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 rotating BTZ black hole. Especially, the contribution to entropy from superradiant modes is carefully incorporated, leading to a very different result from known ones in the literature. That is, it is positive 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. [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 divergences on the entropy of quantum fields has been understood in many ways. In particular, Susskind and Uglum [4] and Jacobson [5] suggested that these divergences can be absorbed as the one-loop renormalization of gravitational coupling constants in the Bekenstein-Hawking formula of the black hole entropy, which was explicitly confirmed by Demers, Lafrance, and Myers [6]. Thus, it indicates that the entropy obtained in the brick-wall model is indeed a quantum correction to the black hole entropy. On the other hand, it has been shown that this “statistical-mechanical” entropy of quantum fields coincides with the black hole entropy as its entanglement entropy [7, 4, 5], and also is connected to counting the states of quantum excitations of the black hole [8]. Therefore, various studies of quantum corrections to the black hole entropy have been a great deal of interest recently [9]. The brick-wall model originally applied to the four dimensional Schwarzschild black hole [3] has been extended to various situations. For instance, Mann, Tarasov, and Zelnikov [10] have shown that, as in t’ Hooft’s, this model works for scalar fields in any nonextremal charged black hole in a spacetime higher than three dimensions( see also Ref. [11]). They, however, found that the brick-wall cutoff in the case of two-dimensional black holes is not universal, but is a function of the mass and charge of the black hole as well as the mass of the quantum field. The application to the case of rotating black holes has also been done for scalar fields in BTZ black holes in three-dimensions [12, 13] and in Kerr-Newman and other rotating black holes in four-dimensions [14, 15]. 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. [13] that the statistical-mechanical entropy of a scalar matter is not proportional to the “area”(i.e., the circumference in this 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. [12], 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 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. [15], the divergence is in the