Assume (for the sake of simplicity) a Schwarzschild black hole (non-rotating, non charged). This black hole has a photon sphere in $r=1.5r_s$, where photons may travel in a circular orbit. Will a gravitational wave propagating tangentially on the boundary of this orbit possibly be caught in a stable or an unstable orbit around the black hole?
I would naively assume the answer is yes, as both light and gravitational waves are supposed to behave the same (namely, propagate along null geodesics).
If so, will such a rotating gravitational wave, a continuously accelerated energy density, produce by itself a gravitational wave (albeit weaker by many magnitudes)? The same question stands for the photons in the photon sphere- will they emit gravitational radiation?
In short: Can a gravitational wave be caught in orbit around a black hole? Will it emit gravitational radiation?