Grant,

There is an indeed an exact solution of a "point mass" against a deSitter (lambda, expanding universe) background, one metric called oddly enough the Schwarzschild-deSitter (SdS) metric. The coordinates here at that of the static deSitter metric with the "black hole" plopped down at r = 0. And I stress this is not in any comoving form, so the typical language doesn't work there.

And another tricky thing is the coordinates don't correspond to any observer and that trips a lot of people up. In static deSitter alone, r = 0 is the location of a co-mover coordinatizing with his own tanget ruler. But in Sds, r = 0 is the singularity. In Schwarzschild alone, the coordinates correspond to an observer at r = infinity. But in SdS, the local frame of the far away observer is not the Sds coordinates.

But it's an r with a clock that you can sort of see how it works, and can certainly plot the geodesics and everything else SdS is nicely static and perfectly spherically symmetrical itself. The magic metric factor neatly is just the sum of deSitter and Schwarzschild:

g_00 = 1/g_11 = (1 - R/r - k*r^2)

Here R is the normal Schwarzschild --BlackShield, I love that -- radius. So Lambda interestingly does not change the event horizon (that holds only for lambda -- other types of "dark energy" models, such as the Big Rip stuff, tha behave differently that "pure" lamdba do indeed perturb the event horizon).

k is the same deSitter alone factor that depends on lambda. IOW, SdS just simply sums the two terms from both. These coordinates sandwich us between two event horizons. There is the regular one at r = R, and then the cosmological horizon at k*r^2 = 1.

There is a balance point, call r0, at the maximum of the g_00 which is the maximum stationary clock rate, where a test particle would hover stationary, with the gravity of the point mass and lambda just cancelling.

There is another interesting result. The lambda part completely cancels for the null geodesics, and they are exactly the same as Schwarzschild alone. This had led many to *erroneously* conclude that the cosmological constant has no effect on gravitational lensing. Wolfgang Rindler, of Rindler metric fame, wrote some recent papers about this. It's a simple coordinate error by assuming the local ruler and clock of a "far away" observer correspond to the coordinates as they do with plain Schwarzschild. They don't here -- a far away observer is different, very different if r is approaching the cosmological horizon, and the lamdba does indeed affect the observed lensing.

Rindler went on to show that distant cluster lensing results were consistent with this effect for the currently accepted value of Lambda.

-Richard