Update: Jacques Distler criticizes this paper. His first complaint is a refined version of an objection I wrote below. Strominger, Hawking, Perry redefine the rules of the game and claim that the BMS symmetry transformations at the horizon aren't redundancies (like gauge symmetries in the bulk should be) but real symmetries, so states don't have to be invariant under them and one may produce new states. Jacques also believes that a certain diagonal BMS subgroup is ill-defined for an evaporating black hole. Finally, Distler agrees with my main general point as well – Hawking et al. work within local field theory where the information loss paradox was born and may be shown to be trouble. Some nonlocal twist is needed to avoid the paradoxes. Jacques insightfully says that local quantum field theory breaks down not only locally at short distances. It has to break at the very long Page time (when the black hole has reduced its entropy to 1/2) because at that time, the entanglement between the early and late Hawking radiation must start to show up.This is a continuation of the story about the BMS supertranslations and their relevance for the black hole mysteries.
Gross, Hawking, Witten, Strominger, Yau. Note how similar the photograph is to these images of 5+ supersymmetric heroes from different movies.
Andy Strominger was just interviewed by a writer for Scientific American – not exactly one of the journals we still respect, to put it mildly (if I have to avoid the term "greasy šit") – about his recent work about soft hair:
co-author from 1978 and 1983 Perry have decided to be Strominger's grad students for a while again.
(However, one must understand that SciAm doesn't write for people who have a clue. It writes for SciAm commenters over there like "essentia" who only asks "what kind of scam is this?" and "naro10" who only says that Hawking has allegedly blamed the State of Israel for the existence of the black holes LOL. This is not what the interview is about, "naro10"! The remaining two commenters offer mushed potatoes involving negative entropy and fractals; and the claim that black holes are just neutron stars. Swines bombarded by pearls.)
But the interview is fun and Andy says lots of things nicely. Why the information loss looks terrible to the physicists (well, the word "determinism" looks too old-fashioned for the unitarity in quantum mechanics given the ability of QM to predict probabilities only but one implicitly hopes that Andy isn't fundamentally misunderstanding something like that) but why it has seemed true at some point, anyway, why you can't escape from a black hole (because the horizon is a sphere expanding outward by the speed of light whose area just happens to be kept fixed thanks to the spacetime curvature), what the Strominger-Vafa derivation has changed, and so on.
Be sure that I agree with some 90% of what Strominger said in the text as a whole.
And I also agree with some mathematical claims about the BMS symmetries, Andy's claims that some of these observations he and pals made are interesting, and about the idea that it should be useful to study them closer and investigate their impact on the black hole mysteries and that they have isolated a couple of these interesting games that should teach us something. However, I disagree with the suggestion that the black hole entropy has been shown to be (partially or completely) carried in the form of the "soft hair" located at the event horizon.
There are several other sentences that raise my eyebrows, however, such as:
And when its energy is zero, there’s a sense in which you can think of it as living on the boundary of spacetime.By the Fourier transform, low energy is linked to low frequencies and therefore long times – which is usually correlated with long distances. But within AdS/CFT and probably other consistent descriptions of the quantum information, if one talks about the locus near the boundary of the spacetime, I think that most people would agree that it is associated with the high-energy (ultraviolet, UV) degrees of freedom in the theory (CFT), not the low-energy ones! On the contrary, the low-energy CFT degrees of freedom "live" near the center of the AdS.
But back to the new message of the text.
Andy has already interpreted the old BMS supertranslations – some diffeomorphisms that act in a simple way on the asymptotic scattering region at infinity so that one gets "a parameter" for the transformation for each point on the sphere \(S^2\) – as a source of some new excitations and degrees of freedom, soft photons and gravitons of some kind which are associated with the horizon for some reasons.
With Hawking and Perry, they argue that these soft photons are excited whenever a charged particle falls to the black hole, and depending on how it does that, the black hole becomes "different". Similarly, soft gravitons should be added to the horizon whenever a particle with the energy-momentum crosses the horizon, and this soft graviton – a component of the soft hair – modifies the microscopic setup of the black hole. The soft hair ultimately influences what the black hole exactly emits as well and they claim to know something about the dependence.
