Figure 1

Schematic diagram of the situation relevant to Corollary 4.1. For simplicity the picture is drawn in flat spacetime, one dimension suppressed. The trapped surface $S$, represented here by a red line, has to bend down in time $t$. $\Sigma$ represents a spacelike hypersurface with positive definite second fundamental form.Reuse & Permissions

Figure 2

Conformal diagram of the Robertson-Walker spacetime (9) for $k=1$ and dust. As usual, null radial lines are drawn at 45° and the future direction is upwards. Closed f-trapped surfaces exist in the colored region (in red, this is $\mathcal{T}$), defined by $\dot{a}<0$. The boundary $\mathcal{B}$ is the maximal hypersurface with $\dot{a}=0$, the recollapsing time, shown here as a dotted horizontal line. A marginally trapped tube—corresponding to the apparent 3-horizon AH to be defined in Sec. 6—foliated by marginally f-trapped round spheres in also shown. The region $\mathcal{Z}$ to the future of the AH is a core of the trapped region, as defined in Sec. 11. If $\mathcal{Z}$ is removed from the spacetime, then there are no remaining closed f-trapped surfaces.Reuse & Permissions

Figure 3

This is the scheme for the vector fields involved in the perturbation analysis. The spherically symmetric null directions are represented at 45°, defined by the radial null vector fields $\overrightarrow{\ell }$ and $\overrightarrow{k}$ given in (11). The vertex at the center represents a marginally f-trapped round sphere in the apparent 3-horizon AH, which is also drawn nearby. The vector $\overrightarrow{m}$, as defined in the main text, is tangent to AH. Thus, perturbing the initial round sphere in the direction of $c\overrightarrow{m}$ with constant $c$ leads to another marginally f-trapped round sphere in AH. By letting $n_{\mu }{n}^{\mu }\in (-\infty ,\infty )$, the general vector $\overrightarrow{n}$ defined in (14) always points towards the shadowed region, though its direction may depend on the point of the sphere, and it can be spacelike, null or timelike. The AH can also have any causal character.Reuse & Permissions

Figure 4

These are conformal diagrams of (10) with (29, 27) when (i) $m(v,r)<M$ everywhere (left) and (ii) $m(v,r)=M$ in some open asymptotic region (right). As usual, null radial lines are drawn at 45° and the future direction is upwards. The discontinuous line marked as $r=0$ is the origin of coordinates. The spacetime is flat until energy starts falling to the center in a spherical manner from past infinity at $v=0$. The hypersurface $\sigma$ separating the flat region from the rest is nonspacelike everywhere, so that material particles travel causally. (The particular case with a null $\sigma$ is depicted in Fig. 5). Thus, the shaded regions are nonflat spherically symmetric spacetimes with nonvanishing energy-momentum. In the figure on the right, the spacetime becomes Schwarzschild with mass $M$ at some other nonspacelike hypersurface (and therefore also for all $v>v_1$, where $v_1$ is the supremum of $v$ on that hypersurface). The connected component of the apparent 3-horizon ${\mathrm{AH}}_1$ approaches the event horizon EH either asymptotically (left) or at some finite value of $v\le v_1$ and $r=2M$ (right). In the latter case, ${\mathrm{AH}}_1$ and EH merge and remain together for $v\ge v_1$ and $r=2M$. In both cases, EH starts developing in the flat region. In the two cases, ${\mathrm{AH}}_1$ is spacelike (ergo a dynamical or future outer trapping horizon) when approaching the EH, but this is not necessarily so in other regions. These possibilities are depicted in Fig. 6. Whether or not the initial collapsing shell $\sigma$ leads to the formation of a singularity when focusing at the center $r=0$ depends on the particular properties of the mass function $m(v,r)$, see the main text. Thus, we have left open the future evolution of the spacetime for the shaded regions.Reuse & Permissions

Figure 5

These are conformal diagrams similar to those of Fig. 4 but with the energy flowing in along null incoming radial hypersurfaces, so that $\sigma$ is now the null hypersurface $v=0$. A particular case of this situation is given by the Vaidya imploding spacetime, analyzed thoroughly in the Appendix. As argued there, the generic situation now leads to the formation of a singularity in the future with $r=0$, as shown, and therefore there is a unique connected component of AH. The case on the left has $m(v,r)<M$ everywhere and that on the right has $m(v,r)=M$ for all $v>v_1$. In the picture on the left, the apparent 3-horizon AH approaches the event horizon EH asymptotically, while in that on the right, AH and EH merge together at $v=v_1$ and $r=2M$. In both cases, EH starts developing in the flat region. Note that the part of flat spacetime that lies inside the event horizon is the intersection of the interiors of two light cones; it is shown without conformal distortion.Reuse & Permissions

