- (Information) Paradox Lost

Tim Maudlin

arXiv:1705.03541 [physics.hist-ph]

Here is the problem. The dynamics of quantum field theories is always reversible. It also preserves probabilities which, taken together (assuming linearity), means the time-evolution is unitary. That quantum field theories are unitary depends on certain assumptions about space-time, notably that space-like hypersurfaces – a generalized version of moments of ‘equal time’ – are complete. Space-like hypersurfaces after the entire evaporation of black holes violate this assumption. They are, as the terminology has it, not complete Cauchy surfaces. Hence, there is no reason for time-evolution to be unitary in a space-time that contains a black hole. What’s the paradox then, Maudlin asks.

First, let me point out that this is hardly news. As Maudlin himself notes, this is an old story, though I admit it’s often not spelled out very clearly in the literature. In particular the Susskind-Thorlacius paper that Maudlin picks on is wrong in more ways than I can possibly get into here. Everyone in the field who has their marbles together knows that time-evolution is unitary on “nice slices”– which are complete Cauchy-hypersurfaces –

*at all finite times*. The non-unitarity comes from eventually cutting these slices. The slices that Maudlin uses aren’t quite as nice because they’re discontinuous, but they essentially tell the same story.

What Maudlin does not spell out however is that knowing where the non-unitarity comes from doesn’t help much to explain why we observe it to be respected. Physicists are using quantum field theory here on planet Earth to describe, for example, what happens in LHC collisions. For all these Earthlings know, there are lots of black holes throughout the universe and their current hypersurface hence isn’t complete. Worse still, in principle black holes can be created and subsequently annihilated in any particle collision as virtual particles. This would mean then, according to Maudlin’s argument, we’d have no reason to even expect a unitary evolution because the mathematical requirements for the necessary proof aren’t fulfilled. But we do.

So that’s what irks physicists: If black holes would violate unitarity all over the place how come we don’t notice? This issue is usually phrased in terms of the scattering-matrix which asks a concrete question: If I could create a black hole in a scattering process how come that we never see any violation of unitarity.

Maybe we do, you might say, or maybe it’s just too small an effect. Yes, people have tried that argument, which is the whole discussion about whether unitarity maybe just is violated etc. That’s the place where Hawking came from all these years ago. Does Maudlin want us to go back to the 1980s?

In his paper, he also points out correctly that – from a strictly logical point of view – there’s nothing to worry about because the information that fell into a black hole can be kept in the black hole forever without any contradictions. I am not sure why he doesn’t mention this isn’t a new insight either – it’s what goes in the literature as a remnant solution. Now, physicists normally assume that inside of remnants there is no singularity because nobody really believes the singularity is physical, whereas Maudlin keeps the singularity, but from the outside perspective that’s entirely irrelevant.

It is also correct, as Maudlin writes, that remnant solutions have been discarded on spurious grounds with the result that research on the black hole information loss problem has grown into a huge bubble of nonsense. The most commonly named objection to remnants – the pair production problem – has no justification because – as Maudlin writes – it presumes that the volume inside the remnant is small for which there is no reason. This too is hardly news. Lee and I pointed this out, for example, in our 2009 paper. You can find more details in a recent review by Chen

*et al*.

The other objection against remnants is that this solution would imply that the Bekenstein-Hawking entropy doesn’t count microstates of the black hole. This idea is very unpopular with string theorists who believe that they have shown the Bekenstein-Hawking entropy counts microstates. (Fyi, I think it’s a circular argument because it assumes a bulk-boundary correspondence ab initio.)

Either way, none of this is really new. Maudlin’s paper is just reiterating all the options that physicists have been chewing on forever: Accept unitarity violation, store information in remnants, or finally get it out.

The real problem with black hole information is that nobody knows what happens with it. As time passes, you inevitably come into a regime where quantum effects of gravity are strong and nobody can calculate what happens then. The main argument we are seeing in the literature is whether quantum gravitational effects become noticeable before the black hole has shrunk to a tiny size.

So what’s new about Maudlin’s paper? The condescending tone by which he attempts public ridicule strikes me as bad news for the – already conflict-laden – relation between physicists and philosophers.

## 89 comments:

Maybe we should stop talking of just "information" -which is more akin to communication engineering- and start thinking of "Information Processes", in which case we have a process transformed into another process after some nice free fall exercise. (Think of a "black hole CPU"!)

https://www.youtube.com/watch?v=7zVeOYlhA78

Tnx Bee! Between the two of you, you have provided much clarity for this reader!

Dear Dr. B.

There is a thing I do not understand in the remnant solutions.

As I understand it, black holes evaporate (almost) completely but a lot of information is left in the remnant. I was taught that information requires a carrier that has energy/mass. In the end, the remnant should have a lot of information but very little mass.

So, I gather one or both of these "understandings" of mine must be wrong, but which one is it?

Cosmic background radiation is blackbody 2.72548 kevins (2.34864 eV). This black hole (BH) temperature is 2.263×10^(-8) solar masses, lifetime 2.432×10^35 Gyr.

No contemporary BH evaporates- by a huge margin. Low mass primordial BH would be exploding. A 13.82 Gyr BH is 8.7×10^(-20) solar masses with radius 2.6×10^(-7) nm, oozing extreme observables.LIGO events GW150914 and GW151226 merged to equilibrium within milliseconds, 4.6% emitted binding energy both, 2D+ε soap bubbles merging with no interior volume and no wildly gyrating singularities therein.

BH theories give exotic predictions because they model unreal constructs. arXiv:1705.01597

Nothingis empirical.For the philosophy it doesn't matter the black hole paradox. It happens to an astronomical scale very far away from human's problems.It is a problem for the gods, not for the humans.

If spacetime were continuous over the whole time of the existance of the black hole, wouldn't that remove the paradox - since information could eventually escape as the event horizon recedes? Wouldn't that move the question to whether or not spacetime is continuous?

Sabine, I'm having trouble understanding what, if any, substantive dispute there is between you and Maudlin on this.

The main point of the blog post seems to be that there's nothing new in Maudlin's paper. But that's also the main point of Maudlin's paper, whose abstract says, "The resources for resolving the "paradox" are familiar and uncontroversial, as has been pointed out in the literature."

Perhaps there's a disagreement whether it's worth writing something that reminds people of something familiar and uncontroversial. But if these are things to which insufficient attention is being paid, and whose significance hasn't been appreciated, then, it seems to me, it is worth doing.

Also, I think there's an important difference between saying that the significance of these familiar facts hasn't been appreciated---which is what Maudlin says---and saying that there are physicists that don't understand them.

Is the main point of disagreement, then, over Maudlin's claim that the significance of the fact that Sigma_2 is not a Cauchy surface has not been widely appreciated?

Theophanis,

It's been said many times before, but clearly not often enough. The reference to information is a red herring. It is entirely irrelevant exactly what is meant by information here.

Rob,

Yes, very little energy, and potentially lots of bits. This means very little energy per bit.

Wayne,

For all I can tell I don't disagree with Maudlin. I merely think the paper lacks some context and makes physicists look rather stupid by leaving out part of the story. I hope to have provided that part of the story here.

Ambi,

Space-time in GR is continuous. I'm not sure what you mean.

"So what’s new about Maudlin’s paper? The condescending tone by which he attempts public ridicule strikes me as bad news for the – already conflict-laden – relation between physicists and philosophers."

I suspect a lot of this is just Tim Maudlin's style. It's pretty much he always writes. In philosopher-philosopher discussions too he often comes across as somewhat condescending - this tone can cross over to contemptuous.

It'd be unfortunate if it aggravated any philosophers vs physicists conflicts. I don't think it's specific to that divide at all.

In my very humble opinion, the conflict-laden relationship between physicists and philosophers is a consequence of the orthogonality of their respective language-based axioms and premises. Hence, conversations between them are hopelessly bogged down by the impedance of their dialog.

Sabine,

I mean, if one has only passed the event horizon, information can still travel into all directions, including outwards. It's just that it cannot escape to infinity anymore. Under the math of ART with classical gravity, information travels outwards slower than spacetime gets stretched by the ongoing collapse, so it would remain in the black hole forever.

If however the stretching of spacetime would be somehow reversed (eg by mass loss due to Hawking radiation), information could continue to travel outwards - removing the paradox.

If one has already passed inside the apparent horizon, information couldn't travel outwards in the first place. But how does one know?

There is a real question in my mind as to whether philosophers have anything useful or interesting about science.

I would recommend you to read the paper by Prof. Dr. Stefan Hofmann from LMU Munich on "Classical versus Quantum Completeness."

https://arxiv.org/abs/1504.05580

Jillur,

I know the paper. Also, by way of a weird coincidence, I talked to Stefan just yesterday. The paper you mention is relevant to the point I made in my paper with Lee, but not to the one discussed here.

