On the Nonlinear Stability of Asymptotically Anti-de Sitter Solutions

TL;DR

This work investigates the nonlinear stability of asymptotically anti-de Sitter spacetimes, arguing that many physically relevant solutions (geons, boson stars, and black holes) are nonlinearly stable in islands near global AdS, despite the known nonlinear instability of AdS itself. By analyzing high angular momentum perturbations through a WKB approach and drawing a parallel with the nonlinear Schrödinger equation on a torus, the authors show that approximate resonances are too weak to trigger instability in most cases and derive universal scaling relations for quasinormal mode deviations: $\mathrm{Re}\,\Delta\omega_n \sim K_d\ell^{-(d-3)/2}$ with long lifetimes $\tau_n \sim e^{\ell}$. They also construct noncoalescing binary black hole solutions allowed by AdS boundary conditions and discuss the nonlinear stability of horizonless backgrounds (geons and boson stars) under appropriate differentiability, identifying dimensional thresholds and islands of stability. The results have implications for gauge/gravity duality, suggesting that large-$N$ field theories can exhibit non-thermal, long-lived excitations and that AdS-like instability is a highly special feature of highly symmetric spacetimes. Overall, the paper provides a coherent framework linking perturbation theory, resonance structure, and nonlinear stability in asymptotically AdS settings.

Abstract

Despite the recent evidence that anti-de Sitter spacetime is nonlinearly unstable, we argue that many asymptotically anti-de Sitter solutions are nonlinearly stable. This includes geons, boson stars, and black holes. As part of our argument, we calculate the frequencies of long-lived gravitational quasinormal modes of AdS black holes in various dimensions. We also discuss a new class of asymptotically anti-de Sitter solutions describing noncoalescing black hole binaries.

Figures (5)

• Figure 1: The structure of solutions near empty AdS. The shaded regions denote islands of stability near 1-parameter families of geons or oscillons, both indicated by black lines with arrows pointing toward increasing amplitude. Because perturbation theory about empty AdS leads to geons only for a measure zero set of seed solutions, each such region has been drawn so that empty AdS lies at a cusp.
• Figure 2: WKB potential $V(z)$ for $r_+ = L$ and WKB allowed ($I,III$) and forbidden ($II$) regions for the corresponding particle orbits.
• Figure 3: Left Panel: Real part of the WKB quasinormal mode frequencies of the Schwarzschild-AdS black hole with respect to the AdS normal mode frequencies as a function of the WKB parameter $\ell_j$, in $d=4$ (and overtone $n=0$, and horizon radius $r_+=0.1 L$). Moving from left/top to right/bottom, the curves describe the scalar gravitational $(S)$, scalar field with BF mass $\mu^2L^2=-9/4$, vector gravitational $(V)$ and massless scalar field ($\mu=0$) cases (there are no regular tensor modes in $d=4$). Right Panel: Evolution of $L \,\Delta \omega_{S,\,n} =L \,\omega_{S,\,n}-L \,\omega_{S,\,n}^{AdS}$ as the dimension $d$ increases, for the scalar gravitational $(S)$ case. From the left/top to right/bottom of the figure we have the lines corresponding to the cases $d=6$, $d=5$, and $d=4$. In these plots the red continuous line is the best fit curve of the data to the curve $\Delta \omega_{j,\,n} = \alpha\, \ell_j^{\,\beta}$. The best fit value of $\beta$ is $-(d-3)/2$ to machine precision.
• Figure 4: The black solid line describes the WKB potentials $V_{\pm}(z)$ for the rotating MP black hole (in the non-superradiant regime, $\Omega_H\, L <1$). For comparison, the red dashed lines describe the same potential when the rotation vanishes, i.e. the Schwarzschild-AdS limit. We also show the WKB allowed ($I,III$) and forbidden ($II$) regions for a given (rescaled) frequency $w$. The left panel describes corotating modes ($m=\ell \gg 1$), while the right panel is for counter-rotating modes ($-m=\ell\gg 1$)
• Figure 5: Two 3-point interactions combine to make an effective 4-point vertex.