Stability of a black hole under the deformation of an extended periodic potential

Abstract

It has been shown that the spectrum of black hole quasi-normal modes is extremely sensitive to localized deformations of the geometry away from the potential peak. The deformation caused by a real astronomical environment will extend throughout the whole space rather than be localized in a finite region. Whether spatially extended fluctuations distributed throughout spacetime can produce similar effects remains unclear. In this work, we construct a toy model to investigate the case that the deformation of the metric is not localized but extends throughout the whole space. We study how it changes the ringdown stage of the black hole in the time domain. Using both the Pöschl-Teller and Regge-Wheeler potentials as representative backgrounds, we show that sufficiently wide spatially periodic perturbations of zero mean can trigger not only the ``spectral instability'' in the frequency domain but also black hole instabilities in the time domain. Through numerical analysis, we uncover a universal scaling relation for the instability threshold and further provide an analytical interpretation of its physical origin.