Figure 1

A Penrose diagram of the OS black hole. The shadowed region is a collapsing part of a closed Friedmann dust model, and the rest is Schwarzschild. The two regions are matched at a comoving timelike hypersurface $\chi ={\chi }_0$. (The coordinates used are explained in the text.) EH denotes the event horizon, and A3H denotes a timelike tube of marginally trapped surfaces. They meet in a sphere at the junction. The simple argument in Sec. 3 shows that there are round trapped surfaces above the past light cone defined by the dashed line. Together with EH it defines a causal diamond in which the boundary of the trapped region must lie. The boundary must lie above the hypersurface $\Sigma$, which has constant Kodama time and forms a past barrier for trapped surfaces [9]. The interior time at the upper vertex of the diamond defines a round 2-sphere on the matching hypersurface whose area coordinate in the exterior is given by $r=\overline{r}=(M/2){cos}^{-2}({\chi }_0/2)$, and this is always less than $M$, also indicated in the figure.Reuse & Permissions

Figure 2

A Penrose diagram of a Vaidya solution. The matter region is filled with null dust. Again there is a tube of marginally round trapped spheres within the matter region, but it is a spacelike tube of spheres that cease to be trapped if deformed outwards within suitable hypersurfaces. For instance, the spacelike hypersurface going out to $i^0$ marked in blue intersects A3H twice, the inner intersection is unstable, while the outer is stable within the given hypersurface. It can be checked that outermost intersections are always stable in this sense. A (nonround) trapped surface going through the center [14] is marked with red dots.Reuse & Permissions

Figure 3

A conformal diagram of the OS black hole. Round spheres centered at the origin are represented by two symmetrically placed points, so that the matching hypersurface here is represented by the two vertical lines with constant $\chi ={\chi }_0$. If the centered round spheres are situated above A3H they are trapped, and they are marginally trapped at A3H itself. However, as the shaded Friedmann region is spatially homogeneous—so that the $\eta =\mathrm{\text{constant}}$ hypersurfaces (horizontal lines in the shaded part of the diagram) are maximally symmetric—we can move these round spheres, as well as A3H, and center them anywhere on the slices as long as they do not enter the Schwarzschild region. This shows that there are trapped round spheres passing through every point of the interior for all $\eta >2\pi -{\chi }_0$. The dashed lines are obtained by shifting the marginally trapped tubes, centering them at values of $\chi \ne 0$, as for example the marginally trapped tube denoted A3H’. The two dots at the end represent a marginally trapped round sphere tangent to the matching hypersurface. The marginally trapped tube denoted A3H” contains a marginally trapped round sphere—represented by the dots—tangent to both the matching hypersurface and the center at $\eta =2\pi -{\chi }_0$.Reuse & Permissions

Figure 4

Three different spatial slices of the OS model embedded in Euclidean space with one dimension suppressed. (a) A constant $\eta$ hypersurface in Friedmann matched to a spacelike hypersurface in Schwarzschild. Horizontal circles represent round spheres. The dashed circle is the sphere at $\chi ={\chi }_0$ where the matching is made. The continuous circles are marginally trapped spheres inside the dust cloud. One of them lies on A3H at a constant value of $\chi$. The other one is obtained by tilting the first one until it becomes tangent to the surface of the star. We see that the tilted circle reaches smaller values of $\chi$. (b) The same spatial slice as in (a) with the equatorial plane drawn as a black curve. This surface reaches the center at $\chi =0$, but it also extends into the Schwarzschild region and it is not clear whether it can be closed or not. (c) The spatial slice on which the surface of Sec. 4a lies. The surface is drawn as a black curve. At a small enough value of $r$ it deviates from the equatorial plane and is closed. Far from the cloud the hypersurface is bent so that it reaches spatial infinity—rather than ending up in the singularity—giving an intuitive definition of an “outer” direction on the surface.Reuse & Permissions

Figure 5

The plots show how to choose $r_0$ and $k$ so that the value of $k_0$ becomes the smallest possible when ${\chi }_0=\pi /4$. The plot to the left helps us identify the allowed choices. Above the dotted curve the condition (23) holds and the surface is spacelike. Below the dashed curve $k$ satisfies the inequality (25), with equality on the curve, and the surface is trapped in the exterior. To the left of the continuous curve the surface is trapped at the center $\chi =0$, and on the curve it is marginally trapped there. We conclude that the allowed region cannot extend past the continuous and dashed curves to the right. The contour plot to the right shows the value of $k_0$ with the forbidden (white) region excluded. The darker the shade, the smaller the value of $k_0$. We see that the best choice of $r_0$ and $k$ is the values they take where the continuous and the dashed curves intersect.Reuse & Permissions

Figure 6

Penrose diagrams of the interior of the dust cloud with ${\chi }_0=\pi /12$, ${\chi }_0=\pi /4$ and ${\chi }_0=5\pi /12$ from left to right. The dotted curves represent the best surfaces we get in the first construction. A point on each curve really represents a sphere in the Penrose diagram, but only the equator of each such sphere is part of the surface. The dashed curves represent the marginally trapped surfaces in the second construction. The two results are barely distinguishable to the eye in this figure.Reuse & Permissions

Figure 7

The OS black hole once again, with some novel decoration. The equatorial planes of the blue hypersurface can be joined to appropriate cylinders in the Schwarzschild region—represented here by the dashed line—such that they can eventually be capped (the final dot) to produce topological spheres that are (weakly) trapped, as has been explicitly demonstrated in the main text. The purple line together with the dashed and blue lines indicates how a possible spacelike and asymptotically flat hypersurface containing the trapped surface looks in the diagram. This hypersurface corresponds to the spatial slice shown in Fig. 4c. The particular hypersurface called $\sigma$ shown in red has minimal equatorial planes and joins the past barrier $\Sigma$ and the EH at the round 2-sphere with $\chi ={\chi }_0$ and $\eta =2(\pi -{\chi }_0)$, as represented. In the main text we prove that $\sigma$ is a past barrier for axially symmetric trapped surfaces which are confined to equatorial planes within the interior part. One wonders if $\sigma$ can be promoted to a new past barrier for more general trapped surfaces, leaving even less room for the boundary $\mathcal{B}$ to be placed.Reuse & Permissions