Emergent Turbulence in Nonlinear Gravity

TL;DR

This work shows that fully nonlinear gravity in 3+1 dimensions supports turbulence-like dynamics when driven by quasi-steady gravitational waves. The authors identify two nonlinear instabilities, a four-mode and a three-mode coupling, which generate inverse cascades toward larger angular and longer temporal scales, linking nonlinear GR phenomena to fluid-turbulence concepts. Growth rates scale as $\dot{A} \sim A_i^{2}$ for four-mode and $\dot{A} \sim A_i^{3}$ for three-mode processes, with higher amplify effects near BH horizons compared to flat spacetime; the system transitions from laminar to turbulent-like regimes as driving strength increases. The findings have implications for gravitational wave modeling, BBH dynamics, and exploring nonlinear GR using fluid-dynamics-inspired intuition, offering a new platform to study the gravity--fluid correspondence beyond AdS/CFT and negative cosmological constant contexts.

Abstract

Gravity in nonlinear and dynamical regimes underpins spectacular astrophysical phenomena and observable consequences, from the early universe to black hole collisions. In these extreme environments, inverse energy cascades - mediated by nonlinear interactions - may help explain the near scale-invariance of cosmic structure and the simplicity of gravitational waves from binary black hole mergers. Yet the presence, characteristics, and generality of such interactions in full General Relativity remain largely unexplored. Here we show that two types of nonlinear interactions - a four-mode and a three-mode interaction - emerge in the fully nonlinear regime, and can indeed channel inverse energy cascades by inducing resonant and anti-damping instabilities. This establishes what was previously only hinted at in highly specialized perturbative contexts. We further demonstrate a ``laminar'' to ``turbulent'' transition for the largest-possible angular structure in General Relativity, whereas finer structures remain persistently turbulent. Our results reveal the impact and generality of these nonlinear interactions (instabilities), which can be key to understanding observations ranging from cosmological to kilometer scales. We anticipate that our work will shed new light on nonlinear gravitational phenomena and their consequences, such as constructing gravitational wave templates and testing General Relativity in the most extreme regime. Moreover, our work is a starting point for addressing nonlinear gravitational interactions using ideas and methods inspired by fluid dynamics.

Figures (13)

• Figure 1: Stirring a non-spinning BH with a $\ell=2$ GW driver at fixed $\omega=0.5/M_i$ and $A_i=1.2\times10^{-3}$. The $\ell=6$ response of $\mathcal{E}$ exhibits three frequency components ($3\omega,2\omega,\omega$). Their amplitude envelopes are shown in the top three panels. The $3\omega$ and $\omega$ modes first arise through third-order couplings (gray), followed by nonlinear growth and saturation of $2\omega$ and $\omega$ (orange). Bottom: time–frequency representation exhibiting an inverse cascade: $3\omega\to 2\omega\to \omega$ (yellow dashed).
• Figure 2: Stirring a non-spinning BH with a $\ell=6$ GW driver at $\omega=0.5/M_i$ and $A_i=4\times10^{-4}$. Response across angular harmonics is measured continuously in time (in blue), with representative time slices shown (dot-dashed). Each harmonic contains $2\omega$ and $\omega$. The $2\omega$ component (top) arises from quadratic couplings and maintains a stable angular spectrum, whereas the $\omega$ mode (bottom) continuously redistributes its spectrum, yielding an inverse cascade toward lower $\ell$.
• Figure 3: Emergent instabilities in a stirred BH induced by $\ell=2$ GWs at $\omega=0.5/M_i$ with various amplitudes (colors). Shown are amplitude envelopes of $\ell=6$ ($2\omega$ and $\omega$) and $\ell=4$ ($\omega$). Modes first undergo laminar excitations via fourth- and third-order couplings (gray). At low amplitudes, the structure persists; at higher amplitudes, new excitations emerge. For intermediate amplitudes, modes grow exponentially (gray dashed) before saturation (open circles). For $2\omega$ in $\ell=6$, the exponential growth rate scales as $A_i^2$ and saturation amplitude as $A_i^4$.
• Figure 4: Emergent instabilities in a stirred BH induced by $\ell=6$ GWs at $\omega=0.5/M_i$ with various amplitudes (colors). The $\omega$ mode in $\ell=2$ and $4$ always grows linearly (dashed lines) regardless of $A_i$. The growth rate scales as $A_i^3$.
• Figure 5: Feynman-type diagram illustrating the four-mode interaction (top) underlying the emergent excitations in Fig. \ref{['fig:injecting_l2_various_amps']}; and the three-mode interaction (bottom) in Fig. \ref{['fig:injecting_l6_various_amps']}.