Energy returns in global AdS_4

TL;DR

This paper investigates the presence of exact or near-exact energy returns in the weakly nonlinear dynamics of gravitational perturbations in global AdS_4, revealing that while approximate returns exist, exact returns are not present in the gravitational case.

Contribution

The study provides a thorough numerical analysis and introduces an analytic approach to demonstrate that gravitational perturbations in AdS_4 do not exhibit exact energy returns, unlike simpler models.

Findings

Numerical analysis shows highly accurate approximate energy returns.

Analytic methods demonstrate the absence of exact returns in gravitational case.

Identification of specific cascade sequences leading to accurate energy returns.

Abstract

Recent studies of the weakly nonlinear dynamics of probe fields in global AdS$_4$ (and of the nonrelativistic limit of AdS resulting in the Gross-Pitaevskii equation) have revealed a number of cases with exact dynamical returns for two-mode initial data. In this paper, we address the question whether similar exact returns are present in the weakly nonlinear dynamics of gravitationally backreacting perturbations in global AdS$_4$. In the literature, approximate returns were first pointed out numerically and with limited precision. We first provide a thorough numerical analysis and discover returns that are so accurate that it would be tantalizing to sign off the small imperfections as an artifact of numerics. To clarify the situation, we introduce a systematic analytic approach by focusing on solutions with spectra localized around one of the two lowest modes. This allows us to demonstrate that in the gravitational case the returns are not exact. Furthermore, our analysis predicts and explains specific integer numbers of direct-reverse cascade sequences that result in particularly accurate energy returns (elaborate hierarchies of more and less precise returns arise if one waits for appropriate longer multiple periods in this manner). In addition, we explain, at least in this regime, the ubiquitous appearance of direct-reverse cascades in the weakly nonlinear dynamics of AdS-like systems.