Thursday, July 06, 2006

Nonsupersymmetric black holes: entropy

For more than twenty years since the Bekenstein-Hawking calculations of black hole thermodynamics, people were unsuccessfully trying to find a microscopic description of the entropy. Strominger and Vafa succeeded in early 1996 and explained the entropy of a special BPS black hole composed of D1-, D5-branes, and momentum. That was a breakthrough. The result agreed including the numerical constant. It had to agree and their main task was to find an example where the agreement can be proved, and check it.

Meanwhile, people have calculated the entropy of many black holes with 3 or 4 types of charges that are supersymmetric (and therefore extremal). An agreement was always found. People have also checked near-extremal black holes, and the microscopic stringy result agreed with the macroscopic prediction in all cases. Near-extremal black holes are black holes that differ from the extremal supersymmetric ones by epsilon and their parameters are expanded up to the leading order in epsilon.

Black holes with up to seven parameters describing various types of charges and angular momentum have been successfully checked. If the black holes are extremal or near-extremal, there are rigorous arguments based on supersymmetry that the entropy had to match if the theory is consistent, and indeed, they do match.

The corrections to the entropy of various black holes have been calculated to all orders; the recent Ooguri-Strominger-Vafa formula plays an important role in this direction of thinking. The successfully checked examples include black holes with unusual topologies such as black rings. The dependence on the charges knows about the exceptional groups and we encounter interesting mathematical structures such as the hyperdeterminant. A lot of other links with interesting mathematical structures and modular functions occured in the heterotic case.

What is interesting is that the full functional dependence on the parameters agrees even for many non-supersymmetric black holes. We have a new example: in their new preprint,

study the 4+1-dimensional Kaluza-Klein black hole embedded in M-theory in the most obvious fashion - by adding a six-torus. The entropy again agrees perfectly. I think that virtually everyone who has studied these things at the technical level would agree that we will no longer be shocked by such checks. The black hole entropy calculations in string theory are a success story and it is completely unreasonable to conjecture that a mismatch will suddenly be found when we look at another example.

Also, more or less everyone who has looked at these things will agree that string theory (with everything that is directly connected to it) is the only known framework - and probably the only mathematically possible framework - in which such a check of the black hole entropy can be successfully made. The analogous attempts to calculate the same things in various "competing theories" fail miserably.

In other words, only ignorants and stupid people are ready to humiliate the highly non-trivial constraints on a theory of quantum gravity imposed by the very consistency of the theory - for example, by an agreement between the microscopic and macroscopic formulae for black hole entropy. Words are easy; calculations are hard. Only ignorants can be preaching about many alternatives to string theory. If they were more than just cheap demagogues and authors of dumb books for undemanding readers, they would at leat try to find such an alternative. If they did it honestly, they would end up with the same conclusion as all of us:

There are no alternatives and the calculations of the microscopic origin of the black hole thermodynamical properties offer one obvious way to see why.

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