CIP,

Ironically I've spent the whole week at the Munich Center for Mathematical Philosophy talking to philosophers, so I'm inclined to say the answer is yes. Even on the bh infloss problem I think that's the case. Unfortunately, Maudlin's paper doesn't address what I think are the relevant points. I'm kind a curious to see if anyone else in the community takes note of it at all. Best,

B.

Ambi,

I don't know what you mean by 'stretching' but, yes, if the black hole evaporate there are cases in which an outwards traveling particle will eventually come out again. I can't see what this has to do with your previous comment thought.

There is a conflict between Philosophers and Physicists only because many students of philosophy (and unfortunately, some professional philosophers also) are of the belief that arguing with words is enough. They therefore find it perplexing that physicists and mathematicians prefer to manipulate symbols. Manipulating symbols does not confer understanding they say. Utterly rubbish.

Hell, philosophers still consider the "Paradox of the Pile" a paradox while those of us who learned mathematics know this is just a failure of proper definition. Humans do not have the cognitive ability to understand nature without the symbolism of mathematics. It's just a fact.

We have already known since the time of Russel that words alone are not sufficient in arguments. Words are self-referential and inherently contradictory. I wish more students of philosophy understand that. Then they can appreciate why manipulating symbols (aka rewriting rules) are very powerful ways of creating understanding.

Sabine, I understand that particles can be in a superposition, so why not spacetime? Isn't it possible that a black hole is in some kind of superposition with a white hole? Only the white hole part has a very low probability amplitude. Can't information escape via the low probability white hole counterpart? Just like the sun has a small probability to teleport 1 lightyear away.

Patat,

Yes, that's possible. It's also exceedingly unlikely though.

I think it is unlikely on a short time interval. On a extremely long time interval I think it is guaranteed. It takes eons for a black hole to evaporate.

Silly question, but what is the status of energy conservation over the lifetime of a black hole?

My understanding is that a physically meaningful conservation law arises from a symmetry of the vacuum (or a nominal background) state (rather than just a mathematical symmetry of the theory).

I see two problems applying this to black holes:

1. It is hard to see how what the time-translation invariant background is that includes a singularity appearing and/or disappearing.

2. For realistic black holes, the universe will expand by a significant factor over their lifetime, and the locality of the black hole deviating from the approximation of a uniformly expanding universe. So presumably we bang our head against the fact that energy is only locally conserved in GR.

Getting back to the information paradox: if there is a 'problem' with conservation of energy for black holes, then loss of information is natural. The lost information can be associated with lost energy, and you can hide both, by either accountancy (put them in a column labeled 'lost') or a philosophy (invent child universes spawned from the black hole).

Bee, I don't see any reason to be annoyed by Maudlin. It is a lucid paper, and if the physicists didn't beat him to writing it, I don't think they can complain, even though they already know everything that Maudlin wrote.

The fragments of Cauchy surfaces we consider for say, analyzing LHC experiments do not extend all the way to a black hole in Andromeda or in the center of the Milky Way and that is why unitarity holds to a sufficiently good approximation in our neighborhood. Because physics is local we don't need Cauchy surfaces that extend all across the universe for our experiments. If I had to know the complete geometry and topology of the universe in order to do experiments in a laboratory, then physics would be impossible to do. Fortunately, nature hasn't put us in that position.

But as to why virtual blackholes don't damage unitarity in the same experiments - that I don't know, and can only wave my hands and say that the effect must be small because of the weakness of gravity. On the other hand maybe it is a significant effect, and the constant degradation of our seemingly complete Cauchy surface by virtual blackholes is what introduces time-asymmetry at our scale.

Arun,

The paper doesn't even mention several points that are relevant to the argument, as I laid out in above blogpost. I wouldn't call that 'lucid'.

Well, I think Tim Maudlin has come to the comments here previously; I hope he does so again. It might be productive.

Bee, you yourself wrote: "It is also correct, as Maudlin writes, that remnant solutions have been discarded on spurious grounds with the result that research on the black hole information loss problem has grown into a huge bubble of nonsense."

It is exactly these huge bubbles of nonsense (blackholes is not the only one) that theoretical high energy particle physicists as an internal community have failed to puncture on their own, even though the antidote to the bubbles is known within the community. Maybe it takes outsiders - complete outsiders like Maudlin, and partial outsiders like Woit - to make the theoretical HEP physics community come to order.

Isn't it possible for all te superpositions of spacetimes to create an escape route from the interior of the black hole to the outside world, without breaking causality? And information can escape this way? A sort of spacetime tunneling.

Arun,

Yes, I agree. That's why I found this article so disappointing. It's easy to dismiss, and I seriously doubt anyone in the field will take it seriously because it's obviously missing key points.

Patat,

yes, and somewhere in the multiverse that's exactly what happens. It's exceedingly unlikely though that this happens in the universe we inhabit.

What I understand from you is that the probability of information tunneling out of a black hole is too small to guarantee that all Hawking radiation contains information about the black hole interior. I imagined Hawking radiation as all the information eventually escaping the black hole via tunneling. But it is too unlikely. But... if information escapes a black hole, isn't it a white hole by definition?

I'm probably missing something here, but isn't Maudlin's point (or Maudlin's construal of Wald's point, or what have you) that the usual reasoning regarding unitarity violation in BH evaporation is invalid? I.e. we can only expect unitarity along complete Cauchy surfaces, which the spacelike surface that is usually invoked in claiming that 'unitarity is violated' (his \Sigma_2) fails to be. So when we calculate that the evolution from some pre-evaporation Cauchy surface to \Sigma_2 is non-unitary, then well, big woop---there's no reason it ought to be, even in vanilla QM.

If that's indeed the point, and the argument is correct, then I think pointing it out is a tremendously useful thing, at least to me, personally---even if it might be clear to every expert in the field (in which case there seem to be lots of papers written by experts that are less than clear on this point), my understanding of the problem always was, basically, 'the evolution from \Sigma_1 to \Sigma_2 is non-unitary, but should be unitary'. If it's in fact correct to say 'the evolution from \Sigma_1 to \Sigma_2 is non-unitary, and there's no reason to expect it to be', then I think the 'problem' as such is far less pressing than usually presented.

Jochen,

Yes, you are missing something. It is correct that the state on the incomplete surface will generically be non-unitarily related to the earlier complete surface because you're leaving behind part behind the horizon. It is incorrect to think that this alone solve the problem. Every experiment that we do is located outside of the black hole horizon. The problem being that for all we can tell unitarity works just fine. Why if, as you said, it shouldn't be so? Now, you could say that maybe it just isn't unitary and we haven't notice, or information comes out after all, etc etc. That's the very story that Hawking started 40 years ago. Let me say this again: Just noting that there is a mathematical reason why it shouldn't be unitary does *not* remove what is normally considered paradoxical.

Thanks for your answer. So, if I understand you correctly, the puzzle is in fact that the evolution between \Sigma_1 and \Sigma_2 appears to be unitary, and thus, that using data of \Sigma_2 we ought to be able to reconstruct the quantum state at \Sigma_1. But is there actually an experiment we can do in the lab that would probe this? Seems to me that unitarity ought to still hold for any experiment where everything between preparation and measurement is kept well away from black holes, even if it's violated 'globally'. Is that not the case?

Jochen,

Black holes can in principle be produced in any particle collision - that's quantum mechanics for you. If they exist at all, they should be there in intermediate states. I actually explained this in the above blogpost. The question is, what does the scattering-matrix look like. Yes, you might say you can just do without unitarity, and people have tried to make that work - some still believe that's the way to go (see Unruh et al), and so on. I'm not saying that accepting non-unitarity is not an option, I am just saying you have to make it work, and people have tried to make it work rather unsuccessfully. In any case, if Maudlin's point was to say we should reconsider non-unitarity, then he should have explained at least how that's not a problem with observation etc. Which is an argument that can be made and has been made - and yes, maybe there's something new to say about this - but it's not the argument he did make.

Sorry to keep pestering you, I'll let it go after this post, but to me, it's not clear that one should expect any non-unitarity in scattering processes even if the evolution between slices \Sigma_1 and \Sigma_2 is non-unitary---after all, 'intermediate' black holes really are just terms in a perturbation expansion for what's itself a unitary operator; that this perturbation series should introduce any non-unitarity seems odd to me. So I don't see that I should worry about the non-unitarity introduced by virtual black holes any more than I should worry about being sucked into them. ;)

I guess what I'd want to see is that if there is some non-unitarity to be expected in ordinary laboratory scattering processes, how big of an effect it would have to be, and whether it should be obvious to present-day experiments, or within reach of experiment, or completely non-accessible. In short, I'd like to know if there is any actual difference in phenomenology between the case where there just isn't any unitary evolution between \Sigma_1 and \Sigma_2, and the case where there is, and thus, an information paradox exists; because if there isn't such a difference, then I think it wouldn't be unreasonable to conclude there's also no problem.

