The effect of a massive object on an expanding universe
  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.
