**...I mean papers on entanglement in quantum gravity theories...**

Yesterday I discussed a paper on the black hole interior that I considered bad but today there's some better news, namely three papers that are interesting and not self-evidently wrong. Let me begin with

Black Holes or Firewalls: A Theory of Horizonsby Nomura, Varela, and Weinberg, three physicists who were previously pointing out that the black hole firewall arguments were flawed because they didn't treat the superpositions of macroscopically distinct states of black holes correctly, among related "interpretational" flaws.

Today, they present an explicit qualitative model of the black hole microstates that is compatible with the unitarity, the locality at long distances, and the equivalence principle. The firewalls are absent and a smooth horizon is present at all times with the probability 100%.

An important component of their construction is a tensor doubling of the Hilbert space to account for the interior modes. In that respect, the new paper is close to Papadodimas' and Raju's paper that is being cited as [28] and described, not too prominently, as a similar construction of the black hole interior operators.

They also use the eternal black hole and the doubling may resemble the second black hole from the Maldacena-Susskind ER-EPR correspondence as well. However, no paper by Maldacena is being cited (neither the eternal black hole, nor his recent paper with Susskind) which seems bizarre. Perhaps to compensate this surprising absence of Maldacena's papers in the list of references, Juan is the only person thanked for conversations in the acknowledgements. ;-)

Note that this is a sign of the typical Maldacena übermodesty. He probably saw the drafts of the paper but he wouldn't mention that it's strange that none of his papers are being referred to. I don't know too many well-known physicists who are this modest. ;-)

OK, what's their construction? They double the Hilbert space for the black hole so instead of \(\exp(A/4G)\)-dimensional Hilbert space, they start with the approximately \(\exp(A/2G)\)-dimensional Hilbert space. The description of all the field-theoretical modes outside the event horizon is conventional; the embedding of the black hole interior modes is subtle.

(I prefer to write \(G\) instead of \(l_P^2\) because the latter is more convoluted and the exponent is actually only right in \(D=4\), so the former notation is even more general. Also, the latter notation is potentially plagued by ambiguous conventions about the (non)rationalized normalization of the Planck length.)

Effectively, they say that a tiny \(\exp(A/4G)\)-dimensional subspace of the larger \(\exp(A/2G)\)-dimensional Hilbert space is relevant for a description of the black hole. Most of the states in the larger space would look like firewalls – a singularity extended up to the horizon – but only the smaller space is relevant which is why no firewalls ever occur.

For a temporary period of time during the evolution of the black hole, they quantify its mass and construct the eternal black hole of the same mass which is said to be a temporarily accurate model of the evolving black hole. Infalling observers don't encounter any firewalls because we're confined to the smaller subspace.

The physical subspace that keeps the horizon smooth is a different, although also \(\exp(A/4G)\)-dimensional, subspace of the doubled \(\exp(A/2G)\)-dimensional space than the subspace considered in a recent wrong paper by Polchinski and Marolf. Those authors had concluded that the probability of encountering a firewall was nearly 100%. The erroneous assumption in that paper was that they assumed that the smaller space may be described by a basis of \(\hat b^\dagger \hat b\) eigenstates (eigenstates of occupation numbers outside the event horizon).

But that can't be assumed because, using your humble correspondent's streamlined arguments, \(\hat b^\dagger\hat b\) doesn't commute with the projection operator onto the physical, smooth-horizon-having smaller subspace, so one can't simultaneously diagonalize them.

A closely related slogan clarifying some mistake in the original AMPS paper is the following:

What is responsible for unitarity of the evolution of the black hole state is not an entanglement between early radiation and modes in \(B\) as imagined in Ref. [6: AMPS], but an entanglement between early radiation and the way \(B\) and \(\tilde B\) degrees of freedom are entangled.Just to be sure, \(\tilde B\) are modes of the stretched horizon while \(B\) are modes in the "next thick shell" outside the stretched horizons that are strongly entangled with \(\tilde B\).

