Double-trace instability of BTZ black holes

TL;DR

The paper analyzes the linear stability of rotating BTZ black holes against massive scalar perturbations under double-trace boundary conditions. Using analytic (gamma-criterion) and fully numerical methods, it identifies a dominant axisymmetric (m=0) instability and maps onset curves in BTZ parameter space, showing BTZ can be unstable even when global AdS3 is stable. The instability originates from energy flux through the asymptotic AdS boundary enabled by negative κ, not from near-horizon physics, and its existence suggests rotating hairy BTZ solutions as endpoints. The work also connects AdS3 double-trace instabilities to Ishibashi–Wald stability criteria and provides a prototype for similar instabilities in higher-dimensional rotating AdS black holes, with implications for AdS3/CFT2 and the landscape of hairy solutions.

Abstract

We perform a comprehensive study of the linear stability of rotating BTZ black holes under massive scalar field perturbations with double-trace boundary conditions. While BTZ black holes are stable under standard Dirichlet and Neumann boundary conditions, we demonstrate that they can develop instabilities when subjected to double-trace boundary conditions. Our key findings are threefold. First, we show that BTZ black holes exhibit instabilities not only for non-axisymmetric modes $\unicode{x2013}$ previously the only known unstable sector $\unicode{x2013}$ but crucially also for axisymmetric modes. Second, we prove that the axisymmetric instability is the dominant and most fundamental: configurations unstable to any non-axisymmetric mode are already unstable to the axisymmetric one. Third, we identify regions in the BTZ parameter space where these black holes are unstable while global AdS$_3$ remains stable, and we map the complete onset curves that determine the corresponding stability boundaries. Unlike conventional superradiant instabilities, the BTZ double-trace instability occurs for angular velocities always satisfying the Hawking-Reall bound. We trace the physical origin of these instabilities to the influx of energy and angular momentum through the asymptotic boundary permitted by double-trace deformations for a particular sign of the coupling, rather than to near-horizon effects. Our results provide a prototype for understanding double-trace instabilities in higher-dimensional rotating AdS black holes and suggest the existence of rotating hairy black hole solutions with scalar condensates, which we construct in a companion paper.

Figures (20)

