Black Hole Thought Experiment

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lgw
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Black Hole Thought Experiment

Postby lgw » Sat Sep 20, 2014 12:12 am UTC

Let's make a black hole - a big one. One light year in radius to be exact. (This requires about ~5x the mass of our galaxy, so such a thing could technically exist.)

Further, let's make it carefully out of only dark matter, all set up in a series of orbitally-stable halos (shells) of as close to uniform density as we can make work, but make sure the rotation is all randomized and even - we want a non-rotating black hole. Some where along the way in adding the outer shell, we'll achive the mass needed to become a black hole. The density is only ~1.8x10-9, and it's all dark matter, so ordinary matter should have next to no interaction with the contents, other than by gravity.

Inside the black hole, I would expect nothing special to change once the outermost shell was added. There's no singularity. We can pack the mass in a bit tighter if needed so that everything can have a stable orbit, without upping the density enough to care. They whole system should be stable, with no crunchy center. The closer you get to the center, the less anything looks like a black hole.

So, now you fall into the black hole. There's no tidal forces to speak of at the horizon (and only 1g acceleration as seen from the infalling frame), so assuming the anti-firewall crowd is right, it should be a pleasant enough trip. Once you're far enough in, there should be no meaningful relativistic effects seen from any slow-moving reference frame that's not close to the edge. The center is not particularly dangerous or interesting.

Well, isn't this an interesting place? What would it look like? I don't see any reason why you'd be dragged to the center by general relativity, or have any swapping of space and time axes, unless you approached the edge. Would you even see the edge as an event horizon, or would you see an expanding universe with a cosmic horizon? Odd idea, no?
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Re: Black Hole Thought Experiment

Postby doogly » Sat Sep 20, 2014 2:49 am UTC

There are no orbits inside a black hole, everything falls in to the center if it's inside the event horizon. That is their deal, basically.
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Re: Black Hole Thought Experiment

Postby Dopefish » Sat Sep 20, 2014 3:27 am UTC

Is everything falling towards the center beyond the event horizon a consequence of every direction being towards the center due to GR weirdness, or are there still spatial 'away' directions and you'd just need infinite thrust or something along those lines to get away?

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Re: Black Hole Thought Experiment

Postby douglasm » Sat Sep 20, 2014 3:33 am UTC

Your construction would (handwaving the manipulation of dark matter) work for a while, but would break down before completing a black hole.

As each shell is added, the particle speed required for the next shell to orbit rather than fall in increases. The required speed will reach the speed of light significantly before you have a black hole, and once you reach that point it is no longer even theoretically possible to add another shell and have it be stable. Particles are still able to not fall in, but only by heading outward rather than orbiting. Any shell you attempt to add at that point will collapse, pass through the next shell in, increase the gravity on that shell causing it to collapse, etc. Density increases in a chain reaction of collapsing shells that are no longer fast enough to orbit around the increased mass inside them, and a black hole considerably smaller than the one you intended forms, with all of the dark matter rushing to the center.

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Re: Black Hole Thought Experiment

Postby Hypnosifl » Sat Sep 20, 2014 3:37 am UTC

Dopefish wrote:Is everything falling towards the center beyond the event horizon a consequence of every direction being towards the center due to GR weirdness, or are there still spatial 'away' directions and you'd just need infinite thrust or something along those lines to get away?

Every direction is "towards" the singularity inside the horizon in the same sense that every direction today is "towards" next Tuesday in SR. This stuff is easier to get a feel for if you use Kruskal-Szekeres coordinates (see "Qualititative features of the Kruskal-Szekeres diagram" at [url]http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates[/url] ), where the time coordinate remains timelike both inside and outside the horizon (unlike with Schwarzschild coordinates where the 'time' coordinate becomes spacelike inside the horizon), light rays always move at 45 degree angles, and the timelike worldlines of slower-than-light observers are always at an angle closer to vertical than 45 (just like in SR diagrams of inertial frames, where all timelike worldlines are confined to the inside of light cones whose sides are at 45 degrees).

