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.