There seem to be some basic confusions here about experimental bounds on violation of unitarity. There have been some papers about such bounds, but the empirical part has nothing to do with collision experiments. A test for unitarity has to look for interference effects, such as neutron interferometry experiments, which are about as far as you can get from particle collisions. Detailed investigations of observable empirical signatures of violation of unitarity have been most extensively studied for the GRW collapse theory, where we have an exact equation to work with. So far, no detectable effect of the non-unitary evolution in that theory has been found, and people have been looking hard. (Similar comments apply to Penrose's gravitational collapse theory, although it is not as well-defined as GRW.) So the idea that any violation of unitarity must have presently noticeable effects is just wrong. And the idea that one should even be looking at particle collisions is probably wrong.

Arun's comments at the start are correct: evaporating black holes outside the earth would lead to a mixed state of the universal wavefunction on Sigma 2. But (to say the least!) no one has ever made or ever will make an empirical prediction based on the universal wave function. So evaporating black holes outside the lab make absolutely no empirical difference for predictions about what happens in the lab. What about evaporating black holes in the lab? Well, in overwhelmingly most labs, there just aren't any. Where would they come from? Sabine suggests that they might be formed in particle collisions, but that would require enough energy to form them. No reason to think it has ever been done. There would obviously be a signature of particles going in and only thermal radiation coming out. (I would also expect offhand that the huge proton decay experiments would have noticed such a signature if there were microscopic evaporating black holes floating around somehow. There aren't any.)

The next confusion concerns "virtual evaporating black holes". Suffice it to say that what are called "virtual" things are not real. They are mathematical fictions, used to make certain calculations. In addition, no one in history has included "virtual black hole evaporation" in any actual calculation ever made. It would, on any view, be a process with, say, particles going in and thermal radiation coming out. They would only show up in a quantum theory of gravity, which of course does not exist yet.

The idea that people have tried to make this idea work and did not succeed is unfounded. No one has been looking specifically at the sort of pure-to-mixed evolution that falls out of this analysis: pure Cauchy-to-Cauchy evolution followed by tracing out. The papers I am aware of are a paper by Ellis, Hagelin, Nanopoulos and Srednicki, which is pretty careful and looks specifically at experiments like neutron interferometry which are actually relevant, and a sort of silly paper by Banks, Susskind and Peskin that does nothing relevant. Again, it is highly relevant to look at the work on the GRW theory, which has the advantage of being an exact theory that violates unitarity. That work is certainly a refutation of any claim that violating unitarity must lead to some obvious, presently existing empirical problem.

(Con't)

Jochen's comment above is right on target. The reason there has not been much work on the consequences of pure-to-mixed evolution is that there has been the completely incorrect claim that such evolution violates quantum theory somehow, and if you want to keep the fundamental postulates of quantum theory intact you have to have pure-to-pure evolution from Sigma 1 to Sigma 2. What my paper points out is that this is the opposite of the truth. Quantum theory only implies pure-to-pure evolution for Cauchy-To-Cauchy evolution. If we take the Penrose diagram that Hawking provides seriously, then Sigma 1-to-SIgma 2 is Cauchy-to-non-Cauchy. Not only does quantum theory not predict that this evolution will be pure-to-pure, it predicts that it will be pure-to-mixed if the state on Sigma 2 (= Sigma 2out) is entangled with the state of Sigma 2in, inside the event horizon. If they are entangled, tracing out over Sigma 2in will yield a mixed state on Sigma 2 out. Finally, Quantum Field Theory implies that the state on Sigma 2in will be highly entangled with the state on Sigma 2out. So not only is the common claim that "quantum mechanics predicts that the state on Sigma 2 will be pure" not well founded, it is the exactly and precisely false. Fundamental principles of quantum mechanics and QFT entail that the state on Sigma 2 will be mixed.

There will be a discussion of the empirical implication of failure of unitarity in the next version of the paper, which should be done soon.

Sabine, thanks for this blog post bringing the matter to my attention. I've been doing a lot of reading and writing on the subject of late, and this is very timely. As for Maudlin's

"condescending tone by which he attempts public ridicule", I rather fear things are going to get worse.Tim,

Three things. First, as I've said a number of times, I am actually quite sympathetic to your take on the matter. Regarding your assertion that no one has ever looked at virtual black holes, however, you might want to check out this (and the long string of references before and after that).

Second, next thing people will come with is the BH entropy, for reasons see blogpost.

Third, most of them don't believe there's a singularity to begin with and they don't believe in remnants (see second point) meaning the final slice is actually complete.

Let me repeat that I am not telling you this because *I* think this is a good argument, but because I've heard this story forwards and backwards 10 million times.

And once you're at that point, the only thing one can conclude is that some people like it this way and some people like it that way and we'll keep on discussing this forever. (Which is pretty much what I wrote in my recent blogpost on the topic if you recall.)

Jochen,

You might want to have a look at this paper for phenomenological consequences. Please note that my point here is not to say that abandoning unitarity is not an option, but merely to say it's an option that has been discussed and I can't see what new has been added to this discussion.

FWIW, I discuss black hole entropy in the next version.

philosophers of physics trying to do physics reminds me of something i once read that goes "when you see a flying pig, you shouldn't critique how well it flies; you should be impressed that it flies at all". there's also the saying "you can't talk the talk unless you've walked the walk". and finally, there's the Nobel laureate Bob Dylan lyric from "Positively 4th Street" that goes

I wish that for just one time you could stand inside my shoes

And just for that one moment I could be you

Yes, I wish that for just one time you could stand inside my shoes

You'd know what a drag it is to see you

This "philosophers vs. physicists" meme is completely off base. Not many people work in the foundations of physics. Very few physicists do. The community that actually works in foundations consists of philosophers, mathematicians, and physicists. If you think there is an error in this paper you are free to point it out. But if there is going to be actual progress, physicists have to stop being so defensive. Respond to the arguments, not ad hominem. (Sorry: a philosopher's phrase.)

Dylan again: There's something happening and you don't know what it is, do you, Mr. Jones?

Sabine and Tim,

what exactly are the bases of your models of black holes: The static black hole? The dynamic collapse towards a black hole? Or yet other models? And in what coordinate systems are you operating in?

Can I add that when I said I fear things are going to get worse, I wasn't referring to Tim Maudlin's tone. I was thinking of public perception of the black hole physics community. The recent inflation hoo-hah is more of the same.

To lard your paper with gratuitous ad hominem comments, and then complain about someone noticing... Well, we have your measure, Mr. Maudlin.

Ambi Valent,

The model is more or less a sequence of static black hole with successively smaller masses and hence smaller event horizons. The basic idea is to use a static black hole as a fixed background space-time, calculate the Hawking radiation, derive an energy flux from that, use a principle of global conservation of energy to argue that the black hole must lose and equivalent mass, then switch to a static background space-time of a black hole with the new, smaller mass, rinse and repeat. There is no reply principled way to deal with the emission of the Hawking radiation and the backreaction of the metric all in one swoop. That is the backreaction problem after which the blog is named.

I should add, although it is not mentioned in this paper, I think that there are conceptual problems with this whole story. But that is the subject of another, even more controversial, paper.

Ambi Valent,

It's a collapsing black hole with the evaporation-part added as a guess since nobody knows exactly what happens. The considerations in Tim's paper only concern the causal structure and the coordinate system is entirely irrelevant for that. This is the usual situation in that kind of discussion.

Araybold,

Please point out any ad hominem comment in my paper, which you say is larded with them. I am certain there is not a single one. Are you sure you know what the phrase means?

Suggestion for the decractors of Tim:

Is a confirm of evaporation of a black hole a lack of content in the article of Tim?

Anyway I think that Tim "flows" with a ratio of a one article per one article!

Joke apart, I think very interesting the article of Tim as ever.

Nonetheless I have to thanks Sabina for posting it.

Great Job!!

I think there is a critical mistake in this Maudlin paper, namely that the crux of the argument -- that Sigma_2 is not a Cauchy slice -- cannot be concluded from the arguments given.

The mistake involves an erroneous over-interpretation of the Penrose diagram for the evaporating black hole. A Penrose diagram suffices only to represent the causal structure of a classical spacetime, i.e. a solution to general relativity or its extensions. It's basically a picture of the metric, the classical field that determines the spacetime geometry. One can attempt to modify the diagram for a particular solution (e.g. a black hole formed from collapse) to account for weak quantum-gravitational effects, such as Hawking radiation, leading to Maudlin's figure 4, but this depiction is just a cartoon, and if read too literally it will lead to incorrect conclusions.

