A nonrelativistic limit for AdS perturbations

TL;DR

This paper explores the nonrelativistic limit of AdS perturbations, revealing a simplified resonant system with integrable dynamics in four dimensions, which aids understanding of stability and evolution of small amplitude solutions.

Contribution

It introduces the Schrodinger-Newton-Hooke system as the nonrelativistic limit of AdS perturbations and derives a resonant approximation that simplifies analysis of small amplitude solutions.

Findings

Resonant system with external harmonic potential derived

In four dimensions, a three-dimensional invariant subspace is integrable

Evolution of low-mode initial data is analytically solvable within this framework

Abstract

The familiar $c\rightarrow \infty$ nonrelativistic limit converts the Klein-Gordon equation in Minkowski spacetime to the free Schroedinger equation, and the Einstein-massive-scalar system without a cosmological constant to the Schroedinger-Newton (SN) equation. In this paper, motivated by the problem of stability of Anti-de Sitter (AdS) spacetime, we examine how this limit is affected by the presence of a negative cosmological constant $\Lambda$. Assuming for consistency that the product $\Lambda c^2$ tends to a negative constant as $c\rightarrow \infty$, we show that the corresponding nonrelativistic limit is given by the SN system with an external harmonic potential which we call the Schrodinger-Newton-Hooke (SNH) system. We then derive the resonant approximation which captures the dynamics of small amplitude spherically symmetric solutions of the SNH system. This resonant system turns out to be much simpler than its general-relativistic version, which makes it amenable to analytic methods. Specifically, in four spatial dimensions, we show that the resonant system possesses a three-dimensional invariant subspace on which the dynamics is completely integrable and hence can be solved analytically. The evolution of the two-lowest-mode initial data (an extensively studied case for the original general-relativistic system), in particular, is described by this family of solutions.