Buzz Bloom said:
If we assume
ΩR = ΩM = Ωk = 0
and
ΩΛ=1,
then
I assume this means you're interested in a case where there is no matter, no radiation, but only a cosmological constant.
This would be represented by the de-Sitter metric. With the addition of matter, so that you have a massive body in a space-time with a cosmological constant and an accelerating expansion, you'd have the De-Sitter Schwarzschild metric,
https://en.wikipedia.org/wiki/De_Sitter–Schwarzschild_metric, henceforth the DSS metric.
This would be a simpler alternative (or more precisely, one I happen to be familiar with) to using the McVitte metric, which is interesting, but I'm not familiar with it.
Using this DSS metric, we can get the answer without any questionable (and probably wrong) assumptions that GR is like Newton's theory.
We can find the r coordinate in this metric where an object stays stationary so that it's r coordinate does not change with time.
Specifically, looking at the Wiki page
https://en.wikipedia.org/w/index.php?title=De_Sitter–Schwarzschild_metric&oldid=959698099#Metric
we have the geodesic equation, and we want to solve for r=constant, which implies that ##\dot{r} = \ddot{r} = 0## and also ##\dot{\theta} = \dot{\phi} = 0##.
The geodesic equation has a solution of this form when ##f'(r)=0##, which implies
$$r = \left( \frac{a}{b} \right)^{\frac{1}{3}}$$
From the wiki, we have
The two parameters a and b give the black hole mass and the cosmological constant respectively.
So the a is proportional to the black hole mass, and b is proportional to the cosmological constant.
Note that the coordinates used for the DSS metric are not necessarily the same as the cosmological coordinates. There is surely some relationship between the two sets of coordinates, but I havean't seen any detailed exposition of what it is. However, we can make useful qualitative statements without the exact knowledge.
The qualitative statement we can make is that there is some critical radius r. At this critical radius, particles in geodesic motion (free fall) exist which do not get further away or closer to the black hole. Below the critical radius, particles in geodesic motion fall inwards, outside the critical radius they accelerate outwards.
There is an intuitive way of looking at this, involving the quantity ##\rho + 3P##. In the de-Sitter space-time, ##\rho## is positive and P = ##-\rho##. So the quantity ##\rho + 3P## is negative. It's a sort of mass density, called the Komar mass, that applies to any stationary space-times. The De-Sitter and the Schwarzschild De-sitter metric are both examples of stationary space-times, as can be seen by the fact that there is no time dependence of the metric coefficients. The more general FRW space-time is not stationary, unfortunately.
My intuitive way of looking at this is that the balance point where r=constant occurs when the positive mass of the black hole is balanced out by the effect of empty space, which has an effective negative mass due to the pressure term in the cosmological constant.
So the gravitational effect of the negative mass cancels out the positive mass at some specific radius, and you have a net zero effect at this particular radius. Inside this critical radius, particles will fall towards the black hole. Outside this critical radius, particles will accelerate away from the black hole.
Note that for this argument to work, we need to rely on the fact that the gravitational effect inside a hollow sphere is zero, which justifies why we only look at the contributions from the "mass" inside the sphere of radius r.
Like all intuitions, this somewhat vague description is a bit suspect, and I can't quote the exact source that gave me this idea, though it was a paper studying the DSS space-time. Nor can I say that my memory is necessarily good on this point.
While the provenance of this intuitive explanation is suspect, I thought it was useful enough an idea to share, even though it is much less rigorous than the argument I gave using the solution of the geodesic equation.
