Schwarzschild half-plane and its geodesics

Question

For some fixed $r_0>0,$ put the semi-Riemannian metric $$ ds^2=\frac{r_0-r}{r}dt^2+\frac{r}{r-r_0}dr^2 $$ on $\{(t,r)\in\mathbb{R}^2:r>r_0\}.$ I would like to show that the $r$-lines are always geodesics. My problem is I don't know the definition of an $r$-line. I've computed all the Christoffel symbols (and I'm sure my computations are correct) and assumed that an $r$-line is simply a vertical line but it doesn't satisfy the geodesic differential equations. I've even assumed that an $r$-line is horizontal line but I failed again in showing that those lines satisfy the geodesic differential equations. Could someone please tell me what I'm doing wrong and/or what the definition of an $r$-line is? Thank you.