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}
x ¨ λ + Γ λ μ ν x ˙ μ x ˙ ν = 0 . {\displaystyle {\ddot {x}}^{\lambda }+\Gamma ^{\lambda }{}_{\mu \nu }{\dot {x}}^{\mu }{\dot {x}}^{\nu }=0\ .}
{ d 2 d τ 2 t − 1 / 2 ( ∂ ∂ t g 00 ) ( d d τ x ) 2 g 33 + ( ∂ ∂ x g 33 ) ( d d τ x ) d d τ t g 33 − 1 / 2 ( ∂ ∂ t g 11 ) ( d d τ y ) 2 g 33 + ( ∂ ∂ y g 33 ) ( d d τ y ) d d τ t g 33 − 1 / 2 ( ∂ ∂ t g 22 ) ( d d τ z ) 2 g 33 + ( ∂ ∂ z g 33 ) ( d d τ z ) d d τ t g 33 + 1 / 2 ( ∂ ∂ t g 33 ) ( d d τ t ) 2 g 33 = 0 d 2 d τ 2 x + 1 / 2 ( ∂ ∂ x g 00 ) ( d d τ x ) 2 g 00 + ( ∂ ∂ y g 00 ) ( d d τ x ) d d τ y g 00 + ( ∂ ∂ z g 00 ) ( d d τ x ( τ ) ) d d τ z g 00 + ( ∂ ∂ t g 00 ) ( d d τ x ) d d τ t ( τ ) g 00 − 1 / 2 ( ∂ ∂ x g 11 ) ( d d τ y ) 2 g 00 − 1 / 2 ( ∂ ∂ x g 22 ) ( d d τ z ) 2 g 00 − 1 / 2 ( ∂ ∂ x g 33 ) ( d d τ t ) 2 g 00 = 0 d 2 d τ 2 y − 1 / 2 ( ∂ ∂ y g 00 ) ( d d τ x ) 2 g 11 + ( ∂ ∂ x g 11 ) ( d d τ x ) d d τ y ( τ ) g 11 + 1 / 2 ( ∂ ∂ y g 11 ) ( d d τ y ) 2 g 11 + ( ∂ ∂ z g 11 ) ( d d τ y ) d d τ z g 11 + ( ∂ ∂ t g 11 ) ( d d τ y ( τ ) ) d d τ t g 11 − 1 / 2 ( ∂ ∂ y g 22 ) ( d d τ z ) 2 g 11 − 1 / 2 ( ∂ ∂ y g 33 ) ( d d τ t ) 2 g 11 = 0 d 2 d τ 2 z − 1 / 2 ( ∂ ∂ z g 00 ) ( d d τ x ) 2 g 22 + ( ∂ ∂ x g 22 ) ( d d τ x ) d d τ z g 22 − 1 / 2 ( ∂ ∂ z g 11 ) ( d d τ y ) 2 g 22 + ( ∂ ∂ y g 22 ) ( d d τ y ) d d τ z g 22 + 1 / 2 ( ∂ ∂ z g 22 ) ( d d τ z ) 2 g 22 + ( ∂ ∂ t g 22 ) ( d d τ z ) d d τ t g 22 − 1 / 2 ( ∂ ∂ z g 33 ) ( d d τ t ) 2 g 22 = 0 {\displaystyle {\begin{cases}{\frac {d^{2}}{d{\tau }^{2}}}t-1/2\,{\frac {\left({\frac {\partial }{\partial t}}{\it {g_{00}}}\right)\left({\frac {d}{d\tau }}x\right)^{2}}{\it {g_{33}}}}+{\frac {\left({\frac {\partial }{\partial x}}{\it {g_{33}}}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}t}{\it {g_{33}}}}-1/2\,{\frac {\left({\frac {\partial }{\partial t}}{\it {g_{11}}}\right)\left({\frac {d}{d\tau }}y\right)^{2}}{\it {g_{33}}}}+{\frac {\left({\frac {\partial }{\partial y}}{\it {g_{33}}}\right)\left({\frac {d}{d\tau }}y\right){\frac {d}{d\tau }}t}{\it {g_{33}}}}-1/2\,{\frac {\left({\frac {\partial }{\partial t}}{\it {g_{22}}}\right)\left({\frac {d}{d\tau }}z\right)^{2}}{\it {g_{33}}}}+{\frac {\left({\frac {\partial }{\partial z}}{\it {g_{33}}}\right)\left({\frac {d}{d\tau }}z\right){\frac {d}{d\tau }}t}{\it {g_{33}}}}+1/2\,{\frac {\left({\frac {\partial }{\partial t}}{\it {g_{33}}}\right)\left({\frac {d}{d\tau }}t\right)^{2}}{\it {g_{33}}}}=0\\{\frac {d^{2}}{d{\tau }^{2}}}x+1/2\,{\frac {\left({\frac {\partial }{\partial x}}{\it {g_{00}}}\right)\left({\frac {d}{d\tau }}x\right)^{2}}{\it {g_{00}}}}+{\frac {\left({\frac {\partial }{\partial y}}{\it {g_{00}}}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}y}{\it {g_{00}}}}+{\frac {\left({\frac {\partial }{\partial z}}{\it {g_{00}}}\right)\left({\frac {d}{d\tau }}x\left(\tau \right)\right){\frac {d}{d\tau }}z}{\it {g_{00}}}}+{\frac {\left({\frac {\partial }{\partial t}}{\it {g_{00}}}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}t\left(\tau \right)}{\it {g_{00}}}}-1/2\,{\frac {\left({\frac {\partial }{\partial x}}{\it {g_{11}}}\right)\left({\frac {d}{d\tau }}y\right)^{2}}{\it {g_{00}}}}-1/2\,{\frac {\left({\frac {\partial }{\partial x}}{\it {g_{22}}}\right)\left({\frac {d}{d\tau }}z\right)^{2}}{\it {g_{00}}}}-1/2\,{\frac {\left({\frac {\partial }{\partial x}}{\it {g_{33}}}\right)\left({\frac {d}{d\tau }}t\right)^{2}}{\it {g_{00}}}}=0\\{\frac {d^{2}}{d{\tau }^{2}}}y-1/2\,{\frac {\left({\frac {\partial }{\partial y}}{\it {g_{00}}}\right)\left({\frac {d}{d\tau }}x\right)^{2}}{\it {g_{11}}}}+{\frac {\left({\frac {\partial }{\partial x}}{\it {g_{11}}}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}y\left(\tau \right)}{\it {g_{11}}}}+1/2\,{\frac {\left({\frac {\partial }{\partial y}}{\it {g_{11}}}\right)\left({\frac {d}{d\tau }}y\right)^{2}}{\it {g_{11}}}}+{\frac {\left({\frac {\partial }{\partial z}}{\it {g_{11}}}\right)\left({\frac {d}{d\tau }}y\right){\frac {d}{d\tau }}z}{\it {g_{11}}}}+{\frac {\left({\frac {\partial }{\partial t}}{\it {g_{11}}}\right)\left({\frac {d}{d\tau }}y\left(\tau \right)\right){\frac {d}{d\tau }}t}{\it {g_{11}}}}-1/2\,{\frac {\left({\frac {\partial }{\partial y}}{\it {g_{22}}}\right)\left({\frac {d}{d\tau }}z\right)^{2}}{\it {g_{11}}}}-1/2\,{\frac {\left({\frac {\partial }{\partial y}}{\it {g_{33}}}\right)\left({\frac {d}{d\tau }}t\right)^{2}}{\it {g_{11}}}}=0\\{\frac {d^{2}}{d{\tau }^{2}}}z-1/2\,{\frac {\left({\frac {\partial }{\partial z}}{\it {g_{00}}}\right)\left({\frac {d}{d\tau }}x\right)^{2}}{\it {g_{22}}}}+{\frac {\left({\frac {\partial }{\partial x}}{\it {g_{22}}}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}z}{\it {g_{22}}}}-1/2\,{\frac {\left({\frac {\partial }{\partial z}}{\it {g_{11}}}\right)\left({\frac {d}{d\tau }}y\right)^{2}}{\it {g_{22}}}}+{\frac {\left({\frac {\partial }{\partial y}}{\it {g_{22}}}\right)\left({\frac {d}{d\tau }}y\right){\frac {d}{d\tau }}z}{\it {g_{22}}}}+1/2\,{\frac {\left({\frac {\partial }{\partial z}}{\it {g_{22}}}\right)\left({\frac {d}{d\tau }}z\right)^{2}}{\it {g_{22}}}}+{\frac {\left({\frac {\partial }{\partial t}}{\it {g_{22}}}\right)\left({\frac {d}{d\tau }}z\right){\frac {d}{d\tau }}t}{\it {g_{22}}}}-1/2\,{\frac {\left({\frac {\partial }{\partial z}}{\it {g_{33}}}\right)\left({\frac {d}{d\tau }}t\right)^{2}}{\it {g_{22}}}}=0\end{cases}}}
