My paper on the subject discusses how the standard synchronisation convention (I argue that it is indeed a mere convention) becomes less useful in situations with a closed topology. This is relevant not only for spatially closed Minkowski spacetimes but also for realistic situations, e.g. on the rim of a rotating disk. My arguments explain in a natural way how the Sagnac effect can be consistent with special relativity.
The lens effect by moving object shows some surprising properties that seem
to contradict common sense at first glance.
One aspect is that radial motion in the same direction as the light
propagation does decrease the light deflection, even though
the "interaction time" seems to be less than for a lens at rest. For
slow test particles, the effect is indeed the opposite. This also implies that
there is a certain critical speed (c/sqrt 3) where the speed of the lens has
For transversal motion of the lens, an additional gravitational redshift is introduced. I explain this effect in the much simplified picture of an elastic collision of photon and lens. The very complicated calculations others used to describe this effect are in fact not necessary.
In a recent publication, my results on the effect of radial motion were disputed. The situation discussed in that paper, however, is not equivalent to the one we had discussed, because they are described in different reference frames. Once the proper translation is applied, the new publication does in fact confirm my result. A conference poster on the topic explains the differences and shows how the concept of angular diameter distances can be generalised to be measured between different reference frames that are moving with respect to each other. This leads to a very clean derivation of the effects of radial motion. A refereed publication was submitted recently.
The main problem is that in order to measure the speed of gravity, one needs a test theory in which this speed is a free parameter but which otherwise is in agreement with general relativity as much as possible. In my opinion their test theory is not the most plausible one. In fact it has not been investigated properly what other deviations from GR would be expected in this theory. If we allow only test theories that obey Lorentz invariance (no preferred frames), then it can be shown easily (by describing the experiment in the reference frame of Jupiter) that the experiment is not sensitive to the speed of gravity it all. The speed of gravity can only be modified if this is at the same time also done with the speed of light. One may thus argue that the experiment is just a complicated way to measure the speed of light. But this is still a matter of discussion.
It has been argued that gravitational lensing may have focusing effects on quasi-static gravitational fields in a similar way as it focuses light. My calculations do not confirm this view. Instead I show that static fields are only affected locally (without any long-range focusing). The situation is different for gravitational waves with wavelengths smaller than the typical scale of the lens. Gravitational lenses can thus focus gravitational waves.