A tetrad-based procedure is presented for solving Einstein's field equations for spherically-symmetric systems; this approach was first discussed by Lasenby et al. in the language of geometric algebra. The method is used to derive metrics describing a point mass in a spatially-flat, open and closed expanding universe respectively. In the spatially-flat case, a simple coordinate transformation relates the metric to the corresponding one derived by McVittie. Nonetheless, our use of non-comoving (`physical') coordinates greatly facilitates physical interpretation. For the open and closed universes, our metrics describe different spacetimes to the corresponding McVittie metrics and we believe the latter to be incorrect. In the closed case, our metric possesses an image mass at the antipodal point of the universe. We calculate the geodesic equations for the spatially-flat metric and interpret them. For radial motion in the Newtonian limit, the force acting on a test particle consists of the usual $1/r^2$ inwards component due to the central mass and a cosmological component proportional to $r$ that is directed outwards (inwards) when the expansion of the universe is accelerating (decelerating). For the standard $\Lambda$CDM concordance cosmology, the cosmological force reverses direction at about $z\approx 0.67$. We also derive an invariant fully general-relativistic expression, valid for arbitrary spherically-symmetric systems, for the force required to hold a test particle at rest relative to the central point mass.Comment: 14 pages, 2 tables, 5 figures; new version, to match the version published in MNRA

The bizarre behaviour of the apparent (black hole and cosmological) horizons of the McVittie spacetime is discussed using, as an analogy, the Schwarzschild-de Sitter-Kottler spacetime (which is a special case of McVittie anyway). For a dust-dominated "background" universe, a black hole cannot exist at early times because its (apparent) horizon would be larger than the cosmological (apparent) horizon. A phantom-dominated "background" universe causes this situation, and the horizon behaviour, to be time-reversed.

We present some astrophysical consequences of the metric for a point mass in an expanding universe derived in Nandra, Lasenby & Hobson, and of the associated invariant expression for the force required to keep a test particle at rest relative to the central mass. We focus on the effect of an expanding universe on massive objects on the scale of galaxies and clusters. Using Newtonian and general-relativistic approaches, we identify two important time-dependent physical radii for such objects when the cosmological expansion is accelerating. The first radius, $r_F$, is that at which the total radial force on a test particle is zero, which is also the radius of the largest possible circular orbit about the central mass $m$ and where the gas pressure and its gradient vanish. The second radius, $r_S$, which is \approx r_F/1.6$, is that of the largest possible stable circular orbit, which we interpret as the theoretical maximum size for an object of mass $m$. In contrast, for a decelerating cosmological expansion, no such finite radii exist. Assuming a cosmological expansion consistent with a $\Lambda$CDM concordance model, at the present epoch we find that these radii put a sensible constraint on the typical sizes of both galaxies and clusters at low redshift. For galaxies, we also find that these radii agree closely with zeroes in the radial velocity field in the neighbourhood of nearby galaxies, as inferred by Peirani & Pacheco from recent observations of stellar velocities. We then consider the future effect on massive objects of an accelerating cosmological expansion driven by phantom energy, for which the universe is predicted to end in a `Big Rip' at a finite time in the future at which the scale factor becomes singular. In particular, we present a novel calculation of the time prior to the Big Rip that an object of a given mass and size will become gravitationally unbound.Comment: 16 pages, 5 tables, 6 figures; new version, to match the version published in MNRA

We present Newtonian and fully general-relativistic solutions for the evolution of a spherical region of uniform interior density ρi(t), embedded in a background of uniform exterior density ρe(t). In both regions, the fluid is assumed to support pressure. In general, the expansion rates of the two regions, expressed in terms of interior and exterior Hubble parameters Hi(t) and He(t), respectively, are independent. We consider in detail two special cases: an object with a static boundary, Hi(t) = 0; and an object whose internal Hubble parameter matches that of the background, Hi(t) = He(t).In the latter case, we also obtain fully general-relativistic expressions for the force required to keep a test particle at rest inside the object, and that required to keep a test particle on the moving boundary. We also derive a generalised form of the Oppenheimer-Volkov equation, valid for general time-dependent spherically-symmetric systems, which may be of interest in its own right.

Abstract.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.