No naked singularities in homogeneous, spherically symmetric bubble spacetimes?

Abstract

We study the evolution of bubble spacetimes in vacuum and electrovac scenarios by numerical means. We find strong evidence against the formation of naked singularities in (i) scenarios with negative masses displaying initially collapsing conditions and (ii) scenarios with negative masses displaying initially expanding conditions, previously reported to give rise to such singularities. Additionally, we show that the presence of strong gauge fields implies that an initially collapsing bubble bounces back and expands. By fine-tuning the strength of the gauge field we find that the solution approaches a static bubble solution.

Figures (5)

• Figure 1: Bubble area vs. proper time at the bubble. In this and the following plots, we set $r_+ = 1$ (by choosing $b$ appropriately). The figure shows four illustrative examples of bubbles whose initial acceleration is positive. As it is evident, the expansion of the bubble continues and the difference is the rate of the exponential expansion. The relative error in these curves is estimated to be well below 0.001%.
• Figure 2: Convergence factor (CF) vs. proper time for the cases illustrated in Fig. \ref{['fig:area']}. This factor, defined as $CF=log_2( ||A(4\Delta)-A(2\Delta)||/||A(2\Delta)-A(\Delta)|| )$ should give the value of $2$ for a second order accurate implementation. Through these runs the grid spacing is defined as $\Delta = 3.75 \times 10^{-3}$ Clearly, the plot shows that second order convergence has been obtained throughout the runs (the peaks are instances with $0/0$ where the expression for $CF$ is ill-defined).
• Figure 3: Rescaled Kretschmann invariant $I R_{AH}^4/12$ vs. asymptotic time for $m=1.1$ (solid line) and $1.99$ (dashed line). The first non-zero values of the lines mark the formation of the apparent horizon. At late times, both lines approach the value of $1$ suggesting a black string has formed.
• Figure 4: Area values vs. proper time at the bubble for different values of $k$ and $m=1.1$. By tuning the value of $k$ appropriately, the amount of time that the area remains fairly constant can be extended for as long as desired.
• Figure 5: Bubble area vs. proper time at the bubble. The figure shows three illustrative examples of bubble with negative mass ($m = -0.1$ each) whose initial acceleration is negative. As it is evident, the collapse of the bubble is halted and the trend is completely reversed. The error in these curves is estimated to be well below 0.001%.