= Timelike geodesics in a warped Poincaré patch
{title2=$e^{-y}=K\cos(s-s_0),\quad x^\mu=x_0^\mu+\frac{p^\mu}{K^2}\tan(s-s_0)$}
For unit-radius metric $dy^2+e^{-2y}\eta_{\mu\nu}dx^\mu dx^\nu$, translations conserve $p^\mu=e^{-2y}\dot x^\mu$. Timelike normalization gives $K^2=-\eta_{\mu\nu}p^\mu p^\nu>0$ and $\dot y^2=K^2e^{2y}-1$. 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.
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