Instability of Flat Space Enclosed in a Cavity

We consider a spherically symmetric self-gravitating massless scalar field enclosed inside a timelike worldtube $\mathbb{R}\times S^3$ with a perfectly reflecting wall. Numerical evidence is given that arbitrarily small generic initial data evolve into a black hole.

Figure 1

Apparent horizon radius $r_{\mathrm{AH}}$ (top) and corresponding formation time $t_{\mathrm{AH}}$ (bottom) as a function of the amplitude of initial data (13). At critical points $\underset{\epsilon \to {\epsilon }_n^{+}}{lim}{r}_{\mathrm{AH}}(\epsilon )=0$, while the horizon formation time exhibits jumps of size $t_{\mathrm{AH}}({\epsilon }_{n+1})-t_{\mathrm{AH}}({\epsilon }_n)\approx 2$ (time in which the pulse traverses the cavity back and forth).

Figure 2

(a) The function ${\Pi }^2(t,0)$ for solutions with initial data (13) for several moderately small amplitudes. For clarity of the plot only the envelopes of rapid oscillations are depicted. (b) The curves from the plot (a) scaled according to ${\epsilon }^{-2}{\Pi }^2({\epsilon }^{2}t,0)$. Plotted curves are labelled by the value of initial data amplitude $\epsilon$.

Figure 3

Plot of the weighted energy norm $\tilde{E}(t)$ for the solution with initial amplitude $\epsilon =8$. The steep bursts of growth occur when the pulse implodes through the center.

Figure 4

Energy spectra at three moments of time (initial, intermediate, and just before collapse) for the solution with initial amplitude $\epsilon =6$. The fit of the power law $E_j\sim j^{-\alpha }$ performed on the interval $j\in [16,128]$ gives the slope $\alpha \approx 1.2$.

Figure 5

The analogue of Fig. 2 for the Neumann boundary condition [for the same initial data (13)]. The time of apparent horizon formation exhibits the same type of scaling $t_{\mathrm{AH}}\sim {\epsilon }^{-2}$.