Wednesday, February 09, 2005

Reall: Higher-dimensional black holes and rings

Harvey Reall gave an excellent job talk at MIT, and we could not have missed it. It was balanced because
  • he figured out the right equilibrium between simple results and non-trivial results - well, the talk was completely comprehensible
  • he divided the time correctly to talk about his own work and the work of others
  • he found the right balance between the string theory motivations and independence of string theory
Harvey is definitely among the world's leading experts in the field of solutions to higher-dimensional gravity and supergravity, especially the black hole solutions. Why do we care about the higher-dimensional solutions? It's harder to answer this question for a string-theoretical infidel.

But the rest of us knows: the Universe is probably higher-dimensional, and therefore it is a research of reality! And as Harvey emphasized many times, comparisons of a microscopic theory of quantum gravity with the predictions of classical (and semiclassical) general relativity is the next best thing after the experiment that we have to test a theory of quantum gravity, i.e. to test string theory.

The higher-dimensional black holes are also important because they can be the holographic dual description of some objects and phenomena in gauge theory - via Maldacena's AdS/CFT correspondence. And gauge theory is of direct physical relevance. For example, there exists the Hawking-Page first order phase transition for thermal gravity in anti de Sitter space: it's a transition between the thermal gas and a black hole. Edward Witten has explained that it corresponds to the confinement-deconfinement phase transition in the gauge theory.

Finally, the microscopic higher-dimensional black holes may be produced at the LHC if the braneworld low-gravity scenarios are right. The chance is epsilon, but it would be as cool as 1/epsilon, and therefore the average benefit is of order one, as Harvey calculated from this scaling law. ;-)

So what black hole solutions - i.e. stationary solutions of GR with horizons - are there?
  • In four dimensions, it's been proved that the the horizon must always have a spherical topology. Pure general relativity in d=4 admits the Kerr solution as the most general one. I believe that the Maxwell-Einstein system has the Kerr-Newman solution as the most general one.
  • This is a particular strong example of a no-hair (or "black hole uniqueness") theorem: once the black hole gets stable (and it does so essentially by the damped quasinormal, ringing modes), it always looks like the same solution that is only parameterized by a few conserved quantities - namely the mass, and the angular momentum (and perhaps the charge, if you couple GR to electromagnetism). The solutions do not have any hair - i.e. choices or parameters that would label different solutions of GR with the same mass and charges.
People - especially the string theorists who were most interested in these questions - used to assume that the same no-hair theorems work in higher dimensions. They assumed that the solution is unique for given conserved charges. In fact, they had known "the" solution for quite some time:
  • The generalization of the Kerr solution to a higher spacetime dimensionality is the Myers-Perry black hole. Its horizon has a (d-2)-sphere as its horizon.
OK, as the advocates of "no innate differences" hypotheses would say: if we have the black hole solution in d dimensions, why should we look for another answer?

The answer to their question is: for example because the no-hair conjecture could simply be wrong in higher dimensions, and we should know it. It turned out that it's indeed the case. Harvey Reall - and I can't cite all his collaborators who include smart people like Roberto Emparan, Harvey Integerr, Harvey Complexx, and Stephen Hawking - showed that already in five dimensions, there can be many black holes with the same value of the mass and angular momentum.

One of them is the Myers-Perry solution and Harvey did not have to look for this one. But he together with Roberto Emparan have found another solution - a black ring whose horizon is topologically not an S^3 but rather a product of S^2 times S^1. You see that the black hole uniqueness must be replaced by the black hole double non-uniqueness, but it still looks fine.

You must realize that in higher dimensions, you need more than just one value of the angular momentum. A localized object in 4+1 dimensions has a little group equal to SO(4), for example, whose rank is 2 - and you can therefore choose two (more generally: the integer part of (d-1)/2) independent angular momenta. Various solutions require various inequalities for them to exist - and avoid the closed time-like curves and naked singularities - but you can see that there is a region in the parameter space (mass vs. angular momenta) in which both the Myers-Perry black hole (that generalizes the Kerr solution, and is therefore also known as the Kerry solution) as well as the black ring exist.

The non-supersymmetric black holes are still much too general, and therefore Harvey focused on the supersymmetric ones. Even in this case, you find the black hole as well as the black ring found by Elvang, Emparan, Mateos, and Reall. But in this case, the regions in the parameter space don't overlap: the black hole only exists if the two angular momenta are equal, while the black ring exists if they're not equal - and the limit in which you send them to the same value is discontinuous. Moreover, the black ring even has continuous parameters which are not the conserved charges - you can call these parameters "dipole moments".

