Globally regular instability of $AdS_3$

Abstract

We consider three-dimensional AdS gravity minimally coupled to a massless scalar field and study numerically the evolution of small smooth circularly symmetric perturbations of the $AdS_3$ spacetime. As in higher dimensions, for a large class of perturbations, we observe a turbulent cascade of energy to high frequencies which entails instability of $AdS_3$. However, in contrast to higher dimensions, the cascade cannot be terminated by black hole formation because small perturbations have energy below the black hole threshold. This situation appears to be challenging for the cosmic censor. Analysing the energy spectrum of the cascade we determine the width $ρ(t)$ of the analyticity strip of solutions in the complex spatial plane and argue by extrapolation that $ρ(t)$ does not vanish in finite time. This provides evidence that the turbulence is too weak to produce a naked singularity and the solutions remain globally regular in time, in accordance with the cosmic censorship hypothesis.

Figures (4)

• Figure 1: Results of convergence tests from runs performed on grids of size $2^n$ for $n$ from 10 to 16 (for the initial data \ref{['data']} with $\varepsilon=0.3$). The convergence factor for the solution $\Phi_n$ computed on the $2^n$-grid is defined by $Q_n=\frac{||\Phi_n-\Phi_{n+1}||}{||\Phi_{n+1}-\Phi_{n+2}||}$, where $||\cdot||$ is the spatial $\ell_2$-norm. By convention, we define the reliability time for the run on the $2^n$-grid as the time when $Q_n$ deviates from the expected value $2^4$ by, say, $7\%$ (depicted by the horizontal dashed line). We find empirically that the reliability time scales linearly with the product $n \,\varepsilon^{-2}$.
• Figure 2: Energy spectra at three instants of time for the initial data \ref{['data']} with $\varepsilon=0.3$ (which gives $M=0.044$). The fit of the formula (\ref{['fit']}) in the interval $10<k<1000$ to the data at $t=230$ is depicted by the black dotted line. The inset displays the same plot in the linear-log scale to better see the exponential decay of the tails.
• Figure 3: Time evolution of the width of analyticity $\rho$, obtained by fitting the formula (\ref{['fit']}) to the energy spectra for the same initial data as in Fig. 2. The result is fairly insensitive to the choice of the fitting interval. The fit of the formula \ref{['delta']} for $70<t<230$ gives $\rho_0=0.09$ and $T=63.4$ (dashed line).
• Figure 4: Time evolution of the $L^2$-norm of the second spatial derivative $\dot H_2=||\phi"(t,x)||_2$. This quantity rapidly oscillates in time so for clarity only the upper envelope of oscillations is plotted. In the inset, the curves corresponding to three different amplitudes $\varepsilon$ are shown to coincide after rescaling.