Weakly Turbulent Instability of Anti–de Sitter Spacetime

We study the nonlinear evolution of a weakly perturbed anti–de Sitter (AdS) space by solving numerically the four-dimensional spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant. Our results suggest that AdS space 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.

Figure 1

Horizon radius vs amplitude for initial data (9). 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 (9) for four moderately small amplitudes. For clarity of the plot only the upper envelopes of rapid oscillations are depicted. After making between about 50 (for $\epsilon =6\sqrt{2}$) and five-hundred (for $\epsilon =3$) reflections, all solutions finally collapse. (b) The curves from the plot (a) after rescaling ${\epsilon }^{-2}{\Pi }^2({\epsilon }^{2}t,0)$.

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 $\varphi (0,x)=\epsilon \mathbf{(}\frac{1}{d_0}{e}_0(x)+\frac{1}{d_1}{e}_1(x)\mathbf{)}$ with $\epsilon =0.088$.