Bound State Perturbations in the Interior of Black Holes

TL;DR

The paper investigates bound state perturbations inside the Schwarzschild black hole, focusing on imaginary-frequency modes that are regular at the center and decay toward the horizon, expressed as ω = i ω_I with ω_I>0. By reformulating the Regge–Wheeler equation in interior Kantowski–Sachs-type coordinates using clocks based on the scale factor, the authors obtain algebraic, Schrödinger-like potentials and study the bound-state spectrum for scalar, vector, and axial perturbations using both analytic (WKB and matching-condition) and numerical methods. They establish a universal lower bound $2 G M \, u_I > 1$ (with asymptotic saturation at large ℓ) and derive an upper bound for axial perturbations that scales as $2 G M \, ω_I^{max} \lesssim 0.042 \, (ℓ+1/2)^4$; the most excited bound states approach $ω_I \to 1$ as ℓ grows, with the ground and low-lying states following $(ℓ+1/2)^4$ scaling. The results reveal that interior bound states can have non-zero profiles on the future horizon, suggesting distinctive interior dynamics and potential connections to exterior QNMs, while opening avenues for exploring polar perturbations and quantum aspects in future work.

Abstract

We revisit our earlier work and investigate the bound state perturbations in the interior of the Schwarzschild black hole. The bound sates are defined as the perturbations in the interior of the black hole with an imaginary spectrum which are regular at the center of black hole while their time-dependent profile falls off exponentially on the event horizon. Using the scale factor in the expanding direction in the interior of the black hole as the clock, we rewrite the corresponding Regge-Wheeler equation and solve it semi-analytically as well as numerically. We confirm that the bound state solutions exist for scalar, vector and axial tensor perturbations. It is shown that for a given value of $\ell >s$, there are total $\ell-s$ such bound states. We obtain the universal lower bound $2 G M ω_I >1$ for the spectrum of bound state which is asymptotically saturated in the large $\ell$ limit. Furthermore, we obtain an upper bound on the spectrum of axial perturbations which for large $\ell$ scales like $2 G M ω_I \lesssim 0.04\, \ell^4 $. As observed recently, these bound states have the curious property that the profile of the total wave function has a non-zero magnitude near the future event horizon.

Figures (8)

• Figure 1: The effective potential $V_\text{eff}(\tau)$ given in Eq. (\ref{['Veff_spin']}) for $s=0$ (Left) and $s=2$ (Right) with $\ell=2$ in both plots.
• Figure 2: The effective potential $U_\text{eff}(N)$ given in Eq. (\ref{['Ueff-N']}) for $s=2$ and $\ell=3$ with $\omega_I= 1.7$ (Left), $\omega_I= 2.5$ (Middle) and $\omega_I= 12$ (Right). As $\omega_I$ increases the global minimum is uplifted so for large $\omega_I$ the bound state does not exist. The asymptotic behaviours are given by Eq. (\ref{['ap-N']}).
• Figure 3: The effective potential $U_\text{eff}(\chi)$ given in Eq. (\ref{['Ueff-chi']}) for $s=2$, $\ell=3$. In the left panel, $\omega_I= \frac{1}{8}$ and the bound state solution does not exit. In the right panel, $\omega_I= 1.7$ and the bound state is allowed.
• Figure 4: The effective potential $U_\text{eff}(N)$ given in Eq. (\ref{['Ueff-N']}) for $s=0$ and various values of $\ell$. Left: $\ell=1, \omega_I= 6.34$. Middle: $\ell=2, \omega_I= 2.06$. Right: $\ell=3, \omega_I= 1.53$. These values of $\omega_I$ correspond to the most excited state, the lowest value of $\omega_I$ for a given $\ell$, as given in table 2.
• Figure 5: The WKB predictions. The vertical axis denotes the ratio $\omega_I/f(\ell, n)$ with $f(\ell, n)$ obtained from the WKB method in Eq. (\ref{['omega-WKB']}) while $\omega_I$ is obtained from the full numerical analysis. From the top row to bottom: $n=0, 1, 2$ and $n=3$ respectively.