Kerr black hole: the CFT entropy works for all M,J

In September 2009, TRF readers could have learned about the work by

Monica Guica et al. about Kerr black hole entropy

calculated using the AdS/CFT methods. Some people may have argued that the correct black hole entropy in the known examples has been linked to supersymmetry and it didn't work in general. Well, we've known since mid 1990s that it wasn't the case - but some people would still claim that.



The extremal Kerr black hole in the conventional 3+1 dimensions has no supersymmetry but Guica et al. could calculate its entropy from the states in a two-dimensional CFT, anyway. However, you may still say that the black holes had to be "extremal" which plays the same role as "supersymmetric" even though these are not supersymmetric. However, in a new paper by Alejandra Fidel Castro, Alexander Maloney, and Andy Strominger (CMS, not to be confused with ATLAS),

Hidden conformal symmetry of the Kerr black hole,

it's been actually shown that the right entropy emerges from the CFT for any value of the mass M and any angular momentum J. If you take the J=0 limit, that actually includes the conventional Schwarzschild black hole! So the entropy of all these "astrophysical" black holes can now be correctly calculated by stringy microscopic methods.




Only in the extremal limit J=M^2, you can derive the conformal symmetry geometrically. But if you assume that the symmetry exists for all M,J, you can see that the the periodicity of the azimuthal "phi" angle makes the Euclidean time periodic so that the left-moving and right-moving temperatures are

T_L = M² / 2 pi J,
T_R = sqrt(M⁴ - J²) / 2 pi J.

The central charges are

c_L = c_R = 12 J

which immediately allows you to compute the entropy via Cardy's formula:

S_micro = (pi² / 3) (c_L T_L + c_R T_R) = ...
... = Area/4.

The story seems to be so self-consistent that you may want to trust it even though they don't actually "derive" the existence of the dual CFT. In fact, even if the CFT doesn't actually "exist", it seems that the calculation based on the assumption that it does leads to the right result.

Of course, I have had no doubts that the microscopic entropy - calculated by sufficiently authentic stringy methods - works correctly for all black holes since the 1990s. But it's still fun to see that some of these calculations may actually be simpler than we may have previously expected.