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 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 $ΔA = 16 πl_{\mathrm{Pl}}^2$.

Figures (3)

• Figure 1: The effective potentials $V^-(r_*)$ and $V^+(r_*)$ given in Eqs. \ref{['V-RW']} and \ref{['V-Z']} with $\ell = 3$.
• Figure 2: The profile of the normalized algebraically special mode $Z^+_{\mathrm{sp}}(r)$ given in Eq. (\ref{['Z-sp']}). Curves from bottom to top correspond to $\ell=2, 3, 4, 5$ respectively.
• Figure 3: $\Re(Z^+)$ for polar perturbations for some values of $\ell$. For a given $\ell$, there are $\ell-1$ bound states with $n=0,1,,..., \ell-2$. Each solution with state level $n$ has $n$ nodes. The ASM is the ground state with no node. For a better visualization, unlike Fig. \ref{['special-fig']}, $Z^+$ are not normalized here.