When strong quantum-gravitational effects are important there is no notion of locality since the metric undergoes large quantum fluctuations, just like any other quantum field. At the end stages of black hole evaporation (or perhaps earlier), quantum-gravitational effects dominate. The Penrose diagram does not capture this physics by its very definition, since the diagram is very literally a depiction of the spacetime's causal structure, which it should be emphasized is not even a well-defined concept in quantum gravity. In the paper, Maudlin gives elementary arguments based on the semiclassical causal structure to argue that Sigma_2 cannot be a Cauchy slice, but these arguments are applied precisely to a situation in which classical GR is not valid, the causal structure receives large quantum corrections. One cannot conclude that the geodesics in question fail to make it to Sigma_2 without knowing the full dynamics of quantum gravity.

In fact, we know from AdS/CFT that evolution from Sigma_1 to Sigma_2 is indeed unitary. Maudlin does address holography briefly at the end of the paper, but unless I have missed something, his argument in the second paragraph of that section is identical to the remnant scenario (the idea that the post-evaporation Cauchy slice contains some degrees of freedom that don't escape the horizon). This scenario has long been ruled out by basic physical considerations, as discussed in most comprehensive reviews of the information paradox. It is also ruled out explicitly by AdS/CFT.

While I don't want to wade into the personal waters, I will remark that the provocative tone used throughout the paper could cause offense in several places, and that Tim shouldn't be surprised when some read the paper as larded with derision.

dark star,

At the beginning of the paper, I state explicitly that the outcome of the paper will be one of two things: either the "paradox" will no longer be considered paradoxical (and in particular it will no longer be claimed that there is a fundamental conflict between quantum theory and general relativity), or the exact nature of the paradox will be clarified. I take it that you are suggesting the latter resolution. Let me make some comments before turning to the proposed resolution.

The first comment is that the Penrose diagram that I am commenting on is universally used in presenting the "paradox", from Hawking onward. Given the conventions for Penrose diagrams, it depicts an exact causal structure (i.e conformal structure) to be analyzed. That is the structure I do analyze. The diagram does not, as you assert, give a picture of the metric, but only of the conformal structure (causal structure). As such, adding the infalling matter does not affect the diagram. A Penrose diagram is not "just a cartoon" that can be variously interpreted, it is a precise depiction of a conformal structure. Of course, I clean up Hawking's diagram, which is a bit vague on certain points (such as whether the EE is in the space-time or not) but I also discuss the situation with and without that specification. So I am taking the diagram seriously.

Now that might be a mistake. Maybe the diagram has never, since Hawking's paper, been meant to be taken seriously. If so, then the clarity of the usual presentations, from Hawking on, has been seriously lacking. There is no warning given that one ought not to take the diagram seriously, of how it might be misleading. So at the very least, there has been an extremely serious breakdown in the clarity and precision with which the paradox has been standardly presented. As a side remark, neither George Ellis nor Robert Wald nor Ted Jacobson, all of whom are prominent experts in General Relativity, have made any complaint about the diagram or the resulting analysis. So even on your account there has been some sort of widespread misunderstanding in the professional community.

(Con't)

How does your presentation of the paradox go? Well, you say that "in quantum gravity" the causal structure is not well-defined, so the Penrose diagram should not be taken seriously. There are several puzzles about this. One is this: since no theory of quantum gravity actually exists, how do you know that causal structure is not well-defined? More particularly, how do you know it is ill-defined in a way that renders the diagram incorrect? You seem to be importing in results from a non-existent theory, whereas the paradox was supposed to provide some clues to discovering that very theory.

This logical structure seems to be exactly backwards. One gets the paradox by trying to take both GR as we have it and quantum theory as we have it and deriving a contradiction from their conjunction. In the normal presentation, the contradiction is supposed to be something like this: GR implies that the evolution from Sigma 1 to Sigma 2 is pure-to-mixed and loses information (is not retrodictable), while quantum theory demands that the evolution from Sigma 1 to Sigma 2 must be unitary, deterministic, pure-to-pure and retrodictable. On this presentation, which I claim is the usual one, one is forced to choose between GR and quantum theory. I address this presentation by pointing out that in the situation as presented in the Penrose diagram this is a false dichotomy and hence a false paradox: sticking to both quantum theory and GR, and taking the diagram seriously, leads to the conclusion that the state on Sigma 2 should be mixed and non-retrodictable. This does not contradict quantum theory but rather is demanded by quantum theory in this setting.

On your understanding, what is the paradox? We start with a space-time whose conformal and causal structure is somehow undefined in certain places, so the whole concept of a Cauchy surface is not well defined. We are not given anything like a Penrose diagram depicting the situation. On what basis, then, are we to conclude anything about the state on Sigma 2? We know that quantum theory, even in plain vanilla Minkowski spacetime, demands Cauchy-to-Cauchy evolutions that are unitary and predictable and retropredictable, and also that it allows for and generically predicts Cauchy-to-non-Cauchy evolutions that are not unitary, are pure-to-mixed, and are not retrodictable. If the correct "quantum gravity" space-time structure does not allow for the definition of a Cauchy surface, then we have no grounds to expect anything, one way or the other, about the Sigma 1 to Sigma 2 evolution. How is that a paradox? However it comes out, we have not violated any principle.

Now about AdS/CFT. How do you think it follows, if AdS/CFT is true, that the evolution from Sigma 1 to Sigma 2 is unitary? If pure states on the boundary always map to pure states in the bulk, then we know that a pure-to-pure transition on the boundary maps to a pure-to-pure transition in the bulk. (This all assumes that the conformal structure on the boundary is unproblematic.) Fine. And let me even grant that the initial pure state on the boundary maps to a pure state on Sigma 1 in the bulk. But by what argument can one conclude that the final pure state on the boundary maps to *the state on Sigma 2* in the bulk? All we know is that it maps to some pure state in the bulk. Why not the state on Sigma 2 U Sigma 2in is the bulk? According to my analysis, this state ought to be pure, and the state on Sigma 2 alone mixed. Somehow you conclude that CFT says it must the the state on Sigma 2 that is mapped to, but I can't see any argument at all to that conclusion. We would need a full dictionary connecting states on the boundary to states in the bulk to conclude anything, and we have no such dictionary. So my solution is not "ruled out explicitly" by AdS/CFT.

(Con't)

In fact, simple dimensional considerations assure us that the map from states on surfaces on the boundary to states in the bulk must be highly non-trivial and non-local: you are connecting states of different dimensionalities to one another. Why can't the state on a connected surface on the boundary map to a state on a disconnected surface in the bulk? These surface-to-surface mappings cannot be continuous, for the dimensional considerations just given.

Finally, you say that remnant scenarios have "long been ruled out by basic physical considerations". I would dearly love to know what those "basic physical considerations" are, as I have not been able to find them and no one will tell me what they are. Or at least point me to a comprehensive review that, in your view, lays out these considerations in a clear way. I have asked many people for where I might find a clear statement of the paradox, and quite simply have never gotten one. But of course I may have missed it. So a suggestion for where to look would be greatly appreciated.

I also don't want to go into detail about the tone of the paper, but I will say two things. At the beginning I say that the paper is a provocation. It is meant to be. I also explain why it is and what sorts of response would be appropriate. The other is that I have had a prominent physicist say that he thought the tone was not provocative save for the last paragraph, which is a parody of Hume that he recognized, but thought that other physicists might not recognize. Maybe people find that passage offensive. Philosophers, knowing the reference, find it amusing. I may well remove it from the final version. But if you can point to anything else in the paper, from beginning to end, that has some offensive tone, I would appreciate pointing it out. I have been told on this blog that the paper is full of ad hominem arguments, when it does not contain a single one. I am explicitly arguing that the theoretical physics community has been on a wild goose chase for forty years. If that claim is true, then there is not going to be a way to make it that does not raise a lot of hackles. And if it isn't, then at least we can get a clear account of why my argument is incorrect and what the paradox really is. But you ought to at least consider the possibility that some of the hostile reaction is attributable not to the tone of the paper, but to the thesis of the paper. I have tried to present the argument deliberately, clearly, and slowly, and have repeated key points. I do that because of long experience having the key points in papers overlooked or misconstrued. If that comes off as pedantic, that is a drawback. But it makes the target of refutation easy to find. If there is a mistake in the paper, it is somewhere explicitly on the page. For the reasons given above, I do not think you have located any error.

dark star,

It's right that the Penrose diagram that Tim has in his paper is only one possible time-evolution since we don't know what happens in the QG regime. However, it happens to be the case that Tim is looking at, so I can't see how that's a mistake. You may question whether it's *relevant*, but nothing wrong with it.

But you state that remnant scenarios are ruled out and that is wrong. Please point me towards the literature that you refer to. I'd be very surprised if you actually have any argument to back up your claim.

