A Proposed Black Hole Entropy Calculation

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Summary:

Discussion of a paper entitled "Islands in Schwarzschild Black Holes" which proposes a calculation of entropy for a black hole emitting Hawking radiation.
The following paper appeared earlier this year on arxiv, entitled "Islands in Schwarzschild Black Holes":

https://arxiv.org/pdf/2004.05863.pdf

First, a bit of background: this paper appears to be part of a larger research effort aimed at resolving the black hole information paradox by showing how all of the information that fell into a black hole does end up coming back out in the Hawking radiation emitted by the hole. The general approach seems to be to show how the information is contained in the entanglements between radiation emitted at early and late times, rather than being contained in individual bits of radiation emitted at particular times, taken by themselves.

In this thread, I'm not intending to discuss that general research effort, but just to look at the particular models and claims made in this particular paper. What the impact would be on the larger research effort if those claims turned out not to be justified is a separate question that I do not want to get into in this thread.

The claims made in the paper generally appear to rest on the idea of an "island", which, in somewhat oversimplified terms, is a kind of connection between the Hawking radiation in two "wedges" of the spacetime, the existence of which limits the entanglement entropy between the inside and outside of the hole at late times. (This is relevant for the larger research effort because, if it is true that all of the information that fell into the hole eventually comes out in Hawking radiation before the hole completely evaporates away, then the entanglement entropy, while it might increase for a while once the hole forms, will eventually, after a time called the "Page time", have to start decreasing again, and ultimately must go to zero by the time the hole finally evaporates, since at that point there is no "inside" the hole any longer--all the information is outside, so the outside must be in a zero entropy pure state just as it was before the hole formed in the first place.)

I have not yet completely understood all of the computations in the paper. However, with what I think I have understood so far, I see several potential issues that I am hoping others might be able to comment on:

(1) The background spacetime used in the paper is Schwarzschild spacetime--more precisely, maximally extended Schwarzschild spacetime. However, this spacetime is a vacuum spacetime--there is no stress-energy anywhere--so I don't see how it can be a suitable model for what the paper is trying to do. The paper keeps talking about "matter fields" being present, but if the spacetime is Schwarzschild, there can't be any matter fields present that contain any energy; they must all be in their ground states, which makes them irrelevant to the analysis being done.

Also, the black hole in this spacetime is eternal--it never evaporates away. Strictly speaking, its mass can't change at all. But the whole point of the analysis is supposed to be to evaluate how the entanglement entropy changes as the hole's mass decreases, not just by a little bit, but from some large finite value all the way down to zero. I don't see how Schwarzschild spacetime can possibly be a valid background against which to make such calculations. Something like the outgoing Vaidya spacetime, at least for a portion of the model, would seem to be necessary in order to capture the decreasing mass and the emitted radiation (and there would also need to be a region to the future of the final evaporation of the hole which was, at least to a good approximation, flat Minkowski spacetime, inside the last spherical wave front of outgoing radiation from the final evaporation).

(2) In addition to all the other issues raised above, there is another issue with using maximally extended Schwarzschild spacetime: only one of the two "wedges" being used in the model will actually exist for a real black hole that forms by gravitational collapse of a massive object. In terms of Fig. 1 in the paper, only a portion of the region marked R+, plus a portion of the black hole region at the top, would exist for a real black hole: the rest of that figure would not be present, with that region instead being occupied by the matter that collapses to form the hole, and that matter region would end at an ##r = 0## line on the left, which would meet the singularity at the top. The spacetime diagram in this Insights article gives a rough idea of what I am describing:

https://www.physicsforums.com/insights/schwarzschild-geometry-part-4/

But the paper's whole analysis depends on both the right and the left "wedges" being present; otherwise the whole construction with the "island" doesn't work. So this analysis, even if it is valid for the case considered (i.e., if the issues raised in #1 above don't invalidate it by themselves), would still seem to me to be irrelevant to a real black hole, since the analysis depends on a region of spacetime being present that is not present for a real black hole.

