Shock waves in classical dust collapse

Abstract

During gravitational collapse of dust in spherical symmetry, matter particles may collide forming shell crossing surfaces (SCS) on which the Einstein equations become indeterministic. We show that there is a unique evolution beyond SCS such that a propagating shock wave forms, the metric remains continuous, and the stress-energy tensor dynamically becomes that of a thin shell. We give numerical simulations that exhibit this result.

Figures (2)

• Figure 1: An example of characteristic crossings for the initial density in the upper frame. The region to the left of the matter distribution is Minkowski spacetime where the characteristics are vertical lines.
• Figure 2: Shock formation: the top frame shows the initial dust density $\rho$, $P_\Lambda$, and apparent horizon function $\Theta$; the middle frame ($t=14.81$) shows shock formation with discontinuity in $P_\Lambda$; the bottom frame ($t=27.41$) shows the shock after entering the event horizon (the outer root of $\Theta$ at $r=10$), with shock located at the inner horizon $r\approx 7.5$.