These solutions are supersymmetric. Therefore they're stable. Matt Headrick as well as I were expecting that an object BR would try to maximize its entropy given the same conserved charges, but Harvey Reall vigorously denies this possibility. Neither a classical instability nor the Hawking radiation seems to be there, and therefore the solutions happily exist regardless of other solutions nearby with the same conserved charges but higher entropy. It's a bit bizarre - intuitively speaking, a "stronger" second law of thermodynamics seems to be "weakly" violated. ;-)

But Harvey discussed various string theory tests of the black hole entropy. The entropy of the supersymmetric black ring was calculated in string theory by Michelle Cyrier, Monica Guica, David Mateos, and Andy Strominger in hep-th/0411187. They used the fact that locally the rotating black ring is indistinguishable from the wrapped black string in M-theory due to Maldacena, Strominger, Witten. In both cases, one can describe the black (st)ring by a (4,0) superconformal field theory describing M5-branes wrapped on four-cycles of a Calabi-Yau manifold. Using the Cardy formula, one obtains a full agreement with the supergravity S=A/4G calculation even though the area depends on seven (!) parameters.

Harvey expressed some skepticism about the calculation by Cyrier, Guica, et al. - especially about the fact that "it is not based on AdS/CFT, the near horizon BTZ-like geometry, and decoupling" - but I don't know what he thinks should exactly be questionable about the calculation. Incidentally, a reader of this blog noted that the entropy of the black rings was first derived from string theory in hep-th/0408186 by Bena and Kraus; this calculation furthermore is "based on AdS/CFT, the near horizon BTZ-like geometry, and decoupling".

At any rate, the understanding of black holes in higher dimensions is more subtle than in d=4. And the microscopic understanding of the non-supersymmetric black holes remains an open, difficult, and technically demanding exercise.


  1. Harvey Reall, Harvey Integerr, Harvey Complexx??? Stop pulling our legs!

  2. I wonder why everyone must always forget about me... Maybe it's because I live nearby, and because I usually don't commute? It's a kind of a bummer.

    Harvey Jeffrey Quaternion, PhD

  3. The universe is probably higher dimansional and that's reality? WOW! I only see four dimensions...

  4. Yeah that's what the multiversity professor said, the multiverse is not 4-D.

  5. Hey--I just saw your comment from ages ago (on to Andelka). I missed the comment because right at the time you left it, our website moved. I'm sure you don't remember. :)

    Sadly, I'm not Czech and don't speak Czech--only enough to understand your questions. How pathetic is that? But I taught English in Prague from 1999 to 2001 (how very . . . mid-20s angsty, I know) and love everything Czech. Now I am awash in nostalgia. *grumble* :)


    (still at

  6. There isn't any evidence for supergravity any more than extra dimensions.

  7. One might note that the entropy of the black rings was first derived from string theory in hep-th/0408186 by Bena and Kraus; this calculation furthermore *is* "based on AdS/CFT, the near horizon BTZ-like geometry, and decoupling".

  8. Totally unrelated, I know, but Lubos, take a look at:

    This is how we get into space. Not by inefficient, expensive government approaches like NASA or the administration's new push to go to the moon and Mars, but by letting capitalism and entrepeneurs do the work! It took 100,000 people to get the space shuttle into space, but only 20 people to get SpaceShipOne into space!

  9. I already pointed out that deriving the form of Hawking blackhole entropy is NOT a big deal at all, since it can be natually and unquely fixed using merely dimentional analysis method.

    Lubos, as you came from the East Bloc, if you had got a good undergraduate general physics education, you've got to know something about dimentional analysis!!! Since Russians are really good at it, as are the people who work on fluid dynamics. See:

    There's a heated debate on that topic, up to 50+ comments!!!


  10. Damn, It's "dimensional", not "dimentional". I know, I know :-)


  11. This comment has been removed by a blog administrator.

  12. Damn Witten has his name on everything. One wonders if theres any area of modern physics that doesn't have one of his seminal contributions (or downright dominating the field)

    Its funny how that always works out that way in math and physics. A few people do the lions share of the publishing, the rest kinda just watch in awe, mere mortals that we are.

