If you consider empty space case (energy-momentum tensor is zero), the curvature of space is Ricci-flat but the Riemann tensor is not necessarily zero, which allows for gravity waves in empty space. It's not difficult to see that the waves would have to be transverse (analogous to EM waves). What I've been trying to wrap my head around is whether this necessarily implies that there is a repulsive component to the wave. In other words (obvious oversimplification here), if you think of the trough of the wave as regular attractive gravity as we know it, is there a necessary implication that the peak of the wave is repulsive? Thoughts please?