Black Hole Attractors and Entropy

On Friday, Atish Dabholkar from the Tata Institute of Fundamental Research, visited Imperial to talk about microscopic entropy counts, small black holes and the use of the attractor mechanism. This is a very interesting topic, and arguably the area where string theory has had its greatest success so far.

So let's recap: In 1973 Bekenstein suggested that the event horizon was proportional to the black hole entropy, shortly after Hawking became convinced of this and produced his famous equation relating the entropy of the black hole to a quarter of the event horizon. However, from statistical mechanics we have become accustomed to being able to understand entropy as a count of the degrees of freedom of a microscopic system, yet from general relativity we expect all material to fall towards a singularity, sometimes a point and sometimes a surface behind the event horizon. All degrees of freedom as we understand them classically are suppressed at the singularity so how to account for the entropy microscopically is quite a puzzle, we are led to believe that there are some hidden degrees of freedom. This is an ideal situation to turn to superstrings (at least if you are predisposed towards string theory), living in ten dimensions gives them six dimensions in which to hide such degrees of freedom from us four-dimensional beings. And indeed Vafa and Strominger were able to make a microscopic count for a five-dimensional black hole using string theory that agreed with the macroscopic entropy coming from the event horizon area. The idea is that we can consider in 10-dimensions coincident D5-branes, a D1-branes and strings going between the two types of brane, and the excitations of the strings account for the degrees of freedom (if you want to read more about this picture at an introductory level then chapter 16 of that wonderful purple book by Zwiebach is recommended). This is then compactified on a T^5 to give a point, corresponding to the singularity in the 5-dimensional spacetime. In fact this construction corresponds to the 5-dimensional Reissner-Nordstrom black hole, as can be seen from the algebraic approach taken by Clifford Johnson in his slightly less purple book D-branes. There are a number of equivalent dual pictures and one of the most illuminating in terms of finding a geometrical reason for picking this combination of branes and strings is described in a very readable paper by Samir Mathur, The fuzzball proposal for black holes: an elementary review. Mathur entices us to consider an M2-brane dimensionally reduced on a spacelike circle in the z-direction to give a NS string in the IIA theory and then further compactify our setting by wrapping the NS string around a circle in the y-direction. Now go back to the 11-dimensional picture and think about the event horizon of the M2, it is shrinking because the M2 tension is wrapped around two closed loops and pulling them tight. The horizon is shrinking to zero, and we find that we have zero macroscopic entropy. We aim to stabilise the 11-dimensional horizon and gain an entropy that isn't disappearing. This will mean that we have a stable extremal, or BPS, black hole, one that isn't radiating and shrinking. Again let's go back to the 11-dimensional M2 picture. The M2 is radiating, so that had there been any other compact dimensions transverse to the M2 it would try and excite them and blow them up. Aha! So let's compactify some of these other dimensions and wrap another brane around those and see if we can't balance the tension of the second brane with the expansion caused by the first brane and vice versa. We pick an M5 brane and place it transverse to z in 11-dimensions, giving us a NS5-brane in IIA, we wrap this around T^4 (transverse to the NS1) and S^1 (in the y-direction). Now this turns out to be enough to stop the shrinking of the z-direction in the 11-dimensional view, but both branes are wrapping the y-circle and it is shrinking. We may excite the y circle by adding momentum charges around the circle which have energy proportional to 1/R, so they have lower energy for larger R and keep the y-circle non-vanishing. Phew. Now we have three charges coming from the NS1-NS5-P system (which may be dualised to D1-D5-P) and a stable horizon. That this is a BPS state means that we can count the degrees of freedom for different values of the coupling constant g and still expect the count to stay the same. So this is a heuristic approach outlined by Mathur for picking this special system. For the actual counting I refer you to some of the literature here, here and here. And what about other black holes, in particular the Schwarzschild black holes: can we find a similar stringy construction for counting the microstates? Well, yes, we can see for example Englert and Rabinovici.

But what about that NS string we considered alone earlier, our arguments told us that it had zero entropy, and yet it still contains microscopic degrees of freedom, so what's going on? Atish Dabholkar started his talk by asking us whether the S(Q)=klog[\Omega(Q)] was absolutely correct and if we could compute corrections to both the macroscopic and microscopic counts of the form:

S=a_0A(Q)+a_1log[A(Q)]+a_2/A(Q)+....
klog[\Omega(Q)]=b_0A(Q)+b_1log[A(Q)]+b_2/A(Q)+....

He pondered whether we could compute the a's and the b's and did they agree, and then told us that for a class of BPS N=4, D=4 black holes this can be confirmed. He said that on the macroscopic side one must take into account higher derivative corrections to the action (i.e. graviton scattering) and work in the thermodynamic limit for the association between entropy and degrees of freedom to carried over exactly from statistical mechanics. If this approach is sensible, then we would find that our NS string would have contributions to the entropy but not at the first order.

Atish outlined his approach, or ingredients as he put it:

1. Action: N=2 sugra + topological string
2. Entropy: Bekenstein-Hawking-Wald formula
3. Solution: via the Attractor mechanism
4. An ensemble: some Ooguri-Strominger-Vafa mix of charges

He told us he would work with small black holes (= only two charges in the ensemble), where the counting can be done exactly and the classical area vanishes (as we saw above), and so corrections are essential. The approach is detailed in his 4-page paper Exact Counting of Black Hole Microstates and in his talk he commenced by telling us about how to regularise black hole backgrounds by using "stringy cloaks" and this is described in his 10-page paper with Renata Kallosh and Alexander Maloney entitled A Stringy Cloak for a Classical Singularity (you can watch a talk by Andrew Maloney on this paper here). Since the details of the talk are not suitable for blogging I will direct the interested reader to the other relevant and much longer papers written with Frederik Denef, Gregory W. Moore and Boris Pioline, the 35-page Exact and Asymptotic Degeneracies of Small Black Holes and the 103-page Precision Counting of Small Black Holes. Also of interest will be Ashoke Sen's Black Hole Entropy Function and the Attractor Mechanism in Higher Derivative Gravity, and you can see the slides and listen to a related talk given by Sen here.

Sorry to trail off without describing the details but one they are tough, and two I am tired. All comments on this approach and joyous sonnets praising (and explaining)the usefulness of the attractor mechanism are welcome. There, and I didn't even mention supersymmetry once, oops.

Update: Check out Jacques Distler's post about David Shih's work on Ooguri-Vafa-Strominger constructions and see also his comments on Dabholkar et al's work and small black holes.