Korteweg-de Vries Integrals for Modified Black Hole Potentials: Instabilities and other Questions

TL;DR

The paper establishes a deep link between hidden integrable structures in Schwarzschild BH perturbations and the spectral properties of the BH, showing that KdV integrals constrain QNM spectra and completely determine greybody factors via a moment problem. By exploring PT fits, environmental bumps, oscillatory corrections, and EFT-induced deformations of the BH potential, it demonstrates that KdV integrals serve as sensitive indicators of (in)stability and isospectrality breaking, with higher-order integrals generally more susceptible to perturbations. It also clarifies how greybody factors can remain robust under certain perturbations even as QNM spectra destabilize, through the mathematical structure of the moment problem and its relation to T(k). The findings suggest a productive path for analytical understanding of BH spectroscopy, potential extensions to other spacetimes, and applications to modeling ringdown signals and environmental effects in gravitational-wave observations.

Abstract

Quasi-normal modes (QNMs) and greybody factors are some of the most characteristic features of the dynamics of black holes (BHs) and represent the basis for a number of fundamental physics tests with gravitational wave observations. It is therefore important to understand the properties of these quantities, naturally introduced within BH perturbation theory, in particular the stability properties under modifications of the BH potential. Instabilities in the QNMs have been recently shown to appear in the BH pseudospectrum under certain circumstances. In this work, we give a novel point of view based on the existence of some recently discovered hidden symmetries in BH dynamics and the associated infinite series of conserved quantities, the Korteweg-de Vries (KdV) integrals. We provide different motivations to use the KdV integrals as indicators of some crucial BH spectral properties. In particular, by studying them in different scenarios described by modified BH barriers, we find strong evidence that the KdV conserved quantities represent a useful tool to look for instabilities in the BH spectrum of QNMs and in their greybody factors.

Figures (29)

• Figure 1: Regge-Wheeler (blue) and Zerilli (red) potentials as functions of the tortoise coordinate $x$ ($-\infty< x < \infty$) in units of the Schwarzschild radius $r^{}_s$, for angular momentum harmonic number $\ell=2$.
• Figure 2: Position of the maximum of the Regge-Wheeler potential (red) and the Zerilli potential (blue), in units of the Schwarzschild radius $r^{}_s$, as a function of the angular momentum harmonic number ($\ell$).
• Figure 3: Comparison of the Pöschl-Teller potential fits to the Regge-Wheeler (left) and Zerilli (right) potentials in units of the Schwarzschild radius $r^{}_s$.
• Figure 4: Comparison of the four potentials considered in this study (Regge-Wheeler, Zerilli and the two Pöschl-Teller fits) for the cases $\ell=2$ (left) and $\ell=3$ (right) in units of the Schwarzschild radius $r^{}_s$.
• Figure 5: Absolute values for the KdV integrals for the Regge-Wheeler potential (RW) and the two Pöschl-Teller fits [PT(RW) and PT(Z)] in logarithmic scale of base $10$. The left figure shows the results for the $\ell=2$ case, whereas the figure on the right shows the results for $\ell=3$. In these plots, each KdV integral $\mathcal{K}_n$ has been multiplied by $r_s^n$ to make them dimensionless.