The more complicated character of entangled described by Nomura, Varela, and Weinberg indicates that the Hilbert space of the early Hawking radiation is "rather close" to the linear space of

*operators*that map \(B\) onto \(\tilde B\) or vice versa. This is sort of natural because the Hawking radiation may be imagined as arising from processes the operate on \(\tilde B\) and produce something in \(B\). The AMPS' idea that \(\tilde B\) may be completely ignored in the ideas about the entanglement isn't right.

The AMPS paper and its uncritical followups are making lots of errors that look like subtleties but these subtleties are extremely important for a proper understanding of the behavior of information around the black holes – because the consistency of principles of quantum gravity, while possible, is indeed a delicate product that requires precision to be demonstrated (there's usually no spare room to waste). AMPS assume that operators that don't commute do commute and/or may be simultaneously diagonalized. They assume oversimplified ways how things are entangled, and so on. They end up with an extraordinary claim – the equivalence principle has to be abandoned – but they just don't have the required extraordinary evidence, just their sloppy one.

The "firewalls aren't there" papers aren't manifestly equivalent to each other yet and maybe they never will. The problems may be difficult but there may also be many seemingly inequivalent yet correct ways how to approach these problems, especially those related to the black hole interior that isn't directly accessible from the exterior or from the CFT in AdS/CFT, for example. However, I think that all the authors of the "firewalls aren't there" papers have understood pretty much the same fine wisdom that allows Nature and mathematics to maintain peace between the principles of quantum gravity (including unitarity, locality at long distances, and the equivalence principle). You simply can't act like a bull in a china shop. The consistency of quantum gravity depends on many "miracles" spiritually analogous to the Green-Schwarz mechanism and the precise numbers and if you are sloppy about them and talk about vague or ill-defined "generic" states and operators, you are likely to end up with many wrong conclusions that are spiritually analogous to the claim that chiral 10D string theories had to be anomalous. They're not and quantum gravity principles aren't inconsistent but sometimes the reasons are bound to look like miracles if not "intelligent design" because the consistency of quantum gravity is indeed a hard (albeit solvable) task.

**Einstein's equations from CFT entropy**

The other two papers I want to mention happen to be on extremely similar topics and appeared on the same day but their wording doesn't suggest that the authors were aware of the work of the other team so the papers may be fully independent and the synchronized timing may be coincidental.

Gravitational Dynamics From Entanglement "Thermodynamics" by Nima Lashkari, Michael B. McDermott, Mark Van RaamsdonkMark and co-authors offer a more technical, AdS/CFT-inspired edition of Ted Jacobson's intriguing 1995 derivation of Einstein's equations from thermodynamics. In that paper, Jacobson managed to obtain not just some misleading conceptual comments about the questions but also a "local" version of the Bekenstein-Hawking relationships between the thermodynamic and geometric properties of the black holes, and Einstein's equations followed.

Entropic Counterpart of Perturbative Einstein Equation by Jyotirmoy Bhattacharya, Tadashi Takayanagi

In the new paper, they start with a property of CFTs, \(\delta S=\delta E\). The variation of the entanglement entropy of a ball-shaped region is equal to the variation of the "hyperbolic" energy (which is dimensionless; it's the energy moving the hyperbolic \(H^n\) slices onto the next ones in the \(H^n\times\RR\) slicing of the AdS space). Now, they reinterpret the entropy geometrically as the areas while the energy remains the energy. So the equation links geometric properties of the spacetime and the energy density in it and they show that this equation is equivalent to an approximate form of Einstein's equations in the AdS space.

Tadashi and his co-author (yes, I picked Mark and Tadashi because I know them but I also believe that they're the senior members of the teams) discuss similar relationships between Einstein's dynamics and the entanglement entropy in the AdS space – especially fluctuations about BTZ black holes – and the setup is clearly quite similar although their results or conclusions are less readable for me than the results by Mark et al.

Tadashi and his co-author claim – if I understand their section titled Conclusions – that they want to find a way to derive a diffeomorphism-invariant formulation of the bulk gravity laws from the CFT, including all the unphysical states.

