Modified Black Hole Potentials and Their Korteweg-de Vries Integrals: Instabilities and Beyond

TL;DR

Problem: characterize how modified BH potentials affect BH spectral signatures (QNMs and GBFs). Approach: apply hidden KdV isospectral flows, with the KdV deformation $V_{,\tau} = 6 V V_{,x} - V_{,xxx}$, and use the trace identities $(-1)^{n+1} 2^{-2n} \pi K_{2n+1} = \int dk\, k^{2n} \ln T(k)$ to interpret $K_n$ as moments of $\ln T(k)$. They test perturbations $V^{\mathrm{odd/even}} = V^{RW/Z} + \epsilon \delta V^{\mathrm{odd/even}}$ (Pöschl–Teller bump and EFT-like polynomials) and quantify $\delta K_n$ and GBF stability via Padé approximants; findings show low-order $K_n$ are infrared-sensitive while high-order $K_n$ are UV-sensitive, and EFT perturbations induce parity isospectrality breaking captured by KdV indicators. Significance: the KdV hierarchy provides a practical diagnostic bridge linking scattering data to BH spectra, with implications for environmental effects and tests of gravity beyond general relativity.

Abstract

Black Hole (BH) Quasi-Normal Modes (QNMs) and Greybody Factors (GBFs) are key signatures of BH dynamics that are crucial for testing fundamental physics via gravitational waves. Recent studies of the BH pseudospectrum have revealed instabilities in QNMs. Here, we introduce a new perspective using hidden symmetries in the BH dynamics, specifically the Korteweg-de Vries (KdV) integrals - an infinite series of conserved quantities. By analyzing modified BH potentials, we find strong evidence that KdV integrals are valuable indicators for detecting instabilities in QNMs and GBFs.

Figures (2)

• Figure 1: (Left) Relative error $\log_{10}\delta \mathcal{K}_{5}$ for $\epsilon = 10^{-5}$ and varying $x_0$ and $\alpha$. The dashed line corresponds to the threshold $\delta \mathcal{K}_{n}=\epsilon$. (Right) Relative error $\delta \mathcal{K}_{2n-1}$ for $n=1,...,30$ for the odd (blue dots) and even (black dots) potentials. The dashed line corresponds to the threshold $\epsilon=10^{-5}$ (orange dashed line).
• Figure 2: (Left) Relative error for the Pöschl–Teller bump corrections for $r^{}_s\,\alpha=1/50$, $x_0=0$ and $\epsilon=10^{-5}$ for diagonal Padé approximants. (Right) Integrated error for the same parameters in the left plot. On the $x$-axis we have $K+L$, tracking the order of the Padé approximants.