Wednesday, March 06, 2013

Black hole monodromies explain why inner horizons matter

...not to mention that they also calculate scattering amplitudes and unmask a hidden conformal symmetry responsible for black hole thermodynamics...
Off-topic, Moriond 2013 (right sidebar has link to live broadcast): ATLAS' diphoton rate is \(1.64\pm 0.30\) times the Standard Model, 2.1 or 2.3 sigma too much...
Two bright Gentlemen whom I know from Harvard, Joshua Lapan and Alex Maloney, and two bright Ladies who are actually at Harvard right now, Alejandra Castro and Maria Rodriguez (yes, their affiliation footnotes are marked as ABBA on the title page, and yes, "A" are females while "B" are males), published a wonderful paper
Black Hole Monodromy and Conformal Field Theory
of the kind that makes one happy, somewhat humble, and somewhat proud to have helped to make similar developments possible. Almost exactly 10 years ago, your humble correspondent and Andy Neitzke wrote a paper (with numerous colorful pictures; I've been almost always using TkPaint to create them: a cool mini-tool) that introduced the "monodromy method" (monodromies around singular point in the complexified \(r\) plane of black hole solutions that mix up the linear space of solutions to the Klein-Gordon and other equations on that background) into the black hole research.

Our goal was to rederive the real parts of asymptotic frequencies of the highly damped, quasinormal (ringing) modes of the black holes, a constant that was known to be proportional to \(\ln 3\), a mysterious constant mostly numerologically guessed by Shahar Hod years earlier and proven by Taylor expansions and recursive relationships for the coefficients by your humble correspondent a month earlier. Wow, that paper looks amazingly rich in comments about loop quantum gravity to me today – at that time, I had to believe that there was a chance that its methods could actually hide a toad of truth. Andy Neitzke extended the monodromy method to an analysis of greybody factors three months after our paper.

For nearly a month at the end of 2002 and at the beginning of 2003, we intensely struggled with some subtle and shocking analytic properties of the Bessel functions (functions we previously wanted to remain bizarre abstract boring references in courses for students, not actual parts of our practical lives!) such as the Stokes phenomenon and with the right formulation of the proof we knew had to exist and we could always see its "foggy skeleton" but hard work was still needed to become sure that "monodromy" was the right word and to figure out what the relevant monodromies were and how they had to be teamed up to create the argument.

Of course, other people were involved in the research of the quasinormal modes and other people after us have used the monodromy method in many ways. Interestingly enough, Alex Maloney and Matt Schwartz, another young man from Harvard, were very helpful when I was writing the December 2002 paper because they created the first Mathematica program in my life (or at least the part I could remember) with cycles and conditions, to analyze the behavior of the recursive maps (I told them what the short program should do). At least my knowledge of Mathematica has slightly improved over the last 10 years! ;-)

On TRF, you may read Quasinormal story on quasinormal modes and some other articles about the topic.

Now, in 2013, Alejandra, Josh, Alex, and Maria use similar methods to clarify some mysterious facts that emerged in the quantum black hole research in recent years. Using the stringy methods, the black hole entropy was microscopically calculated for many types of black holes including objects such as the Kerr black hole in \(d=4\). The dual description exploits a conformal field theory (CFT), by a generalization of the AdS/CFT correspondence. And this two-dimensional CFT incorporates left-moving and right-moving excitations that are different in general (an asymmetry you know from the heterotic strings).

The black hole entropy has separate contributions from the left-movers and the right-movers and the interesting fact is that if you revert the sign of the smaller among these two contributions, the resulting difference still has many thermodynamic properties we associate with the entropy. In fact, it describes the area of the inner horizon. I would say that it has to be so because the radii of both horizons solve an algebraic (quadratic) equation and they differ by a different choice of the sign of the square root. But I have never written it before haha.

More generally, it's interesting that the parameters describing the inner horizon appear at many places when you calculate things like scattering on the black hole background. Now, these four authors probably have not only an explanation why it is so. By having found it, they may also explain why the thermodynamics of seemingly different black holes may be analyzed by similar maths. And they have a new, more conceptual, and independent of approximations, way to unmask the origin of the hidden conformal symmetry of Kerr black holes that Alejandra, Alex, and Andy Strominger clarified using low-energy approximate methods in 2010. (There are dozens of TRF blog entries that have something to do with the Kerr black hole entropy and related research by Andy Strominger et al. in recent years.)

And they may calculate lots of quantities about the scattering purely from the monodromies – in some sense, by "group theoretical methods". One could say that these monodromies may be beginning to play an equally important role as the monodromies in the \(\NNN=2\) gauge theories studied by Seiberg and Witten in the early 1990s.

I don't understand all the details yet; the preprint only appeared yesterday. However, let me sketch some basic tools and steps. They study the Klein-Gordon equation on the background of a black hole; as I mentioned, this may cover many apparently different black holes in many spacetime dimensions. It's a second-order linear equation (the time derivatives matter here) so for each frequency, it will typically have two linearly independent solutions. In this two-dimensional space, one may choose a basis but there is no universally preferred basis.

