Quantum Gravity of Dust Collapse: Shock Waves from Black Holes

We study the quantum gravitational collapse of spherically symmetric pressureless dust. Using an effective equation derived from a polymer quantization in the connection-triad phase space variables of general relativity, we find numerically, for a variety of initial dust configurations, that (i) trapped surfaces form and disappear as an initially collapsing density profile evolves into an outgoing shock wave; (ii) black hole lifetime is proportional to the square of its mass; and (iii) there is no mass inflation at inner apparent horizons. These results provide a substantially different view of black hole formation and subsequent evolution than found from semiclassical analyses.

Figure 1

Dust density $\rho$ (left column—the vertical scale varies by frame) and apparent horizon function ${\mathrm{\Theta }}_{+}$ (right column) at the times indicated for initial data ${\rho }_0(r)=\exp [-2(x-1.5M{)}^2]/5+\exp [-4(x-3.5M{)}^2]/10$ with $M=5$ in (23). The first two rows show the collapse, the middle one shows the bounce, and the last two show the outgoing shock wave. Apparent horizons are at the zeros of ${\mathrm{\Theta }}_{+}$; the outermost horizon forms and stays at $r=x=2M=10$ until the shock wave reaches it at $t\sim 202$.

Figure 2

Black hole lifetime for initial data (21) with $\sigma =0.5$.

Figure 3

Black hole lifetime for initial data (22) with $\sigma =0.5$.

Figure 4

Conformal diagram for dust collapse and bounce. The red lines show the outer and inner dynamical horizons—the outer one becomes null once all infalling matter has passed through; the dashed blue line shows a typical ingoing dust trajectory; the solid blue lines show the outgoing shock wave trajectory according to the interior and exterior metrics, respectively—these metrics differ due to the shock discontinuity; the shaded portion of the diagram is excised and the two solid blue lines are identified. An outside observer sees the outer horizon disappear as the shock wave emerges.