More on McVittie's Legacy: A Schwarzschild - de Sitter black and white
  hole embedded in an asymptotically $\Lambda$CDM cosmology
  Recently Kaloper, Kleban and Martin reexamined the McVittie solution and
argued, contrary to a very widely held belief, that the solution contains a
black hole in an expanding universe. Here we corroborate their main conclusion
but go on to examine, in some detail, a specific solution that asymptotes to
the $\Lambda$CDM cosmology. We show that part of the boundary of the solution
contains the inner bifurcation two - sphere of the Schwarzschild - de Sitter
spacetime and so both the black and white hole horizons together form a partial
boundary of this McVittie solution. We go on to show that the null and weak
energy conditions are satisfied and that the dominant energy condition is
satisfied almost everywhere in the solution. The solution is understood here by
way of a systematic construction of a conformal diagram based on detailed
numerical integrations of the null geodesic equations. We find that the
McVittie solution admits a degenerate limit in which the bifurcation two -
sphere disappears. For solutions with zero cosmological constant, we find no
evidence for the development of a weak null singularity. Rather, we find that
in this case there is either a black hole to the future of an initial
singularity or a white hole to its past.
