Abstract

This talk is concerned with the stability of anti–de Sitter (AdS) spacetime to spherically symmetric perturbations of a real scalar field in general relativity. For small-amplitude initial data, two types of behavior have been observed numerically, depending on the initial data profile: collapse to a black hole via a turbulent cascade of energy, and noncollapse characterized by recurrences that approach the initial state. In this talk, I will first introduce a two timescale approximation that describes the leading nonlinear interactions for small-amplitude perturbations, and is suitable for studying the weakly turbulent instability of AdS—-both noncollapsing and collapsing solutions. Within this framework, I will identify a large class of equilibrium solutions, which are quasiperiodic in time, and I will show that they are stable. I argue that these solutions represent “islands of stability” in AdS and that recurrences observed in generic noncollapsing solutions are orbits about these equilibria. Moreover, the stability analysis gives rise to the measured recurrence times. Alternatively, for initial data far from an equilibrium solution, large amounts of energy are driven to high-frequency modes, and the two timescale approximation breaks down. Depending on the higher order dynamics of the full system, this often signals an imminent collapse to a black hole.