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 \(\ell _\eta \). Finally, we demonstrate the gravitational-wave ringdown transitions between three distinct patterns as the ratio \(m/\ell _\eta \) 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.

广义相对论承认过多的精确紧凑对象解决方案。爱因斯坦作用与非最小耦合项的增强导致了具有丰富结构的修正理论，这反过来又提供了具有有趣现象学的非平凡解决方案。因此，评估它们在一般波动下的生存能力对于引力理论至关重要。我们考虑 Horndeski 子类的静态和球对称解，其中包括与爱因斯坦张量非最小耦合的无质量标量场。这种理论具有二阶场方程，并承认具有标量头发的精确黑洞解。在这里，我们研究了这种解在轴向引力扰动下的稳定性，发现它是线性稳定的。m和非最小耦合强度\(\ell _\eta \)。最后，我们展示了随着比率\(m/\ell _\eta \)的增加，三种不同模式之间的引力波衰荡跃迁；一种以光子球激发为主并保持典型的准正常衰荡的状态，一种表现出引力波回波的中间长寿命状态，最后是一种衰荡和回波迅速耗尽以变成指数尾的状态。