(3) There also seems to me to be an issue with the calculation in Appendix B, of what is called the "distance" between the two wedges (as far as I can tell, this is the distance between points a+ and a- in Fig. 1--or possibly between b+ and b-, it's not entirely clear--or rather how that distance changes as the points move upward and to the left/right in the diagram). This calculation uses Schwarzschild coordinates; but it is calculating something that crosses the horizon--actually it crosses two horizons (since it has to go from the right wedge, through the interior of the hole, out to the left wedge). But Schwarzschild coordinates are singular at the horizon, which would mean that either some other chart should be used (the obvious one would be Kruskal, or else the Penrose chart used to draw Fig. 1 itself), or that the calculation should be done in three pieces, one for each segment (right wedge point a+ to horizon, between horizons in the interior, horizon to left wedge point a-), with appropriate limits being taken as the horizons were approached. Nothing like this appears to be done in Appendix B.

Heuristically, what Appendix B appears to be calculating is a "distance" along the blue line that represents the "island" in Fig. 1, and the purpose of the calculation appears to be to justify the obvious intuition that, as you move "up" the diagram, since the horizons move "further apart", the "distance" along the blue line has to increase. However, this distance is called "geodesic distance" in the paper, but no proof is given that the line along which the "distance" is calculated is actually a spacelike geodesic, and I don't think it is.

All these issues make me skeptical that the calculations in the paper are correct, or that they are relevant to actual evaporating black holes. However, it's quite possible that I am missing something. Any input is appreciated.
 
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Answers and Replies

  • #2
Haelfix
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Hi Peter.

This research direction has been rather lively in the past year or so, and there are a lot of people working in this field. Indeed, the original papers derive the Page curve from first principles for the first time (something many didn't expect to see in our lifetimes) and then somewhat miraculously derived it from a completely different method (the initial papers get it from entropy arguments, and the second derives it by an explicit calculation involving the path integral ). I can't speak for this particular paper, but the original research and rigorous calculations have been done in a rather particular setting. Namely Jackiw-Teitelbom gravity (1+1 Einstein gravity + a dilaton)

arXiv:1905.08255
arXiv:1911.11977
arXiv:1905.08762

With regards to your questions
1) So I might be wrong here for this specific work, but in general the community are not working in the setting of pure GR. They are instead working with the machinery of quantum field theory in curved spacetimes and indeed go further with the full semiclassical apparatus just like the setting where Hawking did his original calculations. Sometimes they 'assume' various equations continue to hold, even in the full nonperturbative theory (for instance if they are justified by entropy considerations). I would point you to various sources for more explicit pedagogy and modern convention (arxiv.org/abs/1409.1231 as well as Hawkings original papers)

So 'matter' acts as small perturbations to a fixed background, and is frequently absorbed into renormalization of various objects such as the stress energy tensor or coupling constants. Backreaction then must be explicitly computed and readjusted by hand. (so think of it like you have one instance of Schwarschild, and then after you calculate various quantities, you have a perturbed solution a small correction away (then there is a second correction that you can compute from the previous set and so forth, the full set to any order is usually what is refered to as the semiclassical expansion). So eg Hawking was able to show to leading order that the classical metric + the first order Ghbar correction yielded a backreaction to the metric, that resulted in a metric with Schwarschild mass M - dm (evaporation).

2) This community has a bad tendency to go back and forth between Schwarschild and Oppenheimer-Synder and sometimes even calls them the same thing. Be sure that they know the difference, it's usually understood by the context and the drawing. With regards to the maximally extended spacetime, there are several interpretations that make it a little less familiar than the usual classical notion (where we usually argue region 3 and 4 away on entropic grounds). The quantum mechanics of these spacetimes are interesting holographically, and have even led some to make claims that *all* blackholes in nature might in some sense be two sided when quantum gravity is properly understood, where the worm hole in some sense connects the left and right region (and there is an analogy to entanglement here). To see how different things are than usual GR, it was shown a few years ago that you can actually explicitly construct a stable and traversable Einstein-Rosen bridge when you introduce quantum shockwaves into the picture.