Just to complete the above:

"If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, "Does it contain any abstract reasoning concerning quantity or number?" No. "Does it contain any experimental reasoning concerning matter of fact and existence?" No. Commit it then to the flames: for it can contain nothing but sophistry and illusion." - David Hume,

An Enquiry Concerning Human Understanding, 1748.Tim Maudlin: What is this dogma "virtual things are not real"? I see it everywhere, so they must teach it in physics graduate schools. I totally disagree. You cannot draw a line between "virtual" particles and "real" particles. Every particle is to some degree off-shell, and so virtual. If real black holes violate unitarity, so do virtual ones.

Does anybody have a theory which draws a distinction between "virtual particles" and "real particles", and shows how "real black holes" can violate unitarity while "virtual black holes" don't? I don't think so, and if you know of one, please give a reference.

Sabine,

Thanks for posting this, I found both your article and the ensuing discussion very interesting. I have to say, though, that I find the

titleof your article more condescending than anything in Tim's article (save, perhaps, the final humorous paragraph, which may be taken amiss by those unfamiliar with the riffed passage in Hume). I don't see that you point to anything that Tim hasfailedto understand; at most you are criticizing him for not saying more about certain issues.As I see it Tim is trying to bring a bit of clarity to an area of discussion that has suffered from, at least, a serious lack of it [clarity]. And I take it that you would agree that greater clarity is sorely needed, since you say "... research on the black hole information loss problem has grown into a huge bubble of nonsense." If things have gone off the rails in this way, all efforts to put things back on track should be lauded.

I look forward to (hopefully) seeing followup articles by Tim on the things he hinted at: the original argument for BH evaporation, AdS/CFT, . . .

Sabine,

It may well be a cartoon of the case that Tim is looking at, but it does not accurately depict the spacetime geometry (which probably does not exist in the sense we are accustomed to) near the "evaporation event", hence it is irrelevant for his discussion of the Cauchyness of Sigma_2.

This may surprise you, but my understanding is that the arguments against remnants are strong. I'll point to [9209058], [9304027], [9412159] and [9501106] as examples of basic physical arguments against. Recent reviews discussing AdS/CFT and remnants include [1409.1231] and especially [1703.02143]. I would be interested to know what issues you take with these arguments.

Tim,

Re: post 1.

As I said in my post, a Penrose diagram depicts only the causal structure of the spacetime (which is determined by the metric). In quantum gravity the causal structure is simply not well-defined and therefore in any situation where quantum gravity is strong, the Penrose diagram is no better than a cartoon.

Taking the diagram seriously is your error, and one that has been made by many physicists over the history of the information paradox. Careful reviews will emphasize that the diagram is not meant to be taken literally, though the point is often not made explicit in papers since our ignorance of what happens in the strong QG regime is common knowledge. I agree that one should be more careful in presenting the picture, at least when there is a risk of confusion, though I disagree that there is a widespread misunderstanding on this point in the community, at least among experts. To your side remark, two of your "prominent experts" have views on the information paradox that lie well outside the mainstream (and that I believe are refuted both by AdS/CFT and the boundary nature of the gravity hamiltonian). The other, if pressed, would likely tell you that the diagram can't be taken too seriously. I think we can have a discussion about the merits of your argument without appeal to their opinions, though.

I'm happy to follow the presentation of the paradox in [1409.1231] for the sake of concreteness. I agree that, absent a theory of quantum gravity, we cannot make definite statements about what happens to causal structure in the quantum gravity regime. The general expectation is that it is not well-defined, but I do not need this for the argument. All I was saying is that the geometry receives large quantum corrections in the quantum gravity regime -- which is what we mean by it being quantum -- hence a large departure from the naive picture in figure 4.

In fact, I can even make an argument without invoking quantum effects, though those are certainly relevant too. Near the singularity the curvature is large, and so higher-derivative terms in the gravitational action become important. This means that the spacetime in the high-curvature regime near the singularity (or near the horizon at the end-stages of evaporation) is modified by these classical, post-Einstein-Hilbert gravitational corrections, so that one cannot hope to use figure 4 as a literal spacetime diagram.

By the way, our theory of perturbative quantum gravity is string theory, which certainly "exists", and while I did not need to import any stringy results (all my claims follow from effective field theory without invoking any details of the UV physics), my statements are consistent with our knowledge of the stringy physics.

Re: "I address this presentation by pointing out that in the situation as presented in the Penrose diagram this is a false dichotomy and hence a false paradox: sticking to both quantum theory and GR, and taking the diagram seriously, leads to the conclusion that the state on Sigma 2 should be mixed and non-retrodictable. This does not contradict quantum theory but rather is demanded by quantum theory in this setting."

This is the point, you cannot stick to GR, it breaks down. The question has always been how, and whether it allows information to escape. The so-called paradox is the conflict between the naive GR(+semiclassical quantum fields) prediction and the constraints of quantum mechanics.

My interpretation of your argument is the following: you point out that GR predicts that the state on Sigma_2 is mixed, and then take issue with the claim that QM implies unitary evolution to Sigma_2, since you argue that Sigma_2 cannot be a Cauchy slice (though your argument involves following geodesics through a high-curvature quantum gravity regime, which you cannot possibly do). Even if I believed your argument was well-justified, it would lead you immediately to the remnant or baby universe scenarios, which I addressed in my response to Sabine. (cont'd)

"We start with a space-time whose conformal and causal structure is somehow undefined in certain places, so the whole concept of a Cauchy surface is not well defined... On what basis, then, are we to conclude anything about the state on Sigma 2? If the correct "quantum gravity" space-time structure does not allow for the definition of a Cauchy surface, then we have no grounds to expect anything, one way or the other, about the Sigma 1 to Sigma 2 evolution. How is that a paradox? However it comes out, we have not violated any principle."

The sharpest answer comes from AdS/CFT. We start with a pure state on the boundary in the vacuum, dual to vacuum in the bulk, then act with sources on the boundary to create a black hole in the bulk (if you prefer, you can think of this as evolving the boundary with a time-dependent hamiltonian). At t=0 in the boundary, before we've turned on the sources, the bulk is just empty AdS and there are no obstructions to picking a Cauchy slice. Much later the black hole will have evaporated, and the gravitational field is weak everywhere in the bulk, so we can construct a bulk Cauchy slice dual to the evolved boundary slice.

"Now about AdS/CFT. How do you think it follows, if AdS/CFT is true, that the evolution from Sigma 1 to Sigma 2 is unitary? If pure states on the boundary always map to pure states in the bulk, then we know that a pure-to-pure transition on the boundary maps to a pure-to-pure transition in the bulk. (This all assumes that the conformal structure on the boundary is unproblematic.)"

The last bit is not an assumption, it is trivially true. Time evolution in quantum field theory is by definition unitary regardless of the manifold on which it lives. The CFTs in the correspondence, for example N=4 SYM, are ordinary, unitary field theories on fixed spacetime backgrounds.

"Fine. And let me even grant that the initial pure state on the boundary maps to a pure state on Sigma 1 in the bulk. But by what argument can one conclude that the final pure state on the boundary maps to *the state on Sigma 2* in the bulk? All we know is that it maps to some pure state in the bulk. Why not the state on Sigma 2 U Sigma 2in is the bulk?"

If the bulk dual to Sigma_2 has some piece behind the horizon at late times, it's a remnant, or baby universe, by definition.

"According to my analysis, this state ought to be pure, and the state on Sigma 2 alone mixed. Somehow you conclude that CFT says it must the the state on Sigma 2 that is mapped to, but I can't see any argument at all to that conclusion. We would need a full dictionary connecting states on the boundary to states in the bulk to conclude anything, and we have no such dictionary. So my solution is not "ruled out explicitly" by AdS/CFT."

Knowing that the dictionary exists is different from knowing the details of the mapping. We have very high confidence that the dictionary exists; the details are irrelevant to the argument that it implies unitary evolution in the bulk, which is implied by its existence.

"In fact, simple dimensional considerations assure us that the map from states on surfaces on the boundary to states in the bulk must be highly non-trivial and non-local: you are connecting states of different dimensionalities to one another."

Agreed, this is why we call it "holography". The bulk-boundary map is indeed highly nontrivial and highly nonlocal, but nobody promised you a rose garden.

I gave some refs for remnants in my response to Sabine.

Re: tone, you represent at least one false statement (that Sigma_2 cannot be Cauchy) as trivially true, and then suggest that failure to recognize it as such has led physicists on a wild goose chase for decades. I personally read this as hubris more than anything else, and would have pushed harder to understand why such a claim has not gained more traction in the community before publication.

