Necessary conditions for an AdS-type instability

TL;DR

This paper investigates the necessary conditions for nonlinear AdS instability in gravity-scalar systems, using KAM theory to explain numerical results and exploring eigenfunction properties influencing black hole formation.

Contribution

It introduces an analytical framework based on KAM theory to identify conditions for AdS instability and applies it to various spacetimes, including the AdS soliton.

Findings

Numerical instability occurs even with asymptotically resonant spectra.

A minimum field amplitude is required to trigger instability.

Eigenfunction properties influence localization and black hole formation.

Abstract

In this work, we analyze the necessary conditions for a nonlinear AdS-type instability in the gravity-scalar field system. In particular, we discuss the necessary conditions for a cascade of energy to higher modes by applying results in KAM theory. Our analytical framework explains numerical observations of instability of spacetimes even when the spectrum is only asymptotically resonant, and the fact that a minimum field amplitude is needed to trigger it. Our framework can be applied for similar stability analyses of fields in (locally) asymptotically AdS spacetimes. We illustrate this with the example of the AdS soliton. Further, we conjecture on the possible reasons for quasi-periodic behaviour observed in perturbations of AdS spacetime. Certain properties of the eigenfunctions of the linear system dictate whether there will be localization in space leading to black hole formation. These properties are examined using the asymptotics of Jacobi polynomials. Finally we discuss recent results on the AdS instability in Einstein-Gauss-Bonnet gravity in light of our work.