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Re: Black Hole Thought Experiment

Postby lgw » Sun Sep 21, 2014 1:16 am UTC

Thanks for the responses everyone!

Let me explain more what motivates this thought experiment: I'm poking at the oddity that GR requires that being inside a distant uniform shell of matter can have an extreme effect on local conditions, in stark contrast to classical expectation (after all, classically that would have 0 effect). In particular, while from the outside the black hole has a center that everything is drawn towards, I'm doubting that from the inside any such preferred location exists (assuming you didn't start with a hyperdense mass), and that instead you'd have a topology that no more has a specific center that everything is drawn towards than our universe has a specific center that everything is expanding away from.

I'm trying to see how to reconcile these, and in fact it doesn't seem far-fetched to me from a GR perspective. As you approach the horizon, the radial and time axes are "twisting" as you move in, an effect that increases steadily as you get close. But once past the horizon, now what? With no hyperdense core in the formation of the black hole, it seems reasonable that the situation would actually be uniform inside. The "twist" of the axes has completed, and you're inside a closed universe now.

In such a case, once past the event horizon you'd see yourself in a uniform universe with a cosmic horizon one light year away (just as in our universe, every point would be the "center" of its observable universe, no point is special), one approaching a "big crunch" only one year in the future.

To further explain why I think this, lets consider another example: take all the contents of our observable universe, and contract all large scale structure about 10-fold, moving the clusters of galaxies closer until it all fit into a sphere of about 4.6 Gly in radius, with appropriately higher CMBR temperature and so on (or we could imagine the contents looking just as they did when the universe was somewhat younger, if you prefer). Per the guess in the Schwarzschild radius article in Wikipedia, we now have a black hole. Again, I'd expect this to form closed universe: a short-lived universe, heading towards a big crunch soon on a cosmological scale, but one with no preferred location. Instead of a center, you'd have a cosmological constant with a sign opposite of our own. This interpretation of the predictions of GR would reconcile the oddity that bothered me above.

Now, one even weirder idea I'll throw out here: we've discussed in other threads the idea that that the sum of universal entropy must increase in the time direction in which the universe is seen as expanding (and CMBR temperature falling), and thus "the future" will always seem to be in the expanding-universe direction. If we accept that for the sake of argument (or argue about in in the old thread), then we'd have a universe that instead had a singularity in its past, and a cosmological constant much like our own - definitely measureable, but inexplicable if one considered only local conditions. Sound like a familiar universe?
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Re: Black Hole Thought Experiment

Postby Hypnosifl » Sun Sep 21, 2014 3:41 am UTC

lgw wrote:Thanks for the responses everyone!

Let me explain more what motivates this thought experiment: I'm poking at the oddity that GR requires that being inside a distant uniform shell of matter can have an extreme effect on local conditions, in stark contrast to classical expectation (after all, classically that would have 0 effect).

Why do you say that? As far as I know, if the observer inside the shell can be considered to have negligible mass, then from what I understanding spacetime is flat inside a spherical shell, see here. It is true that if you depart from a friend far outside the shell, enter the shell, hang out inside it for a while and then exit and return to your friend, you will have aged less than them, but I don't think this is really an "effect on local conditions" inside the shell, more a matter of how the flat region inside the shell is linked up to the approximately flat region far outside the shell.
lgw wrote:I'm trying to see how to reconcile these, and in fact it doesn't seem far-fetched to me from a GR perspective. As you approach the horizon, the radial and time axes are "twisting" as you move in, an effect that increases steadily as you get close.

That "twisting" is purely a matter of the coordinate system you choose, there is no such twisting effect in the Kruskal-Szekeres coordinate system I mentioned.
lgw wrote:But once past the horizon, now what? With no hyperdense core in the formation of the black hole, it seems reasonable that the situation would actually be uniform inside. The "twist" of the axes has completed, and you're inside a closed universe now.

