## Saturday, November 15, 2008 ... /////

### Fuzzballs: a modern review

In July, we discussed a paper by de Boer et al. about Samir Mathur's fuzzball proposal, an attempt to revise our understanding about the black hole interiors and the origin of black hole entropy in terms of many horizon-free configurations.

As a reader has pointed out, de Boer et al. just published a new, 79-page review of the fuzzball science:

Black holes as effective geometries
I am grateful to Mr Phantom for the tip because last week, I missed the preprint because when the paper was released, I was just reinstalling the OS on my desktop PC which is usually not a pleasant story. The message is that when you repeatedly restart your PC because a program badly freezes, you should be careful about the consistency of your hard disk: use CHKDSK.

(It seems to work flawlessly now, at least for a week.)

The work by LLM about the bubbling AdS space is presented as one of the simplest examples of the fuzzball approach - which is an interpretation that is clearly true but already slightly nontrivial. They also review various stringy situations in AdS5 x S5, AdS3 x S3 x T4, and AdS3 x S2 x CY and discuss the known things about the parametrization of the black hole microstates: they only talk about black holes that preserve a lot of supersymmetry. The analogous story in the non-supersymmetric case is likely to be much more complicated.

Besides their discussion of the construction of the horizon-free solutions associated with the black hole microstates, they also include a lot of qualitative comments about the framework. Let me organize the answers to some FAQ starting from the most obviously true ones:

Black holes allow both mixed states and pure states, much like all quantum systems

If you ask whether black holes are pure states or mixed states, the answer copies the answer in all other known quantum systems: black holes can be both, of course. In principle, you can imagine that the exact microstate of a black hole is known and you describe it as a pure state. In reality, the available probes prevent you from knowing everything about the object, so the black hole is represented as a mixed state (e.g. a thermal state). Mixed states (density matrices) are just a technical tool to deal with a situation in which the information about the system is incomplete.

The Hilbert space is a quantization of the phase space

Now, the smooth geometries (and perhaps geometries with some extra configuration of additional stringy stuff) are not "identical" to the black hole microstates (vectors in the Hilbert space). These solutions are points in the phase space (or "a" phase space) and this phase space should be quantized, just like in all other quantum situations. Each cell of the volume "(2.pi.hbar)^N" gives rise to one basis vector of the Hilbert space.

As we explained in July, if some region of the phase space is very thin, its small volume may include a very small number of basis vectors (or none).

If you take a generic pure state - a random linear combination of the basis vectors - you will clearly not obtain a unique classical solution: such generic states resemble Schrödinger's cat that is half-dead, half-alive. Only special pure states whose Wigner distribution (my word here) is quasi-localized on the phase space - states that they aptly call "coherent" - may be associated with unique classical smooth geometries (or their stringy generalizations).

Why is the entropy large?

Because the volume of the phase space is large.

Why should all these microstates be obtained by quantizing a phase space of SUGRA solutions?

Well, this is only true for some highly BPS black holes that classically have a vanishing area (when the higher-derivative terms are neglected). For more complicated (and non-SUSY) black holes, you have to quantize a phase space that is parametrized by other stringy, non-SUGRA degrees of freedom, too.

The fuzzball paradigm is still nontrivial in this case because it leads you to study what the new "classical" stringy degrees of freedom producing the phase space are. (When I was writing the previous sentence, my sister called me and independently asked me what the term "paradigm" meant - what a coincidence.) The point is that you shouldn't treat the "exp(S)" microstates of a black hole as degenerate, almost identical animals. They differ, the differences can be described, and they can be measured by fine enough probes.

Semiclassical probes won't be able to distinguish in between them.

Now, how is it possible that the stringy effects are supposed to change physics a long distance away from the singularity, the only place where the curvature seems to be high?

They argue that this whole "non-local" effect - that allows the stringy physics to modify the whole black hole interior and not just the vicinity of the singularity - is due to the "thin" regions of the phase space that don't admit a large number of microstates which is why it prevents you from taking the usual classical limit.

Note that this "paradox" only arises if the event horizon has actually a macroscopic area. In this case, you need stringy, and not just SUGRA, coordinates to span the whole phase space even though it is often useful to consider just a subspace, too.

However, they seem to believe that the "nonlocality" doesn't arise from having a large number of stringy fields or something like that but purely from the necessity to combine the widely separated points on the "thin" phase space into legitimate pure states.