Carl3,

The purpose of the title is point out he tried to understand it, which is arguably true. For all I can tell the whole purpose of his paper is explain why he doesn't understand why physicists spend time thinking about the problem, so clearly he failed at it. But you're jumping to conclusions about my intention. I'd say that I myself fail to understand why my colleagues discuss a lot of the issues they do discuss (and I wrote about this previously), hence my remark about the bubble. You could have read my title as "philosopher puzzled about insanity in theoretical physics." That you didn't says more about you than about me.

dark star,

If you post 7-digit arxiv numbers, please include the category. But let us take the Preskill review as example, it's a good starting point. It's full with phrases like "it seems" so and "it seems so". If you bother to look at the references quoted, they contain nothing to back up the claims in the paper. The large-volume explanation has never been ruled out. The example mentioned in the paper is a red herring (seriously, go and read the papers). Most troubling though, any such argument implicitly assumes that effective field theory holds *at the Planck scale* which is clearly unwarranted. The pair production "problem" is a non-problem, both because we have all reasons to assume eft to break down and because there's no reason to believe remnants must be long-lived or degenerate at long wave-lengths.

This has been said many times (even Tim debunks this claim), so why do you keep bringing this up?

I'm not sure why I would be interested in remnants in AdS/CFT, can you tell me why the papers you mention are important to the issue? Best,

B.

Peter Shor,

As I understand it, "virtual particles" are just mathematical artifacts that arise in doing perturbation theory. Similarly, Feynman diagrams do not depict any real physical events: they are just a handy mnemonic device for keeping track of a bunch of terms that contribute to the exact solution to an equation. Of course, to go into this properly one would have to be precise about the sense in which any particle is "real". As far as a know, none of this is taught in physics graduate school, where the concept of "physical reality" is not much used. That is why physicists cannot agree about whether the wavefunction of a system is "real", or even what that might mean. It is also why most physicists cannot explain how to solve the measurement problem.

Certainly, the idea that there is no fundamental difference between virtual and real particles would need some strong defense. For example, in the GRW theory, real particles suffer GRW collapses. "Virtual particles" do not, and could not, if the theory is to work.

There is some confused talk that ties "virtual particles" to "fluctuations", and that mentions the Heisenberg time/energy uncertainty principle, as if a virtual particle can exist as a short-term fluctuation, but the longer lived it is the less energy it must have. This talk is confused because of a misunderstanding of the term "fluctuation". There are, for example, "quantum fluctuations" in the Minkowski vacuum state, but the state itself is stationary, and does not fluctuate at all. The so-called "fluctuations" are expectation values for certain quantities if they were measured. No physical change corresponds to them.

But in the end, all of this is not really relevant to the paper. I do not say that real evaporating black holes violate unitarity and virtual ones do not, as you relate. I explicitly say that real black holes, including evaporating ones, do not violate unitarity in the only place where we could expect it, namely for Cauchy-to-Cauchy evolution. The same would be true for virtual evaporating black holes, if there were any.

dark star

OK, a lot to go through here. You continue to insist that the diagram is a "cartoon" because quantum gravity. The diagram has been used since Hawking's original paper, and continues to be used, when discussing the "paradox", and no warnings or disclaimers are given. Certainly, Hawking himself thought that there is a fundamental breakdown of unitarity on the basis of the diagram, so he took it seriously. Your claim that "the point is often not made explicit in papers since our ignorance of what happens in the strong QG regime is common knowledge" is impossible to refute and impossible to prove, of course. Since you yourself say that "Taking the diagram seriously is your error, and one that has been made by many physicists over the history of the information paradox", at the very least the paper shows that physicists have been imprecise and sloppy in ways that have misled other physicists. But, as I said, if one is not to take the diagram seriously, what is one to take seriously? What is the paradox supposed to be?

Perhaps the operative paragraph of your post is this:

"This is the point, you cannot stick to GR, it breaks down. The question has always been how, and whether it allows information to escape. The so-called paradox is the conflict between the naive GR(+semiclassical quantum fields) prediction and the constraints of quantum mechanics."

This shows that you have not understood my argument at all. What I have argued is that there is no conflict between naive GR and the constraints of quantum mechanics, which is why there is no paradox. We certainly agree that quantum mechanics does not require that all evolutions be pure-to-pure and preserve information: Wald's example in plain vanilla Minkowski space-time is a counter-example to that. The only constraint we have from quantum mechanics is that Cauchy-to-Cauchy evolution must be pure-to-pure, and the black hole evaporation scenario suggests no violation of that at all. The evolution from Sigma 1 to Sigma 2 U Sigma 2in can perfectly well be pure-to-pure, unitary, and preserve information. The evolution from Sigma 1 to Sigma 2 arises from this pure-to-pure evolution followed by tracing out over Sigma 2in. This leaves a mixed state on Sigma 2 that fails to preserve information. This does not violate quantum theory: it instantiates it.

We know from Wald's example that not every evolution is pure-to-pure, or unitary, or preserves information. And I have given you a strict criterion about when it must be unitary and preserve information: when it is Cauchy-to-Cauchy If you insist that the space-time of the evaporating black hole does not have a definite causal structure, so that the very notion of a Cauchy surface is not applicable, then you need to replace this criterion with another one that can be applied. Absent such a criterion we have no grounds to expect anything in particular about the evolution from Sigma 1 to Sigma 2 .And the criterion better reduce to being Cauchy-to-Cauchy in regimes where a space-time emerges. Without the criterion there is no paradox.

Con't

You say that by definition my solution is a remnant solution. You are free to define things as you like: I claim that is the right solution. Looking at the literature, I find most "remnants" not to yield disconnected Cauchy surfaces, and the literature says that to get a remnant the evaporation has to stop at Planck scale and not run to completion. In my solution the evaporation does run to completion. But this is all just semantics: I claim that my solution, whatever you call it, is a consequence of quantum theory and General Relativity. It contradicts neither of them.

My point about not having the dictionary in AdS/CFT is simple: without it, why not conclude that the unitary evolution on the boundary maps to the unitary evolution from Sigma 1 to Sgma 2 U sigma 2in? Then there is no paradox. In this sense, the details of the map are critical, not irrelevant.

"Re: tone, you represent at least one false statement (that Sigma_2 cannot be Cauchy) as trivially true, and then suggest that failure to recognize it as such has led physicists on a wild goose chase for decades." But your so-called false statement is trivially true in the Penrose diagram! And I don't see how any imaginable correction of the diagram in the high-curvature regime could render it false.

Carl3: Thank you.

Sabine: Perhaps your ear for English is flawed, but anyone would take your title (and even more your Twitter, which added "unsuccessfully") to be derisive of the paper and, by extension, of philosophers in general. It has universally been taken that way to my knowledge. I think I understand the situation with respect to the paradox perfectly well, that I have unravelled the paradox, as it were. It turns on the error described in the section "But it no longer exists". In any case, Carl3's reaction does not say more about him than about you: it says a lot about the natural understanding of the title. And you say, apparently derisively, that there is nothing new in the paper which, as Wayne pointed out above, is exactly what I say in the abstract. To be clear: none of the principles of quantum theory or of General Relativity that I make use of in the paper is new or unknown. But the consequences of these principles has not been appreciated. Nor, to my knowledge, is the way one of Geroch's theorems breaks down and the other doesn't.

Tim,

"Perhaps your ear for English is flawed, but anyone would take your title (and even more your Twitter, which added "unsuccessfully") to be derisive of the paper and, by extension, of philosophers in general. It has universally been taken that way to my knowledge.No really, lol - I wonder why.

As I see it, Bee has little to no dispute with Tim's physics.

As I see it, Bee's criticism of Tim's paper is that the first version needs to address some more issues.

As I see it Bee's title refers to the fact that the philosopher has not yet understood why physicists spend so much time on this evaporated paradox. That is not a problem of physics but rather perhaps one of sociology. To quote Bee, "For all I can tell the whole purpose of his paper is explain why he doesn't understand why physicists spend time thinking about the problem, so clearly he failed at it."

That is, the problem is: "why is lack of conceptual clarity so acceptable among modern-day physicists?"

The papers below are relevant to some of the issues Maudlin raises. (Apologies for spamming!)

The 2006 paper discusses the Cauchy surface issue. The 2009 paper notes that decoherence (For All Practical Purposes -- FAPP per Bell) mimics pure to mixed evolution. An experiment which can go beyond FAPP to detect BH unitarity violation would also be able to detect Everett branches. Small amounts of pure to mixed evolution are not excluded and perhaps never will be.

Black holes, information and decoherence

https://arxiv.org/abs/0903.2258

We investigate the experimental capabilities required to test whether black holes destroy information. We show that an experiment capable of illuminating the information puzzle must necessarily be able to detect or manipulate macroscopic superpositions (i.e., Everett branches). Hence, it could also address the fundamental question of decoherence versus wavefunction collapse.

Spacetime topology change and black hole information

https://arxiv.org/abs/hep-th/0608175

Topology change -- the creation of a disconnected baby universe -- due to black hole collapse may resolve the information loss paradox. Evolution from an early time Cauchy surface to a final surface which includes a slice of the disconnected region can be unitary and consistent with conventional quantum mechanics. We discuss the issue of cluster decomposition, showing that any violations thereof are likely to be unobservably small. Topology change is similar to the black hole remnant scenario and only requires assumptions about the behavior of quantum gravity in planckian regimes. It does not require non-locality or any modification of low-energy physics.

