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In classical general relativity, one can think of an event horizon a tube of light cones in (3+1)D Minkowski space-time, in which the future-cone of each lightcone is inside the 4D region and the past-cone of each lightcone is outside the 4D region. Thus creating a one-way surface that light can enter but can't escape.

But we know that on very small scales, the light-cones are unlikely to line up quite so neatly.

Is it possible to prove that no smooth perturbation of Schwarzschild event horizon will ever create an escape route for light to escape? (One could imagine jiggling the light cones near the event horizon such that they look fairly chaotic). But in a smooth manner so neighbouring light-cones line up. i.e. the metric still has to be differentiable.

On the other hand, is it possible to prove that a discrete jump in the direction of two light-cones at neighbouring points in space near the horizon, i.e. the metric is not a continuous function at some point, would provide a place on the event horizon where light could escape?

In other words is it possible to prove that event horizons require smoothly differentiable metrics?

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    $\begingroup$ smooth perturbations or any diffeomorphism isn't going to remove the already existing event horizon. May be you should ask whether these smooth perturbations can actually prevent the formation of trapped surfaces in the first place $\endgroup$
    – KP99
    Commented Oct 11, 2022 at 18:49
  • $\begingroup$ I guess, yes, if light could escape at some point then it would no longer be called an event horizon! Perhaps what I was imagining was some kind of pseudo event horizon which looks 99.9% like a normal event horizon except some light managed to escape. But I think such a thing does not exist. $\endgroup$
    – user84158
    Commented Oct 12, 2022 at 1:54
  • $\begingroup$ Unless one modified the definition to instead of light being trapped forever, it was trapped for "a very long time". $\endgroup$
    – user84158
    Commented Oct 12, 2022 at 2:01
  • $\begingroup$ At least, experimentally as of now no one can make a difference b/w an ideal BH and 99.99% BH, one could say wait long enough and see how it behaves. Also the area theorem ensures that the horizon can only monotonically increase under reasonable energy conditions. I feel that there could be some criterions based on known physics which can dismiss space time having ideal BH to be unphysical $\endgroup$
    – KP99
    Commented Oct 12, 2022 at 6:11
  • $\begingroup$ if it's close to but not inside the horizon, it's "possible" (i mean highly ONLY in theory) - if its inside the event horizon, no. $\endgroup$
    – William Martens
    Commented Jan 9, 2023 at 10:38

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By definition, no. An event horizon is defined to be the boundary of a region of spacetime from which light cannot escape to infinity. This means that if a photon gets to infinity from a particular spacetime event, that spacetime event was not inside the event horizon.

A counterintuitive property of this definition, by the way, is that it's non-local. The definition of where an event horizon is depends on the entire future of the spacetime; if something happens "in the future" that allows light to escape to infinity from an event $A$, then that event is "retroactively" defined to not be in the event horizon. (The quotes are because the whole notion of "past" and "future" are a bit hand-wavy here.)

An extreme example of this is Oppenheimer-Snyder collapse, a toy model in which a thin shell of dust collapses into a black hole. It is possible to show that the spacetime inside the shell is completely flat (as we might expect from the Newtonian analogue.) But there are regions of spacetime inside the collapsing shell, at times well before the shell collapses to a point, which are inside the event horizon — because in the future, the dust shell is going to fall past these points and pull any light rays back in to the central singularity that will eventually form.

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  • $\begingroup$ This makes sense. I guess this is related to the Penrose black hole theorems and the no naked singularity conjecture and the Hawking area theorem. $\endgroup$
    – user84158
    Commented Oct 12, 2022 at 1:58

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