Strominger is happy that the region where they place some "special new stuff" is the event horizon so it has the area of the event horizon and one could get the right entropy. Except that they can't get the coefficient \(1/4\) in this case. He suggests that it is not a big defect.
I think that Andy isn't too familiar with the non-stringy literature claiming to explain the black hole entropy. We used to agree that those papers were garbage but I think that he couldn't have known what we were exactly talking about because, you know, the main message of this new paper isn't "radically" different from the totally idiotic claims e.g. in loop quantum gravity that they can calculate the black hole entropy (also without the right numerical coefficient) by surgically removing the interior and adding some special Chern-Simons stuff at the horizon etc. But there are tons of sloppy papers "calculating" the black hole entropy without the numerical coefficient. Almost all of them are garbage and all the remaining ones are using a mathematically uncontrollable formalism.
You know, if you know the right result, \(S=A/4G\), it's very easy to fool yourself into thinking that "the information must be carried by something at the event horizon", so one invents some exception for the horizon, claims that the typical length scale is the Planck scale, and the happy end follows, \(S\sim A/G\).
The Bekenstein picture with qubits above is a typical example of that way of thinking. Qubits – quantum degrees of freedom described by 2-dimensional Hilbert spaces – are placed all over the horizon. And that's how the black hole "works". The picture is a nice one optimized for pretty popular talks but my discussions with Bekenstein made it clear that he had really believed this kind of discreteness. So he actually believed in the heavily discrete spectrum of areas – integer multiples of \(4G \ln 2\). If this discrete spectrum were translated to a discrete spectrum of mass/energy, black holes with such a heavily discrete energy spectrum would have no chance to emit anything like a thermal radiation because the energy of the quanta would basically have to be integer multiples of the Hawking temperature or so.
You may be happy that the Bekenstein "model" of the black hole on the picture reproduces the right entropy if you choose the triangles' areas to be \(4G\ln 2\), an unnatural value (which was why Bekenstein, like the loop quantum gravity crackpots, got so excited about my and then my+Neitzke paper about the quasinormal modes). But if a model may reproduce one piece of data, it doesn't mean that it's the right model. One agreement isn't enough to prove a hypothesis; but one disagreement is enough to disprove it. The picture above is actually wrong for many reasons.
First, as I said, the spectrum of the horizon areas etc. simply cannot be discrete, at least not with this huge spacing. Second, the information in Nature is almost never encoded in qubits. Qubits are man-made concepts that are helpful for computers but natural systems simply don't work like that. They carry the information in much more general degrees of freedom. A highly excited state in 2D conformal field theory carries the information in the question "which microstate at the given level we pick". A given excited level of a CFT isn't a tensor product of many qubits; the degeneracy is almost never a power of two. The degeneracy may only approximately computed from Cardy's formula and Cardy's formula has an exponential form – but the natural base of the exponent is \(e\), not \(2\), of course. Nature stores the information in nats, not bits. That's why we use the term "natural logarithm" for the base-\(e\), not base-2, logarithm.
So this picture involving quantum binary digits is just naive – partly inspired by people who may have listened to the common computer architecture much more than to Nature. But a more serious problem is that the idea that the information is carried strictly by some "new structure at the horizon", some Planckian vicinity of the strict event horizon, is wrong, too. I am sure that many people believe it once they learn about the \(S=A/4G\) formula and maybe I believed it years ago, too. But there simply cannot be any special structure that "must" be placed on the event horizon and nowhere else.
As I previously argued, the location of the event horizon is only known in the future once you know the whole spacetime geometry. The event horizon is a spherical shell around the stellar center expanding by the speed of light, as Andy also said. But when the black hole (and its event horizon) begins to form, it starts at a seemingly random moment when the center of the star becomes the "first" point that belongs to the event horizon. It's followed by tiny and then larger spheres around the stellar center. There is still no singularity there. At some moment, the growing sphere devours the surface of the star and the black hole is basically "complete". The curvature of the spacetime is so huge that even though the sphere keeps on expanding along a null surface, the area of the sphere stops growing. When the oscillations get stabilized a little bit, you get a nearly static black hole spacetime and, if you were lucky to have picked the initial point of the event horizon correctly, you may see that the the interior side of the event horizon indeed contains all the points from which you can't escape to infinity (to the black hole exterior).
