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...

Horizon on Hawking

Last night Horizon was on the well chosen topic of information loss paradox in black holes. It chose to cover the story as Stephen Hawking's "greatest ever mistake", drawing unbidden parallels with Einstein's greatest mistake. It was hard not to get excited about a popularization of such a technical nature, and I felt somewhat let down when we were treated to the usual combination of ominous voice-over (why, oh why, must science be presented as if it's a horror movie?), violins playing purposeful music, dazzling graphics, and vague presentation of the story. In fact the story was shifted away from physics to one questioning Stephen Hawking's scientific reputation leading the Horizon team to entitle their programme not "The Information Loss Paradox" but "The Hawking Paradox".

I admit though that, before I watched the programme, I was excited about it all day; in the best of all worlds I was hoping to hear some commentary on Hawking's most recent paper, perhaps even some insights that might help me understand it. But alas not. The programme aired at 9pm on BBC2 yesterday, and my spirits immediately sank when the announcer introduced it as "the reputation of the world's most famous scientist at stake". The first images we see are from a beach and there's a wall with at least 6ft high graffiti on it, and the largest graffito of all is of Hawking's equation relating entropy and event-horizon area. Perhaps this was also meant to draw parallels with graffiti of E=mc^2, who knows? The voice over begins, and the camera ranges over the beach to an astrologer:

"What if the world were so strange we could never hope to understand it and science was wasting its time? It sounds like the sort of thing a mystic might say but this was a suggestion made three decades ago by the most famous scientist in the world, Stephen Hawking."

From then on one had the idea that the scientific story was going to lag behind the human story, but, for pity's sake, why?

The introduction focussed on Hawking's celebrity, with Kip Thorne saying of him:

"He's absolutely unique, and I think he has been a very important person in both the intellectual and the cultural life of the past century."

These fair comments were countered by the voice-over's,

"But recently doubts have been expressed by some physicists about Hawking's scientific reputation."

Thereby initiating the main story being addressed by Horizon, that perhaps Hawking ain't so great. Frankly this appeared as unfounded, unsupported and scurrilous journalism used to appeal to a wider audience, and at no stage were any of Hawking's conributions to physics not related to information loss discussed.

The information loss paradox was described as the result of a black-hole evaporating to nothing leaving behind only thermal radiation, i.e. carrying no information. That there would be a problem even if the black-hole didn't disappear was not made clear
(There is clarification about where this is a concern from Christophe Galfard in the comments). Without recourse to quantum mechanics this was described as a violation of one of the most fundamental principles of physics, that information is never destroyed. The voice over spookily summarised,

"Effectively bits of the universe are missing...nothing science knows not even our memories could be trusted."

Lawrence Krauss commented, probably to the delight of the Council for making Science Scary who seem to be in charge of Horizon,

"...at its most extreme scale what it means is everything you come to know and love would ultimately disappear."

This scary comment went without any guiding timescales.

Cue Leonard Susskind who was presented as Hawking's adversary in an immense intellectual battle. Susskind described how he felt a need to resolve how it was that one could watch someone cross into an event horizon and potentially be pulled apart, while the same person would feel no great change as they themselves fell across the horizon. Ah, a good old-fashioned change to Eddington-Finkelstein coordinates, at last some firm ground. The voice-over's interpretation of this coordinate change:

"The same equations were saying that someone could be both dead and alive."

Hmph. Susskind described his resolution that allowed both of these possibilities to coexist and resolved the information loss paradox: holography. That the black hole acts like an information projector and that anything that falls into it has its information "beamed" onto the lower-dimensional event-horizon, thus avoiding losing information inside the event horizon. Although quite what was going to happen to the information when the black hole evaporated was not addressed. Susskind's belief's were presented as being vindicated by Maldacena's paper, Eternal Black Holes in AdS. And finally the programme concluded after the Dublin conference where Hawking conceded that information was preserved. So, no chance of an explanation of the sum-over-topologies approach, ho-hum. There is some commentary by Lubos Motl on the paper Information Loss in Black Holes, and if you are interested in holography you could read TASI lectures on the Holographic PrincipleTASI lectures on holography by Bigatti and Susskind.

Of course despite my disappointment with the lack of theory, the human story was appealing. There were some very nice pieces of footage of Hawking working with his students. In one shot, prior to adopting his synthesiser, Hawking is seen talking to what looks like a seminar with the help of a student, Chris Hull. Chris Hull says that they happen to have a model of the universe with them and pulls out a cylinder and puts it on a table in front of the audience. Hawking makes some comments and grins, the student scratches his head and then turns the cylinder the other way up. It is identicle both ways.

Also, there were some encouraging comments about the trials of a student from Christophe Galfard, who, contrary to my earlier scandalous comments, has pointed out that he does not work in the signature (+---), described the start of his PhD with Hawking, saying:

"For the first year-and-a-half every sentence of Stephen's took me about six months to understand."

And of reading Maldacena's paper:

"I took a little while to read it, a little while being about a year-and-a-half."

Again: here, here!

The show ended with Hawking saying:

"I have no intention of stopping anytime soon. I want to understand the universe and answer the big questions, that is what keeps me going."

At no point did the programme mention the word string, nor topology! Is this really the best way to promote science to a popular audience? Could it not be done with at least a little humour, and less of the portentous voice over? Maybe even less of the human-interest story? After all the science is fascinating if communicated well. Oh for a more perfect world.

