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Stability of black holes with non-minimally coupled scalar hair to the Einstein tensor

General relativity admits a plethora of exact compact object solutions. The augmentation of Einstein's action with non-minimal coupling terms leads to modified theories with rich structure, which, in turn, provide non-trivial solutions with intriguing phenomenology. Thus, assessing their viability under generic fluctuations is of utmost importance for gravity theories. We consider static and spherically-symmetric solutions of a Horndeski subclass which includes a massless scalar field non-minimally coupled to the Einstein tensor. Such theory possesses second-order field equations and admits an exact black hole solution with scalar hair. Here, we study the stability of such solution under axial gravitational perturbations and find that it is linearly stable. The qualitative features of the ringdown waveform depend solely on the ratio of the two available parameters of spacetime, namely the black hole mass $m$ and the non-minimal coupling strength $\ell_η$. Finally, we demonstrate the gravitational-wave ringdown transitions between three distinct patterns as the ratio $m/\ell_η$ increases; a state which is dominated by photon-sphere excitations and maintains a typical quasinormal ringdown, an intermediate long-lived state which exhibits gravitational-wave echoes and, finally, a state where the ringdown and echoes are depleted rapidly to turn to an exponential tail.