On the generalization of McVittie's model for an inhomogeneity in a
  cosmological spacetime
  McVittie's spacetime is a spherically symmetric solution to Einstein's
equation with an energy-momentum tensor of a perfect fluid. It describes the
external field of a single quasi-isolated object with vanishing electric charge
and angular momentum in an environment that asymptotically tends to a
Friedmann--Lemaitre--Robertson--Walker universe. We critically discuss some
recently proposed generalizations of this solution, in which radial matter
accretion as well as heat currents are allowed. We clarify the hitherto
unexplained constraints between these two generalizing aspects as being due to
a geometric property, here called spatial Ricci-isotropy, which forces
solutions covered by the McVittie ansatz to be rather special. We also clarify
other aspects of these solutions, like whether they include geometries which
are in the same conformal equivalence class as the exterior Schwarzschild
solution, which leads us to contradict some of the statements in the recent
literature.
