More on lensing by a cosmological constant

The question of whether or not the cosmological constant affects the bending of light around a concentrated mass has been the subject of some recent papers. We present here a simple, specific and transparent example where Lambda bending clearly takes place, and where it is clearly neither a coordinate effect nor an aberration effect. We then show that in some recent works using perturbation theory the Lambda contribution was missed because of initial too stringent smallness assumptions. Namely, our method has been to insert a Kottler (Schwarzschild with Lambda) vacuole into a Friedmann universe, and to calculate the total bending within the vacuole. We assume that no more bending occurs outside. It is important to observe that while the mass contribution to the bending takes place mainly quite near the lens, the Lambda bending continues throughout the vacuole. Thus, if one deliberately restricts one's search for Lambda bending to the immediate neighbourhood of the lens, one will not find it. Lastly, we show that the Lambda bending also follows from standard Weyl focusing, and so again, it cannot be a coordinate effect.