The Large Dimension Limit of a Small Black Hole Instability in Anti-de Sitter Space

TL;DR

This paper investigates the instability of large black holes in anti-de Sitter space as the number of dimensions grows, deriving analytical descriptions and analyzing the nonlinear evolution of the instability.

Contribution

It provides the first analytical derivation of the quasinormal mode dispersion relation and a nonlinear analysis of black hole instability in the large dimension limit.

Findings

Derived the dispersion relation for the instability in a 1/d expansion.

Found solutions resembling black spots and belts that break rotational symmetry.

Tracked the time evolution of the instability, noting it does not settle into stationary states.

Abstract

We study the dynamics of a black hole in an asymptotically $AdS_d \times S^d$ space-time in the limit of a large number of dimensions, $d \to \infty$. 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 $S^3$ in the microcanonical ensemble and in a strong coupling and large number of colors limit.