Static sphere observers and geodesics in Schwarzschild-de Sitter spacetime

We analyze null- and spacelike radial geodesics in Schwarzschild-de Sitter spacetime connecting two conjugate static sphere observers, i.e. free-falling observers at a fixed radius in between the two horizons. We explicitly determine the changes in the causal structure with respect to these natural observers as a result of the inward bending of the black hole singularity, as well as the outward bending of asymptotic infinity. Notably, the inward and outward bending changes as a function of the black hole mass, first increasing towards a maximum and then decreasing to vanish in the extreme Nariai limit. For a generic mass of the black hole this implies the existence of finite size (temporal) windows for the presence of symmetric radial geodesics between the static sphere observers probing the interior region of the black hole, as well as the exterior de Sitter region. We determine the size of the interior (black hole) and exterior (de Sitter) temporal windows in $4$, $5$ and $6$ spacetime dimensions, finding that they are equal in $D=5$, and compute the proper lengths of the symmetric radial geodesics. We comment on the implications for information exchange and the potential role of the symmetric radial geodesics in a geodesic approximation of static sphere correlators in Schwarzschild-de Sitter spacetime.