You can choose surfaces of simultaneity (3D slices through 4D spacetime that represent space at a particular "instant" in some coordinate system, see relativity of simultaneity if you're not familiar with the idea that simultaneity depends on the choice of coordinate system) such that the spatial curvature is indeed uniform inside the horizon horizon. Just look at the dotted line "F" in the diagram below from p. 528 of the textbook Gravitation by Misner/Thorne/Wheeler, drawn over the [url=http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates#Qualitative_features_of_the_Kruskal.E2.80.93Szekeres_diagram]Kruskal-Szekeres diagram[/url] which I mentioned in a previous comment (I can't emphasize enough how much getting a feel for Kruskal-Szkekeres diagrams will help your intuitions about black holes!) If you look at the corresponding embedding diagram for F to the right, it's a universe which is closed along one "short" axis but extends forever along the other "long" axis, like an infinite cylinder (like all embedding diagrams, one of the three spatial coordinates is kept fixed so we can represent the curvature of space on a 2D slice through 3D space where the other two dimensions are allowed to vary).

Spoiler:
Image


F is a hyperbola bounded by the two black hole event horizons in the Kruskal-Szkeres diagram, so if you pick different hyperbolas with the same boundary, they represent different simultaneity surfaces where the closed spatial dimension has different sizes--and ones below F have a larger size for the closed dimension, and any above F would have a smaller size, and any infalling observer will pass through each surface in order from bottom to top. So, an observer inside the horizon can be seen as living in a universe where space is "collapsing" along one spatial axis ("one spatial axis" in the embedding diagram anyway, in terms of the curvature of 3D space I believe each simultaneity surface is a hypercylinder which is shrinking in two different directions, while remaining infinitely extended in the third). In this picture the singularity is a future event, experienced by everyone living in this hypercylinder, where the closed dimension(s) shrink to zero length, similar to the Big Crunch in cosmology (where the universe would shrink to zero along all spatial dimensions).
lgw wrote:To further explain why I think this, lets consider another example: take all the contents of our observable universe, and contract all large scale structure about 10-fold, moving the clusters of galaxies closer until it all fit into a sphere of about 4.6 Gly in radius, with appropriately higher CMBR temperature and so on (or we could imagine the contents looking just as they did when the universe was somewhat younger, if you prefer). Per the guess in the Schwarzschild radius article in Wikipedia, we now have a black hole.

That claim in the wiki article has a "citation needed" so you shouldn't put too much trust in it. As pointed out in this entry from the usenet physics FAQ, in an expanding universe you can no longer use the Schwarzschild radius to determine whether a given collection of matter will become a black hole:
Sometimes people find it hard to understand why the Big Bang is not a black hole. After all, the density of matter in the first fraction of a second was much higher than that found in any star, and dense matter is supposed to curve spacetime strongly. At sufficient density there must be matter contained within a region smaller than the Schwarzschild radius for its mass. Nevertheless, the Big Bang manages to avoid being trapped inside a black hole of its own making and paradoxically the space near the singularity is actually flat rather than curving tightly. How can this be?

The short answer is that the Big Bang gets away with it because it is expanding rapidly near the beginning and the rate of expansion is slowing down. Space can be flat even when spacetime is not. Spacetime's curvature can come from the temporal parts of the spacetime metric which measures the deceleration of the expansion of the universe. So the total curvature of spacetime is related to the density of matter, but there is a contribution to curvature from the expansion as well as from any curvature of space. The Schwarzschild solution of the gravitational equations is static and demonstrates the limits placed on a static spherical body before it must collapse to a black hole. The Schwarzschild limit does not apply to rapidly expanding matter.

lgw wrote:Now, one even weirder idea I'll throw out here: we've discussed in other threads the idea that that the sum of universal entropy must increase in the time direction in which the universe is seen as expanding (and CMBR temperature falling), and thus "the future" will always seem to be in the expanding-universe direction. If we accept that for the sake of argument (or argue about in in the old thread), then we'd have a universe that instead had a singularity in its past, and a cosmological constant much like our own - definitely measureable, but inexplicable if one considered only local conditions. Sound like a familiar universe?

