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General relativity admits a plethora of exact compact object solutions. Theaugmentation of Einstein's action with non-minimal coupling terms leads tomodified theories with rich structure, which, in turn, provide non-trivialsolutions with intriguing phenomenology. Thus, assessing their viability undergeneric fluctuations is of utmost importance for gravity theories. We considerstatic and spherically-symmetric solutions of a Horndeski subclass whichincludes a massless scalar field non-minimally coupled to the Einstein tensor.Such theory possesses second-order field equations and admits an exact blackhole solution with scalar hair. Here, we study the stability of such solutionunder axial gravitational perturbations and find that it is linearly stable.The qualitative features of the ringdown waveform depend solely on the ratio ofthe two available parameters of spacetime, namely the black hole mass $m$ andthe non-minimal coupling strength $\ell_\eta$. Finally, we demonstrate thegravitational-wave ringdown transitions between three distinct patterns as theratio $m/\ell_\eta$ increases; a state which is dominated by photon-sphereexcitations and maintains a typical quasinormal ringdown, an intermediatelong-lived state which exhibits gravitational-wave echoes and, finally, a statewhere the ringdown and echoes are depleted rapidly to turn to an exponentialtail.

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