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.

Hawking on Richard & Judy

One day's notice for those of you in the UK that Stephen Hawking will be appearing on tomorrow's (12th October) Richard & Judy show, probably to talk about his new book. This is one of the most unlikely pairings I can imagine and should be fun to watch.

Nobel Prize 2005

The Nobel prize for physics this year has been awarded to:

(1/2 of the prize) Roy J. Glauber for "for his contribution to the quantum theory of optical coherence",

(1/4 of the prize each) John L. Hall and Theodor W. Hänsch "for their contributions to the development of laser-based precision spectroscopy, including the optical frequency comb technique."

Many hearty congratulations to them!

Update: the BBC have a short story on the award, and there are also comments from Lubos Motl and Peter Woit.

Topological Mass Generation

So perhaps you thought the Higgs mechanism was the only mechanism for mass generation in four dimensional spacetime? Probably it is, but Roman Jackiw, co-discoverer of the axial chiral anomaly with Bell and Adler, spoke at Imperial College yesterday about an alternative and elegant method to generate a 4-dimensional mass term. Roman first described to us the three-dimensional model where a Chern-Simons term can be added to the Lagrangian to generate mass, but told us that his motivation would come from the Schwinger model in two-dimensions. In the Schwinger model massless Dirac fermions are added and then eliminated in order to generate a mass. With hindsight Topological Aspects of Gauge Theories by Jackiw, which is to appear in the Encylopaedia of Mathematical Physics would have been a good article to read before attending this seminar.

Roman went through the original model and then repeated the analysis using a number of dualised terms, he referred to this as going "towards the topological model". In particular he highlighted that the field acquires a mass due to the presence of a chiral anomaly in the axial vector current and leads to a massive pseudoscalar; the pseudoscalar being dual to the two-index field strength as well as being proportional to the divergence of the axial vector current.

Now Roman's aim was to take this two-dimensional model, made out of purely topological terms, and then write out the equivalent expression using the four dimensional topological objects. He said that he would call this topological mass generation since now he would refer to the terms we had before with their topological names.

In two dimensions, using the dualised terms, a pseudoscalar crops up that is the Chern-Pontryagin density, P, and the dual of the potential field, C^\mu=\epsilon^\mu\nu A_\nu, is the two-dimensional Chern-Simons current. These are suitable quantities to take across to four dimensions, however it turns out to be a requirement of the method that the dual of the axial current must be a conserved quantity, and this can be guaranteed to occur by adding two fields, added in the form of Lagrange multipliers to the dual Lagrangian (I omit the details here unfortunately because I still haven't opted for a way of putting TeX in these posts). Surprisingly when one does this in order to conserve the dual axial current, one obtains a gauge invariant dual Lagrangian - the two go hand in hand. The generalisation of the Schwinger model to four dimensions is now straightforward, and is carried out by using the four-dimensional topological terms. Roman finalised by mentioning two shortcomings of this approach, the first being that the anomaly producing dynamics has not been specified and as such this model presents a phenomenological model of mass generation. The second shortcoming was the resulting dual Lagrangian was a dimension eight operator, and this, I am told, presents difficulties for renormalization. However on the positive side the specific contribution needed for the anomaly appears in the expansion of the Born-Infeld action to quadratic terms. Furthermore Roman pondered whether it might not present a phenomenological description for the elusive \eta'. Roman reminded us that the \eta' is the ninth goldstone boson suspected to arise by promoting an SU(3)xSU(3) symmetry to a U(3)xU(3) symmetry. This topological mass generation if it were indeed applicable to the \eta' would give a numerical prediction of the \eta' mass. For some notes about the mysterious \eta' see here. Apologies for any mistakes, one day I will read the much-recommended book by Zee and learn some QCD. Now I've written it here I just have to do it...