ( g 1 α , β + g 1 β , α − g α β , 1 ) + g μ 2 ( g 2 α , β + g 2 β , α − g α β , 2 ) + g μ 3 ( g 3 α , β + g 3 β , α − g α β , 3 ) x ˙ α x ˙ β + g μ 0 ( g 0 α , β + g 0 β , α − g α β , 0 ) + g μ 1 ( g 1 α , β + g 1 β , α − g α β , 1 ) + g μ 2 ( g 2 α , β + g 2 β , α − g α β , 2 ) + g μ 3 ( g 3 α , β + g 3 β , α − g α β , 3 ) x ˙ α x ˙ β + g μ 0 ( g 0 α , β + g 0 β , α − g α β , 0 ) + g μ 1 ( g 1 α , β + g 1 β , α − g α β , 1 ) + g μ 2 ( g 2 α , β + g 2 β , α − g α β , 2 ) + g μ 3 ( g 3 α , β + g 3 β , α − g α β , 3 ) x ˙ α x ˙ β + g μ 0 ( g 0 α , β + g 0 β , α − g α β , 0 ) + g μ 1 ( g 1 α , β + g 1 β , α − g α β , 1 ) + g μ 2 ( g 2 α , β + g 2 β , α − g α β , 2 ) + g μ 3 ( g 3 α , β + g 3 β , α − g α β , 3 ) x ˙ α x ˙ β + ) = 0 {\displaystyle {\ddot {x}}^{\mu }+{\frac {1}{2}}\left(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 }+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 }+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 }+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 }+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 }+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 }+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 }+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 }+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 }+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 }+\right)=0}
