Using the coordinates $\alpha=\log \frac{1+r}{1-r}$ and $\theta$ where $(r,\theta)$ are the usual polar coordinates, show that the segment of the y axis between $(0,0)$ and $(0,r)$ where $0<r<1$ is the curve of shortest length between the two points.
Note: I have calculated the metric components in the new coordinates to be: $g_{11}=1$, $g_{12}=0$ and $g_{22}=\sinh^2\alpha$. Is the best way to proceed via the standard Geodesic Equation? Is it sufficient to show the existence of a straight line geodesic passing through this point and then claim that it is unique?