Arun,

Yes, excellent summary.

Dear Stephen,

Thanks so much for the references. They are both highly relevant, and I will cite them in the next version.

There is a terminological question that I have, which I think (from your paper) you might help me with. dark star above writes: "If the bulk dual to Sigma_2 has some piece behind the horizon at late times, it's a remnant, or baby universe, by definition." Now there is a little confusion here since Sigma 2 would be in the bulk, not on the boundary, so the claim is really about the bulk dual of the final Cauchy surface on the boundary. But the terminological question is this: is there a standard meaning of "remnant" and "baby universe"? I have sometimes been told that the solution in my paper is a remnant solution, but my impression is that remnants require that the evaporation not "run to completion", and hence leave a connected Cauchy surface. I suppose the solution would be a "baby universe", but I'm just not sure how these terms are used. You say the solution is not a remnant solution, so I infer you have something like this criterion in mind. Can you confirm that?

Thanks,

Tim

Sabine,

Sorry about that, everything I listed was hep-th :)

I agree that parts of the Preskill paper are vague. Absent a theory of nonperturbative quantum gravity, it's hard to know the rules of the game, as you say. Perhaps I should not have cited the classic papers against remnants as those are well-known to you, but I did cite two modern reviews that I believe refute the arguments you gave with Smolin (especially the Marolf review I highlighted). For example, we know that there must exist an EFT description of any would-be remnant from AdS/CFT.

You should care about remnants in holography since we can sharply formulate questions about quantum gravity in that context. Many of the lessons also extrapolate to asymptotically flat black holes, by undoing the decoupling limit. Also, small black holes in AdS are almost identical to AF black holes. I really don't understand why one would simply ignore what we've learned from the correspondence. Do you think that the resolution of the info. paradox is fundamentally different in other circumstances?

I also don't understand why you say that remnants can be short-lived or non-degenerate. My understanding of remnants is that they store all the information left over after BH evaporation. If they were short-lived it seems like they would be equivalent to the ordinary evaporation-to-completion scenarios, and if non-degenerate they would not store the required information so would not solve the paradox.

Would you mind explaining the large-volume proposal to me?

My main interest in joining this conversation was to point out the flaw in Tim's naive GR argument, and that his holographic example invokes remnants. I would be happy to have convinced on these points; the viability of remnants is a separate (but obviously very interesting) issue.

Tim,

I understand your argument just fine, but it seems like mine hasn't made it across. I am pointing out that you are erroneously using classical GR to describe the end-stages of black hole evaporation. It does not matter if you reconcile some naive GR prediction with QM, GR breaks down towards the end of evaporation (if not before), and so its predictions are both irrelevant and inapplicable to any resolution of the paradox.

Maybe another wording would be helpful. Whether or not a statement is true or false in the Penrose diagram could not matter less, since the Penrose diagram is not an accurate description of the physics in the end stages of BH evaporation. It then follows logically that any arguments based on the Penrose diagram are irrelevant.

"And I don't see how any imaginable correction of the diagram in the high-curvature regime could render it false."

I trust this was not meant as an argument -- nothing follows logically from one's lack of imagination. I described two types of corrections that can wildly change the spacetime geometry: quantum, and higher-curvature, both of which are relevant in this situation. Unless you have a physical argument that these corrections do not change the spacetime geometry, it's hard to take this seriously.

I addressed your concerns about the existence of Cauchy surfaces with the holographic example, which you may want to spend some time with. I was careful to stick to a physical setup where we know the details of the map. Also, just to keep things clear: the flaw in your argument based on the Penrose diagram is independent of this point; I only mentioned holography since it furnishes an explicit counterexample.

As for remnants, all the problems I alluded to manifest whether the Cauchy surface is connected or disconnected. The thing that leads to problems is having a lot of entropy in a very low-mass object.

As for AdS/CFT, as I said before, if some piece of the evolved Cauchy surface is stuck behind the horizon, it's a remnant. This is true whether or not we know the details of the map, it's a definition, and yes, it's just semantics. However, this is not: we know enough about the dictionary to conclude that there would be a low-energy state in the field theory for every state of the remnant, while there is strong evidence that this is not the case in any holographic field theory.

dark star,

AdS/CFT presumes the solution that's why I'm not interested in it as an approach to information loss (though it is interesting for other reasons). It may be self-consistent, but that doesn't help because, to state the obvious, we don't live in AdS. I don't know what you mean by 'results extrapolate'. It's a non-continuous limit from a space with to a space without (conformal) boundary.

"I also don't understand why you say that remnants can be short-lived or non-degenerate. My understanding of remnants is that they store all the information left over after BH evaporation. If they were short-lived it seems like they would be equivalent to the ordinary evaporation-to-completion scenarios,"Sure that's exactly what they are, except that the strong interpretation of the BH entropy doesn't hold. Maybe one shouldn't call them remnants in this case, you are right. I do that just because most people in the community have no idea what the weak interpretation of the BH entropy is, but they know remnants.

"Would you mind explaining the large-volume proposal to me?"The proposal is that the volume is large. More seriously, it's explained in the review I mentioned better than I can possibly do here. The point is simply that if the volume is large, there's no reason why the remnant's information should decouple in the EFT limit, hence they're not indistinguishable.

Sabine,

I don't understand at all why you say that AdS/CFT presumes the solution. The nature of the resolution is implied by AdS/CFT, which we believe for a host of completely unrelated reasons, both from the bottom-up and top-down. I would appreciate clarification on your stance here.

By reintroducing the coupling of the supergravity fields in the asymptotically flat region to the CFT on the branes, one undoes the near-horizon limit. Of course the theory that one gets from this is a theory with dynamical gravity, but the dynamics near the horizon are still described by the CFT, though now the radiation can escape to infinity. This is what I mean by extrapolation.

Whether or not we live in AdS seems completely irrelevant to me. If you propose to ignore the resolution of the paradox in AdS, where the nature of the resolution is a consequence (not a presumption) of the duality, you must believe that the resolution of the info. paradox is fundamentally different depending on the boundary conditions at infinity. To my knowledge there is no positive evidence to suggest this, and in addition there's the negative evidence I gave above. Let's suppose it was the case, though. There would have to be some mechanism that would tell the black hole (in its end stages of evaporation when it is far from the boundary and much smaller than the AdS curvature scale) whether or not it lived in AdS. This mechanism would have to be very nonlocal, and stretch into regions with low curvature. Do you have a proposal for it?

I'll spend some more time with your review later but I don't see how distinguishability of the remnant states affects the EFT argument. They would still have to be there in the theory and show up in transition amplitudes as well as the spectrum.

I also posted a separate response to Tim when I posted my response to you, which I don't see above. Let me know if that bit didn't make it through.

dark star,

Sorry, I had missed one of your comments, it should appear now.

In AdS/CFT you only look at fields that can be expanded around the boundary. If you'd want to say something about information loss/preservation, you should look at fields that have *no* expansion around the boundary. (And I know there's been some discussion about this. I am not aware though anything conclusive came out of it.)

In fact I do believe that the solution of the paradox is fundamentally different whether or not you assume you only have fields that can be expanded around the AdS boundary. Hence my reminder that the limit \Lambda \to 0 isn't continuous, and there's no reason to believe it is.

Be that as it may, it doesn't matter what I believe or you believe or anyone believes. There are different mathematically consistent solutions to this problem and we can discuss this forever back and forth and write papers about it and we'll not agree on anything. I don't think this is science any more. Best,

B.

dark star

I asked some questions about your argument, and the supposed flaw in my paper, but I cannot see any answer to them. So let me address what you say directly.

You complain that the Penrose diagram that is universally used when explicating the "paradox" is not to be taken seriously, so nothing can come of analyzing it. That is a very odd position to take. After all, the idea is to actually present a paradox. If some contradiction with basic principles arises from analysis of the diagram, then one can say that there must be something wrong with it, or else abandon a basic principle.Certainly, the "paradox" is often presented this way, as if fundamental GR principles (taking GR as exact) entail that the state on Sigma 2 must be mixed while fundamental QM principles say it must be pure. If this were correct, then we could conclude that either GR or QM has to be modified to deal with this case, and it becomes an important test case for the character of quantum gravity. But what I show is that there is no such conflict between GR and QM, taking the diagram seriously as a representation of the conformal structure. I take it you do not disagree with any of that.

I take it that Wald's simple case also establishes to your satisfaction that sometimes pure states evolve into mixed states even in the complete absence of any exotic or extreme space-time structure. And also that where there is a well-defined conformal structure, the criterion for the different sorts of evolution is clear: Cauchy-to-Cauchy is always pure-to-pure and information-preserving, while Cauchy-to-non-Cauchy always loses information and typically is pure-to=mixed. Do you dispute any of that?

