Recently, Dr. H.E. Dowdye pointed me to his web page http://www.extinctionshift.com/SignificantFindings08.htm where he gives examples of astrophysical objects where gravitational lensing should be expected but is not observed (I noticed that he posted the link also in this forum a while ago, but I didn't want to revive a thread almost two years old).
The case of the Sombrero nebula addressed on this page is not really that conclusive in my opinion as the effect should be too small to be obvious, but the stars near the supermassive object at the centre of our galaxy should be dramatically affected by the corresponding gravitational lensing:
the stars are orbiting the supermassive object (which has a mass of 4*10^6 solar masses) within a distance of about 10 light days, which corresponds to a distance of 2.6*10^11 km or 3.7*10^5 solar radii. This means that the gravitational deflection for these stars should be 4*10^6/ 3.7*10^5 /2 = 5 times larger than the gravitational deflection near the sun's limb i.e. about 9" (arcseconds) (the additional factor 1/2 in the ratio comes from the fact that the light is produced within the gravitational field and does not come from outside like for the solar case). Since 10 light days at a distance of 26,000 light years corresponds to an angle of 0.2" (see also http://www.astrophysicsspectator.com/tables/MilkyWayCentralStars.html ). This means that, according to GR, we should see the stars actually at a distance 45 times further from the galactic centre than they appear to be. This clearly falsifies not only GR, but they idea of light being affected by masses altogether.
I just wonder whether anybody knows if this point has been addressed somewhere within the framework of GR. Or has this just been ignored so far? Despite a lot of searching on the web, I could not find any reference to this at all.
Just an update and correction on this:
I did recently a more thorough derivation of the gravitational lensing effect for objects close to the lensing mass, and it turned out that my estimate above is actually incorrect: the value I gave holds strictly speaking only if the if the lensed star is far behind the lensing mass (as compared to our distance from the latter). If it is much closer on the other hand, it has to be multiplied by a factor r/d where r is the distance of the lensed star from the galactic center and d our distance from the latter (this is straightforward to show by using the cosine and sine laws for the corresponding triangle).
With this, the lensing effect on the stars surrounding the galactic center mass becomes obviously much too small to be observable, but it still leaves the possibility that the lensing (or its absence) could be observed for stars a suitable distance behind the galactic center (i.e. far enough to get a large enough lensing effect, and close enough to have sufficiently short orbital periods so that the effect can be observed within a reasonable time period (e.g. a few years)).