[Paper Review] Small black holes in $AdS_5 imes S^5$

Alex Buchel, Luis Lehner|arXiv (Cornell University)|Feb 5, 2015

Black Holes and Theoretical Physics8 references5 citations

TL;DR

This paper investigates the Gregory-Laflamme (GL) instability of small, $SO(6)$-symmetric black holes in $AdS_5 \times S^5$ by computing quasinormal mode spectra up to $\ell=10$. It confirms that the instability onset coincides with the normalizability of the Hubeny-Rangamani $\ell=1$ zero mode at $r_+/L \approx 0.4259$, and demonstrates that higher-$\ell$ modes become unstable at smaller black hole sizes, indicating a cascade of instabilities as black holes shrink.

ABSTRACT

We consider small black holes in $AdS_5 imes S^5$, smeared on $S^5$. We compute the spectrum of $\ell \in [1,10]$ $S^5$-quasinormal modes corresponding to fluctuations leading to localization of these black holes on $S^5$. We recover the zero mode found by Hubeny and Rangamani (HR) previously \cite{Hubeny:2002xn}, and explicitly demonstrate that a Gregory-Laflamme type instability is at play in this system. The instability is associated with the expectation value of a dimension-5 operator.

Motivation & Objective

• To investigate the stability of small black holes in $AdS_5 \times S^5$ under $SO(5)$-invariant fluctuations.
• To determine whether a Gregory-Laflamme-type instability occurs in these black holes as they shrink.
• To compute the quasinormal mode spectrum for $\ell \in [1,10]$ and identify the onset of instability.
• To clarify the holographic interpretation of the instability in terms of dual $\mathcal{N}=4$ SYM operators.

Proposed method

• Derives the $SO(4)\times SO(5)$ symmetric ansatz in type IIB supergravity with metric and 5-form flux, imposing duality constraints to reduce degrees of freedom.
• Solves the linearized Einstein and Maxwell equations for $SO(6)$-symmetric black holes, focusing on $\ell=1$ and higher modes on $S^5$.
• Imposes boundary conditions at the horizon and asymptotic $AdS_5$ boundary to extract quasinormal mode frequencies.
• Performs numerical computation of the quasinormal mode spectrum for $\ell=1$ to $10$, identifying the onset of instability via complex frequency poles.
• Fits the instability onset $\rho_+^2$ as a function of $\ell$ to extract asymptotic behavior in the large-$\ell$ limit.
• Verifies the leading-order $1/s$ dependence of the instability threshold via near-boundary analysis of the mode equations.

Experimental results

Research questions

• RQ1 At what black hole size does the Gregory-Laflamme instability first appear in small $AdS_5 \times S^5$ black holes?
• RQ2 How does the instability threshold depend on the spherical harmonic index $\ell$ of the fluctuation?
• RQ3 Is the $\ell=1$ Hubeny-Rangamani zero mode the precursor to the GL instability, and when does it become normalizable?
• RQ4 What is the holographic interpretation of the unstable modes in terms of dual $\mathcal{N}=4$ SYM operators?
• RQ5 Does the instability cascade to higher-$\ell$ modes at progressively smaller black hole sizes?

Key findings

• The Gregory-Laflamme instability in small $AdS_5 \times S^5$ black holes is confirmed via explicit computation of quasinormal modes.
• The instability onset for $\ell=1$ modes occurs precisely when the Hubeny-Rangamani zero mode becomes normalizable, at $r_+/L \approx 0.4259$.
• Higher-$\ell$ modes become unstable at progressively smaller black hole sizes, with $\ell=10$ modes destabilizing at $\rho_+^2 \approx 0.105962$.
• The instability threshold $\rho_+^2$ for large $\ell$ follows a $1/s$-dependent behavior with a leading coefficient $q_0 \approx 1.61015$, matching the analytical fit.
• The radial profile of the unstable modes suggests a dual $\mathcal{N}=4$ SYM operator of dimension $\Delta = \ell + 4$, consistent with a massive Kaluza-Klein graviton.
• The results suggest a potential cascade of instabilities toward smaller black holes, possibly leading to horizon pinch-off and naked singularities, analogous to black strings in flat space.