The spectrum and dynamical instability, as well as the transient effect of the tensor perturbation for the so-called Maeda–Dadhich black hole, a type of Kaluza–Klein black hole, in Einstein–Gauss–Bonnet gravity have been investigated in framework of pseudospectrum. We cast the problem of solving quasinormal modes (QNMs) in AdS-like spacetime as the linear evolution problem of the non-normal operator in null slicing by using ingoing Eddington–Finkelstein coordinates. In terms of spectrum instability, based on the generalized eigenvalue problem, the QNM spectrum and ε-pseudospectrum has been studied, while the open structure of ε-pseudospectrum caused by the non-normality of operator indicates the spectrum instability. In terms of dynamical instability, we introduce the concept of the distance to dynamical instability, which plays a crucial role in bridging the spectrum instability and the dynamical instability. We calculate such distance, named the complex stability radius, as parameters vary. Finally, we show the behavior of the energy norm of the evolution operator, which can be roughly reflected by the three kinds of abscissas in context of pseudospectrum, and find the transient growth of the energy norm of the evolution operator.

中文翻译：

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爱因斯坦-高斯-邦内引力中 Kaluza-Klein 黑洞的伪光谱和瞬态


在爱因斯坦-高斯-邦内引力的框架中，所谓的前田-达迪奇黑洞（一种卡鲁扎-克莱因黑洞）的光谱和动力学不稳定性，以及张量扰动的瞬态效应已经进行了研究。我们利用传入的 Eddington-Finkelstein 坐标，将求解类 AdS 时空准正态模态 （QNM） 的问题转化为零切片中非正规算子的线性演化问题。在谱不稳定性方面，基于广义特征值问题，研究了 QNM 谱和 ε-pseudospectrum ，而算子的非正态性引起的 ε-pseudospectrum 的开放结构表明了谱的不稳定性。在动力学不稳定性方面，我们引入了动力学不稳定性距离的概念，它在弥合频谱不稳定性和动力学不稳定性方面起着至关重要的作用。我们计算这样的距离，称为复稳定半径，因为参数会发生变化。最后，我们展示了进化算子的能量范数的行为，在伪谱的背景下，这可以大致反映出三种横坐标，并找到进化算子能量范数的瞬态增长。