In fact, the space is still two-dimensional near the singular points in the \(r\)-plane, the complexification of some radial ("tortoise"...) coordinate describing the black hole geometry. And around these points, the two-dimensional space gets mixed up by a monodromy transformation in \(GL(2,\CC)\). The monodromies around the inner horizon, outer horizon, and the point at infinity satisfy conditions such as\[

M_- M_+ M_\infty = {\bf 1}

\] which are the key to their calculations. They play with the eigenstates and eigenvalues of these monodromies and they reformulate the scattering problem – a transformation of incoming and outgoing waves between the initial and final state – as something similar to a monodromy, too. There are various technical subtleties they must be careful about – some of those were known to me and Andy Neitzke, most of them were probably not.

But an outcome is that some bilinear function of the scattering amplitudes may be extracted from the monodromies regardless of the type of the black hole. The fact that so many seemingly different black holes carry entropies that admits a two-dimensional CFT approximation is interpreted as the consequence of their having two singular points at finite values of \(r\) and one singular point at \(r=\infty\). The monodromies around these points know about all the basic things in black hole thermodynamics, including the question how the central charges etc. should be divided between the left-movers and the right-movers!

For black holes with a greater number of singular points, the situation is harder and the relevant quantities aren't analytically calculable from the monodromies (right now) but an interesting generalization could exist for those black holes, too. I won't add too many comments here because this is meant to be just an advertisement for the paper and you may read it yourself. The same four authors are preparing a longer paper on similar topics, "Black Hole Scattering from Monodromy".

I still suspect, just like I did at the end of this 2005 blog entry, that the analytic continuation will play an increasingly important role in our understanding of quantum gravity. The Euclideanized black hole solutions according to Hawking and Gibbons have previously allowed an elegant calculation of the black hole entropy and temperature. The requirement was that the solution had no deficit/excess angle at the Euclideanized horizon.

However, the irregularities – e.g. monodromies – that actually do exist might turn out to be even more useful. They could know about lots of things. In this paper, the monodromies are acting on the low-energy fields only. But maybe one should think about the action of the monodromies on the space of all possible microstates, too. A black hole itself may be identified with a mathematical structure defined by purely group-theoretical constraints of a sort and all thermodynamic features of the black hole may be encoded in them.

And of course, I can't forget to recall that analytic continuation is also likely to play an important role in the right explanation of the black hole information loss, firewalls, quantum xeroxing, and other non-existent anomalies, as we discussed in the context of the Papadodimas-Raju paper.

Off-topic, soccer

Tomorrow, my hometown may very well look like this:

That's how the fans of Fenerbahce Instanbul reacted in May 2012 when they lost to another Turkish team. The UEFA home matches over there are closed for fans because UEFA managed to notice that the latter are wild animals. (The ban of fans didn't help; they were shooting parachuting suicide bombers by ballistic missiles into the stadium from outside when they played BATE Borisov.) However, Pilsen won't enjoy this kind of protection. At least 800+ Turkish fans will be present on the stadium and many more may be around. The reward for our having defeated Naples, currently the 2nd team of the top Italian league, 5-to-0 aggregate may be somewhat bittersweet.

Fenerbahce is by far the most popular Turkish soccer club, counting 1/3 of the Turkish population as its fans. They include this jerk in the middle. Note that the very fact that they play the Europa League is a playful act of political correctness. A fast look at the map of Instanbul reveals that Fenerbahce is an Asian club.

Well, the match carries the highest risk rating. Our police has prepared 300 heavily dressed cops, cops on horses, a helicopter, ready teams of traffic cops, immigration cops, even some German and Austrian cops, public order cops, many other subspecies of cops, and the anti-conflict teams. Let's pray that these wild Asians won't destroy our elegant city. ;-)


  1. More on blackhole paradox

  2. I noticed. From their "press release"

    This hype looks silly to me relatively to what i see in the paper. After all, it's not addressing any of the arguments on either side of the firewall debate - for various reasons, including a simple causal one: this York paper was submitted in April 2012, three months before the firewalls were even proposed!

    So the paper actually talks about "energetic curtains" and not "firewalls" which are only added at the end.

    Even if this causal problems weren't there, I wouldn't think it's a paper that seems to solve something important. Why don't you look at more important papers, such as the paper this blog entry is about, instead of rubbish that is being promoted among gullible people by press releases?

  3. Ok, I read the paper and have a question about monodromies. I'll apologize in advance if my understanding is still weak, but monodromy is similar in concept to holonomy except monodromy is defined on complex manifolds. So a monodromy group is similar to the concept of the fundamental group and defines an algebra associated with a particular manifold. Complex manifolds with certain topological features will have a monodromy group associated with it. The monodromy matrix measures the meromorphicity (or lack of) of the complex manifold, and effectively gives us information about the number of poles (or singularities) associated with the manifold.

    This is confirmed by the statement "monodromy matrices form a representation of the fundamental group of the complex r -plane with punctures at the singularities of the wave equation".

    Monodromies refer to the elements of their associated Monodromy matrices. For a each type of black hole, there is a monodromy matrix associated with each singularity in the black hole.

