In essence I want to know if GW can affect the momentum of baryonic matter. I know that massive objects like planets, stars, galaxies, etc will not be "ripped apart" or even disturbed by even the most massive GW from a SMBH major merger. But what about gas in the ISM or IGM? Is there any way a particle, subatomic or larger, can be "carried" away by GW as they radiate outward? Any sources or resources that show the mathematical description would also be appreciated.
The classic thought experiment showing that GW carries real energy is the "sticky bead argument" due to Feynman. Two beads on a rigid rod will be moved relative to each other and the rod by the wave as it passes, and if there is friction some energy will be released.
The question of how much energy there is in a wave is a bit tricky, since energy is not well-defined in spacetimes lacking time translation symmetry, and the weak field limit is approximate anyway. See Carroll p.160- for a discussion and math.
The field doesn't give particles net acceleration, but rather makes them oscillate relative to each other. That means gas and dust will not absorb any energy (since they do not touch each other), but extended rigid bodies can turn some of that energy into mechanical vibrations. They still don't get any momentum, though.