Question Number 197344 by MathedUp last updated on 14/Sep/23
$$\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\centerdot\frac{\mathrm{d}{x}^{\boldsymbol{\lambda}} }{\mathrm{d}{t}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{g}^{\boldsymbol{\lambda\alpha}} \left(\partial_{\boldsymbol{\mu}} ^{\:} \mathrm{g}_{\boldsymbol{\alpha\nu}} +\partial_{\boldsymbol{\nu}} ^{\:} \mathrm{g}_{\boldsymbol{\alpha\mu}} −\partial_{\boldsymbol{\alpha}} ^{\:} \mathrm{g}_{\boldsymbol{\mu\nu}} \right)\frac{\mathrm{d}{x}^{\boldsymbol{\mu}} }{\mathrm{d}{t}}\centerdot\frac{\mathrm{d}{x}^{\boldsymbol{\nu}} }{\mathrm{d}{t}}=\mathrm{0} \\ $$
Commented by TheHoneyCat last updated on 20/Sep/23
I'm guessing you're looking for a paramedic expression of a geodesic, but if you don't give us the metric, we can't really do much.