But if something were changed about the future evolution of the collapsing star (some new star would arrive to speed up or slow down the formation of the black hole), the locus of the horizon could be elsewhere, or it could start to grow earlier or later. There exists no "local condition" by which you could identify the place of the event horizon around you at a given moment. To find the position of the event horizon around you really means to make a successful prediction of the distant future.
That is why it is completely wrong to imagine that something materially different is taking place (or some objects or degrees of freedom are located) at the horizon "in real time" although they can't be located anywhere else. Such a claim is wrong for a simple reason – the laws of physics that would imply such special "objects" at the event horizon would be acausal. They would know something about the evolution of the spacetime in the distant future. They would have to know it, otherwise they couldn't know where the "new horizon-bound objects" should be supplemented.
So the event horizon may only play a role in the laws of physics if you describe this physics from the viewpoint of an observer who ultimately knows about the whole future. Clearly, it must be an exterior observer who didn't fall into the black hole. But because he didn't, he doesn't see into the black hole, either. For him, the horizon is the boundary separating the accessible part of the world from the inaccessible one.
Now, there is some sense in which the horizon must store all the information etc. from the viewpoint of the exterior observer. For the exterior observer, the black hole is covered by a conducting warm membrane etc. and nothing beyond the membrane (beneath the horizon) exists; it's the membrane paradigm. In which form this information is stored? AdS/CFT has an answer. In AdS/CFT, the CFT really doesn't easily see inside the black holes. Some extra work is needed to figure out what's happening in the interior and people doing that (like Papadodimas and Raju) aren't quite sure whether the answer has to be unique. But what we do know is in what form the CFT encodes the information stored in the black hole: the microstate is just some random near-thermal microstate composed of the gluons or whatever the CFT contains.
But within the CFT, it is very hard to say "where" objects are in the AdS bulk. The locality in the bulk, especially when it comes to the precise localization of objects and events according to the new radial (holographic) direction, is an extremely tough business. I am confident that almost all AdS/CFT folks who have also studied the black hole information puzzles will agree with me that the information described by the "thermal state of the CFT" isn't in any canonical way localized strictly at the event horizon of the black hole. It's more sensible to say that it's localized "almost anywhere". If you change the thermal state of the CFT by one particle, this particle may be reconstructed to be nearly anywhere in the bulk, certainly at distances comparable to the black hole radius from the event horizon (but outside).
So I think that the idea that the degrees of freedom of the black hole "should" be located strictly at the event horizon doesn't really follow from any of the BMS derivations and any of the new (or resuscitated) legitimate GR stuff that these papers by Strominger and pals have covered. Instead, this idea of an "exceptional structure at the event horizon", the "soft hair", comes from the old desire to explain the Bekenstein-Hawking entropy in an easy way. But this easy explanation is acausal. And equally importantly, it is indefensible from the viewpoint of an infalling observer. Once we embrace the infalling observer's perspective and accept the existence of the black hole interior, there simply cannot be anything special whatsoever about the location of the horizon. The event horizon isn't a metallic (or glass) shell in any sense. There's nothing to keep you there. If you get too close to the horizon (so that your rocket jets simply can't have enough energy to save you), you're basically guaranteed to be sucked in.
In 4D etc., the no-hair theorem is really true. (Fuzzballs as well as black holes of assorted topologies may be said to be counterexamples in higher spacetime dimensions.) One may try to engineer a philosophy in which the no-hair theorem is wrong because of some "fine Planckian stuff" and Hawking-Perry-Strominger are probably an example but I think that this whole effort to invalidate the no-hair theorem in 4D is just misguided.
While the BMS-related GR equations in these papers are probably right, I think that the ways how they connect them with the black hole information and quantum gravity are ultimately deeply flawed. If some degrees of freedom are assigned strictly to the event horizon, it should be just a particular choice of a "gauge", one of many methods to integrate the interior out, and so on. One can't imagine that something special is "really" happening at the event horizon.
