Collapsing shells, critical phenomena, and black hole formation

We study the gravitational collapse of two thin shells of matter, in asymptotically flat spacetime or constrained to move within a spherical box. We show that this simple two-body system has surprisingly rich dynamics, which includes prompt collapse to a black hole, perpetually oscillating solutions or black hole formation at arbitrarily large times. Collapse is induced by shell crossing and the black hole mass depends sensitively on the number of shell crossings. At certain critical points, the black hole mass exhibits critical behavior, determined by the change in parity (even or odd) of the number of crossings, with or without mass-gap during the transition. Some of the features we observe are reminiscent of confined scalars undergoing “turbulent” dynamics.

Figure 1

Evolution of a “free” double-shell system under their own gravitational field. The solid (red) curve denotes the outermost shell and dotted (blue) points the innermost. We take the total gravitational mass $M_1=1$, and the equation of state to be specified by $w=0.86$. Both shells have identical mass parameter $m=2.9$; the inner gravitational mass was tuned to $M_2=0.663$. The crossing point is at $R=2.10533719$.

Figure 2

Evolution of a double-shell spacetime, when confined to a box of radius $R_{\mathrm{ext}}=24$. The upper panel shows shell positions as function of coordinate time $t$ as measured between shells, for $M_1=1$, $w=0.2$, $m=0.9$, $R_i=4.45825863$. Lower panel shows the lowest value of $1-2M_{1,2}/R_{1,2}$ computed at the surface of both shells.

Figure 3

Evolution of a double-shell spacetime with type A initial conditions, when confined to a box of radius $R_{\mathrm{ext}}=24$. We find no collapse (or then it occurs only after a very large number of crossings not covered by our numerical results) for $7.2\lesssim R_i\lesssim 14.2$. We find arbitrarily large number of crossings at each critical point, where the number of crossings changes.

Figure 4

BH mass as a function of the initial energy content $\delta$. From right to left, before each peak in the BH mass, the number of shell crossings is 1, 3, 5, 7, 9. We do not observe any mass gap in the transitions. For $\delta \lesssim 0.13267$ we find no collapse.