Analytical Criteria for Black Hole Instability in Non-Minimally Coupled Scalar-Tensor Theories
In this paper, we show that the robustness of black hole stability is not preserved when perturbations are subjected to critical values of the coupling constant in two non-minimally coupled scalar-tensor models, particularly for several regular black holes. Our main result is the derivation of a general near-horizon analytical criterion for the onset of instability, which yields analytical expressions for the critical coupling constants in both coupling models. Numerical time-domain analysis shows that these critical values are coincident with the threshold points of the field perturbation profiles. At the critical coupling, the effective potential of the Regge-Wheeler equation develops an extremum exactly at the location of event horizon. We further show that, in the near-horizon regime, the real parts of the quasi-normal frequencies vanish, giving rise to purely imaginary modes. Finally, we recover the universal area quantization of spherically symmetric black holes at the critical coupling without resorting to the highly damped-mode approximation, and show that the result is independent of the particular non-minimal coupling model.