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.
Highlights
We study the dynamics of a black hole in an asymptotically AdSd × Sd spacetime in the limit of a large number of dimensions, d → ∞
3.1 Effective equations In the large d limit, planar black holes are found to behave like a viscous fluid described by effective hydrodynamic-like equations [24]
The global effective equations near the equator (3.9) are simple enough that they can be solved numerically with prepackaged differential equation solvers, such as the NDSolve function in Mathematica.1. For these equations, provided we begin with a black hole radius rc2 < 1/3 with a perturbation that preserves the u → −u symmetry, we can evolve to an approximate steady-state that closely matches the Gaussian profile (3.15)
Summary
We begin with the formulation of the small black hole solutions in an arbitrary number of dimensions. + 2 e−iωtχ3(r)Y (Ωd)dtdr + r2(1 + e−iωtχ4(r)Y (Ωd))dΩ2d−2 + L2dΩ2d, where ω is the quasinormal mode frequency This fluctuation is subject to the linearized Einstein equations. In the analysis of quasinormal modes, one solves the χ equation (2.9) with physical boundary conditions and obtains the dispersion relation ω = ω(r+, ). This relation contains information about how quickly a perturbation to the black hole (or equivalently the dual nonzero energy state of the field theory) dies off. The zero mode, ω = 0, will give the threshold radius r∗ of instability Since these quantities can be readily computed numerically, we shall obtain some insight into the quality of the large d approximation for our setup. The analytic expression for ω will serve as a consistency check for the hydrodynamic equations that we derive later
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