I don't understand, what would be inexplicable? Our universe already has a singularity in its past, that's what the Big Bang is (of course quantum gravity might change that, but I'm just talking about what's predicted by general relativity).

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Re: Black Hole Thought Experiment

Postby Tyndmyr » Mon Sep 22, 2014 4:45 pm UTC

Dopefish wrote:Is everything falling towards the center beyond the event horizon a consequence of every direction being towards the center due to GR weirdness, or are there still spatial 'away' directions and you'd just need infinite thrust or something along those lines to get away?


The latter. The event horizon is merely the point at which escape velocity hits the speed of light. If, somehow, FTL worked, you could escape from a black hole by simply driving away very fast. FTL would break down a lot of assumptions, of course...but that very incompatibility with the known laws of physics is sort of an issue.

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Re: Black Hole Thought Experiment

Postby Hypnosifl » Mon Sep 22, 2014 5:20 pm UTC

Tyndmyr wrote:
Dopefish wrote:Is everything falling towards the center beyond the event horizon a consequence of every direction being towards the center due to GR weirdness, or are there still spatial 'away' directions and you'd just need infinite thrust or something along those lines to get away?


The latter. The event horizon is merely the point at which escape velocity hits the speed of light. If, somehow, FTL worked, you could escape from a black hole by simply driving away very fast. FTL would break down a lot of assumptions, of course...but that very incompatibility with the known laws of physics is sort of an issue.

Although it should be noted that according to relativity, moving FTL would be equivalent to moving backwards in time according to some local inertial observers, and there are spacelike (FTL) worldlines that would allow the traveler to meet up with their earlier self at a past point on their worldline before they crossed the horizon, so thinking about this just in terms of moving away fast enough in the right spatial direction might be a bit misleading.

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Re: Black Hole Thought Experiment

Postby Tyndmyr » Mon Sep 22, 2014 9:34 pm UTC

It's simplified, to be sure. But with FTL, one could theoretically take a path out of an event horizon that either does or does not cross their path into the event horizon. From your perspective, it'd all be pretty straightforward, but yeah, an observer could see apparently impossible events taking place.

The most elegant explanation is simply "there isn't any FTL", naturally.

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Re: Black Hole Thought Experiment

Postby lgw » Tue Sep 23, 2014 11:43 pm UTC

Hypnosifl wrote:
lgw wrote:Let me explain more what motivates this thought experiment: I'm poking at the oddity that GR requires that being inside a distant uniform shell of matter can have an extreme effect on local conditions, in stark contrast to classical expectation (after all, classically that would have 0 effect).

Why do you say that? As far as I know, if the observer inside the shell can be considered to have negligible mass, then from what I understanding spacetime is flat inside a spherical shell, see here. It is true that if you depart from a friend far outside the shell, enter the shell, hang out inside it for a while and then exit and return to your friend, you will have aged less than them, but I don't think this is really an "effect on local conditions" inside the shell, more a matter of how the flat region inside the shell is linked up to the approximately flat region far outside the shell.


We may be saying the same thing here and below, but let me clarify and maybe you can tell if so. Let's say you make a black hole where all the needed mass is in the shell (even if only momentarily). From the outside frame, everything must inevitably rush together towards the center - even two test particles inside. From the inside frame there's no sensible reason for those particles to be forced together. To resolve this tension to the best of my know-how, I take note of the fact that, inside, the singularity does not lie in any spatial direction, but instead along the time axis. So if every particle has the singularity in its future, not in the "center" from the inside frame, and no matter which way you go you can't escape to rejoin your friend, then I interpret this as a closed space, and one that shrinks over time (space itself shrinking, as it would in a Big Crunch).

Hypnosifl wrote:F is a hyperbola bounded by the two black hole event horizons in the Kruskal-Szkeres diagram, so if you pick different hyperbolas with the same boundary, they represent different simultaneity surfaces where the closed spatial dimension has different sizes--and ones below F have a larger size for the closed dimension, and any above F would have a smaller size, and any infalling observer will pass through each surface in order from bottom to top. So, an observer inside the horizon can be seen as living in a universe where space is "collapsing" along one spatial axis ("one spatial axis" in the embedding diagram anyway, in terms of the curvature of 3D space I believe each simultaneity surface is a hypercylinder which is shrinking in two different directions, while remaining infinitely extended in the third). In this picture the singularity is a future event, experienced by everyone living in this hypercylinder, where the closed dimension(s) shrink to zero length, similar to the Big Crunch in cosmology (where the universe would shrink to zero along all spatial dimensions).


