Schwarzschild Black Hole Turbulence: Scalar Probe

TL;DR

This work investigates how small perturbations of a Schwarzschild black hole redistribute energy among scalar quasinormal modes and seed turbulence-like cascades. It employs the van der Pol–Krylov–Bogoliubov averaging method to derive a two-mode reduction capturing near-resonant couplings between neighboring multipoles under a monochromatic $L=2,M=0$ pump. Two instability routes are analyzed: off-diagonal difference-frequency three-wave mixing and diagonal Mathieu self-modulation; in the eikonal limit the difference-frequency channel dominates, producing a unidirectional energy cascade from high to low frequencies. Higher-order harmonics from nonlinear GR perturbations create additional tongues at $\mu=1/n$, with strengths $O(\varepsilon^n)$, providing a simple, quantitative mechanism for energy transfer in black hole ringdowns and clarifying when turbulent signatures may arise in linear probes of a weakly perturbed background.

Abstract

We explore how perturbations of a Schwarzschild black hole can redistribute energy among scalar modes and seed turbulent like cascades. We make use of the van der Pol-Krylov-Bogoliubov averaging method and derive coupled mode equations that describe near-resonant interactions between neighbouring multipoles. We compare two routes to instability, namely the difference-frequency mixing between adjacent modes and the diagonal (Mathieu) self-modulation channel. We show that, at high multipole number (eikonal limit), the difference-frequency route dominates and drives a one-way cascade from higher to lower frequencies. We chart the corresponding instability regions ("tongues") and quantify their detuning dependence. The framework provides a simple, quantitative mechanism for energy transfer in black hole ringdowns and clarifies when and how turbulent signatures can arise within linear probes on a weakly perturbed background.

Figures (2)

• Figure 1: Arnold instability tongue for the difference–frequency (three–wave) channel. The solid curve is the normalized threshold $\tilde{\varepsilon}_{\mathrm{thr}}(\Omega_R)=\sqrt{\alpha^{2}+\tfrac{1}{4}[\Omega_R-(\omega_\ell^R-\omega_{\ell-2}^R)]^2}$, and the horizontally hatched region denotes instability ($\tilde{\varepsilon}\!\ge\!\tilde{\varepsilon}_{\mathrm{thr}}$). The vertical dashed line marks the resonance center $\Omega_R=\omega_\ell^R-\omega_{\ell-2}^R$. For this plot we used the eikonal spacings $\omega_\ell^R-\omega_{\ell-2}^R=2\omega_c$ and $\alpha=\omega_c/2$, where $\omega_c=1/3\sqrt{3}M$. In practice, even low $\ell$ give the same Arnold tongues so the plot changes only by a small horizontal shift with a new minimum and a vertical adjustment. For example, even $\ell=6$ scalar QNMs are already very close to eikonal values.
• Figure 2: Mathieu–type parametric–instability map in the scaled plane $\mu=\Omega_R/(2\omega_\ell)$ and $\varepsilon_{\mathrm{eff}}=\varepsilon |q_\ell|/(4\omega_\ell)$. Black curves are the instability boundaries; horizontal grey lines indicate the unstable region. The primary lobe is centered at $\mu=1$; higher–order tongues at $\mu=\tfrac{1}{2},\tfrac{1}{3},\tfrac{1}{4}$ are shown narrower, as expected from higher–order averaging. Vertical dashed lines mark the tongue centers.