Schwarzschild-de Sitter spacetime

The Schwarzschild-de Sitter spacetime is the spherically symmetric vacuum solution with positive cosmological constant. In static coordinates its metric function is $f(r)=1-2m/r-H^2{r}^2$, with a black-hole horizon and a cosmological horizon when $mH<1/\sqrt{27}$.

For $d{s}^2=Ad{t}^2-d{r}^2/A-r^{2}d{\Omega }^2$, $A=1-2GM/r-\Lambda r^2/3$, and $c=1$, a radial timelike geodesic has ${\dot{r}}^2=k^2-A$. The radial geodesic equation gives $\ddot{r}=-A^{\prime }/2=-GM/r^2+\Lambda r/3$, 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 $d^{2}r/d{t}^2=-GM/r^2+\Lambda c^{2}r/3$.