{ d 2 d τ 2 t − 1 / 2 ( ∂ ∂ t g 11 ) ( d d τ x ) 2 g 44 − ( ∂ ∂ t g 12 ) ( d d τ x ) d d τ y g 44 + ( ∂ ∂ x g 44 ) ( d d τ x ( τ ) ) d d τ t g 44 − 1 / 2 ( ∂ ∂ t g 22 ) ( d d τ y ) 2 g 44 + ( ∂ ∂ y g 44 ) ( d d τ y ( τ ) ) d d τ t g 44 − 1 / 2 ( ∂ ∂ t g 33 ) ( d d τ z ) 2 g 44 + ( ∂ ∂ z g 44 ) ( d d τ z ( τ ) ) d d τ t g 44 + 1 / 2 ( ∂ ∂ t g 44 ) ( d d τ t ) 2 g 44 = 0 , d 2 d τ 2 z − 1 / 2 ( ∂ ∂ z g 11 ) ( d d τ x ) 2 g 33 − ( ∂ ∂ z g 12 ) ( d d τ x ) d d τ y g 33 + ( ∂ ∂ x g 33 ) ( d d τ x ( τ ) ) d d τ z g 33 − 1 / 2 ( ∂ ∂ z g 22 ) ( d d τ y ) 2 g 33 + ( ∂ ∂ y g 33 ) ( d d τ y ( τ ) ) d d τ z g 33 + 1 / 2 ( ∂ ∂ z g 33 ) ( d d τ z ) 2 g 33 + ( ∂ ∂ t g 33 ) ( d d τ z ( τ ) ) d d τ t g 33 − 1 / 2 ( ∂ ∂ z g 44 ) ( d d τ t ) 2 g 33 = 0 , d 2 d τ 2 x + 1 / 2 ( g 22 ∂ ∂ x g 11 − 2 g 12 ∂ ∂ x g 12 + g 12 ∂ ∂ y g 11 ) ( d d τ x ) 2 g 11 g 22 − ( g 12 ) 2 + ( g 22 ∂ ∂ y g 11 − g 12 ∂ ∂ x g 22 ) ( d d τ x ) d d τ y ( τ ) g 11 g 22 − ( g 12 ) 2 + ( g 22 ∂ ∂ z g 11 − g 12 ∂ ∂ z g 12 ) ( d d τ x ) d d τ z g 11 g 22 − ( g 12 ) 2 + ( g 22 ∂ ∂ t g 11 − g 12 ∂ ∂ t g 12 ) ( d d τ x ) d d τ t ( τ ) g 11 g 22 − ( g 12 ) 2 − 1 / 2 ( − 2 g 22 ∂ ∂ y g 12 + g 22 ∂ ∂ x g 22 + g 12 ∂ ∂ y g 22 ) ( d d τ y ) 2 g 11 g 22 − ( g 12 ) 2 + ( g 22 ∂ ∂ z g 12 − g 12 ∂ ∂ z g 22 ) ( d d τ y ) d d τ z g 11 g 22 − ( g 12 ) 2 + ( g 22 ∂ ∂ t g 12 − g 12 ∂ ∂ t g 22 ) ( d d τ y ( τ ) ) d d τ t g 11 g 22 − ( g 12 ) 2 − 1 / 2 ( g 22 ∂ ∂ x g 33 − g 12 ∂ ∂ y g 33 ) ( d d τ z ) 2 g 11 g 22 − ( g 12 ) 2 − 1 / 2 ( g 22 ∂ ∂ x g 44 − g 12 ∂ ∂ y g 44 ) ( d d τ t ) 2 g 11 g 22 − ( g 12 ) 2 = 0 , d 2 d τ 2 y − 1 / 2 ( g 12 ∂ ∂ x g 11 − 2 g 11 ∂ ∂ x g 12 + g 11 ∂ ∂ y g 11 ) ( d d τ x ) 2 g 11 g 22 − ( g 12 ) 2 − ( g 12 ∂ ∂ y g 11 − g 11 ∂ ∂ x g 22 ) ( d d τ x ) d d τ y g 11 g 22 − ( g 12 ) 2 − ( g 12 ∂ ∂ z g 11 − g 11 ∂ ∂ z g 12 ) ( d d τ x ) d d τ z ( τ ) g 11 g 22 − ( g 12 ) 2 − ( g 12 ∂ ∂ t g 11 − g 11 ∂ ∂ t g 12 ) ( d d τ x ) d d τ t g 11 g 22 − ( g 12 ) 2 + 1 / 2 ( − 2 g 12 ∂ ∂ y g 12 + g 12 ∂ ∂ x g 22 + g 11 ∂ ∂ y g 22 ) ( d d τ y ( τ ) ) 2 g 11 g 22 − ( g 12 ) 2 − ( g 12 ∂ ∂ z g 12 − g 11 ∂ ∂ z g 22 ) ( d d τ y ( τ ) ) d d τ z g 11 g 22 − ( g 12 ) 2 − ( g 12 ∂ ∂ t g 12 − g 11 ∂ ∂ t g 22 ) ( d d τ y ) d d τ t g 11 g 22 − ( g 12 ) 2 + 1 / 2 ( g 12 ∂ ∂ x g 33 − g 11 ∂ ∂ y g 33 ) ( d d τ z ( τ ) ) 2 g 11 g 22 − ( g 12 ) 2 − 1 / 2 ( − g 12 ∂ ∂ x g 44 + g 11 ∂ ∂ y g 44 ) ( d d τ t ( τ ) ) 2 g 11 g 22 − ( g 12 ) 2 = 0 } {\displaystyle \left\{{\frac {d^{2}}{d{\tau }^{2}}}t-1/2\,{\frac {\left({\frac {\partial }{\partial t}}g_{11}\right)\left({\frac {d}{d\tau }}x\right)^{2}}{g_{44}}}-{\frac {\left({\frac {\partial }{\partial t}}g_{12}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}y}{g_{44}}}+{\frac {\left({\frac {\partial }{\partial x}}g_{44}\right)\left({\frac {d}{d\tau }}x\left(\tau \right)\right){\frac {d}{d\tau }}t}{g_{44}}}-1/2\,{\frac {\left({\frac {\partial }{\partial t}}g_{22}\right)\left({\frac {d}{d\tau }}y\right)^{2}}{g_{44}}}+{\frac {\left({\frac {\partial }{\partial y}}g_{44}\right)\left({\frac {d}{d\tau }}y\left(\tau \right)\right){\frac {d}{d\tau }}t}{g_{44}}}-1/2\,{\frac {\left({\frac {\partial }{\partial t}}g_{33}\right)\left({\frac {d}{d\tau }}z\right)^{2}}{g_{44}}}+{\frac {\left({\frac {\partial }{\partial z}}g_{44}\right)\left({\frac {d}{d\tau }}z\left(\tau \right)\right){\frac {d}{d\tau }}t}{g_{44}}}+1/2\,{\frac {\left({\frac {\partial }{\partial t}}g_{44}\right)\left({\frac {d}{d\tau }}t\right)^{2}}{g_{44}}}=0,{\frac {d^{2}}{d{\tau }^{2}}}z-1/2\,{\frac {\left({\frac {\partial }{\partial z}}g_{11}\right)\left({\frac {d}{d\tau }}x\right)^{2}}{g_{33}}}-{\frac {\left({\frac {\partial }{\partial z}}g_{12}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}y}{g_{33}}}+{\frac {\left({\frac {\partial }{\partial x}}g_{33}\right)\left({\frac {d}{d\tau }}x\left(\tau \right)\right){\frac {d}{d\tau }}z}{g_{33}}}-1/2\,{\frac {\left({\frac {\partial }{\partial z}}g_{22}\right)\left({\frac {d}{d\tau }}y\right)^{2}}{g_{33}}}+{\frac {\left({\frac {\partial }{\partial y}}g_{33}\right)\left({\frac {d}{d\tau }}y\left(\tau \right)\right){\frac {d}{d\tau }}z}{g_{33}}}+1/2\,{\frac {\left({\frac {\partial }{\partial z}}g_{33}\right)\left({\frac {d}{d\tau }}z\right)^{2}}{g_{33}}}+{\frac {\left({\frac {\partial }{\partial t}}g_{33}\right)\left({\frac {d}{d\tau }}z\left(\tau \right)\right){\frac {d}{d\tau }}t}{g_{33}}}-1/2\,{\frac {\left({\frac {\partial }{\partial z}}g_{44}\right)\left({\frac {d}{d\tau }}t\right)^{2}}{g_{33}}}=0,{\frac {d^{2}}{d{\tau }^{2}}}x+1/2\,{\frac {\left(g_{22}{\frac {\partial }{\partial x}}g_{11}-2\,g_{12}{\frac {\partial }{\partial x}}g_{12}+g_{12}{\frac {\partial }{\partial y}}g_{11}\right)\left({\frac {d}{d\tau }}x\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+{\frac {\left(g_{22}{\frac {\partial }{\partial y}}g_{11}-g_{12}{\frac {\partial }{\partial x}}g_{22}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}y\left(\tau \right)}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+{\frac {\left(g_{22}{\frac {\partial }{\partial z}}g_{11}-g_{12}{\frac {\partial }{\partial z}}g_{12}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}z}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+{\frac {\left(g_{22}{\frac {\partial }{\partial t}}g_{11}-g_{12}{\frac {\partial }{\partial t}}g_{12}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}t\left(\tau \right)}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-1/2\,{\frac {\left(-2\,g_{22}{\frac {\partial }{\partial y}}g_{12}+g_{22}{\frac {\partial }{\partial x}}g_{22}+g_{12}{\frac {\partial }{\partial y}}g_{22}\right)\left({\frac {d}{d\tau }}y\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+{\frac {\left(g_{22}{\frac {\partial }{\partial z}}g_{12}-g_{12}{\frac {\partial }{\partial z}}g_{22}\right)\left({\frac {d}{d\tau }}y\right){\frac {d}{d\tau }}z}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+{\frac {\left(g_{22}{\frac {\partial }{\partial