Tim Maudlin has an interesting paper in which he criticizes the importance given to the black hole information paradox, and even brings arguments that it is not even a problem: (Information) Paradox Lost. I agree that the importance of the problem is perhaps exaggerated, but at the same time many consider it to be a useful benchmark to test quantum gravity solutions. This led to decades of research made by many physicists, and to many controversies. I wrote a bit about some of the proposed solutions to the problem in some older posts, for example [1,2,3]. Maudlin's paper is discussed by Sabine here.

One of the central arguments in Maudlin's paper is that the well-known spacetime illustrating the information loss can be foliated into some 3D spaces (which are Cauchy hypersurfaces that are discontinuous at the singularity). These hypersurfaces have a part outside the black hole, and another one inside it, which are not connected to one another. Cauchy hypersurfaces contain the Cauchy data necessary to solve the partial differential equations, so the information should be preserved if we consider both their part inside and their part outside the black hole.

I illustrate this with this animated gif:

I made this gif back in 2010, when I independently had the same idea and wanted to write about it, but I don't think I made it public. Probably the idea is older. The reason I didn't write about it was that I was more attracted* to another solution I found, which led to an analytic extension of the black hole spacetime, and has Cauchy hypersurfaces but no discontinuities. I reproduce a picture of the Penrose diagram from an older post in which I say more about this:

**A.**The standard Penrose diagram of an evaporating black hole.

B The diagram from the analytic solution I proposed.

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* The reason I preferred to work at the second solution is that it allows the information to become available after the evaporation to an external observer. The solution which relies on completing the Cauchy hypersurface with a part inside the black hole doesn't restore information and unitarity for an external observer. I don't know if this is a problem, but many physicists believe that information should be restored for an external observer, because otherwise we would observe violations of unitarity even in the most mundane cases, considering that micro black holes form and evaporate at very high energies. I don't think this argument, also given by Sabine, is very good, because there is no reason to believe that micro black holes form at high energy under normal conditions. People arrive at high energies for normal situations because they use perturbative expansions, but this is just a method of approximation. And even so, I doubt anyone who sums over Feynman diagrams includes black holes. But nevertheless, I wouldn't like information to be lost for an outside observer after evaporation, but this is just personal taste, I don't claim that there is some experiment that proved this. And the solution I preferred to research allows recovery of information and unitarity for an external observer, and other things which I explained in the mentioned posts and my PhD thesis.

## 2 comments:

I also don't find the 'virtual black holes' argument terribly convincing. Ultimately, these things are really just terms in the perturbative expansion approximating a process that is itself by construction unitary---so that if we were able to work without the approximation, we wouldn't ever notice any hint of the possible non-unitarity introduced by evaporating virtual black holes. I think that hastily reifying such virtual objects is about as misleading as claiming that a particle is 'in two places at once' in a superposed state---perhaps useful as a figure of speech among those that know what they really mean, but if taken too seriously, implying a sort of pseudo-classical picture that distorts what's actually happening.

That said, I'm also not sure Maudlin's paper proposes an answer to the information loss problem as most see it---after all, the 'whole universe' at a certain point in time after the evaporation of the BH is given by a (non-Cauchy) hypersurface stretching from r=0 to spatial infinity, and if it makes sense to call this the whole universe, then there is in fact information missing from it---that pertaining to all the stuff that fell into the black hole (that is, all inextensible timelike curves ending at the singularity).

But that may itself be naive, in taking such a hypersurface to describe the state of the universe at any particular time. Maybe that kind of talk should just as well be regarded as a façon de parler as talk about particles being in two places at once.

In the end, I think there is potential for an interesting debate here; unfortunately, I think that's unlikely to manifest, mainly due to the 'physicists-vs.-philosophers'-kind of sociology that's unfortunately still too pervasive. It's true that Maudlin's tone, which is going to seem needlessly confrontational to most physicists, doesn't help the issue, but there's really no reason to respond in kind. But eh, that's just me tilting at windmills I suppose.

Yes, such arguments taking too seriously virtual stuff needed just because we don't know to do better math unfortunately plagues many discussions about QFT and quantum gravity.

Regarding unitarity, on the one hand it is something I think it must be preserved, so that's why I preferred to extend the solutions through the singularities. And also that's why I am interested to save unitary evolution in the quantum measurement problem as well. And I think it is strange that the same people who want to save unitarity in black holes often adopt a wavefunction collapse position (which is not unitary and doesn't ensure the conservation laws - https://arxiv.org/abs/1607.02076). Sometimes they say that QM is still unitary because of decoherence, but this only works with many worlds, and a single world still has these problems.

On the other hand, given that Einstein's equation is local, like the other classical equations, I think that unitarity in QM and QFT is forced upon us mainly because quantization is made in phase space, so usual quantization is global. But I think that the theory can be local in the sense of the PDEs involved, and at the same time nonlocal in the Bell sense and also contextual, and also unitary. But in order to be like this, I think that conervation laws and conservation of information should be local (I think this is also required by relativity). So I find a bit meaningless the approaches trying to restore information lost at the singularity by looking at the horizon. So it is this belief I have in unitarity and the locality of the PDEs, quantized or not, that I think it is satisfied either if singularities don't really exist, or if they don't pose a problem but to the standard mathematical description (and not to other alternative equations, like those I propose here https://arxiv.org/abs/1301.2231.

So I have more reasons, merging into a sort of not-yet-formal view, which made be dissatisfied with the broken Cauchy hypersurfaces idea. And I think that Maudlin himelf is not satisfied, given that he advocates the position that time is real. I am not sure what this means, given that he discusses it in a second volume which is still work in progress, but I find hard to see how time can be real (hence has something absolute in it), and at the same time work well with broken Cauchy hypersurfaces, whose time coordinate is clearly assigned arbitrarily and ad hoc to get a foliation of spacetime into Cauchy hypersurfaces.

About the dispute between physicsts and philosophers, I agree with you here too. We need each other, and we need to simultaneously be physicists, mathematicians, and philosophers (at least the critical thinking part from philosophy is something we need more as physicists). We always need philosophers to try to poke holes in our theories. If they are doing it right, we can improve the theory. If they only show they didn't understand, we need to improve the conceptual and explanatory part. So I was disappointed when I saw a blog article about Maudlin paper, which I won't mention here, in which the tone was kind of elitist and condescending both towards Maudlin, and towards philosophers of physics in general. Maybe it was just an impression I had, but the blog comments around that article, and the facebook comments, proved that the other readers understood this as a green light to become aggressive against Maudlin and philosophers in general, without justification (not that aggressiveness and rudeness can be justified even when one is right :) )

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