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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?

Qmechanic
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Sean E. Lake
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  • Possible duplicate of (I ask for more details, though): https://physics.stackexchange.com/questions/238721/vibrating-black-holes?rq=1 – Sean E. Lake Nov 13 '20 at 01:38
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    This paper considers the scattering of gravitational waves by a Schwarzschild hole. Unfortunately, it is behind a paywall. My guess is that the effect is most significant when the wavelength of the wave is comparable to the Schwarzschild radius, but I’m not sure of that. – G. Smith Nov 13 '20 at 01:53

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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.)

TimRias
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  • To complete the answer: the photon sphere has radius $r = \frac{3GM}{c^2}$, thus the angular frequency is $\frac{c^3}{3GM}$. For reference, at 10 solar masses, this gives about $6800,\text{rad},\text{s}^{-1}$, and $0.02,\text{rad},\text{s}^{-1}$ for the supermassive black hole at the Milky Way's center. – Sean E. Lake Nov 13 '20 at 08:43
  • @SeanE.Lake That is for a non-spinning black hole. More generally the radius of the photon sphere depends on the spin (and inclination of the orbit). – TimRias Nov 13 '20 at 09:02