# Geodesic equation

1. Apr 1, 2012

### rbwang1225

1. The problem statement, all variables and given/known data
If a general parameter $t=f(s)$ is used to parameterize a straight line in Euclidean space, then the geodesic equation takes the form $\frac{d^2u^i}{dt^2}+\Gamma^i_{jk}\frac{du^j}{dt}\frac{du^k}{dt}=h(s)\frac{du^i}{dt}$, where $h(s)=-\frac{d^2t}{ds^2}{(\frac{dt}{ds})}^{-2}$. Show that this reduces to the simple form $t=f(s)$ is used to parameterize a straight line in Euclidean space, then the geodesic equation takes the form $\frac{d^2u^i}{dt^2}+\Gamma^i_{jk}\frac{du^j}{dt}\frac{du^k}{dt}=0$ if and only if $t=As+B$, where$A, B$ are constants ($A$≠$0$)

3. The attempt at a solution
I can not prove the inverse statment, i.e., if the geodesic equation is of the form $\frac{d^2u^i}{dt^2}+\Gamma^i_{jk}\frac{du^j}{dt}\frac{du^k}{dt}=0$, then $t=As+B$.

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