A note on geodesics in inhomogeneous expanding spacetimes

There are several solutions of Einstein field equations that describe an inhomogeneity in an expanding universe. Among these solutions, the McVittie metric and its generalizations have been investigated through decades, though a full understanding of them is still lacking. In this note, we explore the trajectories of photons and massive particles in generalized McVittie spacetimes. In the case of massless particles, we show that no circular orbits are possible for those models that admit cosmological singularities. We also analyze the trajectory of particles for a specific generalized McVittie spacetime that is conformal to the Schwarzschild metric. By integrating the equations of motion in the Newtonian approximation, we show that particles behave in quite distinctive ways in different cosmological black hole solutions. We conclude that the analysis of the geodetic motion in inhomogeneous expanding metrics can help to discriminate those solutions that represent real cosmological black holes in the universe.