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 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 brickwall 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. [email protected] [email protected] Since Bekenstein [1] suggested that black holes carry an intrinsic entropy proportional to the surface area of the event horizon, and Hawking [2] provided a physical basis for this idea by considering quantum effect, there have been various approaches to understanding the black hole entropy. One of them is the so-called “brick-wall model” introduced by ’t Hooft [3]. He has considered a quantum gas of scalar particles propagating just outside the event horizon of the Schwarzschild black hole. The entropy obtained just by applying the usual statistical mechanical method to this system turns out to be divergent due to the infinite blue shift of waves at the horizon. ’t Hooft, however, has shown that the leading order term on the entropy has the same form as the Bekenstein-Hawking formula for the black hole entropy by introducing a brick-wall cutoff which is a property of the horizon only and is the order of the Planck length. The appearance of this divergence [4, 5] and relationships of this “statistical-mechanical” entropy of quantum fields near a black hole with its entanglement entropy [6] and quantum excitations of the black hole [7] have been studied, leading a great deal of interest recently [8]. The brick-wall model originally applied to the four dimensional Schwarzschild black hole [3] has been extended to various situations. The application to the case of rotating black holes has also been done for scalar fields in BTZ black holes in three-dimensions [9, 10] and in Kerr-Newman and other rotating black holes in four-dimensions [11, 12]. In a background spacetime of rotating black holes, it is well known that scalar fields have a special class of mode solutions, giving superradiance. It is claimed in Ref. [10] that the statistical-mechanical entropy of a scalar matter is not proportional to the “area”(i.e., the circumference in the three-dimensional case) of the horizon of a rotating BTZ black hole and that the divergent parts are not necessarily due to the existence of the horizon. Contrary to it, in Ref. [9], the leading divergent term on the entropy is proportional to the “area” of the horizon, and it is possible to introduce a universal brick-wall cutoff which makes the entropy equivalent to the black hole entropy. Moreover, it is claimed in Ref. [9] that the contribution from superradiant modes to entropy is negative and its divergence is in a subleading order compared to that from nonsuperradiant modes. On the other hand, for the case of Kerr black holes in Ref. [12], the divergence is in the leading order but the entropy contribution is still negative. One may expect that the leading contribution to the entropy comes from the region very near the horizon as in the case of nonrotating black holes. Since the vicinity of a rotating horizon can also be approximated by the Rindler metric, it is seemingly that the essential feature of the leading contribution will be same as that in nonrotating cases. Our study in detail shows this naive expectation is indeed true. That is, we point out that previous erroneous results appeared in the literature were mainly related to an incorrect quantization of superradiant modes. For the case of rotating BTZ black holes, we have explicitly shown that superradiant modes also give leading order divergence to the entropy