Nereid,

Wonderful. Yes, indeedy do, I'm enjoying that. Here is the earlier paper by Rindler and Ishak showing that Lamda does indeed contribute to the observered gravitational bending of light:

http://arxiv.org/abs/0709.2948

This is correcting a previous notion that lamba would have no effect on the bending angle. I'm really enjoying this because it is a perfect example of how coordinates can trip you up in our favorite metric theory of gravity. That still gets even the high priests.... This is a classic example of that. *Classic*.

Consider Schwarzschild. What is that? That is the space-time of a spherically symmetric mass, and the limit of that as that mass goes to a "point". It comes from plopping down a mass into flat space-time. What do you get with zero source terms in the EFE? You get Mr. Minkowski's metric, the playground of SR.

So Schwarzschild is basically dropping a "point mass" (massenpunkt as Schwarzschild himself called it) against a flat Minkowski background.

What if the background weren't Minkowski? Well, if you add Lamba to the EFE, and still set the mass source to zero, you get something different from Minkowski. That's the deSitter space-time, an expanding space-time, although it has a *static* form, which blows your mind. That's the simplest "lamba vacuum" you can have. How can expanding space have a static form. That is coordinates for you! That metric simply says that if you drop something, it flies away from you, and the farther it gets, the faster it goes, until it gets to a cosmological horizon thingy. However, it doesn't see anything funny, it just sees you flying away and thinks you hit a cosmological horizon. Your own ruler and clock's notion of space-time is completely static. Stuff just tends to fly away from you in all directions if you drop it.

Now, what happens when you drop a massenpunkt down in that thing? That's the Schwarzschild-deSitter space-time. Pondering that thing, you'll notice it looks very similiar to the original. The magic metric factor there, 1 - R/r, gets another term on it. -Lr^2/3, where L is lamba.

Now, note that gives a local maximum to clock rates, which then drop off to zero again at some high r. But it still goes to zero at R, just like the original. That means a black hole still has the same event horizon, even in a Lamba vacuum! A black hole remains unperturbed by Lamba (but that doesn't hold for "dark energy" type models that expand "harder than lamba", mind you).

However, its influence is opposed. Get far enough away, right where the magic metric factor is maximum, and you are at a balance point. Stay there, and you stay a constant notion of distance away from the black hole.

Note this is a very different space-time from Schwarzschild alone. That sucker is asymptotically flat. This thing is NOT. Get far enough away, and you hit another horizon, that cosmological one. Schwarzschild coordinates correspond to the ruler and clock of an observer, stationary to the massenpunkt, but infinitely far away.

There ain't no such observer here! So just whose coordinates are these, anyway? Well, you can use coordinates that don't correspond to any observer, actually. These coordiantes, this r, theta, phi and t, are sort of imaging an inertial observer in deSitter space-time, giving him some spherical coordinates, then plopping the mass down at his r = 0. Don't hold me to this, because I just know enough to be dangerous, but I think those coordinates will correspond to the local ruler and clock of an observer sitting right at the balance point.

Now, the error everyone made apparently in thinking Lamba didn't matter for gravitational lensing was assuming these coordinates were just like Schwarschild, and applied to someone "very far away". They don't at all.

It turns out that lamda cancelled for null geodesics in a cute way, so the path of light was exactly the same as it is in Schwarszchild. So, the far away observer should see exactly the same thing.

That was wrong, because those far away observers have very different ruler and clocks.

We can thank Rindler for straightening that out, among his many other accomplishments.

-Richard