3) With respect to island calculations, I don't know this particular papers setup (it appears to be a much stronger claim as the setting appears to be in 3+1 Schwarschild). But the setup in JT gravity usually involves a couple different gravitational systems (a thermal bath system coupled to the JT blackhole) where there are junction conditions that are applied. Then there is a holographic quantum extremal surface prescription that's applied to the blackhole, where you can see that the correct sum of the bath + the region within the entanglement wedge correctly outputs the Page curve. There are some really nice intro lectures by Netta Englehart and Douglas Stanford on this subject that are on youtube (look for a session on may19, 2020 by Netta). See also some lectures by Alhmeiri.
 
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  • #3
PeterDonis
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think of it like you have one instance of Schwarschild, and then after you calculate various quantities, you have a perturbed solution a small correction away
This makes sense, yes. It still leaves an issue, though, it seems to me, for the final evaporation of the hole, since I'm not sure all of the assumptions underlying what you describe will hold there (heuristically, this is because there is a "corner" in spacetime there, where the singularity ends, and I'm not sure how that "corner" works in terms of having an open neighborhood for every event).

The quantum mechanics of these spacetimes are interesting holographically, and have even led some to make claims that *all* blackholes in nature might in some sense be two sided
Would "two-sided" mean that both the left and the right wedges belong to the same global spacetime, just two different regions of it?

it was shown a few years ago that you can actually explicitly construct a stable and traversable Einstein-Rosen bridge when you introduce quantum shockwaves into the picture
Can you give a specific reference? I have heard a bit about this but have not seen an actual paper.
 
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  • #4
Haelfix
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This makes sense, yes. It still leaves an issue, though, it seems to me, for the final evaporation of the hole, since I'm not sure all of the assumptions underlying what you describe will hold there (heuristically, this is because there is a "corner" in spacetime there, where the singularity ends, and I'm not sure how that "corner" works in terms of having an open neighborhood for every event).
I don't think anyone really understands that. Certainly not for 4d Schwarschild. The actual 'state' of the spacetime is still very much a mystery in this business. What they have are entropy calculations and path integral calculations that give the 'plausibly correct' answer in 1+1d, but it's not entirely clear how to interpret them or what they mean exactly. In the 2d JT gravity case, the right answer seems to require including wormhole contributions to the PI, it seems to involve holography in the form of those islands (which involve a souped up Ryu-Takayanagi prescription), and it seems to require a reinterpretation of quantum mechanics (where you have to take an ensemble average of quantum systems). This might be a peculiarity of 2d, or it might be a hint that there is something strange going on for the real case.

Would "two-sided" mean that both the left and the right wedges belong to the same global spacetime, just two different regions of it?
Yea, I mean where there are two independent asymptotic regions. In this business, usually controlled by two distinct CFTs.

Can you give a specific reference? I have heard a bit about this but have not seen an actual paper.
Sure the original solution/trick was by Gao, Jafferis and Wall here
arXiv:1608.05687

Maldacena et al clarified and generalized it to higher dimensions, eternal Ads and to 4d Schwarschild..

arXiv:1704.05333, arXiv:1807.04726,

The original paper might be more to your liking (albeit technical) b/c Wall is a classically trained GR expert. If you're anything like me, I have difficulty understanding Juan's papers and talks, but Douglas Stanford on the other hand is extremely clear so I actually recommend his talks. This material is considerably less opaque when it's explained live, so I recommend the talk 'diving into traversable wormholes' by Stanford that you can find on eg youtube..
 
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  • #5
PeterDonis
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I mean where there are two independent asymptotic regions.
Ok, good. That sharpens the issue with what happens when the hole finally evaporates, because it would seem that those two separate regions would have to turn into one, since what was separating them was the hole and it's gone. But that one region can't be "two-sided", since that makes no sense in the absence of a hole between the two; and yet it's not clear how it would become "one-sided" either if it was "two-sided" to start with.
 