t}}g_{12}-g_{12}{\frac {\partial }{\partial t}}g_{22}\right)\left({\frac {d}{d\tau }}y\left(\tau \right)\right){\frac {d}{d\tau }}t}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-1/2\,{\frac {\left(g_{22}{\frac {\partial }{\partial x}}g_{33}-g_{12}{\frac {\partial }{\partial y}}g_{33}\right)\left({\frac {d}{d\tau }}z\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-1/2\,{\frac {\left(g_{22}{\frac {\partial }{\partial x}}g_{44}-g_{12}{\frac {\partial }{\partial y}}g_{44}\right)\left({\frac {d}{d\tau }}t\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}=0,{\frac {d^{2}}{d{\tau }^{2}}}y-1/2\,{\frac {\left(g_{12}{\frac {\partial }{\partial x}}g_{11}-2\,g_{11}{\frac {\partial }{\partial x}}g_{12}+g_{11}{\frac {\partial }{\partial y}}g_{11}\right)\left({\frac {d}{d\tau }}x\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-{\frac {\left(g_{12}{\frac {\partial }{\partial y}}g_{11}-g_{11}{\frac {\partial }{\partial x}}g_{22}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}y}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-{\frac {\left(g_{12}{\frac {\partial }{\partial z}}g_{11}-g_{11}{\frac {\partial }{\partial z}}g_{12}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}z\left(\tau \right)}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-{\frac {\left(g_{12}{\frac {\partial }{\partial t}}g_{11}-g_{11}{\frac {\partial }{\partial t}}g_{12}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}t}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+1/2\,{\frac {\left(-2\,g_{12}{\frac {\partial }{\partial y}}g_{12}+g_{12}{\frac {\partial }{\partial x}}g_{22}+g_{11}{\frac {\partial }{\partial y}}g_{22}\right)\left({\frac {d}{d\tau }}y\left(\tau \right)\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-{\frac {\left(g_{12}{\frac {\partial }{\partial z}}g_{12}-g_{11}{\frac {\partial }{\partial z}}g_{22}\right)\left({\frac {d}{d\tau }}y\left(\tau \right)\right){\frac {d}{d\tau }}z}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-{\frac {\left(g_{12}{\frac {\partial }{\partial t}}g_{12}-g_{11}{\frac {\partial }{\partial t}}g_{22}\right)\left({\frac {d}{d\tau }}y\right){\frac {d}{d\tau }}t}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+1/2\,{\frac {\left(g_{12}{\frac {\partial }{\partial x}}g_{33}-g_{11}{\frac {\partial }{\partial y}}g_{33}\right)\left({\frac {d}{d\tau }}z\left(\tau \right)\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-1/2\,{\frac {\left(-g_{12}{\frac {\partial }{\partial x}}g_{44}+g_{11}{\frac {\partial }{\partial y}}g_{44}\right)\left({\frac {d}{d\tau }}t\left(\tau \right)\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}=0\right\}}