    The fact the Monodromy matrices can be combined allows for intermixing of interior and exterior degrees of freedom, allowing inner horizon information to influence the scattering amplitudes. The angular momentum and energies are also related to the monodromies, which allow for mixing, thus allow for developing a relationship between the inner and outer horizon entropies.

    In any case, it is the analytic structure of solutions which are driven by the properties of black holes the determine the entropy of the black hole and help explain mixing of internal and external degrees of freedom.

  4. Dear anony, all the notions - monodromy, holonomy, fundamental group - have some relationships but none of them are really the same and you're making things more complicated than you are.

    The fundamental group contains elements that tell you how you can encircle topologically inequivalent paths of a manifold, or around special marked/removed points in the manifold.

    The fundamental group of the sphere (compactified complex plane) with 3 singular points is generated by the circles around these three points. A,B,C are literally elements of the group. There's no information.

    Monodromy is something else.. It's the transformation induced on something else - in this black hole case, the 2-dimensional space of solutions to the KG equation - by making a round trip around the particular loop i.e. closed path. This closed path may be identified with a particular element of the fundamental group but the monodromy carries some more information - the information about the transformation acting on something else, more complicated.

    Holonomy of a manifold is the whole group - not just a single monodromy - generated by all possible monodromies that are acting on a particular thing, namely the tangent space at a point in the manifold, and induced by a particular operation, namely parallel transport.

    The differences between the notions are substantial. I think that if you don't understand the informal definitions above, you have to study the rigorous formal definitions of these concepts because the way how you mix them up suggests that your understanding of the concepts is way too vague for you to meaningfully read papers such as this one.

  5. Well these are helpful comments. As far as complicated, such things are often subject dependent. I squeeze in maybe a few hours out of year so I won't claim to have the same amount of time invested in these matters as perhaps I'd like and I can certainly measure progress during those hours in such areas better then anyone. I make the effort despite what others may say.

    To make a criticism and light of such things is surely anyone's prerogative, if you find me simple than I apologize for giving you reason to find that contemptible. It is surely such a despicable condition that one should be worthy of such admonishment. Hopefully such a presence will not offend in the future, and certainly such things will not matter in a few hundred years anyway. However, if people wish to think me stupid, I know otherwise.

  6. Ha ha Lumo, these are cool ABBA :-D

    I had to open the paper just for the purpose to verify the comments about the affiliation footnotes :-)
    And thanks for reporting about this, your comments give me a feeling that this ABBA paper is great even though I understand only a tiny bit of it. I alway like it when things, such as thermodynamics of black holes for example, can be derived theoretically by more fundamental underlying structures and relationships.

    The abstract of your "LQG-paper" I had to read thrice to believe what I read ... :-P

    Dumb question:
    I am not sure if I have already heard about an "inner horizon" of a lack hole. What is the physical meaning of this? I would like and appreciate it if you could say a bit more about it...


  7. Dear Dilaton, this ABBA pattern is funny, almost more funny than what was overheard in a British restaurant:

    Sir, are you sure that this is beef?
    Yes, of horse!

    When you write down the solution for a charged or rotating black hole and you ask for what values of "r", the time stops, i.e. g_{00}=0, so that there is a horizon, you will typically get a quadratic equation (or a harder one) which has numerous solutions. The largest value of "r" that obeys it is the event horizon but there are also
    smaller values of "r" that are solutions, for which g_{00}=0. So they're horizons of a sort. They're inside the black hole - already the outer, largest "r" horizon behaves like the normal Schwarzschild horizon, so they modify what an infalling observer may see (like new black-hole phenomena inside the large black hole). But they're unphysical for various reasons, unstable, Cauchy surfaces, and other expletives. Still, as discussed in this paper (but also previously), the inner horizon gets indirectly imprinted to various things.

  8. Dear Lumo,

    do you mean something like this by the quote ;-) ...?

    And thanks for the nice explanations about the inner horizons :-)

  9. Yup, exactly, except that I think that the Britons are more sensitive about horsemeat because they really consider horses to be our friends and peers...

  10. OT, but I just discovered a semi-popular introduction to the mathematics of general relativity by Lillian Lieber (The Einstein Theory of Relativity) that Lubos might want to recommend to his lay readers. It's a long-out-of-print classic, recently reissued. I learned for the first time in my life what a tensor is and how tensor calculus is done, complete with derivation of Einstein's equations.

  11. "the Britons are more sensitive about horsemeat because they really consider horses to be our friends and peers..."

    --nope, our betters and superiors. Consider the Houyhnhnms.....:)

  12. Dear Lumo, slightly off topic:

    This is another string theory question of a user who seems to be seriously interested in STuff ;-)

    Maybe you's like to give him an answer, if you still post on the site (?)

  13. Dear Dilaton, I've given an answer of a sort to the question. It's a part of the tricks in bosonization which I am no world sheet expert in but I hope that the proof via the stress-energy tensor extra term is valid even if it is not the most complete proof of the equivalence.

  14. Thanks Lumo :-)

    and good to see that you are still in the "business", I though maybe you are definitely too annoyed now about the very basic level of questions (since the eclat last December)...

    And I am always worried that such technical questions about advanced topics go unanswered, if you do not see them ...