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 non-extremal 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 non-extremal 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 non-zero Love numbers corresponds to the non-axisymmetric dynamical tidal perturbations of Kerr black holes (to arbitrary order of angular momentum). We also demonstrate that the non-zero dynamical Love numbers, for both non-extremal and extremal Kerr black holes, get modified under the change of the sign of the spin-weight, for electromagnetic and gravitational tidal perturbations.