Do black holes resonate? Merging black holes undergo a phenomenon called "ring-down" after the merger. If a black hole were to be perturbed by a hypothetical gravitational wave, would it exhibit a resonance type phenomenon that depends on the frequency? If so, how would that frequency be related to the black hole mass?
 A: Yes, black holes resonate when perturbed by gravitational waves (or otherwise). The characteristic frequencies with which a black hole does this are called the quasinormal modes (QNMs) of the black hole and are determine by the mass and angular momentum (and charge) of the black hole.
These frequencies are closely linked to the lightring/photonsphere, a set of unstable circular lightlike orbits around a black hole. To first approximation the fundamental QNMs are given by the frequency with which a "photon" would circle on this orbit, while the decay time of the mode is related to the Lyapunov exponent of the lightring.
Because the frequency of the lightring is higher than of any stable circular orbit, the gravitational waves generated during the quasicircular inspiral of a binary black hole cannot excite any of the QNMs of the black holes. However, it has recently been shown that if the binary is eccentric some of the QNMs maybe excited the during the inspiral. (However, since the decay time of the QNMs is typically short compared to the orbital period, there is no time for substantial "resonant" effects to accumulate.)