• Figure 1: Top panel: Real part of the frequency, $\text{Re}(\omega L)$, as a function of $\theta = \frac{2}{\pi}\arctan(-\kappa)$ for scalar modes of global AdS$_3$ with $\mu^2 L^2 = -15/16$ and $m=0$, for the first five radial overtones $n=0,1,2,3,4$ (bottom to top curves). All overtones start at the Dirichlet green square on the left ($\kappa=+\infty$, i.e., $\theta=-1$), connect continuously to the Neumann red dot ($\kappa=0$), and for $n\ge 1$ (black curves) end at the Dirichlet green square at $\kappa\to -\infty$ ($\theta=+1$). These overtones have ${\rm Im}\,\hat{\omega}^{\rm AdS}_{n\ge1} = 0$ for all $\kappa$. (Frequencies $-\hat{\omega}$ are also solutions, not shown.) Bottom panel: Detailed view of $n=0$ (blue curve, also in the top panel). Left: $\text{Re}(\omega L)$; right: $\text{Im}(\omega L)$, both as functions of $-\kappa$ (restricted to $\kappa \in \mathbb{R}^-$). For $0 \ge \kappa \gtrsim -0.49$, $\text{Re}(\omega L) >0$ and $\text{Im}(\omega L) =0$. For $\kappa \lesssim -0.49$, $\text{Re}(\omega L) =0$ and $\text{Im}(\omega L) >0$, indicating that this AdS$_3$ mode becomes unstable, with $\underset{{\kappa \to -\infty}}{\lim} {\rm Im}(\omega L) \to +\infty$.
• Figure 2: Top panel: Real part of the frequency, $\text{Re}(\omega L)$, as a function of $\theta = \frac{2}{\pi}\arctan(-\kappa)$ for scalar modes of global AdS$_3$ with $\mu^2 L^2 = -15/16$ and $m=1$, for the first five radial overtones $n=0,1,2,3,4$ (bottom to top curves). All overtones start at the Dirichlet green square on the left ($\kappa=+\infty$, $\theta=-1$), connect continuously to the Neumann red dot ($\kappa=0$), and for $n\ge1$ (black curves, excluding the purple $n=0$ curve) end at the Dirichlet green square at $\kappa \to -\infty$ ($\theta=+1$). Overtones with $n\ge1$ have vanishing imaginary part, ${\rm Im}\, \hat{\omega}^{\rm AdS}_{n\ge1} = 0$ (frequencies $-\hat{\omega}$ are also solutions, not shown). Bottom panel: Detailed view of the $n=0$ mode (purple curve, also in the top panel). Left: $\text{Re}(\omega L)$; right: $\text{Im}(\omega L)$, as a function of $-\kappa$ (restricted to $\kappa \in \mathbb{R}^-$). For $0 \ge \kappa \gtrsim -1.01$, the mode is stable with $\text{Re}(\omega L) >0$ and $\text{Im}(\omega L)=0$, while for $\kappa \lesssim -1.01$ it becomes purely imaginary, $\text{Re}(\omega L) =0$, $\text{Im}(\omega L) >0$, indicating an $m=1$ instability. The instability grows as $\kappa \to -\infty$, $\underset{{\kappa \to -\infty}}{\lim} {\rm Im}(\omega L) \to +\infty$.
• Figure 3: Left panel:$\text{Im}(\omega L)$ vs. $\theta = \frac{2}{\pi}\arctan(-\kappa)$ for radial overtones $n=0,\dots,10$ of $m=0$ modes ($\mu^2 L^2 = -15/16$) for a BTZ black hole with $\{\hat{M},\hat{J}\} = \{5/16,1/4\}$ ($\{y_-,y_+\}=\{1/4,1/2\}$). Red discs at $\kappa=0$ ($\theta=0$) denote $\hat{\omega}^{\rm BTZ}_n|_{\rm Neu}$\ref{['BTZ:wNeu']}, and green squares at $\kappa = \pm\infty$ ($\theta = \mp 1$) denote $\hat{\omega}^{\rm BTZ}_n|_{\rm Dir}$\ref{['BTZ:wDir']}. The $n=0$ mode (blue curve) becomes unstable at $\kappa \sim -0.33$. Modes $n=0$ (blue) and $n=3,4,7,8,11,12,15,16,\dots$ (grey) have $\text{Re}(\omega L)=0$, unlike $n=1,2,5,6,9,10,13,14,\dots$ (black, see right panel). Right panel:$\text{Re}(\omega L)$ vs. $\theta(\kappa)$ for the black $n=1,2$ overtones. These modes have $\text{Re}(\omega L)=0$ when $\text{Im}(\omega L)$ is distinct; when $\text{Im}(\omega L)$ coincides, $\text{Re}(\omega L)$ is symmetric. The $n=0$ blue curve always has $\text{Re}(\omega L)=0$; for $0 \ge \kappa \gtrsim -0.33$, $\text{Im}(\omega L)<0$, while for $\kappa \lesssim -0.33$, $\text{Im}(\omega L)>0$, indicating instability with $\underset{{\kappa \to -\infty}}{\lim} \text{Im}(\omega L) \to +\infty$ (details in top panel of Fig. \ref{['fig:BTZw-kappa:m1_m5']}). See Fig. \ref{['fig:BTZw-kappa:m0-OtherMasses']} (Appendix \ref{['sec:AppOtherMasses']}) for other masses $\mu^2 L^2=-8/9,-3/4,-1/10$.
• Figure 4: Top-left:$\text{Im}(\omega L)$ vs. $\theta(\kappa)$ for radial overtones $n=0,\dots,9$ of co-rotating$m=1$ modes ($\mu^2 L^2=-15/16$) for a BTZ black hole with $\{\hat{M},\hat{J}\} = \{5/16,1/4\}$. Top-right: Double-trace BTZ modes in the complex plane, $\text{Im}(\omega L)$ vs. $\text{Re}(\omega L)$, including co-rotating $n=0,\dots,12$ (curves pass through red Neumann solutions with ${\rm Re}\,\hat{\omega}=+1$ on the right) and counter-rotating $n=0,1,2,3$ (curves pass through red Neumann solutions with ${\rm Re}\,\hat{\omega}=-1$ on the left). All curves are parametrized by $\kappa$; arrows indicate decreasing $\kappa$. Green squares at $\kappa\to \pm \infty$ mark $\hat{\omega}^{\rm BTZ}_n|_{\rm Dir}$\ref{['BTZ:wDir']}, and red discs at $\kappa=0$ mark $\hat{\omega}^{\rm BTZ}_n|_{\rm Neu}$\ref{['BTZ:wNeu']}. Bottom-left: Detail of the interaction/repulsion between the counter-rotating $n=0,1$ modes (dark-red and cyan) near $\kappa \sim -1.3$, where ${\rm Im}(\omega L)$ coincides but ${\rm Re}(\omega L)$ repels. Bottom-right: Zoom-in on the top-right panel showing the co-rotating $n=0,1,2$ curves. See Fig. \ref{['fig:BTZw-kappa:m1-OtherMasses']} (Appendix \ref{['sec:AppOtherMasses']}) for results with $\mu^2 L^2=-8/9,-3/4,-1/10$.
• Figure 5: $\text{Im}(\omega L)$ (diamonds) and $m\Omega_+ L - \text{Re}(\omega L)$ (gray circles) as a function of $\kappa$ for the unstable $n=0$ mode of a BTZ black hole with $\{\hat{M},\hat{J}\} = \{5/16,1/4\}$ and $\mu^2L^2=-15/16$. Top panel:$m=0$. Bottom-left panel:$m=1$. Bottom-right panel:$m=5$. The instability onset occurs at $\kappa=\kappa^{\rm BTZ}_{m,\hat{\mu}^2}$, where ${\rm Im}\,\hat{\omega}=0$ and $m\Omega_+ - {\rm Re}\,\hat{\omega}=0$. The magnitude of $|\kappa^{\rm BTZ}_{m,\hat{\mu}^2}|$ increases with $m$, approximately $-0.33$, $-0.87$, and $-1.99$ for $m=0,1,5$, respectively.