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  • #6
PeterDonis
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the original solution/trick was by Gao, Jafferis and Wall here
arXiv:1608.05687

Maldacena et al clarified and generalized it to higher dimensions, eternal Ads and to 4d Schwarschild..

arXiv:1704.05333, arXiv:1807.04726,
Thanks for the links, I'll take a look.
 
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  • #8
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As @Haelfix says, this work (mostly? entirely?) originates in a 2d model (JT gravity) that has been very popular in the past few years. There's a summary at Quanta Magazine, and you can also read the thoughts of a young string theorist about the big picture.

I haven't read the papers yet, but I will ask @Haelfix a naive question or two:
In the 2d JT gravity case, the right answer seems to require including wormhole contributions to the PI
In the Quanta article, halfway down there's a five-part diagram of stages of black hole evaporation. At stage 3, the new extremal surface appears, and there seems to be Hawking radiation that disappears inward.

Naively, it's almost as if part of the black hole interior has become a separate "baby universe", and there's Hawking radiation that ends up there, rather than in the original universe. Is this a viable interpretation of the role of the wormholes? But is there a problem of conservation of energy (mass lost to the baby universe), or is it somehow countered by space-time curvature of the baby universe?

@Haelfix also mentions that this 2d gravity involves a sum over ensembles. McNamara (the young string theorist mentioned above) and Vafa suggest that it's actually an incomplete theory - as I understand it, something like a theory of worldsheet gravity for a 1-brane embedded in a larger space. In that case, the "emission of a baby universe" would just be the 1-brane budding another 1-brane. This would also make the inwardly disappearing Hawking radiation less mysterious - they are quanta in the new 1-brane that pinches off. But if all that were true, it makes me wonder if all these calculations only work for "black holes embedded in branes". Probably I am jumping to wrong conclusions...

(I hope discussion of stringy complications doesn't drown out discussion of the original paper posted by @PeterDonis, which is about the 4d case.)
 
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  • #10
PeterDonis
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Surprisingly good for a popular-science paper.
In terms of the description of the research, yes. But the headline and tag line are, IMO, way too strong (which is par for the course at a place like Quanta magazine).
 
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  • #11
Demystifier
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In terms of the description of the research, yes. But the headline and tag line are, IMO, way too strong (which is par for the course at a place like Quanta magazine).
I couldn't agree more.
 
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That's great! Surprisingly good for a popular-science paper.
In terms of the description of the research, yes. But the headline and tag line are, IMO, way too strong (which is par for the course at a place like Quanta magazine).
A necessary evil, especially if it is able to lure in a few students into becoming passionate about physics again. For example, my last protégé left mathematical physics because there was a better offer on the table (namely making money in business, a huge waste of potential); he is also part of the target audience and his eyes lit up when he heard this news... one can hope.

The most important thing is that seasoned experts as well as most other theoretical scientists are able to take away the right message from such an article without having to read a systematic review per se because they remain critical and steadily keep on asking the hard questions. In any case, @Demystifier, this work makes me want to work on your non-holographic AdS/CFT even more.
 
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  • #13
PeterDonis
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A necessary evil
I think that's a matter of opinion. See below.

especially if it is able to lure in a few students into becoming passionate about physics again.
The most important thing is that seasoned experts as well as most other theoretical scientists are able to take away the right message from such an article
IMO, unfortunately, the first of these, while it's great if it happens, happens rarely. The second, IMO, is wrong: seasoned experts won't even read the article, they'll read the actual paper.

Also, you left out a huge additional negative effect: all the members of the lay public who read the headline, don't read the article (and probably wouldn't have the background to correctly understand what it says), and who therefore only get one message: Science claims to be an authority. And when that authority is found to be wrong (as it always is sooner or later), the public then thinks they are free to disbelieve science all the time.
 
