Nonlinear dynamics and the instability of Anti-de Sitter space

The study of stability of gravitational systems under small perturbations dates back to the days of celestial mechanics and the problem of stability of our solar system over long periods of time when the gravitational effects of the planets are taken into account and therefore deviations from the Keplerian orbits are expected. This seemingly simple question has unveiled very rich underlying dynamics and has led to tremendous discoveries in Physics and Mathematics with the most prominent one being perhaps the Kolomogorov-Arnold-Moser (KAM) theory. Of similar, if not greater, importance is the stability of the vacuum solutions of General Relativity (GR), Einstein’s theory of Gravity. GR has three such solutions known as de Sitter (dS), Minkowski and Anti-de Sitter (AdS) spacetimes. In this work we will deal with developments regarding the (in)stability of the AdS space. More specifically, we will study the evolution of small excitations in the AdS background, both from the spacetime point of view, as well as using Fourier analysis. Our aim is to understand when a singularity forms in the spacetime and what is the underlying dynamics that leads the system towards (in)stability.