Figure 6

Once again, these are conformal diagrams of (10) with (29, 27) when (i) $m(v,r)<M$ everywhere (left) and (ii) $m(v,r)=M$ in some open asymptotic region (right). All features are essentially the same as in Fig. 4 except that EH does not intersect the flat region and that the connected component ${\mathrm{AH}}_1$ of the apparent 3-horizon may be timelike in some region (in the figures, close to the left upper corner). An explicit example of the latter behavior is provided by the matching of a $p=\gamma \varrho$ Robertson-Walker metric with $\Lambda =0$ and $\gamma <1/3$ to the outwardly radiating Vaidya spacetime, see 40 (time reversal of Fig. 7 therein). See also 13.Reuse & Permissions

Figure 7

These are enlargements of relevant regions in Fig. 5; similar drawings could be performed for the cases of Figs. 4, 6. On the left, we picture the cases with $m(v,r)<M$ everywhere, and on the right with $m(v,r)=M$ for all $v>v_1$. The two pictures on top describe the cases when $\Sigma$ penetrates the flat region, for instance if $8{\dot{m}}_0>(1-2m_0^{\prime }{)}^2$, see Sec. 9a, while the two bottom pictures depict the cases where $\Sigma$ never penetrates the flat region. Pertinent $\tau =\mathrm{\text{const}}$ hypersurfaces are shown for all cases. Observe that these are spacelike everywhere (and approaching spacelike infinity $i^0$) if they have $\tau <{\tau }_{\Sigma }$ (example shown as $\tau ={\tau }_1$), while they are partly spacelike and partly timelike, becoming null at AH, if $\tau >{\tau }_{\Sigma }$ (example shown as $\tau ={\tau }_2$.) The hypersurface $\Sigma$ separating these two cases is nontimelike everywhere, and spacelike where it differs from EH: thus, it is spacelike everywhere for the left drawings, while it becomes null and identical with EH for all $v>v_1$ for the cases on the right. The closed set $\mathcal{K}$ used in the proof of Theorem 9.2 limits to the future with $\Sigma$ and to the past with EH, and is shown in green in all cases.Reuse & Permissions

Figure 8

A Penrose diagram —corresponding to the case on the right of Fig. 5, but any other choice would be similar—showing a nonspherically symmetric closed f-trapped surface $S$ penetrating into the flat region of the spacetime. Such f-trapped surfaces were explicitly constructed in 16. $S$ is shown as a dotted line emerging into the Schwarzschild part as a finite part of a line of constant $r$. The intersection of $S$ with AH occurs at $r=\widehat{r}$ and $\tau ={\widehat{\tau }}_m$, using the notation of Theorem 10.1. The hypersurfaces $r=\widehat{r}$ and $\tau ={\widehat{\tau }}_m$ are explicitly shown. They meet at AH where both of them become null and then change their causal character. Theorem 4.3 in 3 implies that $S$ cannot penetrate into the region $J^{-}(\mathrm{AH}{|}_{r>\widehat{r}})$. However, Theorem 10.1 provides an improvement on that restriction, since $S$ cannot actually penetrate into the region with $\tau \le {\widehat{\tau }}_m$ below AH. Thus, the region shown in purple, allowed in principle by the former restriction, becomes a forbidden region for $S$.Reuse & Permissions

