Shell-crossings and shock formation during gravitational collapse in effective loop quantum gravity

Effective models of gravitational collapse in loop quantum gravity for the Lemaître-Tolman-Bondi spacetime predict that collapsing matter reaches a maximum finite density, bounces, and then expands outward. We show that in the marginally bound case, shell-crossing singularities commonly occur for inhomogeneous initial profiles of the dust energy density; this is the case in particular for all profiles that are continuous and of compact support, including configurations arbitrarily close to the Oppenheimer-Snyder model. When a shell-crossing singularity occurs, it is necessary to seek weak solutions to the dynamics; we argue that weak solutions typically contain shock waves.

Figure 1

Examples of characteristic curves for infalling dust from solutions of (10): from left to right, these show no crossing, crossing before the bounce, and crossing of a curve that has bounced with an ingoing one. At the point where characteristics cross, it becomes necessary to consider weak solutions. For ${\partial }_{R}m\ne 0$, shell-crossing singularities and characteristic crossing coincide, showing that these are not coordinate artefacts. The initial conditions for the curves shown here (with $\mathrm{\Delta }=1$) are: for the red curve, $r(0)=8,m(8)=0.2$ (for all three plots) and from left to right $\mathcal{E}(8)=2$, 1, 3; for the blue curve, $r(0)=10,m(10)=0.4$ (for all three plots), and from left to right $\mathcal{E}(10)=3$, 3, 2.