Gravitational Bound State Perturbations Inside Black Holes and Isospectrality

Abstract: We study the bound state solutions for the polar perturbations in the interior of the Schwarzschild black hole. It is shown that for a given value of the spherical harmonic index $\ell$, there are a total of $\ell-1$ bound states for polar perturbations. We show both analytically and numerically that the spectrum of $\ell-2$ of these perturbations agrees identically with the spectrum of axial perturbations. Correspondingly, the isospectrality between the bound states of the axial and polar perturbations in the interior of the black hole does hold. This is significant since the effective potential is singular at the center of the black hole, and the existence of the isospectrality is far from obvious. Furthermore, we demonstrate that the additional mode found in the spectrum of polar perturbations is the algebraically special mode, which is the ground state of polar perturbations as well.