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 $Ad{S}_d\times S^d$ 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.

arXiv comment20 pages + 10 pages of appendices, 7 figures

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• Phases and Stability of Non-Uniform Black Strings
• Submatrix deconfinement and small black holes in AdS
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• Black holes in presence of cosmological constant: Second order in 1/D
• Holographic Bjorken Flow at Large-D
• Large D holography with metric deformations
• Stationary Solutions from the Large D Membrane Paradigm
• Large D gravity and charged membrane dynamics with nonzero cosmological constant
• The large D Membrane Paradigm For Einstein-Gauss-Bonnet Gravity
• The Fate of Instability of de Sitter Black Holes at Large $D$
• Fluid-gravity and membrane-gravity dualities - Comparison at subleading orders
• Black Rings in Large $D$ Membrane Paradigm at the First Order
• Black hole one-loop determinants in the large dimension limit
• Stress Tensor for Large-$D$ membrane at subleading orders
• Squashed black holes at large $D$