Outer trapped surfaces in Vaidya spacetimes

It is proven that in Vaidya spacetimes of bounded total mass, the outer boundary, in spacetime, of the region containing outer trapped surfaces, is the event horizon. Further, it is shown that the region containing trapped surfaces in these spacetimes does not always extend to the event horizon.

Figure 1

A spacetime diagram of the collapse of a thin null dust shell in flat space. Two angular dimensions are suppressed. Points in this diagram are, therefore, 2-spheres. The region to the left of the shell is a portion of flat space. The region to the right of it is a portion of a Schwarzschild spacetime. The event horizon, a null 3-surface, is shown. The point on the left represents a single event (as part of a 2-sphere). It lies inside the event horizon but is deep in the flat region. Is there an outer trapped surface that contains this point?

Figure 2

Left, a narrow tube shaped like a worm’s skin. The ingoing and outgoing directions are shown. Right, a similar tube closed off differently. Now it is closed inside-out, like the face of Pacman. As a result the ingoing and outgoing directions of the tube on the left, are interchanged. Therefore, a portion of an inner trapped surface (the worm) can be used in constructing an outer trapped surface (the Pacman). The specific closing off inside-out will be done in the far future, inside the apparent horizon, where 2-spheres are outer trapped.

Figure 3

Top: A spacetime diagram of a Vaidya spacetime. Two angular dimensions are suppressed. Points in this diagram are, therefore, 2-spheres. A spherically symmetric spacelike 3-surface is shown. Bottom: A schematic of this 3-surface. One angular dimension is not shown. The desired outer trapped surface is the line traced on this 3-surface.

Figure 4

A spacetime diagram of the collapse of null dust in flat space. The coordinate $t$ is such that $(\frac{\partial }{\partial t}{)}^a$ is pointing upwards in the flat region in the diagram. The surfaces $v=v_{\mathrm{flat}}$ and $t=t_0$ are shown. The excluded region is the region inside the event horizon satisfying $t<t_0$. This is the shaded triangular region in the diagram. It follows from the proposition that trapped surfaces cannot extend into this region. As a result, the boundary of the region containing trapped surfaces, is not, in general, the event horizon.