Andy seems to think that these claims about the information's being carried by the soft hair have been "derived". I suspect that the problem is that he isn't interpreting the status of the BMS transformations correctly. He views them as global symmetries that are capable of producing new physical states. The BMS transformations nontrivially act at points at infinity – like the rigid translations of the Minkowski or AdS space – so they might look like global transformations.
Still, they are diffeomorphisms and they seem to depend on too many parameters – like the generic diffeomorphisms acting purely in the bulk. The latter are gauge symmetries so all physical states have to be invariant under them. Their generators (at least if they act trivially at infinity) have to annihilate the physical states. I suspect that this will partly be true for the BMS supertranslations, too. A part, many, most, or all of the states that they consider physical, distinguishable, and nonzero will be unphysical, indistinguishable, or zero.
When Strominger says
And so there are infinitely many vacua, which can be thought of as being different from one another by the addition of soft photons or soft gravitons.you should see an obvious can of worms, I think. For a given Minkowskian or AdS compactification, the vacuum state is unique. You may use a formalism in which the state is multiplied and many gauge copies etc. are created but at the end, you shouldn't forget that this degeneracy was a trick and you must remove the clones (like in the usual gauge symmetry). It seems to me that they're at least partly failing to do this part of the work so many of their "new states" are unphysical bogus.
I don't know the precise rules but I think that the correct qualitative description of the status of the BMS supertranslations and the new "soft hair" states will resemble e.g. the recent work on the intercontinental Wilson lines. In that paper, the operators would be trivial or equivalent if they only acted on one of the AdS boundaries but one can define a "comparison" between two AdS boundaries and that's where new operators (like the Wilson lines) may arise. Note that in that paper, the operators are "nonlocal", associated with lines going through the whole spacetime. I believe that when the qualitative naive conceptual mistakes are fixed in the Strominger-Perry-Hawking setup, they will have a similar picture. Nothing will "force us" to think that some new degrees of freedom are located exactly at the horizon. It will be at most one "gauge", a parameterization to describe the behavior of Wilson lines or similar nonlocal or "comparative" objects.
There are several other angles from which I look at their claims and that look seriously incorrect to me. The very claim that the information should depend on some thin "Planckian shells of exceptional behavior" seems to go against the results of many years of recent QG research – especially the entanglement-glue duality.
I think that one of the broader lessons we have learned is that
whenever we describe physics and the information in terms of the bulk, we are allowed to assume that the local spacetime geometry is perfectly smooth.So there is never any "need" to imagine that we must be careful about shapes at a better-than-Planckian accuracy. The ER=EPR correspondence – which I view as a special subclass of solutions to the Papadodimas-Raju constraints – goes further and says that if you describe the information carried by two similar black holes, you have at least two smooth geometries you can use. You may either assume that the black holes are separate; or they are connected via an Einstein-Rosen bridge.
Note that the smoothness comment above doesn't say that the metric tensor cannot fluctuate. But the fluctuations may always be viewed as extra excitations added on top of a perfectly smooth spacetime and these excitations are basically composed of gravitons whose number is finite or small for low-energy states.
The point of my comment about ER=EPR is that these are two smooth spacetime geometry that "dramatically", topologically differ. But both of them are still perfectly fine. Consistent quantum theories of gravity never force you to imagine that the precise shape of the spacetime has some complicated Planckian or sub-Planckian wiggles you must carefully adjust. There are no wiggles. You may and you should allow the background spacetime to be any reasonable enough smooth geometry (perhaps one obeying some classical/effective equations) and if your theory of quantum gravity is consistent, it will allow you to describe the degrees of freedom in terms of objects or fields on top of this smooth geometry.
To say that some Planckian or sub-Planckian accuracy is needed to recover physical information about the microstates seems to contradict the most successful lessons from the recent decade or two of QG research. Well, fuzzballs do want to describe black hole microstates in terms of some fine solution but at least these are solutions in the whole black hole interior, keeping a smooth spacetime with no preferred measure-zero loci.
Strominger, Hawking, and Perry envision some potential of the full quantum theory of gravity to store a huge amount of information in some volume – the shell near the horizon. But why couldn't similar degrees of freedom be "everywhere"? Don't you predict, just like the qualitatively analogous (as I mentioned) loop quantum gravity papers, that there should be a volume-extensive entropy of the black hole, and perhaps an infinite entropy?
