 Timelike geodesics in a warped Poincaré patch

For unit-radius metric $d{y}^2+e^{-2y}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$, translations conserve $p^{\mu }=e^{-2y}{\dot{x}}^{\mu }$. Timelike normalization gives $K^2=-{\eta }_{\mu \nu }{p}^{\mu }{p}^{\nu }>0$ and ${\dot{y}}^2=K^2{e}^{2y}-1$. Setting $w=e^{-y}$ yields ${\dot{w}}^2+w^2=K^2$, hence the displayed proper-time solution where the cosine is positive. Its two patch-horizon crossings are separated by proper time $\pi$, although the globally extended geodesic is complete.