Abstract
We assess the role of a resonant spectrum in the AdS instability, and quantify the extent to which breaking the resonant spectrum of AdS can restore stability. Specifically, we study non-collapsing `multi-oscillator' solutions in AdS under various boundary conditions that allow for both resonant and non-resonant spectra. We find non-collapsing two mode, equal amplitude solutions in the non-resonant Robin case, and that these solutions vanish in the fully resonant Dirichlet case. This is consistent with non-resonant stability, and with the idea that stable solutions in the Dirichlet case are all single-mode dominated. Surprisingly, when the boundary condition is Neumann, we find non-collapsing solutions arbitrarily close to AdS that are not single-mode dominated, despite the spectrum being fully resonant.
Highlights
Because of the presence of a reflecting boundary, global anti–deSitter space (AdS) may allow dynamical black hole formation starting from arbitrarily small excitations, which would imply that anti–de Sitter (AdS) is nonlinearly unstable, a possibility which was tentatively suggested in [1,2]. (See [3] for early work on the issue of the AdS instability.) The first evidence of this instability was presented in [4] where the numerical evolution of a massless scalar field in AdS, initially in a Gaussian configuration, eventually forms an event horizon even for very small amplitudes
We find that the space of these solutions appears to vanish as the boundary condition approaches the resonant Dirichlet case, in agreement with a nonlinear instability for two-mode initial data in this case
A Dirichlet boundary condition corresponds to κ 1⁄4 1 with frequencies ωn 1⁄4 2n, and a Neumann boundary condition corresponding to κ 1⁄4 0 with frequencies ωn 1⁄4 2n þ 1
Summary
Because of the presence of a reflecting boundary, global anti–deSitter space (AdS) may allow dynamical black hole formation starting from arbitrarily small excitations, which would imply that AdS is nonlinearly unstable, a possibility which was tentatively suggested in [1,2]. (See [3] for early work on the issue of the AdS instability.) The first evidence of this instability was presented in [4] where the numerical evolution of a massless scalar field in AdS, initially in a Gaussian configuration, eventually forms an event horizon even for very small amplitudes. Dirichlet case (which has a resonant spectrum), two-mode initial data is well studied, and accumulated evidence suggests that such data always leads to collapse, and lies outside the islands of stability [7,8,9,10,11,12,13,14,19,42,43,53,54,55,56].
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