Globally regular instability of $AdS_3$

TL;DR

This paper investigates the nonlinear stability of $AdS_3$ spacetime under small scalar perturbations, revealing a turbulent energy cascade that remains globally regular and supports cosmic censorship.

Contribution

It demonstrates that, unlike in higher dimensions, $AdS_3$ does not form black holes from small perturbations, showing turbulence does not lead to singularities.

Findings

Energy cascades to high frequencies in $AdS_3$

No black hole formation from small perturbations

Solutions remain globally regular over time

Abstract

We consider three-dimensional AdS gravity minimally coupled to a massless scalar field and study numerically the evolution of small smooth circularly symmetric perturbations of the $AdS_3$ spacetime. As in higher dimensions, for a large class of perturbations, we observe a turbulent cascade of energy to high frequencies which entails instability of $AdS_3$. However, in contrast to higher dimensions, the cascade cannot be terminated by black hole formation because small perturbations have energy below the black hole threshold. This situation appears to be challenging for the cosmic censor. Analysing the energy spectrum of the cascade we determine the width $\rho(t)$ of the analyticity strip of solutions in the complex spatial plane and argue by extrapolation that $\rho(t)$ does not vanish in finite time. This provides evidence that the turbulence is too weak to produce a naked singularity and the solutions remain globally regular in time, in accordance with the cosmic censorship hypothesis.