Instability of Flat Space Enclosed in a Cavity

TL;DR

A spherically symmetric self-gravitating massless scalar field enclosed inside a timelike worldtube R×S(3) with a perfectly reflecting wall is considered.

Abstract

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

Figures (5)

• Figure 1: Apparent horizon radius $r_{AH}$ (top) and corresponding formation time $t_{AH}$ (bottom) as a function of the amplitude of initial data \ref{['eq:idg']}. At critical points $\lim_{\varepsilon\rightarrow\varepsilon_{n}^+} r_{AH}(\varepsilon)=0$, while the horizon formation time exhibits jumps of size $t_{AH}(\varepsilon_{n+1})-t_{AH}(\varepsilon_{n})\approx 2$ (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 \ref{['eq:idg']} 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 $\varepsilon^{-2}\Pi^{2}(\varepsilon^{2} t,0)$. Plotted curves are labelled by the value of initial data amplitude $\varepsilon$.
• Figure 3: Plot of the weighted energy norm $\widetilde{E}(t)$ for the solution with initial amplitude $\varepsilon=8$. 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 $\varepsilon=6$. 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. \ref{['fig:ricci']} for the Neumann boundary condition [for the same initial data \ref{['eq:idg']}]. The time of apparent horizon formation exhibits the same type of scaling $t_{AH}\sim\varepsilon^{-2}$.