Perturbative and non-linear analyses of gravitational turbulence in spacetimes with stable light rings

TL;DR

This paper investigates the nonlinear dynamics of gravitational waves in spacetimes with stable light rings, demonstrating that small perturbations grow in higher-order energies but do not cause instabilities, indicating weak radiation hair formation.

Contribution

It introduces a perturbative framework using a nonlinear wave equation to model gravitational perturbations in spacetimes with stable light rings, confirming energy growth without instability.

Findings

Higher-order energy norms grow for small initial fluctuations

Late-time mode spectrum shows a direct energy cascade

Spacetimes develop weak, high-frequency radiation hair

Abstract

Some black hole mimickers, as well as black strings and other higher-dimensional spacetimes, exhibit stable light rings-regions where light or high-frequency gravitational waves can be trapped. In these regions, linear perturbations decay slowly, raising the possibility of nonlinear instability mechanisms. In this work, we study the cubic nonlinear wave equation as a proxy for Einstein's equations, using a four-dimensional model geometry that allows stable trapping. By employing a perturbative approach, we show that the nonlinear wave equation on the sphere with dissipative terms captures several features of the full nonlinear problem. This framework allows us to confirm a previous conjecture: all higher-order energy norms grow for arbitrarily small initial fluctuation amplitudes. Additionally, we analyze the system's mode spectrum at late times, revealing an inertial range dominated by a direct energy cascade. These findings further support the notion that spacetimes with stable light rings develop weak, high-frequency radiation hair, which will not generically lead to instabilities.