= Geodesics of two-dimensional de Sitter spacetime
{title2=$X''=\varepsilon X,\quad \varepsilon=X'^2\in\{1,0,-1\}$}
A <geodesic> on the embedded <two-dimensional de Sitter spacetime> is the intersection with a two-plane through the ambient origin. For initial point $P$ and tangent $V$ satisfying $P^2=-1$, $P\cdot V=0$, $V^2=\varepsilon$, its affine parametrization is $P\cosh s+V\sinh s$ when timelike, $P+sV$ when null and $P\cos s+V\sin s$ when spacelike. These solve $X''=\varepsilon X$. The <spacelike geodesics> are closed; <timelike geodesics> and <null geodesics> extend to both conformal infinities with infinite affine length.
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