d 2 d τ 2 t − 1 / 2 ( ∂ ∂ t g 11 ) ( d d τ x ) 2 g 44 − ( ∂ ∂ t g 12 ) ( d d τ x ) d d τ y g 44 + ( ∂ ∂ x g 44 ) ( d d τ x ( τ ) ) d d τ t g 44 − 1 / 2 ( ∂ ∂ t g 22 ) ( d d τ y ) 2 g 44 + ( ∂ ∂ y g 44 ) ( d d τ y ( τ ) ) d d τ t g 44 − 1 / 2 ( ∂ ∂ t g 33 ) ( d d τ z ) 2 g 44 + ( ∂ ∂ z g 44 ) ( d d τ z ( τ ) ) d d τ t g 44 + 1 / 2 ( ∂ ∂ t g 44 ) ( d d τ t ) 2 g 44 = 0 , {\displaystyle {\frac {d^{2}}{d{\tau }^{2}}}t-1/2\,{\frac {\left({\frac {\partial }{\partial t}}g_{11}\right)\left({\frac {d}{d\tau }}x\right)^{2}}{g_{44}}}-{\frac {\left({\frac {\partial }{\partial t}}g_{12}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}y}{g_{44}}}+{\frac {\left({\frac {\partial }{\partial x}}g_{44}\right)\left({\frac {d}{d\tau }}x\left(\tau \right)\right){\frac {d}{d\tau }}t}{g_{44}}}-1/2\,{\frac {\left({\frac {\partial }{\partial t}}g_{22}\right)\left({\frac {d}{d\tau }}y\right)^{2}}{g_{44}}}+{\frac {\left({\frac {\partial }{\partial y}}g_{44}\right)\left({\frac {d}{d\tau }}y\left(\tau \right)\right){\frac {d}{d\tau }}t}{g_{44}}}-1/2\,{\frac {\left({\frac {\partial }{\partial t}}g_{33}\right)\left({\frac {d}{d\tau }}z\right)^{2}}{g_{44}}}+{\frac {\left({\frac {\partial }{\partial z}}g_{44}\right)\left({\frac {d}{d\tau }}z\left(\tau \right)\right){\frac {d}{d\tau }}t}{g_{44}}}+1/2\,{\frac {\left({\frac {\partial }{\partial t}}g_{44}\right)\left({\frac {d}{d\tau }}t\right)^{2}}{g_{44}}}=0,}
d 2 d τ 2 z − 1 / 2 ( ∂ ∂ z g 11 ) ( d d τ x ) 2 g 33 − ( ∂ ∂ z g 12 ) ( d d τ x ) d d τ y g 33 + ( ∂ ∂ x g 33 ) ( d d τ x ( τ ) ) d d τ z g 33 − 1 / 2 ( ∂ ∂ z g 22 ) ( d d τ y ) 2 g 33 + ( ∂ ∂ y g 33 ) ( d d τ y ( τ ) ) d d τ z g 33 + 1 / 2 ( ∂ ∂ z g 33 ) ( d d τ z ) 2 g 33 + ( ∂ ∂ t g 33 ) ( d d τ z ( τ ) ) d d τ t g 33 − 1 / 2 ( ∂ ∂ z g 44 ) ( d d τ t ) 2 g 33 = 0 , {\displaystyle {\frac {d^{2}}{d{\tau }^{2}}}z-1/2\,{\frac {\left({\frac {\partial }{\partial z}}g_{11}\right)\left({\frac {d}{d\tau }}x\right)^{2}}{g_{33}}}-{\frac {\left({\frac {\partial }{\partial z}}g_{12}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}y}{g_{33}}}+{\frac {\left({\frac {\partial }{\partial x}}g_{33}\right)\left({\frac {d}{d\tau }}x\left(\tau \right)\right){\frac {d}{d\tau }}z}{g_{33}}}-1/2\,{\frac {\left({\frac {\partial }{\partial z}}g_{22}\right)\left({\frac {d}{d\tau }}y\right)^{2}}{g_{33}}}+{\frac {\left({\frac {\partial }{\partial y}}g_{33}\right)\left({\frac {d}{d\tau }}y\left(\tau \right)\right){\frac {d}{d\tau }}z}{g_{33}}}+1/2\,{\frac {\left({\frac {\partial }{\partial z}}g_{33}\right)\left({\frac {d}{d\tau }}z\right)^{2}}{g_{33}}}+{\frac {\left({\frac {\partial }{\partial t}}g_{33}\right)\left({\frac {d}{d\tau }}z\left(\tau \right)\right){\frac {d}{d\tau }}t}{g_{33}}}-1/2\,{\frac {\left({\frac {\partial }{\partial z}}g_{44}\right)\left({\frac {d}{d\tau }}t\right)^{2}}{g_{33}}}=0,}