A paragraph indicates that Strominger is totally aware of the problem but he sweeps it under the rug:
One thing that bothered us about this right from the very beginning is: Why doesn’t this allow an infinite amount of information? We don’t want an infinite amount of information. Ultimately we’d like to somehow use this to recover the Hawking-Bekenstein area entropy law. It looked like we were getting an infinite amount of hair because you seem to be able to have these soft photons that had an angular localization that was arbitrarily small. But there’s no physical way to excite one of those. So those are not physically realizable states of the black hole.Yes, he says, they seem to predict an infinite amount of bogus information that may be carried by that type of soft hair. But except for a finite subset, Strominger wants to believe, these degrees of freedom cannot be excited. It still doesn't explain why there are not tons of these new degrees of freedom associated with other null surfaces that also have the potential to be identified as event horizons in the future. And he doesn't explain why they cannot be excited – for what reasons their classical description breaks down. But these are not the biggest problems here.
The biggest problem is that the only way how quantum gravity may protect us against this invalid proliferation of states (and from volume-extensive entropy) is very different and requires some UV/IR mixing and therefore nonlocality. Why can't we probe sub-Planckian distances? Because when we accelerate the future-LHC protons at the trans-Planckian energies, their collisions start to produce black holes that are ever larger. The recipe to "shorten the distances by increasing the energies" breaks down because at some point (the Planck energy – but in string theory, it may be in some sense earlier), the intermediate states visually grow bigger so they involve some nonlocal behavior (and thermalization) of an object that is extended to distances larger than the Planck length and increasing.
It's very important that we say that some new QG/stringy physics breaks the QFT recipe "higher energies means shorter distances". But this recipe is a QFT recipe as I said – it directly follows from the picture of local fields. String theory (or any consistent theory of quantum gravity if there were another one) can only truncate the hunt for ever higher energies (it truncates it at the string or Planck scale) by not being quite local. The information can only get out of the black hole by a nonlocal process. You can use local effective field theories as long as no black holes appear etc. But for black holes, especially once they grow bigger, the overwhelming part of the information that the black holes carry must be stored in "nonlocal" degrees of freedom. To say that they should result from local degrees of freedom in the normal spacetime would mean to claim that a local description is valid even in the presence of the black hole – and that would bring us back all the bogus volume-intensive degrees of freedom etc.
And Papadodimas-Raju have reformulated the problem of co-existence of the black hole microstates and the local fields as an inverse problem to reconstruct (the algebra of) the fields that may be defined on a given Hilbert space of microstates. This is a complicated inverse problem because the whole algebra of the fields must obey some given conditions. In this way, Papadodimas-Raju really see that the spacetime is emergent. It's emergent because the field operators are only a possible extra structure added on a Hilbert space – a structure (solution) that isn't guaranteed to exist; but a structure (solution) that may also be non-unique. T-dualities, U-dualities, string-string duality, AdS/CFT, and ER-EPR correspondences admit two or more seemingly different spacetimes whose physics is equivalent. This simply means that there are many solutions (in a certain category obeying some conditions such as the "shared" Hamiltonian) to approximately or exactly define all the field operators on a certain Hilbert space.
It seems to me that Hawking, Perry, and Strominger are working within the conceptual framework in which everything, including the black hole microstates, must be constructed out of some (quantized) local fields, basically a straightforwardly quantized GR. They want to believe it's possible after all because they have rediscovered a previously understudied and potentially underestimated feature of classical GR, the BMS supertranslations and similar objects, and they want to believe that this new addition may be enough to make the innately local description of quantum gravity possible. Within this intrinsically local framework, it's always possible to say "where" the information resides, and they have to claim that the black hole entropy is actually stored at the horizon. One may confidently say that if the Hawking, Perry, Strominger picture were right, the spacetime wouldn't be emergent at all.
Nevertheless, I think that this picture of the old-fashioned, non-emergent spacetime – which many non-experts believe as well – represents an attempt to return physics by decades, perhaps before the holographic principle, and it has been debunked and superseded by a new paradigm that makes much more sense. And I am confident that most experts in the black hole information business actually agree with my general criticism.