I get the choice of coordinates, though it seems to me it would have to be closed along each axis, possibly because I can't see what the now-space-once-time axis would look like from inside, but since the lifetime of the black hole is limited, surely that spatial axis must also be? Or have I just misunderstood the diagram?


Hypnosifl wrote:Sometimes people find it hard to understand why the Big Bang is not a black hole. After all, the density of matter in the first fraction of a second was much higher than that found in any star, and dense matter is supposed to curve spacetime strongly. At sufficient density there must be matter contained within a region smaller than the Schwarzschild radius for its mass. Nevertheless, the Big Bang manages to avoid being trapped inside a black hole of its own making and paradoxically the space near the singularity is actually flat rather than curving tightly. How can this be?

The short answer is that the Big Bang gets away with it because it is expanding rapidly near the beginning and the rate of expansion is slowing down. Space can be flat even when spacetime is not. Spacetime's curvature can come from the temporal parts of the spacetime metric which measures the deceleration of the expansion of the universe. So the total curvature of spacetime is related to the density of matter, but there is a contribution to curvature from the expansion as well as from any curvature of space. The Schwarzschild solution of the gravitational equations is static and demonstrates the limits placed on a static spherical body before it must collapse to a black hole. The Schwarzschild limit does not apply to rapidly expanding matter.
....
lgw wrote:Now, one even weirder idea I'll throw out here: we've discussed in other threads the idea that that the sum of universal entropy must increase in the time direction in which the universe is seen as expanding (and CMBR temperature falling), and thus "the future" will always seem to be in the expanding-universe direction. If we accept that for the sake of argument (or argue about in in the old thread), then we'd have a universe that instead had a singularity in its past, and a cosmological constant much like our own - definitely measureable, but inexplicable if one considered only local conditions. Sound like a familiar universe?

I don't understand, what would be inexplicable? Our universe already has a singularity in its past, that's what the Big Bang is (of course quantum gravity might change that, but I'm just talking about what's predicted by general relativity).


IIRC, inflation is about rapidly expanding space, not matter - expanding FTL in fact (which is legit for space to do, apparently), but that's not where I was going.

We think of two kinds of singularity - one kind that has an event horizon, and exists only in the future (in the infalling frame) , while the other kind exists only in the past, and has no event horizon (but does have a cosmic horizon). I'm saying these are probably the same, if you accept the sign flip on the time axis. Take the inside of a black hole as described (space contracting, singularity in the future of all points in space), and in the other time direction you see space expanding and a singularity in the past of all points in space. Further, time is not unbounded in the future in this view, as it ends in a Big Rip. So in this view, the inside-frame "past" direction is the outside-frame radial direction towards the center, bounded by a singularity. (Motivation for the "time sign flip" was discussed in that other thread, but it does seem to me that at a large-enough scale, entropy must increase in the time direction in which the universe expands.)

If that's all consistent, then our universe could very well be a large black hole in a parent universe. However, for that to be worth anything as a theory, it would need to add something - for example, explaining the extreme fine-tuning of the density of the early universe needed to have it be so long-lived. Oh, well, my understanding of the math will get there eventually and maybe I'll disabuse myself of this notion.
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Re: Black Hole Thought Experiment

Postby Goemon » Wed Sep 24, 2014 1:43 am UTC

lgw wrote:Let's say you make a black hole where all the needed mass is in the shell...


Just FYI, this really isn't possible.

If we somehow move all the matter in the Earth into a thin shell at the surface, then as you note, spacetime in the hollow interior is basically flat. But we do have a small engineering problem in that the structural strength needed to keep the shell from collapsing is pretty significant - imagine trying to build a brick ceiling over your house whose supports are not just over the horizon, they're ten thousand kilometers distant!

