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In gravitational collapse, it seems that there is no close or simple logical relationship between the formation of an event horizon (absolute horizon) and formation of a trapped surface (which implies an apparent horizon).

Modeling gravitational collapse is a specialized and highly technical field, and I don't know much about it. IIRC, simulations can most easily detect the formation of an apparent horizon, whereas the formation of an absolute horizon may not even be easy to pick out, since it's a global notion.

If we observe an object that has formed an event horizon, then I automatically imagine that it has also formed a trapped surface, and therefore must have a singularity due to the Penrose singularity theorem. But what basis do we have for this implication? Is it just a fact that arises from simulations, or is there some theorem that guarantees it?

Question: In realistic gravitational collapse, does the formation of an absolute horizon imply a trapped surface? If there are exceptions, are there strong reasons to believe these are unphysical or not generic?

The following, from Wald and Iyer, doi:10.1103/physrevd.44.r3719 , seems relevant:

...no general theorems require the presence of trapped surfaces in the collapse to a black hole. (The event horizon of the black hole must "settle down" to an outer marginally trapped surface at late times, but will normally have a positive expansion at any finite time. ) Nevertheless, the usual physical arguments concerning why black holes rather than naked singularities should be formed by collapse strongly suggests that outer trapped surfaces always should accompany black-hole formation.

From Hawking and Ellis, p. 321, it looks like "marginally" means the expansion scalar $\theta=0$. If I'm understanding the definition on p. 319 correctly, then "outer" means $\theta\ge0$. Their figure 59 on p. 321 shows an example of astrophysical collapse in which the apparent horizon forms later than the event horizon.

I'm not clear on how to interpret this and would appreciate further explanation. The gist of it seems to be that there are strong reasons to expect a trapped surface in realistic cases. The sentence in parens seems to be saying that due to no-hair theorems, the exterior spacetime has to approach a Kerr-Newman spacetime. I guess this would lead us to expect that the interior would also approach Kerr-Newman spacetime, but maybe this is not an absolute implication. Would the exception be unphysical, e.g., it has to be non-generic?

The final sentence from the Wald paper seems to be making a link with cosmic censorship, but that seems vague to me. Weak cosmic censorship just says there's an absolute horizon, but doesn't say there's a trapped surface.

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  • $\begingroup$ Spacelike slaces of the event horizon are trapped surafes. So if you have an event horizon you must have trapped surfaces. $\endgroup$
    – MBN
    Jan 14, 2020 at 10:18
  • $\begingroup$ It is probably more acurate to say that Penrose theorem implies that there will be a singularity, not that there is one right after the formation of a trapped surface. $\endgroup$
    – MBN
    Jan 14, 2020 at 10:19
  • $\begingroup$ @MBN: Maybe that should be an answer. Does it matter which spacelike slice of the event horizon you take? I.e., are you saying that there exists a spacelike slice that is a trapped surface, or that all spacelike slices of the horizon are trapped surfaces? $\endgroup$
    – user4552
    Jan 14, 2020 at 13:19
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    $\begingroup$ related: physics.stackexchange.com/q/93909 $\endgroup$
    – user4552
    Jan 14, 2020 at 17:48
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    $\begingroup$ Relevant paper: The spatial relation between the event horizon and trapping horizon, by A.B. Nielsen. $\endgroup$
    – A.V.S.
    Jan 20, 2020 at 5:16

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Great question! So, we will need two Penrose diagrams to describe the process of a collapsing start and when a horizon begins and then a singularity forms. The first image is my own I made for a portion of my research (disregard the particle creation), and the second is Penrose's original diagram taken from his $\sim$ 1964 paper.

Now, consider the first diagram which could describe a neutron with another star orbiting around it from which it is taking matter in (so it is still growing). Once this neutron star begins to pass the point of its Schwarzschild radius, a small horizon begins to form in the center of the neutron star, as seen from the diagram, but no singularity yet! Switching to Penrose's diagram, we can see that once the horizon is formed, we have a trapped surface (and if you read any more literature, it is called a Cauchy surface).

Now, the horizon then begins to grow as the neutron star begins to take in more mass from its orbiting partner, and once the neutron star grows large enough over some period of time, the horizon will also grow to become an apparent horizon, at the same time a physical singularity begins to form. In the first diagram it is denoted by a jagged line, in Penrose's diagram it can be seen as that dark line within the horizon.

As to whether or not there are exceptions to this rule/process: no, there can not be. This is known as a naked singularity, which according to Penrose, can not exist, and so far he appears to be correct.

For a better explanation, you will want to look into the Penrose-Hawking singularity theorems. There are more, but those are the most important.

I hope this helps!

Penrose diagram for a collapsing star

Penrose's Original Diagram

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