Gravitational Bound State Perturbations Inside Black Holes and Isospectrality

TL;DR

This paper investigates the bound state spectra of polar and axial perturbations inside Schwarzschild black holes, revealing isospectrality and implications for black hole area quantization.

Contribution

It demonstrates the exact isospectrality of polar and axial bound states inside black holes and links the spectrum to black hole area quantization.

Findings

Polar perturbations have $ ext{l}-1$ bound states.

Spectrum of $ ext{l}-2$ polar states matches axial states.

Highly excited states are equally spaced, supporting area quantization.

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 coincides exactly with the spectrum of axial perturbations. Consequently, the isospectrality between the bound states of axial and polar perturbations in the interior of the black hole is preserved. Furthermore, the additional mode found in the spectrum of polar perturbations is the algebraically special mode, which also furnishes the ground state of polar perturbations. It is shown that the spectrum of the highly excited states is equally spaced, which, in the semi-classical approximation, yields the black hole area quantization $\Delta A = 16 \pi l_{\mathrm{Pl}}^2$.