  13. Yes he is truly remarkable. His name keeps cropping up. Newton once solved a very difficult problem anonomously but everyone knew who had done it. A statement another mathematician made was "we recognise the lion by his claw"
    Steve m

  14. Lubos, I noticed you deleted my comment above. Perhaps a slight misunderstanding: when I said "your errors and erronous" statements" I was referring to those made by quantoken on his blog and I did not mean you. It was off topic anyway but sorry about that.

  15. Anonymous said: "Lubos, I noticed you deleted my comment above. Perhaps a slight misunderstanding: when I said "your errors and erronous" statements" I was referring to those made by quantoken on his blog and I did not mean you. It was off topic anyway but sorry about that."

    I think Lubos is much more smarter than that! Mr. Anonymous, I am sorry, but I think you have greatly insulted Lubos's intelligence by even suggesting that he could mis-read your message. No, I think he is better than that.


  16. quantoken, what did anonymous actually say?

  17. To the anonymous poster who doubts the existence of higher dimensions and supergravity, there is indeed a very valid reason for you skepticism. Interacting quantum field theories in higher dimensions are nonrenormalizable, with the exception of phi^3_5, which is unstable. Supergravity is also nonrenormalizable, even in four dimensions.

  18. I apologize for having deleted a meaningful comment. I try to follow the policy of reducing the number of useless messages that are nothing more than not-too-illuminating personal attacks.

    Sometimes I click on a wrong trash can, and sometimes I click on a different one that attacks someone else, even though I agree with the attacker - for the sake of neutrality. This one was an error, of course. ;-)

  19. Lee Smolin, Surely You Were Wrong!

    There have been a heated discussion by Marcus et al:
    Which is about these paper:
    And an earlier one by Lee Smolin:

    What interested those people are the claim that they obtained a dimensionless coupling constant, which equals G*Lambda, i.e., gravity constant times the cosmological constant, the number is claimed to be very small, 10^-120, so that "it becomes amenable for perturbation theory which doesn't break general covariance".

    These folks have been completely wrong in this and I am going to talk about why. See it here:

  20. I dont want to get off topic and will try to be be brief. I should also have something technical to say later about the actual (interesting) topic here which is higher-dimensional black holes. But Quantoken said "I think
    you have greatly insulted Lubos's intelligence...I think he is better than that" That's rich coming from someone who actually uses every little opportunity on this blog to try and undermine Lubos's intelligence and his
    competance. Recently for example quantoken tried to make Lubos appear foolish by commenting that he (Lubos) forgot to mention the LHC in his recent entry entitled "Peace in Iran, LHC and other News". Of course, after quite some time you (quantoken)figured out that the "Compact Muon Solenoid" reference was actually about the LHC and retracted your foolish backfiring statement.

    I regards Lubos's intelligence highly, but I realized the remark I made could easily be
    misunderstood by anyone, thinking I was talking to the host(since it is his blog) and not "quantoken". Everyone makes mistakes no matter how brilliant they are. As for you saying "I think he is better than that", well he is certainly better and deserves better than being pestered every day by your annoying and pathetic heckling comments and weird lame "theories".
    (If you delete this Lubos it is ok, it is off topic, but I think I speak for quite a few people here;)

  21. Has research in higher dimensional black holes been limited to 5D so far? I can't seem to find anything in 6D. Are 6D black holes perhaps less natural since they don't appear from any simple compactifications of M-theory (as far as I can tell)?

  22. Nope, there are hundreds of papers about black holes in any number of spacetime dimensions up to 11 and even beyond. This very article is about some of the higher-dimensional ones.

    There are black holes in any number of (large) dimensions in any gravitational theory. It's inevitable. So it's also true that there are many black holes in 6D compactifications of string theory, both extremal and non-extremal.

  23. Huh, what strange things are here going on below this nice post involving a "quantoken" and one or several "anonymous" ... :-/ ?

  24. Fascinating: 2 months ago, Quantoken ended his 7.5-year-long blogging vacations

    and wrote two sentences again! ;-)

  25. Yeah, it looks slightly yellowed ;-)

  26. Thanks for the reply. When I originally commented I was having trouble finding specific nontrivial metrics for higher dimensional black holes (D>5) because I wasn't sure what to "look up".

    I'm currently working with a mathematician at my University (I'm still an undergraduate) who would like to study some of the areas where string theory and geometry interact.

    Originally we were going to try to find metrics for black holes in D>6 but it seems like all nontrivial metrics have already been discovered.

    Another topic we're considering is finding metrics for new Calabi Yau manifolds but he's worried that it might be too difficult of a topic to tackle over just the summer. (I'm graduating soon)