= Radial geodesic acceleration in Schwarzschild-de Sitter spacetime
{title2=$d^2r/d\tau^2=-GM/r^2+\Lambda r/3$}
For $ds^2=A\,dt^2-dr^2/A-r^2d\Omega^2$, $A=1-2GM/r-\Lambda r^2/3$, and $c=1$, a radial timelike <geodesic> has $\dot r^2=k^2-A$. The radial <geodesic equation> gives $\ddot r=-A'/2=-GM/r^2+\Lambda r/3$, including at a <turning point>. This is a second <derivative> of the <areal radius> with respect to <proper time>; the <geodesic>'s <four-acceleration> still vanishes. In the slow-motion weak-field limit, restoring units gives $d^2r/dt^2=-GM/r^2+\Lambda c^2r/3$.
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