A Critique of the Fuzzball Programby Suvrat Raju and his graduate student Pushkal Shrivastava. They criticize the whole paradigm of the fuzzballs – an approach to understand black hole thermodynamics – which is mostly associated with the name of Samir Mathur, see his 2007 guest blog and dozens of other posts.

I know Suvrat from Harvard, he's been an amazing T.A. in a course of mine, and I still love his numerous papers such as the papers about the black hole interior written along with Kyriakos Papadodimas. There is always some degree of perfectionism in his work and reliance on explicit equations.

(Lots of people in politicized science disciplines should appreciate what kind of a clean praise the radical Marxist Suvrat Raju may receive on a right-wing blog. Is that possible in your field? What about the climate science? I added this paragraph later – because I just couldn't resist.)

Mathur's fuzzballs are proposed models for black hole microstates – effectively some precise star-like solutions of the supergravity field equations (or, more speculatively, solutions that involve non-local degrees of freedom in string/M-theory).

In the SUGRA case, these star-like equations have been explicitly written down for some highly supersymmetric black holes whose horizon area is only nonzero because of the nonzero quantum corrections. In these solutions, the black hole interior is replaced with lots of fuzz – some hair – which may be constructed from ALE-like spaces for every shape of a string (which contributes to the wave functional of a string model of a black hole). Black holes with a decreasing amount of supersymmetry or with the increasing role of the classically nonzero horizon areas are realized as increasingly speculative and non-explicit fuzzballs that are increasingly likely to involve some non-local degrees of freedom which are less understood.

What you may find cool is that both sides of this fuzzball debate are Indian – India is really full of sultans of string. China is expanding in many industries but I think it's completely accurate to say that the Indians are more important players than the Chinese in formal theoretical high-energy physics. At any rate, the fuzzball is an important enough idea so that quantum gravity experts should have some research or statements about it in the whole world. It may look surprising that India is still leading – and it's leading both sides. India has harbored some amazing civilizations thousands of years ago – but I still think that when it comes to mathematics and the modern theoretical physics, the Indians should be grateful to the British imperial overlords for having arranged the environment and providing role models. Even Jamsetji Tata (the tata of the Tata industrial family) had to study at the Elphinstone College first.

OK, Suvrat and Pushkal argue that exactly when the fuzzballs are said to be relevant for the black hole microstates, black hole interior etc., the quantum corrections become large so the fuzzballs can't be trusted. For this reason, these fuzzballs shouldn't be able to say anything about the black hole interior (and therefore about the black hole information puzzle).

I tend to agree with these statements. It seems to me that even Mathur would have to agree. He has talked about the "fuzzball complementarity" which should imply, if I correctly assume that he preserves some common sense, that the fuzzball geometry – which is nontrivial and hairy (violating the lore or "invalid theorems" about the absence of the black hole hair in the relevant contexts) – should have nothing to do with "what the infalling observer sees in the black hole interior".

So I think that the "black hole complementarity" promoted by Mathur must include the statement that the fuzzball and the experience of an infalling observer in the black hole interior have very little to do with one another – they're some dual descriptions of similar degrees of freedom. So the fuzzballs are just some unusual, dual way to parameterize the microstates.

But Suvrat and his student go further. They claim that because the interior is described "incorrectly" by the fuzzballs, it follows that these fuzzball solutions only describe some multiplicity of the possible geometries "slightly outside" the event horizon. For that reason, the entropy coming from the fuzzballs shouldn't be identified with the black hole entropy – the entropy of the interior. Instead, they should be

*added*.

I have some trouble with this statement because in Mathur's most famous example, that would seem to imply that the total entropy is twice as large as it should be.

It seems to me that just because some quantum corrections are large doesn't imply that the uncorrected degrees of freedom obeying the uncorrected equations can't describe the right degrees of freedom. Some of the equality of the two entropies could be guaranteed by the supersymmetric nonrenormalization theorems. If something agrees, one should still know whether some agreement only holds when it's protected by supersymmetry; or whether it holds more generally even in the absence of sufficient supersymmetry.

In the Strominger-Vafa black hole, you may think of the system as some curved black holes or D-branes with lots of excited open strings stretched between them. The latter picture includes no quantum corrections. But the two entropies are still the same and should be identified. You may extrapolate the flat-space states with D-branes and open strings to a strong coupling. It's an example of the open-closed and/or holographic duality – but honestly, in this case, the extrapolation of the entropy is only possible due to the supersymmetric nonrenormalization theorems. Does the paper by Raju and Shrivastava contain a proof that the fuzzball solutions can't be another dual description of the same Hilbert space of microstates? Maybe it does. My real problem is that I actually believe that the fuzzballs are correct in this sense.

But for the black hole interior phenomena, I do agree that the fuzzballs are wrong or irrelevant. Mathur would envision some kind of "averaging over the fuzzballs" that is needed to reconstruct the observations made by an infalling observer. Does he actually have a quantitative rule how this "averaging" is supposed to be made? What are the observables that one is averaging? Are there any? I doubt it. There are no background-independent local field observables in quantum gravity, we believe – but those are exactly the observables you would need to "exist" that should be averaged. So there's really nothing well-defined to take the average over fuzzball solutions from.

So again, I think that there are two different questions – whether fuzzballs and their generalizations may be used to parameterize the black hole microstates; and whether they can answer questions about the (infalling observers') observations in the black hole interiors. I think that the answers are Yes and No, respectively.

Update: Suvrat wrote a fun mail to me. He agrees that there are two questions and they mostly discussed the inapplicability of the fuzzball solutions for questions about the black hole interior. Suvrat adds that the claim about "addition of two entropies" was just a quote taken from Ashoke Sen – an Indian, of course.

Most amusingly, I was informed that the comment about Marxism is relevant because their title is a parody of The Critique of the Gotha Program, some Marxist rant. (I couldn't have guessed it even if I had one million attempts and if I were protected from the Marxist stuff by 5,000 condoms.) There's a problem, however: the Marxist rant is "the" critique while theirs is "a" critique! ;-)

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