If we make the shell denser - more mass at the same radius, then pretty soon the structural problems stop being engineering problems and start being problems of theoretical physics. LONG before you approach a density that would cause an event horizon to form, the force required to keep the structure from collapsing becomes infinite. Literally, infinite. The shell MUST collapse, according to relativity.

So if you're still wondering how to reconcile the ideas of "the interior of a thin shell is a tranquil place where objects are not pulled in any direction" with "the interior of an event horizon requires all objects to be pulled toward the center", well - the latter dictates that the former cannot be massive enough to generate an event horizon. There's nothing to reconcile.
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Re: Black Hole Thought Experiment

Postby gmalivuk » Wed Sep 24, 2014 6:21 pm UTC

Is there a handy formula to figure out the compressive forces in a shell like that? I remember doing the calculations for a ring once, and as I recall that's a fairly straightforward limit of polygons. But since there's not a nice progression of polyhedra like that, I don't even know how to begin.
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Re: Black Hole Thought Experiment

Postby Nicias » Wed Sep 24, 2014 6:48 pm UTC

It should work if you cut it in half, like a plastic easter egg:

http://www.aleph.se/Nada/dysonFAQ.html#STRENGTH

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Re: Black Hole Thought Experiment

Postby lgw » Wed Sep 24, 2014 10:11 pm UTC

Goemon wrote:
lgw wrote:Let's say you make a black hole where all the needed mass is in the shell...


Just FYI, this really isn't possible.

If we somehow move all the matter in the Earth into a thin shell at the surface, then as you note, spacetime in the hollow interior is basically flat. But we do have a small engineering problem in that the structural strength needed to keep the shell from collapsing is pretty significant - imagine trying to build a brick ceiling over your house whose supports are not just over the horizon, they're ten thousand kilometers distant!

If we make the shell denser - more mass at the same radius, then pretty soon the structural problems stop being engineering problems and start being problems of theoretical physics. LONG before you approach a density that would cause an event horizon to form, the force required to keep the structure from collapsing becomes infinite. Literally, infinite. The shell MUST collapse, according to relativity.

So if you're still wondering how to reconcile the ideas of "the interior of a thin shell is a tranquil place where objects are not pulled in any direction" with "the interior of an event horizon requires all objects to be pulled toward the center", well - the latter dictates that the former cannot be massive enough to generate an event horizon. There's nothing to reconcile.


For this thought experiment, the bricks can be in free-fall the entire time, they only need to form the required shell momentarily. All inside their collective Schwarzschild radius, but none very near the center. Obviously, you'd need some very dense bricks, or a very large black hole indeed (or both) for the math to work. Or as per my original approach - a uniform distribution of bricks throughout the volume at the moment you pass the critical mass (so to speak). I only chose dark matter so that you don't get everything in motion colliding and sticking together.

And again, I'm pretty sure that there is no a priori "center" in the frame internal to the black hole (or at least that's what I'm trying to illustrate). The direction called "center" in a distant frame of reference is the direction called "future" in an internal frame, and objects inside are of course dragged inevitably towards the future, but that's no longer any spatial direction. The only way I see everything inevitably coming together in that future is that space itself shrinks in this closed topology - same crunch, but no center.
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Re: Black Hole Thought Experiment

Postby gmalivuk » Thu Sep 25, 2014 12:31 am UTC

Using the Schwarzschild solution to figure out what happens inside a forming black hole may not give you a lot of meaningful results, as it is a static vacuum solution for what's going on outside a spherically symmetric mass-energy distribution.

I *suspect* that what happens locally continues to depend only on how matter is distributed "below" you, as it does in the low-density limit. In other words, if you're sitting in flat space as an event horizon forms around a dark matter shell collapsing around you, you won't notice anything amiss in your local bit of spacetime until the shell falls past you (after which point you have no choice but to fall in after it since by hypothesis it's dense enough to have already formed a black hole).
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