It is known that the Poincare disc $\mathbb D=\{z:|z|<1\}$ with with the usual Riemannian metric $\frac{\mathrm dz^2}{(1-|z|^2)^2}$ is geodesically complete. Also for any $0\neq x\in\mathbb D\cap\mathbb R$ $\gamma_0(t)=tx$ is a geodesic joining $0$ and $x$. Can any one show explicitly how the corresponding maximal geodesic extends to $\mathbb R$?
It seems that from the above discussion that $\gamma_0(t)=t$ is a geodesic which reaches the boundary in finite time! What is going wrong here?