On weakly turbulent instability of anti-de Sitter space

TL;DR

The results suggest that AdS space is unstable under arbitrarily small generic perturbations, and it is conjecture that this instability is triggered by a resonant mode mixing which gives rise to diffusion of energy from low to high frequencies.

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.

Figures (3)

• Figure 1: Horizon radius vs amplitude for initial data \ref{['idata']}. The number of reflections off the AdS boundary before collapse varies from zero to nine (from right to left).
• Figure 2: (a) $\Pi^2(t,0)$ for solutions with initial data \ref{['idata']} for four moderately small amplitudes. For clarity of the plot only the upper envelopes of rapid oscillations are depicted. After making between about fifty (for $\varepsilon=6 \sqrt{2}$) and five-hundred (for $\varepsilon=3$) reflections, all solutions finally collapse. (b) The curves from the plot (a) after rescaling $\varepsilon^{-2} \Pi^2(\varepsilon^2 t,0)$.
• Figure 3: Evolution of the fraction of the total energy contained in the first $(k+1)$ modes $\Sigma_k:=\frac{1}{M}\sum_{j=0}^k E_j$ for the two-mode initial data $\phi(0,x)=\varepsilon \left(\tfrac{1}{d_0} e_0(x)+\tfrac{1}{d_1} e_1(x)\right)$ with $\varepsilon=0.088$.