I believe that the best hep-th paper today is the last one,

Insightful D-branes (PDF)

by Gary Horowitz, Albion Lawrence, and Eva Silverstein. The authors decided to say something new about the fate of information near the black hole horizons.

In order to do so, they had to find an appropriate model of a black hole with many descriptions. So they picked an orbifold of AdS5 x S5 with the orbifold group identifying points at some hyperbolic purely space-like slice of the past light cone of a point in AdS5 x S5.

Much like for the four-dimensional Schwarzschild and other black holes, one can use Schwarzschild-like coordinates that don't see inside the horizon; these coordinates are often natural in unitary descriptions where quantum mechanics is respected. And one can also use other coordinates that do see beneath the horizon; such descriptions take classical general relativity seriously. Many of the black hole paradoxes can be reduced to the mysterious relationship between these two types of a description.

Because they want to study how physics changes at different radial distances from the singularity, they use spherical D3-brane probes - the same kind of spherical D3-brane shells whose collapse could have created the black hole in the first place. In the dual AdS/CFT description, their position is given by the six adjoint scalars.

They investigate the dynamics of these scalars i.e. these D3-branes. In some approximation, they're governed by DBI-like actions. And the eigenvalues of these positions themselves are primarily affected by the inverted harmonic oscillator potential. For different energies, starting from the very low energies that are well below zero (well below the maximum of the potential), this degree of freedom can either

- (A) classically bounce; the bulk sees D3-branes that never reach any horizons; they also think that naked singularities are removed by some repulson-like dynamics
- (B) bounce with large quantum corrections; the bulk image includes time-like singularities cloaked by outer and (unstable) inner horizons
- (C) reach the origin of the configuration space; the bulk theory interprets it as the space-like singularity

This is a nice and simple map that links qualitatively different causal diagrams with different types of motion in the system that is as easy as the inverted harmonic oscillator. Great. That's the kind of a true insight that is important in physics: a classification of difficult things is correlated with a classification of simpler things.

The pictures where the horizon is crossed or not crossed are related by time-dependent field redefinitions for the Yang-Mills scalars. These words may sound satisfactory to someone but I simply don't see how these words actually solve any of the problems. There are still two possible pictures: in one of them, the probe never gets inside; in the other picture, it does.

These two pictures differ by the amount of information that can live in space. According to black hole complementarity, the information about the black hole interior should not be independent. It seems that the authors think that they have confirmed this principle but the details how this occurs are not clear to me.

After all, a field redefinition is a trivial operation. But on both sides, one should have some consistent rules about the boundary conditions, normalization conditions, and domains where the scalars (and time) are allowed to take their values. Most importantly, how many "places" on the D3-branes (where excitations can live) are there?

The two pictures, with the black hole interior and without the black hole interior, seem to have different answers to all these questions. And so far I don't see how the different answers are being reconciled by the paper. They say that in the Schwarzschild coordinates, the infalling D-branes are slowing down and the corresponding adjoint scalar fields become heavily fluctuating.

Well, it is almost certainly the case. But if they believe that they have a well-defined, essentially integrable system to analyze the situation, shouldn't it be possible to say some details about this transition? How do these large quantum fluctuations look like and how do they imply - via the field redefinition - the mostly "empty space" seen by the infalling observer and/or the nearly thermal radiation leaving the black hole? How is the information about the black hole interior (and the black hole entropy) really stored in these highly fluctuating degrees of freedom?

My feeling of confusion is not being helped by Figure 4, either. They reproduce the topologically trivial "Penrose diagram" of a black hole by Ashtekar and Bojowald, featuring a blob called "nongeometric region" but no horizon. The moral purpose of this picture is to say that in quantum gravity, there are no strict causality rules. Indeed, that must be the case because the information can get out. So all "black hole interiors" are just artifacts of the classical approximation and don't exist at the quantum level.

That's almost surely the case. But is there something more about the picture that should be taken seriously? What does it mean to draw a geometric picture (namely a Penrose diagram) of a nongeometric region? Isn't it obviously impossible? Where is the boundary of the nongeometric region? Is it near the singularity, near the horizon, or is the whole spacetime slightly nongeometric - with the "effect" and "degree of nongeometricity" decreasing as we are going away from the singularity? If these questions can't be answered, I think it is misleading to draw any new Penrose diagrams. Penrose diagrams are meant to tell us something the classical causal structure of a geometry. If the latter doesn't exist, Penrose diagrams shouldn't exist, either.

Although the situation was probably nicely chosen to allow them to do a relatively complete analysis, I don't feel that this analysis has been quite made so far. The term "field redefinition" seems to be a vacuous buzzword. As a physical operation, it seems to lack the beef (although it may look complicated if the Yang-Mills symmetry is being gauge-fixed differently on both sides) and it seems that the basic questions - e.g. whether the degrees of freedom inside and outside in some slices (that see inside) exist independently at the same moment - have not yet been answered here.

There is one more way to explain why I feel that the gap hasn't quite been removed. There are essentially two basic descriptions. One of them is that of classical GR; it includes the black hole interior as an independent piece of space; nothing special happens near the horizon; the information gets lost. The other naturally talks about the black hole exterior only; it is manifestly unitary and quantum-mechanics-friendly; but it doesn't see anything beneath the horizon.

The basic gap is in between these two descriptions. It is clear that the classical GR description is only correct in the classical limit; while the second, pure quantum description a priori fails to imply the correct limit with the empty and seemingly independent black hole interior. From a solution to those black hole puzzles, I expect the authors to have a picture that satisfies the quantum principles - including unitarity and complementarity; but also can be used to derive all the features of the classical physics of GR; and tell us something beyond that, too.

That's why I am probably going to continue to think about these matters because they have not been fully resolved by these toy models.