Response of a Kerr black hole to a generic tidal perturbation

We derive the response, the real part of which provides the tidal Love numbers, for nonextremal as well as extremal Kerr black holes under generic tidal perturbations. Our results suggest that the static as well as dynamical (linear-in-frequency) Love numbers vanish for both Schwarzschild and slowly rotating (linear-in-angular momentum) Kerr black holes, under generic perturbations. The vanishing of static and dynamical Love numbers also holds for axisymmetric tidal perturbations of nonextremal and extremal Kerr black holes. In fact, even under generic tidal perturbations, the static Love numbers of Kerr black holes vanish identically. The only case with nonzero Love numbers corresponds to the nonaxisymmetric dynamical tidal perturbations of Kerr black holes (to arbitrary order of angular momentum). We also demonstrate that the nonzero dynamical Love numbers, for both nonextremal and extremal Kerr black holes, get modified under the change of the sign of the spin weight, for electromagnetic and gravitational tidal perturbations.