d 2 d τ 2 x + 1 / 2 ( g 22 ∂ ∂ x g 11 − 2 g 12 ∂ ∂ x g 12 + g 12 ∂ ∂ y g 11 ) ( d d τ x ) 2 g 11 g 22 − ( g 12 ) 2 + ( g 22 ∂ ∂ y g 11 − g 12 ∂ ∂ x g 22 ) ( d d τ x ) d d τ y ( τ ) g 11 g 22 − ( g 12 ) 2 + ( g 22 ∂ ∂ z g 11 − g 12 ∂ ∂ z g 12 ) ( d d τ x ) d d τ z g 11 g 22 − ( g 12 ) 2 + ( g 22 ∂ ∂ t g 11 − g 12 ∂ ∂ t g 12 ) ( d d τ x ) d d τ t ( τ ) g 11 g 22 − ( g 12 ) 2 − 1 / 2 ( − 2 g 22 ∂ ∂ y g 12 + g 22 ∂ ∂ x g 22 + g 12 ∂ ∂ y g 22 ) ( d d τ y ) 2 g 11 g 22 − ( g 12 ) 2 + ( g 22 ∂ ∂ z g 12 − g 12 ∂ ∂ z g 22 ) ( d d τ y ) d d τ z g 11 g 22 − ( g 12 ) 2 + ( g 22 ∂ ∂ t g 12 − g 12 ∂ ∂ t g 22 ) ( d d τ y ( τ ) ) d d τ t g 11 g 22 − ( g 12 ) 2 − 1 / 2 ( g 22 ∂ ∂ x g 33 − g 12 ∂ ∂ y g 33 ) ( d d τ z ) 2 g 11 g 22 − ( g 12 ) 2 − 1 / 2 ( g 22 ∂ ∂ x g 44 − g 12 ∂ ∂ y g 44 ) ( d d τ t ) 2 g 11 g 22 − ( g 12 ) 2 = 0 , {\displaystyle {\frac {d^{2}}{d{\tau }^{2}}}x+1/2\,{\frac {\left(g_{22}{\frac {\partial }{\partial x}}g_{11}-2\,g_{12}{\frac {\partial }{\partial x}}g_{12}+g_{12}{\frac {\partial }{\partial y}}g_{11}\right)\left({\frac {d}{d\tau }}x\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+{\frac {\left(g_{22}{\frac {\partial }{\partial y}}g_{11}-g_{12}{\frac {\partial }{\partial x}}g_{22}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}y\left(\tau \right)}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+{\frac {\left(g_{22}{\frac {\partial }{\partial z}}g_{11}-g_{12}{\frac {\partial }{\partial z}}g_{12}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}z}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+{\frac {\left(g_{22}{\frac {\partial }{\partial t}}g_{11}-g_{12}{\frac {\partial }{\partial t}}g_{12}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}t\left(\tau \right)}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-1/2\,{\frac {\left(-2\,g_{22}{\frac {\partial }{\partial y}}g_{12}+g_{22}{\frac {\partial }{\partial x}}g_{22}+g_{12}{\frac {\partial }{\partial y}}g_{22}\right)\left({\frac {d}{d\tau }}y\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+{\frac {\left(g_{22}{\frac {\partial }{\partial z}}g_{12}-g_{12}{\frac {\partial }{\partial z}}g_{22}\right)\left({\frac {d}{d\tau }}y\right){\frac {d}{d\tau }}z}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+{\frac {\left(g_{22}{\frac {\partial }{\partial t}}g_{12}-g_{12}{\frac {\partial }{\partial t}}g_{22}\right)\left({\frac {d}{d\tau }}y\left(\tau \right)\right){\frac {d}{d\tau }}t}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-1/2\,{\frac {\left(g_{22}{\frac {\partial }{\partial x}}g_{33}-g_{12}{\frac {\partial }{\partial y}}g_{33}\right)\left({\frac {d}{d\tau }}z\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-1/2\,{\frac {\left(g_{22}{\frac {\partial }{\partial x}}g_{44}-g_{12}{\frac {\partial }{\partial y}}g_{44}\right)\left({\frac {d}{d\tau }}t\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}=0,}