## snail feedback (6) :

well Maldacena does not need to be citated, his work speaks for himself, I think that sometimes science needs more top scientists like him

Hola Jaume, agreed - except that the verb is "cited", not "citated". ;-)

Duh, concerning the last two similar papers from reading the abstracts I thought I would rather dare to open the second one if anything, I dont know why ... :-P

I just want to make a remark, but I don't know if it is relevant in this context.

According to Robert Wald: "For linear fields in curved spacetime, a fully satisfactory, mathematically rigorous theory can be constructed."

My naive reaction is, that then there should not be a problem with classical BH backgrounds - thus no firewall problem.

BUT, when lifting QFT to a curved background all but one of the Wightman axioms fail (Axiomatic Quantum Field Theory in Curved Spacetime). Could this be the reason for the malaise ?

What I find suspicious is that I cannot find any mentioning of "Haag's Theorem", the "Stone-Von Neumann theorem", "unitary inequivalence", etc. in this context.

Lubos, For more than a year you have written better, deeper, and with more insight than any other on the multiple flaws in AMPS.. I would unnecessarily parrot you on the problems or questions to express why I feel it puzzling that more papers still assert or need address firewalls. Your posts in this stringy quantum gravity topic long seem to answer more fully and with less complexity than any other source.

Thank you for the interesting and challenging materials.

Most quantum paradoxes disappear when looked at in the Feynman-sum-over histories formulation. We don't yet have a detailed understanding of how to compute this for a black hole including its central singularity, but I think we know enough to resolve this paradox. In order to look at the firewall paradox, we're going to need to do a sum over histories in the spacetime of the Kruskal diagram of the collapsing and then evaporating black hole, including bothe the inside and the outside of the singularity (and likely also other tolopologies, but by a well known aDS complementarity the result from higher topologies give a negligible contribution to the sum over histories). By unitarity, at the spacelike singularity at the center of the black hole there has to be a no-loss of information boundary condition. When the black hole finally (that is, finally in outside time terms) evaporates, the information has to all be gone (up to the of-the-order-of-a-plank-mass-worth carried away by the last evaporation quantum). So the singularity has to propagate an amount of entropy equal to the well-known block hole entropy result, decreasing as it evaporates, and the information (restrictions on Feynman histories we can sum over given specific spacelike boundary conditions before the collapse of the black hole) from the material that collapsed to form the black hole propagates along the singlarity in a locally space-like direction. This is of course a breach of normal causality, but this is a singularity: the space-time approximation to quantum gravity has broken down, so spacelike propagation of information isn't forbidden. The no-loss of information constraints at the singularity puts strong constraints on the information flow (allowed histories to sum over) of the infalling negative energy (negative in external terms, locally their energy is positive) Hawking-pair-created particles, such that as they reach the singularity and decreases the mass-energy (in external terms) of the singularity they also carry away entropy. So the information flow carried by the negative (external) energy particles must be radially *outwards*. This is a timelike direction, but it's the opposite direction to the radially inwards information flow for an infalling observer, whose information and mass-energy are going to increase the mass-energy-entropy of the singularity. Information flows radially (time-like) inwrds into the black hole when things fall into it, in a spacelike direction along the sigularity, then radially (time-like) outwards during black-hole evaporation. Where the radially outwards information flow of the Hawking particles inside reaches the event horizon, Hawking pair creation happens. Since the particle pairs are fully entangled, the information flow in the exterior Hawking radiation is then forwards in external time, as you'd expect. Looked at in a Feynmann sum-over-histories approach, the interior and exterior Hawking radiation from a Hawking pair creation pair are the same particle, following a path that (semi-classically) goes radially outwards from the singularity to the event horizon then forwards in time and outwards from the event horizon. The propagation of information (from spacelike boundary conditions before the collapse of the black hole) flows in those directions along this semi-classical path. So inside the black hole, the causal flow is bidirectional,

So the solution to the firewall paradox has to be that the no-cloning result from quantum information theory doesn't apply in a situation like that inside an evaporating black hole with occasional infalling observers, where there are causal counterflows in opposite directions in local time with a spacelike no-loss-of-information boundary condition. I'm pretty sure that conditions like these weren't considered when the no-cloning result was derived: when information reflects off the singularity between travelling radially inwards and radially outwards, you can't avoid getting two copies of it.

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