Geodesic Equation for Straight Lines in Euclidean Space

[rbwang1225]

Homework Statement

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 ($A$≠$0$)

The Attempt at a Solution

I can not prove the inverse statement, 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 $t=As+B$.

 

[jbsteele]

However, I can prove the forward direction. If $t=As+B$, then $\frac{dt}{ds}=A$ and $\frac{d^2t}{ds^2}=0$. Thus, $h(s)=-\frac{d^2t}{ds^2}{(\frac{dt}{ds})}^{-2}=0$, and the geodesic equation reduces to the desired form.