d 2 d τ 2 y − 1 / 2 ( g 12 ∂ ∂ x g 11 − 2 g 11 ∂ ∂ x g 12 + g 11 ∂ ∂ y g 11 ) ( d d τ x ) 2 g 11 g 22 − ( g 12 ) 2 − ( g 12 ∂ ∂ y g 11 − g 11 ∂ ∂ x g 22 ) ( d d τ x ) d d τ y g 11 g 22 − ( g 12 ) 2 − ( g 12 ∂ ∂ z g 11 − g 11 ∂ ∂ z g 12 ) ( d d τ x ) d d τ z ( τ ) g 11 g 22 − ( g 12 ) 2 − ( g 12 ∂ ∂ t g 11 − g 11 ∂ ∂ t g 12 ) ( d d τ x ) d d τ t g 11 g 22 − ( g 12 ) 2 + 1 / 2 ( − 2 g 12 ∂ ∂ y g 12 + g 12 ∂ ∂ x g 22 + g 11 ∂ ∂ y g 22 ) ( d d τ y ( τ ) ) 2 g 11 g 22 − ( g 12 ) 2 − ( g 12 ∂ ∂ z g 12 − g 11 ∂ ∂ z g 22 ) ( d d τ y ( τ ) ) d d τ z g 11 g 22 − ( g 12 ) 2 − ( g 12 ∂ ∂ t g 12 − g 11 ∂ ∂ t g 22 ) ( d d τ y ) d d τ t g 11 g 22 − ( g 12 ) 2 + 1 / 2 ( g 12 ∂ ∂ x g 33 − g 11 ∂ ∂ y g 33 ) ( d d τ z ( τ ) ) 2 g 11 g 22 − ( g 12 ) 2 − 1 / 2 ( − g 12 ∂ ∂ x g 44 + g 11 ∂ ∂ y g 44 ) ( d d τ t ( τ ) ) 2 g 11 g 22 − ( g 12 ) 2 = 0 {\displaystyle {\frac {d^{2}}{d{\tau }^{2}}}y-1/2\,{\frac {\left(g_{12}{\frac {\partial }{\partial x}}g_{11}-2\,g_{11}{\frac {\partial }{\partial x}}g_{12}+g_{11}{\frac {\partial }{\partial y}}g_{11}\right)\left({\frac {d}{d\tau }}x\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-{\frac {\left(g_{12}{\frac {\partial }{\partial y}}g_{11}-g_{11}{\frac {\partial }{\partial x}}g_{22}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}y}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-{\frac {\left(g_{12}{\frac {\partial }{\partial z}}g_{11}-g_{11}{\frac {\partial }{\partial z}}g_{12}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}z\left(\tau \right)}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-{\frac {\left(g_{12}{\frac {\partial }{\partial t}}g_{11}-g_{11}{\frac {\partial }{\partial t}}g_{12}\right)\left({\frac {d}{d\tau }}x\right){\frac {d}{d\tau }}t}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+1/2\,{\frac {\left(-2\,g_{12}{\frac {\partial }{\partial y}}g_{12}+g_{12}{\frac {\partial }{\partial x}}g_{22}+g_{11}{\frac {\partial }{\partial y}}g_{22}\right)\left({\frac {d}{d\tau }}y\left(\tau \right)\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-{\frac {\left(g_{12}{\frac {\partial }{\partial z}}g_{12}-g_{11}{\frac {\partial }{\partial z}}g_{22}\right)\left({\frac {d}{d\tau }}y\left(\tau \right)\right){\frac {d}{d\tau }}z}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-{\frac {\left(g_{12}{\frac {\partial }{\partial t}}g_{12}-g_{11}{\frac {\partial }{\partial t}}g_{22}\right)\left({\frac {d}{d\tau }}y\right){\frac {d}{d\tau }}t}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}+1/2\,{\frac {\left(g_{12}{\frac {\partial }{\partial x}}g_{33}-g_{11}{\frac {\partial }{\partial y}}g_{33}\right)\left({\frac {d}{d\tau }}z\left(\tau \right)\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}-1/2\,{\frac {\left(-g_{12}{\frac {\partial }{\partial x}}g_{44}+g_{11}{\frac {\partial }{\partial y}}g_{44}\right)\left({\frac {d}{d\tau }}t\left(\tau \right)\right)^{2}}{g_{11}g_{22}-\left(g_{12}\right)^{2}}}=0}
