Conditionally Extended Validity of Perturbation Theory: Persistence of AdS Stability Islands

TL;DR

This paper establishes conditions under which perturbation theory remains valid over long times in AdS stability studies, demonstrating that non-collapsing solutions persist at small amplitudes contrary to previous conjectures.

Contribution

It provides a rigorous framework for extending the validity of truncated perturbation theory to long times and proves the persistence of non-collapsing solutions in AdS stability at zero amplitude.

Findings

Non-collapsing solutions persist as amplitude approaches zero.

Perturbation theory validity can be extended to long times under certain conditions.

Collapse solutions' persistence remains more challenging to prove.

Abstract

Approximating nonlinear dynamics with a truncated perturbative expan- sion may be accurate for a while, but it in general breaks down at a long time scale that is one over the small expansion parameter. There are interesting occasions in which such breakdown does not happen. We provide a mathematically general and precise definition of those occasions, in which we prove that the validity of truncated theory trivially extends to the long time scale. This enables us to utilize numerical results, which are only obtainable within finite times, to legitimately predict the dynamic when the expansion parameter goes to zero, thus the long time scale goes to infinity. In particular, this shows that existing non-collapsing solutions in the AdS (in)stability problem persist to the zero-amplitude limit, opposing the conjecture by Dias, Horowitz, Marolf and Santos that predicts a shrinkage to measure-zero [1]. We also point out why the persistence of collapsing solutions is harder to prove, and how the recent interesting progress by Bizon, Maliborski and Rostoworowski is not there yet [2].