Superradiance and the Statistical-Mechanical Entropy of Rotating BTZ Black Holes

We have considered the divergence structure in the brick-wall model for the statistical mechanical entropy of a quantum field in thermal equilibrium with a black hole which {\it rotates}. Especially, the contribution to entropy from superradiant modes is carefully incorporated, leading to a result for this contribution which corrects some previous errors in the literature. It turns out that the previous errors were due to an incorrect quantization of the superradiant modes. Some of main results for the case of rotating BTZ black holes are that the entropy contribution from superradiant modes is positive rather than negative and also has a leading order divergence as that from nonsuperradiant modes. The total entropy, however, can still be identified with the Bekenstein-Hawking entropy of the rotating black hole by introducing a universal brick-wall cutoff. Our correct treatment of superradiant modes in the ``angular-momentum modified canonical ensemble'' also removes unnecessary introductions of regulating cutoff numbers as well as ill-defined expressions in the literature.Comment: 13 pages, plain latex, no figure, version published in Phys. Lett. B, Emphasis is given on the quantization of superradiance mode

Comments on `` Black Hole Entropy from Conformal Field Theory in Any Dimension ''

In a recent letter, Carlip proposed a generalization of the Brown-Henneaux-Strominger construction to any dimension. We present two criticisms about his formulation.Comment: 4 pages, Enriched version for the accepted one (Phys. Rev. Lett.

Entropy in the Kerr-Newman Black Hole

Entropy of the Kerr-Newman black hole is calculated via the brick wall method with maintaining careful attention to the contribution of superradiant scalar modes. It turns out that the nonsuperradinat and superradiant modes simultaneously contribute to the entropy with the same order in terms of the brick wall cutoff $\epsilon$. In particular, the contribution of the superradiant modes to the entropy is negative. To avoid divergency in this method when the angular velocity tends to zero, we propose to intr oduce a lower bound of angular velocity and to treat the case of the angular momentum per unit mass $a=0$ separately. Moreover, from the lower bound of the angular velocity, we obtain the $\theta$-dependence structure of the brick wall cutoff, which natu rally requires an angular cutoff $\delta$. Finally, if the cutoff values, $\epsilon$ and $\delta$, satisfy a proper relation between them, the resulting entropy satisfies the area law.Comment: 16 pages, Latex, no figures, References are included, Subsection A and B are reduced to subsection A, Abstract is rewritten, Minor corrections are include

Einstein Structure of Squashed Four-Spheres

It is known that the moduli space of Einstein structures is an isolated point so that an Einstein structure has no infinitesimal Einstein deformations. We examine the rigidity of the Einstein structure by considering deformations of the round four-sphere $\mathbb{S}^4$. We show that any deformation of the round four-sphere $\mathbb{S}^4$ causes it to deviate from the Einstein structure, except trivial deformations at most only changing the size of the sphere.Comment: v2: 22 pages, 1 figure, references adde