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  • #14
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IMO, unfortunately, the first of these, while it's great if it happens, happens rarely.
Agreed
The second, IMO, is wrong: seasoned experts won't even read the article, they'll read the actual paper.
The one doesn't exclude the other. On the contrary, assuming one isn't a relativist, one isn't mainly working on AdS/CFT or one isn't even a theorist, it's somewhat likelier that one would come across the popular article first, feel it out and then only search for the actual paper after.

I know far more active physicists who aren't theorists and who therefore mostly or even only proactively read technical papers within their own field (e.g. condensed matter, photonics, advanced optics, plasma physics, etc) and occasionally get lured into reading theory papers that they have heard about by word of mouth or through articles, blogs, tweets and forums.
Also, you left out a huge additional negative effect: all the members of the lay public who read the headline, don't read the article (and probably wouldn't have the background to correctly understand what it says), and who therefore only get one message: Science claims to be an authority. And when that authority is found to be wrong (as it always is sooner or later), the public then thinks they are free to disbelieve science all the time.
After complaining and worrying about that for years, I've since learned to stop worrying about that. Worrying and complaining is clearly a failed intervention for solving that problem; all the amount of effort put into correcting articles is mostly completely wasted energy and worse still, can when brought in front of the public's eyes usually be undone by a single politician or journalist: the public, generally speaking, simply does not seem to care about intellectual honesty.

I think a much better intervention would be if academia and the scientific education institutes themselves started focusing more on producing scientists that specialise as science popularisers who can function as whistleblowers/vocal critics of other groups of scientists or even entire research programs, in the style of say Sabine Hossenfelder; given the current state of academia however, this seems to be a mere pipedream.

My diagnosis is that the root problem is that scientists tend to prematurely or reflexively feign strong certainty, where actually there is immense uncertainty; they do this as a premature sense of seeking approval and wanting to appear more as if the science has more consensus than it actually has, aside from lobbying for their own work. Doing this has no actual penalty in science, while in other fields one is usually punished accordingly, sometimes even with prison time. The risk is that stifling theoretical science in such a way would probably be throwing out the baby together with the bathwater.
 
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  • #15
Demystifier
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In any case, @Demystifier, this work makes me want to work on your non-holographic AdS/CFT even more.
Great! If you get some new ideas how my conjecture could be better justified, I would like to discuss it with you. My e-mail is in the paper so ...
 
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  • #16
Haelfix
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Naively, it's almost as if part of the black hole interior has become a separate "baby universe", and there's Hawking radiation that ends up there, rather than in the original universe. Is this a viable interpretation of the role of the wormholes? But is there a problem of conservation of energy (mass lost to the baby universe), or is it somehow countered by space-time curvature of the baby universe?
@Haelfix also mentions that this 2d gravity involves a sum over ensembles. McNamara (the young string theorist mentioned above) and Vafa suggest that it's actually an incomplete theory -
I don't quite know how to answer your first question. That requires detailed knowledge of bulk physics and a spacetime perspective that is seemingly absent from the calculations at this stage (at least I don't understand it). All they can say is that cutting up spacetime in exactly the way they mention in JT gravity, leads to a pure state (as desired) as well as the desired unitary evolution. What that exactly means, and what exactly the 'blackhole hairs' are precisely doing will likely be a hot topic for the next few years. The difference is that there is now a tangible and concrete calculation that is present, rather than a bunch of handwaving.

Your second question is also difficult to answer. I believe Vafa's point is that the 2d gravity under consideration is in the swampland (which is itself a conjecture), and therefore that the ensemble average interpretation is likely an artifact of this pathology. It is indeed plausible that a more realistic theory with microstates does not suffer the same problems. However, since that hasn't been constructed yet to my knowledge, the status of this statement remains to be seen.
 
