AdS Instability and Nonlinear Perturbations in Confined Geometries

Abstract

In 2011, Bizon and Rostworowski, through their seminal work, gave numerical evidence that Anti-de Sitter spacetime with reflecting boundary conditions is nonlinearly unstable against black hole formation for a wide range of initial conditions, under arbitrarily small perturbations. The results were particularly interesting because AdS has been known to be linearly stable. Numerical and perturbative analysis showed that the mechanism behind the instability was weak turbulence i.e. transfer of energy from low frequency to high frequency modes. The resonant nature of the spectrum was considered to be a crucial ingredient in driving this instability. Setups, which mimic AdS were also studied numerically, an example is---fields trapped in a spherical cavity in flat spacetime. While the mechanism which drives an instability in such systems were similar, there were important differences as well---specifically in those set-ups whose linear spectra were non-resonant. In such set-ups, a threshold amplitude was detected, below which the system remained stable for very long times. The value of the threshold amplitude would be too small, which would make its detection in numerical studies very difficult. The main objective of our thesis is to study such gravitational systems with confined geometries and obtain a deeper understanding of the nature of instabilities observed in them, by using the results in nonlinear dynamics. In the first part of the thesis, we study the necessary conditions for a nonlinear instability to occur in similar gravity-scalar field systems. We also, take up the case of the AdS soliton and demonstrate that these results in nonlinear dynamics could be applied to (locally) asymptotically AdS spacetimes as well. In the second part, we study the gravitational perturbations of Minkowski enclosed in a spherical Dirichlet wall. We use the formalism by Ishibashi, Kodama and Seto to classify the metric perturbations according to their tensorial behavior on a sphere as the scalar, vector and tensor-type. We simplify the perturbation equations upto all orders of perturbation theory. We then apply the arguments developed in the first part to comment upon the nonlinear stability of the system. Finally, we study the gravitational perturbations of AdS spacetime in (n+2) dimensions, with n>2. We obtain the solutions to metric perturbations and render them asymptotically AdS at each order. We, for the first time, perform the higher order perturbative analysis of the tensor sector. As an example, we take a special case where the initial data contains only a single-mode tensor-type seed at the linear level. Interestingly, we find that there are no resonances at the second order.