Not with a whimper...

The BBC news website has a story entitled Black holes start with many bangs. Observation of multiple gamma ray bursts by the Swift observatory, designed to detect very short bursts, improves upon previous recordings of a single decaying burst. The bursts are expected to be associated with black hole formation, radiation from infalling material. But from the tone of the article it seems the astronomers are not clear about the causes yet. While you're thinking about black holes go and look at Jillian's Guide to Black Holes, if you haven't already done so, it is a beautiful site.

Also in the news recently is a new "three line" putative proof of Fermat's last theorem. Alexander Ilyin, a "doctor of technical science" who works in automated data processing in Omsk, unveiled his proof at a press conference on 23rd August, and according to this Pravda article

"colleagues in Omsk believe Alexander's proof is flawless and simple"

Furthermore the article continues,

"Omsk-based scientists and journalist have not found any errors so far"

Journalists are obviously of a much higher calibre in Russia. A follow-up article that fails to make it plain whether or not the proof has been withdrawn, but covers the popular history of Fermat's last theorem very nicely is here, and a discussion thread here.

Thanks to the Mighty Emperor of Room 102, Peter McKeag for pointing out this story.

The Joy of Ping

I have been learning about trackbacking, and as the observant reader will note there is now a new trackback link beneath each post. So for those as uninformed as I was "trackbacking" is the name given to creating links to blog entries containing content relevant (hopefully) to the post you have just read. The purpose is really to keep a record of links that didn't make it into the post, principally because the links were created after the post. If you find that last sentence confusing, then don't think twice about going to see the movie Primer (incidentally, I watched it yesterday and found it very confusing, but I'm enjoying thinking thorugh its time-travel paradoxes, so forget my comments and go and see it and then tell me what happened). In short it is a blogger's duty to trackback to articles that have inspired comment.

How does one do this? Well, first you can only trackback to an article if you include a link to that article in your main post. And second you can only do this on blogs that support trackbacking - so standalone blogger blogs which do not support trackbacking have to use an external program, I am using haloscan. If trackback is supported then beneath a post will be a trackback link that gives a URL to ping. In essence, pinging means sending some information to another server to let it know you are there. One pings and is pinged, but one is never punged nor panged, one briefly can be pinging and one can certainly pong. If you keep a trackback-supported blog, then you can use your trackback software to ping other blog articles to let the world know you have something to say about that post. Some blogs have automated trackbacking (e.g. WordPress) where the software automatically pings every link in a post. This is almost enough to motivate a change of blog software...

Why all the sudden fuss about trackbacks and pings I hear you cry/sob? Well as you may have read here, here, here, here, here and here the arxiv is trying out trackbacking. On each abstract page there is now a trackback link, if any exist, so that it would seem that anyone in the blogosphere can comment on any paper. There is some manual checking of trackbacks so that supposedly only bloggers with legitimate comments about papers can add a trackback. It remains unclear how a comment may be judged legitimate, but probably a commen-sense test will suffice. Apart from this it would seem there are few safeguards and I can't decide if this is a good thing or a bad thing. For example, in the past I have attended seminars and attempted to understand a particular paper from the arxiv. Often I have understood a little of the beginning of a paper and have posted comments about that on this page. Now it seems I, along with others, will face the dilemma of whether sparse comments on a paper warrant a trackback to the abstract page of the paper. My feeling is that all legitimate discussion is positive and merits a trackback to the arxiv. I imagine that, if it takes off, trackbacking on the arxiv will offer a connection to debates about the papers content as well as earnest readers' descriptions of their attempts to understand (parts of) papers. To me it sounds utopian, but we'll have to see how it works out. Furthermore it may make the physics blogosphere lose its orbit, so to speak, after all will anyone who wants to make a comment about a paper and have it recorded on the archive also have to keep a blog? Recently there have been several bloggers sending posts from the midst of a conference, and at least one case of specific conference blog. Inclusion of comments from such blogs on the archive would seem to be a very exciting development since these are usually fairly technical, and hopefully useful. Perhaps this will motivate conferences to keep such blogs, and give an indirect link from the archive, via a blog entry, to footage or slides from relevent talks at a conference. Well let's hope so.

Elsewhere, the debate about the greatest physics paper ever trundles on, and still no-one has argued on grounds of simple beauty in favour of Kaluza's 1921 paper about the fifth dimension! Well except yours truly that is. Nevertheless similar to the BBC's series of greatest ever lists, cosmicvariance's lists have begun to hyperbole. Now the debates for the greatest ever physics textbook as well as the greatest popular science book are in full swing. Also Lubos has posted an updated link to the video footage of the talks from Sydneyfest. Also as reported here, here and here a second physics blog has become a book(let's not forget this trend was started by Lieven Le Bruyn)! Peter Woit's Not Even Wrong will be available in UK bookshops from 16th March, 2006. It promises to be an honest review of the toughest problems faced by string theory, and probably a critique of studying string theory with blinkers on. There has been some discussion about the merits of a non-string theorist writing a popular science book about string theory, but you can read them on Not Even Wrong the blog and, probably, the book in due course. Have no fear, I foresee no situation where this blog will become a book, although Tangent Space wouldn't be an awful title for a sci-fi novel (all suggestions for plot are welcome, in return for an earnest acknowledgement in the foreward)...