On weakly turbulent instability of anti-de Sitter space

Abstract: We study the nonlinear evolution of a weakly perturbed anti-de Sitter (AdS) spacetime by solving numerically the four-dimensional spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant. Our results suggest that AdS spacetime is unstable under arbitrarily small generic perturbations. We conjecture that this instability is triggered by a resonant mode mixing which gives rise to diffusion of energy from low to high frequencies.

• The paper establishes that even infinitesimally small perturbations trigger instability in AdS space through resonant mode mixing.
• It employs fourth-order finite-difference simulations of the Einstein-massless-scalar field equations to analyze energy diffusion to higher frequency modes.
• Findings indicate that cascading instabilities can delay collapse, implying a fundamental turbulent behavior unique to AdS geometries.

Weakly Turbulent Instability of Anti-de Sitter Space

This paper investigates the nonlinear dynamical stability of anti-de Sitter (AdS) space under the impact of weak perturbations. The research utilizes numerical simulations to explore the four-dimensional spherically symmetric Einstein-massless-scalar field equations with a negative cosmological constant. The principal finding of this study suggests an intrinsic instability in the AdS space when subjected to infinitesimally small, generic perturbations. The authors propose that this instability arises due to resonant mode mixing that facilitates energy diffusion from lower to higher frequency modes, leading to a potentially turbulent response within the structure of the AdS space.

Numerical Model and Results

The authors conduct numerical simulations using a fourth-order finite-difference code to analyze the scalar field's dynamic progression in the AdS settings. The numerical model employs Gaussian-type initial data with varying amplitudes to simulate initial perturbations. It is observed that solutions to these perturbations result in a cascading instability that is characterized by energy migration towards high-frequency modes. The results demonstrate that when the amplitude of perturbation decreases, the development of instability can be delayed, increasing the number of reflections the wave packet undergoes before collapsing into a black hole. A noteworthy scaling behavior indicating no lower threshold for instability was observed.

Mechanism of Instability

The theoretical framework employed involves the use of weakly nonlinear perturbation theory to dissect the mechanisms at play. At first order, solutions exhibit the linear stability that is common in such negatively curved spaces. However, as perturbations progress, resonant triads manifest within the scalar field's evolution equations, instigating secular terms that grow over time, eventually invalidating linear stability and requiring second-order corrections. A significant conclusion drawn is the failure to eliminate these secular terms through any multiscale or modulation approach, pointing towards a fundamental resonant interaction intrinsic to AdS spacetime under perturbations.

Implications and Theoretical Considerations

The study's implications are profound for theoretical physics, particularly in understanding the physical and mathematical properties of AdS and its relationship to phenomena such as gravitational collapse. The continuation and acceleration of energy towards high modes is indicative of weak turbulence—a phenomenon more typically observed in fluid dynamics and nonlinear systems, but here extending to the Einstein equations. The potential end-result, a black hole formation, underscores critical differences between AdS behavior and other spacetimes, pointing to the unique influence of negative cosmological curvature.

Future Perspectives

This research opens several inquiries for future study, especially in dissecting the distinction between dynamical effects due to negative cosmological constants and the inherently bounded nature of AdS spaces. Further exploration into higher dimensions, as mentioned for $AdS_5$ within the paper, would expand understanding of the geometric influence and might yield significant insights relevant to the AdS/CFT correspondence. Moreover, there lies an opportunity to extend these findings towards defining and exploring stability within other non-integrable nonlinear wave equations across bounded domains, possibly informing new theoretical frameworks or even observational parallels in modern cosmology.

In conclusion, this paper significantly contributes to the understanding of nonlinear instabilities in AdS spaces, backed by robust numerical experiments and theoretical analysis, paving the path for nuanced questions around the stability of spacetime under weak perturbations.