Note on trapped surfaces in the Vaidya solution

The Vaidya solution describes the gravitational collapse of a finite shell of incoherent radiation falling into flat spacetime and giving rise to a Schwarzschild black hole. There has been a question whether closed trapped surfaces can extend into the flat region (whereas closed outer trapped surfaces certainly can). For the special case of self-similar collapse we show that the answer is yes, if and only if the mass function rises fast enough.

Figure 1

A closed trapped surface we want to construct. The vertical coordinate is $t=v-r$, while the horizontal one is $r$. One dimension has been suppressed, and the 2-surface appears as a curve. The Vaidya region has been shrunk to a very thin shell, causing the trapped curve to have kinks as it passes through it.

Figure 2

A $v-r$-diagram, drawn for the case $\mu =1/2$. The trapped surfaces discussed in the text are represented by curves that become vertical at $v=M/\mu =2M$, and lie to the left of the curve representing the marginally trapped surface. From the companion paper [10] we know for certain that no closed trapped surfaces can extend below the dashed line. Readers who prefer Penrose diagrams are referred to Fig. 4.

Figure 3

“Capping”: In the $\Theta -V$-plane the straight line coming in from the left represents a cylinder, and the dashed line a round sphere. The curve interpolates between them with a quadrant of a circle, resulting in the intrinsic geometry shown on the right.

Figure 4

A Penrose diagram of the Vaidya solution for a mass function rising quickly enough for a closed trapped surface to penetrate into the flat region. It is shown as a dotted line emerging into the Schwarzschild part as a finite part of a line of constant $r$. Closed trapped surfaces cannot extend below the dashed line [10]. The part of flat space that lies inside the event horizon is shown without conformal distortion.