If you don't dispute it, then here is the problem with your position. If, as you say, there is no well-defined causal structure in the high curvature area, then there is no longer a distinction between Cauchy and non-Cauchy surfaces. And without such a distinction, we don't have any criterion for when to expect pure-to-pure evolution vs. pure-to-mixed. If the causal structure is not well-defined, then we have no reason to expect any evolution to be pure-to-pure as opposed to pure-to-mixed.So then there just isn't any paradox at all. What, on your telling, is the paradox supposed to be?

Con't

You also have not understood my objection to the relevance of AdS/CFT. Again, I will grant that the boundary has an unproblematic conformal/causal structure, so we can identify Cauchy surfaces there. We begin with a pure state on a Cauchy surface C1 on the boundary. Let's grant that this maps to a pure state on Sigma 1 in the bulk. The pure state on the boundary evolves to a pure state on another Cauchy surface C2 on the boundary. Let's grant that this in turn represents some pure state in the bulk. So I am granting you things left and right that have not been proven, including that there is a correspondence at all. But all of this granting still does not get you to your conclusion. The question now is: what state in the bulk corresponds to the new pure state on C2 on the boundary? As far as I can tell, you just assert that it must be a pure state on Sigma 2 in the bulk rather than, say, a pure state on a disconnected such as Sigma 2 U Sigma 2in in the bulk. But by what principle are you entitled to that conclusion? Without having a detailed translation manual, you simply cannot make such a conclusion.In fact, what you have is not a paradox but just a pile of ignorance about what is going on in the bulk.

As I understand it, my solution is not a "remnant" solution but a "baby universe" solution. You seem to think that there is some physical argument against such solutions. The only thing that looks like an argument is this claim: "The thing that leads to problems is having a lot of entropy in a very low-mass object." If that is your objection, then it is answered in the next version of the paper. Short answer: the arguments that attempt to connect entropy to mass and to information are all invalid. You will like that part of the paper even less than this. But it's not hard to show.

Sabine,

I agree about the irrelevance of our beliefs but I'm disappointed that you take this perspective, every positive argument I've given is supported by calculations in a UV-complete theory of quantum gravity. Apart from that, most everything else I've said has been either a logical deduction or a question. I'm not sure at all why you think this isn't science, but if that's your stance then it's probably not productive to discuss further.

Tim,

The idea is to understand what happens in the end stages of black hole evaporation, i.e. resolve whether the information escapes or not, and how. The idea is not to present a paradox for the sake of presenting a paradox -- we're physicists, not philosophers. I agree that taking the diagram seriously as a representation of the causal structure (which is emphatically not correct) would lead one to conclude that there is no paradox.

I also agree that in the absence of a well-defined Cauchy surface, such as in the high-curvature regime near the singularity, it's impossible to use Wald's criterion for pure-to-pure evolution (which itself is certainly true in non-gravitational theories). However this does not imply that pure-to-pure evolution does not occur, only that that justification is lost. This is why I keep repeating my AdS/CFT example: in that case, we do have Cauchy slices on the boundary, and since the boundary evolution preserves information the bulk must too. No paradox, just a lack of understanding of how the information gets out. I believe this is the consensus view in the community. (If you still want to harp on the well-definedness of bulk Cauchy surfaces, go back to my first example: in that case, one can explicitly construct bulk Cauchy surfaces at early and late times dual to the initial and final boundary Cauchy surfaces, since all grav. fields are weak at those times. In this example one can also see explicitly that there is no piece of the bulk Cauchy behind the horizon.)

You say you are granting me things left and right that have not been proven. It is often true that things are unproven, but nevertheless there is an overwhelming amount of evidence they are true, and no evidence that they are false. This is the case with AdS/CFT, and if it weren't very few people would take it seriously.

Let me briefly address baby universe vs remnant scenarios. I think the former are even easier to rule out holographically. You seem to be defining a baby universe as something with a disconnected Cauchy surface. This implies that once the baby universe forms, the bulk Cauchy surface must remain disconnected at all future times, otherwise one could retrodict from some point on the future Cauchy surface to some point between the two pieces of the past surface, in contradiction with the initial surface being Cauchy. Now, since the exterior piece of the Cauchy surface is connected to the boundary, the interior piece must be disconnected from the boundary. Therefore no signal can travel from the interior piece to the boundary, so the state on the boundary must be independent of the state of the baby universe. But the boundary state was dual to *everything* in the bulk before the black hole formed, and in this scenario after evaporation it is only dual to the degrees of freedom outside the baby universe, and this implies that the boundary evolution is not pure-to-pure.

If instead there's something more like a remnant that stays in contact with the boundary, then it's ruled out by the entropic reasoning you're about to disprove. It may be worth mentioning that the Bekenstein-Hawking formula has been proven in string theory, via the Strominger-Vafa counting of D-brane states that become black hole microstates at strong coupling. I hope you will also point out their error, or the problem with string theory at least, in your followup :)

dark star,

It seems like we are in agreement on some points. Let's see if we can make further progress.

You say: "This is why I keep repeating my AdS/CFT example: in that case, we do have Cauchy slices on the boundary, and since the boundary evolution preserves information the bulk must too. No paradox, just a lack of understanding of how the information gets out." This is, of course, question-begging since in the scenario implied by the Penrose diagram information is not lost and the Cauchy-to-Cauchy evolution is unitary, but the information doesn't get out. I know you don't want to take the diagram seriously, but it is at least worthy of note that if you do take it seriously there is no paradox, unitarity is not violated dynamically, and information is not lost. So even if the Penrose diagram is inaccurate, it provides an example of a certain kind of solution that you seem to be ignoring. That is, you are equating "Information is not lost" with "information escapes", but you can have one without the other. Maybe the correct diagram, or whatever replaces a diagram in the strong gravity regime, afford the same solution.

In any case, if the Cauchy-to-Cauchy criterion cannot be applied in the strong gravity regime, it is worthwhile to figure out what could take its place.

I cannot follow your supposed refutation of the baby universe scenario. The idea, as I understand it, is that there is a duality between the surface and the bulk: for every state on one there is a state on the other such that the dynamics between the one set is isomorphic to the dynamics between the other. There is no need, in implementing such a duality, that any signals "travel from the interior piece to the boundary". It is not that the boundary and the bulk communicate with each other but that at the appropriate level of abstraction they mimic each other. As I have said, the relation between bulk states and boundary states must be extremely complicated and non-intuitive, because you are mapping between spaces of different dimensionality. There won't even exist any 1-to-1 continuous map between points on the boundary and points in the bulk. So there is no reason at all to think that the geometrical features of a set of points on the boundary (such as connectedness) must be carried by the map to similar features in the bulk. Even at early and late times, the dimensionality problem remains.

I must say, the more I look into the original Bekenstein papers that people cite the worse it becomes. FWIW, I was just raising some of these objections and Rovelli said that one shouldn't pay attention to Bekenstein, since the papers are so confused. But when I asked him for a clear, accurate account of the BH entropy/ area law, he said he could not think of one off the top of his head. This seems to be a field where a lot of things are taken as well established, but not in the papers that first announced them, and no where else either. It is very curious.

I know it's a joke about string theory. But if there was no coherent conceptual foundation for the area/entropy law in the first place, I am not going to think it at all plausible that it is confirmed clearly by string-theoretic arguments.

>> in principle black holes can be created and subsequently annihilated in any particle collision as virtual particles

Indeed, but in any experiment we can do, e.g. at the LHC, the energy of such a virtual b.h. (with any reasonable contribution) would be well below the Planck mass i.e. far from the quasi-classical limit where the information loss problem is discussed.

Wolfgang,

1) Quantum effects of black holes are more pronounced the *lighter* the black hole is, not the heavier it is.

2) If it's virtual it can have any energy/mass.

dark star,

"Bekenstein-Hawking formula has been proven in string theory, via the Strominger-Vafa counting of D-brane states that become black hole microstates at strong coupling. I hope you will also point out their error, or the problem with string theory at least, in your followup :)"Which presumes a bulk-boundary correspondence. As I said above, it's a circular argument. You put in x, you get out x. Please have a look at this paper. Hyperentropic cases exist in GR. Where are they in AdS/CFT? Answer: They aren't there because you have assumed they aren't there - they're not states that can be expanded around the boundary.

Having said that, that the BH entropy counts microstates also leads to the firewall problem. That's a major issue because it means you'll have to give up the equivalence principle or quantum mechanics, or both, whereas unifying them was what string theory was supposed to do in the first place. But giving up string theory is of course not an option, so it has to be fixed somehow.

Sorry, link to paper got lost, it's here: https://arxiv.org/abs/0706.3239v2

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