Extremal Black Hole Weather

TL;DR

This paper investigates the non-linear evolution of gravitational perturbations near extremal Kerr black holes, revealing a stable mode distribution that resembles weather phenomena and is driven by weakly non-linear effects.

Contribution

It introduces a non-linear dynamical system for quasinormal mode amplitudes and finds a stable, exponentially suppressed mode distribution as an endpoint of evolution.

Findings

Existence of a time-independent equilibrium solution for mode amplitudes.

Mode amplitudes decay exponentially with angular momentum number ^{-rac{ ext{l}}{2}}.

Evolution approaches a mode distribution akin to atmospheric weather patterns.

Abstract

We consider weakly non-linear gravitational perturbations of a near-extremal Kerr black hole governed by the second order vacuum Einstein equation. Using the GHZ formalism [Green et al., Class. Quant. Grav. 7(7):075001, 2020], these are parameterized by a Hertz potential. We make an ansatz for the Hertz potential as a series of zero-damped quasinormal modes with time-dependent amplitudes, and derive a non-linear dynamical system for them. We find that our dynamical system has a time-independent solution within the near horizon scaling limit. This equilibrium solution is supported on axisymmetric modes, with amplitudes scaling as $c_\ell \sim C^{\rm low} 2^{-\ell/2} \ell^{-\frac{7}{2}}$ for large polar angular momentum mode number $\ell$, where $C^{\rm low}$ is a cumulative amplitude of the low $\ell$ modes. We interpret our result as evidence that the dynamical evolution will approach, for a parametrically long time as extremality is approached, a distribution of mode amplitudes dyadically exponentially suppressed in $\ell$, hence as the endpoint of an inverse cascade. It is reminiscent of weather-like phenomena in certain models of atmospheric dynamics of rotating bodies. During the timescale considered, the decay of the QNMs themselves plays no role given their parametrically long half-life. Hence, our result is due entirely to weakly non-linear effects.