Instability of nonlinear scalar field on strongly charged asymptotically AdS black hole background

TL;DR

This study analyzes the nonlinear dynamics of a cubic conformal scalar field on RNAdS spacetimes with Robin boundary conditions, spanning defocusing and focusing nonlinearities. By linearizing and examining static solutions, it maps a charge- and boundary-condition–dependent phase space that separates stable and unstable regimes, including the extremal limit. The main results show a transition near high charges: defocusing dynamics tend toward nontrivial static attractors or maintain stability, while focusing dynamics exhibit finite-time blow-up above a critical charge, with a coexisting mechanism where horizon energy influx and unbounded energy play roles. The work highlights two distinct instability channels, draws connections to extremal and rotating black holes, and outlines natural extensions to Kerr–AdS and self-gravitating settings as future directions with potential broad implications for holographic scalar dynamics.

Abstract

The conformally invariant scalar equation permits the Robin boundary condition at infinity of asymptotically-AdS spacetimes. We show how the dynamics of conformal cubic scalar field on the Reissner-Nordström-anti-de Sitter background depend on the black hole size, charge, and choice of the boundary condition. We study the whole range of admissible charges, including the extremal case. In particular, we observe the transition in stability of the field for large black holes at the specific critical value of the charge. Similarities between Reissner-Nordström and Kerr black hole let us suspect that a similar effect may also occur in rotating black holes.

Figures (8)

• Figure 1: Phase plots $(y_{H},\beta)$ for increasing charge $\sigma$. For $\beta$ below the critical curve $\beta_*(y_H,\sigma)$ (black line) the trivial solution $\phi=0$ is linearly unstable. This line also separates regions with distinct behaviour for nonlinear problem.
• Figure 2: Phase plots for $\sigma$ approaching the extremality, cf. Fig. \ref{['fig:phase_plots_small']}. The critical curve separates the phase space into regions (denoted by various shades of gray) characterised by different sets of static solutions of the nonlinear equation, see Sec. \ref{['sec:DefocusingStaticSolutions']} and \ref{['sec:FocusingStaticSolutions']}. New regions emerge at charges $\sigma_{w}^{(n)}$, see Tab. \ref{['tab:critical_sigmas']}. For $y_{H}<\sqrt{2}$ the winding number goes to infinity as $\sigma\rightarrow 1$.
• Figure 3: Phase plot for extremal case $\sigma=1$. The critical curve $\beta_{*}(y_{H},1)$ terminates at $y_{H}=\sqrt{2}$. The white and gray regions correspond to the analogous parts of the phase plots shown in Figs. \ref{['fig:phase_plots_small']} and \ref{['fig:phase_plots_large']}, whereas the black part is the unique feature of the extremal case.
• Figure 4: Values of $\beta$ as a function of $1-\sigma$ for various $y_H$. For $y_H\geq\sqrt{2}$ we observe convergence with $(1-\sigma)\rightarrow 0$, whereas for $y_H<\sqrt{2}$ the Robin parameter $\beta$ appears approximately periodic in $\log(1-\sigma)$.
• Figure 5: Phase space for $\sigma=1-10^{-12}$ without the periodic identification of the Robin parameter $\beta$. Each point of this diagram represents a static solution. Static solutions for parameters $(y_{H},\beta)$ on the dashed lines satisfy Dirichlet boundary condition. Those lines separate regions populated by static solutions with different numbers of zeroes. Solutions with small amplitude $c$, see \ref{['eqn:static']} and \ref{['eqn:initial']}, bifurcate from $u=0$ at the critical curve $\beta_{*}$ in directions presented by the arrows, depending on the type of the nonlinearity. Hence, the critical curve $\beta_{*}$ separates the diagram into two regions, gray and white, in which there exist static solutions to the defocusing and focusing cases respectively. As $c$ increases static solutions in the defocusing case eventually become singular at $\beta=-\pi/2$, while for the focusing nonlinearity the Robin parameter $\beta$ increases indefinitely and static solutions with more and more zeroes appear. The original phase plot, cf. Fig. \ref{['fig:phase_plots_extremal']}, can be recovered by cutting this plot along the dashed lines and stacking the obtained fragments one on another.