  • #17
Haelfix
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For an example of what I was thinking of here, see this paper:

https://www.groundai.com/project/a-linear-mass-vaidya-metric-at-the-end-of-black-hole-evaporation/1

The "vanishing point" in Fig. 2 is the "corner" I was referring to.
I think I follow. I will say there are a few different points of view about this sort of thing.. One point of view is to be a little careful with putting too much dogma into the absolute veracity of Penrose diagrams in quantum gravity. There has been a lot of work from the GR community going back half a century where people will cut out various problematic or poorly understood regions in semi classical models, then glue on various 'plausible' spacetimes so as to see what that implies about the stress energy content of the tiny cutout region where (quantum gravity dragons reside). People like Bill Unruh and Wald have made many such models. Indeed it makes a good deal of physical sense when viewed from that lense.

However I think the other side of the community takes a somewhat different but perhaps complementary tact. Of course they use Penrose diagrams to setup the calculations, but the interpretations are often much more subtle.. For instance in the traversable wormhole paper I mentioned, the descriptions of the two asymptotic regions really corresponds to an analogy with two quantum computers, with emergent gravitational duals and how they can share quantum information together (and how the mechanism propagates)...

Indeed, in general holography can do quite a number on what we think of as locality and how information is transferred, even on the macroscopic scale and its not entirely clear how completely classical concepts can hope to fully account for this.
 
  • #18
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Thanks to @Haelfix for his reply.

I don't know if I'll ever get time to think this through, but here are some references I would use, to start trying to understand what's going on in the 2d case.

My focus would be on two preprints from a year ago, described by Quanta magazine as from East Coast USA (Princeton-Cornell) and West Coast USA (Stanford). They both study a "pinwheel geometry" with multiple asymptotic regions (see East Coast page 10 figure 7, West Coast page 32 figure 4.9).

If I have understood correctly, the branches of the pinwheel are "replicas" of the 2d black hole being analyzed, and the Page curve comes about as a transition in which wormholes to the replicas dominate the dynamics.

The replica trick is a mathematical technique for calculating statistical properties of a system, by calculating the property for n replicas of the system, something which involves a power of a partition function, and then taking the limit as the exponent goes to zero. This takes us back to n=1, but it is as if we are passing through non-integer numbers of copies of the system to get there.

Mathematically, the validity of the replica trick has not been proven; but Wikipedia helpfully hints that it could be proven, if Carlson's theorem could be shown to apply to the functions. ("Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers.")

Under West Coast figure 4.9, we are asked to consider, not the full pinwheel, but rather a quotient of the pinwheel to just two of its arms. Possibly this is a context in which the replica calculation can be understood, without actually needing all the replicas? Taking the limit here involves something like varying a solid angle in the interior of the gravitational path integral.

Meanwhile, here is a close analysis of how the replica trick is applied in another system with one space dimension.

So one strategy of interpretation would be to say, the pinwheel isn't real, it's just a calculation method that works because Carlson's theorem applies, and the core issue is "one-sided" versus "two-sided", as in #5... This "strategy" could be entirely wrong, but in trying to make sense of things, one must start somewhere...
 
  • #19
Demystifier
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Leaving technical issues aside, I think the best conceptual explanation of the big picture is given in the semi-technical review https://arxiv.org/abs/2006.06872

In my opinion, the only reasonable intuitive picture of what all this means is that the black hole is really a wormhole connected to a parallel universe. Otherwise, I don't see how to make sense of black hole islands semi-classically.
 
  • #21
PeterDonis
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The wormhole connected to the parallel/baby universe is starting to make more and more sense to me ...
I agree that if we assume there is a true black hole (i.e., a true event horizon and region of spacetime inside it that cannot send light signals to future null infinity), this model avoids a number of issues that would otherwise arise when the hole finally evaporates.

My personal preference for avoiding those issues would be for a model in which a true event horizon never forms in the first place, i.e., in which quantum gravity effects end up allowing only apparent horizons to form, which eventually evaporate away but in which every event in the spacetime is still in the causal past of future null infinity. However, of course we don't know if that's the way things will actually work out when we find a theory of quantum gravity.
 
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