Scalar Collapse in AdS

TL;DR

This paper demonstrates that the AdS scalar collapse instability persists for a complex scalar field, showing that arbitrarily small initial data can evolve to a black hole via repeated boundary reflections, even in a finite reflecting domain. The authors formulate the dual gravity–bulk system, implement a stable numerical scheme with adaptive mesh refinement, and analyze boundary CFT observables derived from asymptotic scalar behavior. Through a weakly nonlinear analysis in the oscillon basis, they show that if all linearized oscillons are excited, resonant instabilities at order $\mathcal{O}(\epsilon^3)$ can be removed by a resonance-frequency shift, independent of global charge. They further reveal an exponential growth of the boundary spectral bandwidth and a Kolmogorov-like energy spectrum in the bulk, linking gravitational focusing in AdS to turbulence-like dynamics and providing a phase-space-like diagnostic via bandwidth growth.

Abstract

Recently, studies of the gravitational collapse of a scalar field within spherically symmetric AdS spacetimes was presented in \cite{Bizon:2011gg,Jalmuzna:2011qw} which showed an instability of pure AdS to black hole formation. In particular, the work showed that arbitrarily small initial configurations of scalar field evolved through some number of reflections off the AdS boundary until a black hole forms. We consider this same system, extended to include a complex scalar field, and reproduce this phenomena. We present tests of our numerical code that demonstrate convergence and consistency. We study the properties of the evolution as the scalar pulse becomes more compact examining the asymptotic behavior of the scalar field, an observable in the corresponding boundary CFT. We demonstrate that such BH formation occurs even when one places a reflecting boundary at finite radius indicating that the sharpening is a property of gravity in a bounded domain, not of AdS itself. We examine how the initial energy is transferred to higher frequencies --which leads to black hole formation-- and uncover interesting features of this transfer.

Figures (11)

• Figure 1: Mass $M$ of initial configurations \ref{['phiQz']} as a function of $\epsilon$ for $d={3,4,5,6}$ (blue/red/green/orange curves). Note that $M\propto \epsilon^2$ for $\ln\epsilon\lesssim 4$. While the coordinate extent of the scalar profile is the same in all $d$, higher dimensional $AdS_{d+1}$ more efficiently "localizes" it, resulting in smaller $M_d$ for a fixed $\epsilon$.
• Figure 2: Mass $M$ of initial configurations \ref{['phiQnz']} as a function of $\epsilon$ for $d=3\cdots6$ (left panel, blue$\cdots$orange) and corresponding ratios $\frac{M}{Q}$ (right panel). Here, $M\propto \epsilon^2$ scaling of mass extends only up to $\ln\epsilon \sim 1$. Note that $\frac{M}{Q}$ approaches a constant as $M\to 0$, while $\ln\frac{M}{Q}\propto M$ for large $M$, effectively making the charge negligible. The rescaling of the horizontal axis of the right panel by $5^{d-3}$ is made for readability of the graph.
• Figure 3: Convergence test for two bounces with $d=3$. The initial data is of the form \ref{['phiQz']} with $\epsilon=20.01$ and $\sigma=1/16$. ( top) The order of convergence obtained from comparisons of the metric function $A(x,t)$ at successively doubled resolutions. The convergence order is computed from three different resolutions, a base resolution and runs with half and one-quarter the base grid solution. Thus, run "32" has a grid spacing $2^{-5}$ that of the run labeled "1."( middle) Order of convergence obtained for $\Phi_1(x,t)$. Both these results indicate convergence is better than third order convergent. ( bottom) The logarithm of the L2-norm of the momentum constraint residual for just the three best resolutions. That it decreases with increasing resolution suggests the results are converging to a proper solution of Einstein's equations. Note that this residual is computed only at first order of accuracy because it involves a time derivative and therefore one should not estimate the order of convergence from it.
• Figure 4: Demonstration of the increasingly compact gravitational potential with successive bounces of initial data of form \ref{['phiQz']} with $\epsilon=20$ and $\sigma=1/16$. Six bounces are shown before a black hole forms. ( top) The minimum of $A(x,t)$ as a function of time. At the final time, the minimum approaches zero, signaling black hole formation. ( middle) The $x$-coordinate where $A(x,t)$ achieves its minimum. In these coordinates, the speed of light is unity, and the line segments roughly indicate the motion of the scalar pulse back and forth across the grid. That compaction increases with each bounce is quite hard to observe on the linear scale, and so the logarithm of this data is shown at bottom. The evolution terminates at the last time shown as the solution approaches black hole formation.
• Figure 5: The behavior of the scalar field at the boundary with successive bounces for the same evolution as in Fig. \ref{['fig:bounces']}. At left, the asymptotic value $\phi_3^{(1)}(t)$ from Eq. (\ref{['eq:asymptotics']}) is shown for segments of time of length $\pi$. The pulse sharpens with each "implosion" through the origin. On the right is shown the Fourier transform of the signal on the left. Shown is the natural logarithm of the amplitude of the Fourier component squared as a function of (the log of) angular frequency $\omega$. The vertical scale is arbitrary but fixed for all bounces. The transform shows that energy is shifted to higher frequencies.