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.
