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MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
50/2024

Extremal Black Hole Weather

Claudio Iuliano, Stefan Hollands, Stephen R. Green and Peter Zimmerman

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.

Received:
16.12.24
Published:
19.12.24