Interplay between cosmological expansion and massive objects

We have derived a metric for a point mass in an expanding universe. In the spatially flat case, a simple coordinate transformation relates our metric to that derived by McVittie. Nonetheless, our use of non-comoving (physical) coordinates greatly facilitates physical interpretation. We have also derived a coordinate-free expression for the force required to keep a particle at rest in this spacetime. For redshift z > 0.67, we have identified two important time-dependent physical radii; the largest possible circular orbit rF(t), and the largest stable circular orbit rS(t), which lies inside rF(t), either of which could be interpreted the egde of an object of mass m. In the case of a dynamical background dominated by a phantom fluid, we can use our force expression to predict the time of the 'Big Rip' when the universe ends. For open and closed universes, our metrics describe different spacetimes to McVittie's metrics; we believe the latter to be incorrect. In the closed case, our metric possesses an image mass at the antipodal point of the universe.