AdS nonlinear instability: moving beyond spherical symmetry

Anti-de Sitter (AdS) is conjectured to be nonlinear unstable to a weakly turbulent mechanism that develops a cascade towards high frequencies, leading to black hole formation (Dafermos and Holzegel 2006 Seminar at DAMTP (University of Cambridge) available at https://dpmms.cam.ac.uk/∼md384/ADSinstability.pdf, Bizon and Rostworowski 2011 Phys. Rev. Lett. 107 031102). We give evidence that the gravitational sector of perturbations behaves differently from the scalar one studied by Bizon and Rostworowski. In contrast with Bizon and Rostworowski, we find that not all gravitational normal modes of AdS can be nonlinearly extended into periodic horizonless smooth solutions of the Einstein equation. In particular, we show that even seeds with a single normal mode can develop secular resonances, unlike the spherically symmetric scalar field collapse studied by Bizon and Rostworowski. Moreover, if the seed has two normal modes, more than one resonance can be generated at third order, unlike the spherical collapse of Bizon and Rostworowski. We also show that weak turbulent perturbative theory predicts the existence of direct and inverse cascades, with the former dominating the latter for equal energy two-mode seeds. (letter)