Study of the nonlinear instability of confined geometries

The discovery of a “weakly turbulent” instability of anti–de Sitter spacetime supports the idea that confined fluctuations eventually collapse to black holes and suggests that similar phenomena might be possible in asymptotically flat spacetime, for example in the context of spherically symmetric oscillations of stars or nonradial pulsations of ultracompact objects. Here we present a detailed study of the evolution of the Einstein-Klein-Gordon system in a cavity, with different types of deformations of the spectrum, including a mass term for the scalar and Neumann conditions at the boundary. We provide numerical evidence that gravitational collapse always occurs, at least for amplitudes that are three orders of magnitude smaller than Choptuik’s critical value and corresponding to more than ${10}^5$ reflections before collapse. The collapse time scales as the inverse square of the initial amplitude in the small-amplitude limit. In addition, we find that fields with nonresonant spectrum collapse earlier than in the fully resonant case, a result that is at odds with the current understanding of the process. Energy is transferred through a direct cascade to high frequencies when the spectrum is resonant, but we observe both direct- and inverse-cascade effects for nonresonant spectra. Our results indicate that a fully resonant spectrum might not be a crucial ingredient of the conjectured turbulent instability and that other mechanisms might be relevant. We discuss how a definitive answer to this problem is essentially impossible within the present framework.

Figure 1

Left panels: Example of numerical initial data. The initial profile is given by Eq. (24a) with $AR=0.125$ and $w=0.125R$. The left top panel shows the source $\mathrm{\Pi }$ and the conformal factor as computed numerically with resolution $dr=R/1000$. The left bottom panel shows the normalized error of the Hamiltonian constraint at $t=0$. Right panels: Comparison between analytic and numerical initial data. In the right top panel we compare the numerical and analytic solution for a source described in Eq. (24b) with $AR=0.1$ and $w=0.125R$. The bottom right panel shows the source profile including the conformal factor.

Figure 2

Collapse time as a function of initial amplitude $A$ with Dirichlet boundary conditions imposed at $r=R$, pulse width $w=0.125R$ and various scalar field masses. For this initial data (24b), Choptuik’s critical amplitude for prompt collapse corresponds to $AR\sim 8$.

Figure 3

Collapse time versus ratio of Compton wavelength to the width of initial pulse ($\mu w$) for Dirichlet boundary. Initial data are described in (24a) with $dr=R/2000$.

Figure 4

Collapse time as a function of initial amplitude with Neumann boundary conditions and compared with Dirichlet boundary conditions for a massless scalar field. Initial data are the same in both cases: we considered the initial pulse (24b) with two choices of the width, namely $w=0.125R$ and $w=0.15R$.

Figure 5

The evolution of the Kretschmann scalar for three representative cases with resolution $dr=R/3200$: a massless scalar field with Dirichlet boundary (top panel) and with Neumann boundary (middle panel) and a massive scalar field with $\mu R=4$ and Dirichlet boundary. In all cases initial data are prescribed through Eq. (24a), $AR=0.04$ and $w=0.125R$. The curves are truncated at the time of formation of an apparent horizon. Note that the case corresponding to a fully resonant spectrum is the one for which collapse takes longer. Our results show that the Kretschmann scalar at the origin scales with the fourth-power of the initial amplitude, $A^4$, at least for small amplitudes and well before horizon formation.

Figure 6

Fourier-transform ${\tilde{\mathrm{\Pi }}}_j$ of the conjugate momentum $\mathrm{\Pi }$ at the origin for different time intervals. We consider three intervals corresponding to the first 10 reflections ($j=1$, black curves, left panels), last 10 reflections before the collapse ($j=11$, blue curves, right panels), and for 10 reflections roughly centered at $t=t_{\mathrm{BH}}/2$ ($j=7$, red curves, middle panels). The top, central and bottom panels show the case of massless fields with Dirichlet boundary, massless field with Neumann boundary and massive field with Dirichlet boundary, respectively. All cases refer to initial data (24a) with $AR=0.04$ and $w=0.125R$.

Figure 7

Direct and inverse cascade effects. The difference $\mathrm{\Delta }{\tilde{\mathrm{\Pi }}}_{j,j-1}\equiv {\tilde{\mathrm{\Pi }}}_j-{\tilde{\mathrm{\Pi }}}_{j-1}$ at the center between two consecutive time intervals is shown in the high-frequency region $40\le \omega R\le 100$. Left, middle and right panels correspond to different time intervals close to the formation of the apparent horizon (from left to right, $j=9,10,11$). The top, middle and bottom panels show the case of massless fields with Dirichlet boundary, massless field with Neumann boundary and massive field with Dirichlet boundary, respectively. In the massless Dirichlet case the difference is always positive and the eigenmodes are clearly excited, confirming a direct-cascade effect. In the Neumann and Dirichlet massive cases we observe both direct and inverse cascade. All cases refer to initial data (24a) with $AR=0.04$ and $w=0.125R$.

Figure 8

Example of the time of apparent horizon formation as a function of the amplitude in the range of moderately large amplitudes with initial profile (24b). The function displays an upward-stairway behavior for decreasing amplitudes which is connected to the power-law behavior shown in Fig. 2 at smaller amplitudes.

Figure 9

Code tests with different boundary conditions and different mass terms. The top left plots, the top right plots and the bottom left plot correspond to Dirichlet boundary conditions with $\mu R=0$, Neumann boundary conditions with $\mu R=0$ and Dirichlet boundary conditions with $\mu R=4$, respectively. Initial data correspond to Eq. (24a). For each plot, the upper and middle panels show the convergence test for $\mathrm{\Pi }$ at the center as a function of time, using different resolutions: $dr=R/1600,R/2000,R/2600$ and $R/3200$. Our results show second-order convergence as expected by the initial data solver. For each plot, the lower panel shows the Hamiltonian constraint as a function of coordinate $r$ on different time slices with $dr=R/3200$ and normalized by the sum of the absolute value of each term in the Hamiltonian constraint. Bottom right plot: convergence factor computed through Eq. (b1). The upper panel shows the convergence factor $|{\mathrm{\Pi }}_{2000,2600}|/|{\mathrm{\Pi }}_{2600,3200}|$ using number of grid points $N=2000,2600$ and 3200, while the lower panel shows $|{\mathrm{\Pi }}_{2600,2800}|/|{\mathrm{\Pi }}_{2800,3200}|$ using $N=2600,2800$ and 3200. All the plots refer to the initial data set by Eq. (24a) with $AR=0.04$ and $w=0.125R$.