Figure 9

Conformal diagram for a particular example where the AH has a nonvanishing portion of isolated-horizon type different from EH. The energy flows in along null incoming radial hypersurfaces, starting at $v=0$ but suddenly stopping at $v=v_2$. There is no inflow of energy between $v_2$ and $v_3$, and then the energy enters again from past null infinity until the radiation stops at $v=v_1$, and then forever. A particular case of this situation is given by the Vaidya imploding spacetime (see Appendix) with $m(v)=$ constant for $v\in (v_2,v_3)$. In the picture, the AH is spacelike for $v\in (0,v_2)\cup (v_3,v_1)$, but it has two portions which are isolated horizons, one given by part of EH (for $v\ge v_1$) and the other one for $v\in (v_2,v_3)$, represented by ${\mathrm{AH}}^{(\mathrm{iso})}$. The past barrier $\Sigma$ merges with AH and EH as usual. The boundary $\mathcal{B}$ will also merge with them at $r=2M$ and $v=v_1$, and it must be strictly below AH everywhere on $\mathrm{AH}\setminus {\mathrm{AH}}^{(\mathrm{iso})}$, but it is not guaranteed that it cannot touch, or even coincide with, ${\mathrm{AH}}^{(\mathrm{iso})}$.Reuse & Permissions

Figure 10

This figure illustrates the proof of Proposition 10.3.Reuse & Permissions

Figure 11

These are enlargements of some previous conformal diagrams showing the possible location of the boundary ${\mathcal{B}}_1$. The red regions are part of the f-trapped region ${\mathcal{T}}_1$, but this region actually extends further down and includes ${\mathrm{AH}}_1$, as follows from Theorem 7.2. On the other hand, Corollary 10.2 informs us that ${\mathcal{B}}_1$ can never touch $\Sigma$ outside EH. Thus, the yellow zones are the allowed regions for ${\mathcal{B}}_1$, keeping in mind that there is always a red zone below ${\mathrm{AH}}_1$, and ${\mathcal{B}}_1$ must be placed strictly above $\Sigma$. In the left upper picture, $\Sigma$ and also ${\mathcal{B}}_1$ extend to the flat region of the spacetime.Reuse & Permissions

Figure 12

The phase plane $\{v,r\}$ around the critical point (0, 0) for the Eq. (a5) in the spacetime (a1) subject to (a2, a3). Only the case with $m(v)=M$ for all $v\ge v_1$ is represented. It should be noticed that horizontal lines ($v=\mathrm{\text{const}}$) are null in the spacetime, while vertical lines ($r=\mathrm{\text{const}}$) are timelike below AH, null at AH, and spacelike above it. Several $\tau =\mathrm{\text{const}}$ hypersurfaces are represented. They are spacelike while they remain below AH, null when they meet AH, if they do, and timelike above AH. Here, we have represented the case with $m_0>1/8$ (see main text), so that the origin is a focus, which implies that all $\tau =\mathrm{\text{const}}$ hypersurfaces, and in particular $\Sigma$, never reach the origin.Reuse & Permissions

Figure 13

Same phase plane as in Fig. 12 but with $m_0<1/8$. The origin is now a node, with two special solutions given by $\tau ={\tau }_{\pm }$ and represented by dotted lines. All solutions but $\tau ={\tau }_{+}$ emerge from the origin with the same slope given by that of the solution $\tau ={\tau }_{-}$. This is represented in the right picture, which is a magnification of a small neighborhood of (0, 0). If the mass function $m(v)$ is such that both solutions $\tau ={\tau }_{\pm }$ cross the AH, as depicted, then $\Sigma$ will never reach the origin. The other possibilities are given in Fig. 14.Reuse & Permissions

Figure 14

Same case $m_0<1/8$ as in Fig. 13 so that the origin is a node, with their corresponding magnifications on the right. If the mass function $m(v)$ is such that at least one of the solutions $\tau ={\tau }_{\pm }$ crosses the AH, then $\Sigma$ will reach the origin tangent to $\tau ={\tau }_{-}$. This implies that $\Sigma$ will never penetrate into the flat part of the spacetime, which is the half-plane $v<0$. The upper figure represents the case when only one of the two solutions crosses AH, while the bottom does the same when both $\tau ={\tau }_{\pm }$ are spacelike everywhere, never crossing AH.Reuse & Permissions

Figure 15

These are conformal diagrams of spacetime (a1) with (a2, a3) when $m(v)=M$ for all $v>v_1$. The notation is as in Fig. 5. Here, on the left picture we have represented the case in which the hypersurface $\Sigma$ has a portion in the flat region of the spacetime (e.g. if $m_0>1/8$, but not only). On the right picture, the other possibility is represented, when $\Sigma$ never touches the flat region. These two pictures are the global Penrose diagrams corresponding to the pictures on the right of Fig. 7.Reuse & Permissions