Interpretation of the Raychaudhuri equation and the attractive nature of gravity
In many books it is stated that the Raychaudhuri equation is a sort of "proof" that in general relativity gravity is attractive.
I would be interested in seeing a quote from a book that actually says this. It just sounds wrong to me. A more reasonable statement would be the following. Many forms of matter that we know of (but not dark energy) obey certain energy conditions. One of these energy conditions is the strong energy condition (SEC), which basically says that gravity is attractive. General relativity does not claim that gravity is always attractive, and in fact we now know from cosmological observations that gravity is not always attractive.
The Raychaudhuri equation describes the result of the SEC. For example, if the SEC holds, then the Raychaudhuri equation says that cosmological expansion must always decelerate, never accelerate. And because the SEC is false for dark energy, cosmological expansion is actually currently accelerating.
Or, if it doesn't, doesn't this example show that gravity can be attractive and yet have a diverging effect?
Both the strong energy condition and the Raychaudhuri equation describe a parcel of test particles in three dimensions. You can't probe that with only two test particles.