The large dimension limit of a small black hole instability in anti-de Sitter space

Abstract

We study the dynamics of a black hole in an asymptotically AdSd × Sd space-time in the limit of a large number of dimensions, d → ∞. Such a black hole is known to become dynamically unstable below a critical radius. We derive the dispersion relation for the quasinormal mode that governs this instability in an expansion in 1/d. We also provide a full nonlinear analysis of the instability at leading order in 1/d. We find solutions that resemble the lumpy black spots and black belts previously constructed numerically for small d, breaking the SO(d + 1) rotational symmetry of the sphere down to SO(d). We are also able to follow the time evolution of the instability. Due possibly to limitations in our analysis, our time dependent simulations do not settle down to stationary solutions. This work has relevance for strongly interacting gauge theories; through the AdS/CFT correspondence, the special case d = 5 corresponds to maximally supersymmetric Yang-Mills theory on a spatial S3 in the microcanonical ensemble and in a strong coupling and large number of colors limit.